This is vendored copy of the pure-Go version of math/big.
To update, run vendor.bash in place.
This will permit the use of the new big.Float functionality in
gc (which is not available in 1.4, the version used for bootstrapping).
Change-Id: I4dcdea875d54710005ca3fdea2e0e30422b1b46d
Reviewed-on: https://go-review.googlesource.com/7857
Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>
Run-TryBot: Robert Griesemer <gri@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Russ Cox <rsc@golang.org>
--- /dev/null
+// generated by stringer -type=Accuracy; DO NOT EDIT
+
+package big
+
+import "fmt"
+
+const _Accuracy_name = "ExactBelowAboveUndef"
+
+var _Accuracy_index = [...]uint8{0, 5, 10, 15, 20}
+
+func (i Accuracy) String() string {
+ if i < 0 || i+1 >= Accuracy(len(_Accuracy_index)) {
+ return fmt.Sprintf("Accuracy(%d)", i)
+ }
+ return _Accuracy_name[_Accuracy_index[i]:_Accuracy_index[i+1]]
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file provides Go implementations of elementary multi-precision
+// arithmetic operations on word vectors. Needed for platforms without
+// assembly implementations of these routines.
+
+package big
+
+// A Word represents a single digit of a multi-precision unsigned integer.
+type Word uintptr
+
+const (
+ // Compute the size _S of a Word in bytes.
+ _m = ^Word(0)
+ _logS = _m>>8&1 + _m>>16&1 + _m>>32&1
+ _S = 1 << _logS
+
+ _W = _S << 3 // word size in bits
+ _B = 1 << _W // digit base
+ _M = _B - 1 // digit mask
+
+ _W2 = _W / 2 // half word size in bits
+ _B2 = 1 << _W2 // half digit base
+ _M2 = _B2 - 1 // half digit mask
+)
+
+// ----------------------------------------------------------------------------
+// Elementary operations on words
+//
+// These operations are used by the vector operations below.
+
+// z1<<_W + z0 = x+y+c, with c == 0 or 1
+func addWW_g(x, y, c Word) (z1, z0 Word) {
+ yc := y + c
+ z0 = x + yc
+ if z0 < x || yc < y {
+ z1 = 1
+ }
+ return
+}
+
+// z1<<_W + z0 = x-y-c, with c == 0 or 1
+func subWW_g(x, y, c Word) (z1, z0 Word) {
+ yc := y + c
+ z0 = x - yc
+ if z0 > x || yc < y {
+ z1 = 1
+ }
+ return
+}
+
+// z1<<_W + z0 = x*y
+// Adapted from Warren, Hacker's Delight, p. 132.
+func mulWW_g(x, y Word) (z1, z0 Word) {
+ x0 := x & _M2
+ x1 := x >> _W2
+ y0 := y & _M2
+ y1 := y >> _W2
+ w0 := x0 * y0
+ t := x1*y0 + w0>>_W2
+ w1 := t & _M2
+ w2 := t >> _W2
+ w1 += x0 * y1
+ z1 = x1*y1 + w2 + w1>>_W2
+ z0 = x * y
+ return
+}
+
+// z1<<_W + z0 = x*y + c
+func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
+ z1, zz0 := mulWW_g(x, y)
+ if z0 = zz0 + c; z0 < zz0 {
+ z1++
+ }
+ return
+}
+
+// Length of x in bits.
+func bitLen_g(x Word) (n int) {
+ for ; x >= 0x8000; x >>= 16 {
+ n += 16
+ }
+ if x >= 0x80 {
+ x >>= 8
+ n += 8
+ }
+ if x >= 0x8 {
+ x >>= 4
+ n += 4
+ }
+ if x >= 0x2 {
+ x >>= 2
+ n += 2
+ }
+ if x >= 0x1 {
+ n++
+ }
+ return
+}
+
+// log2 computes the integer binary logarithm of x.
+// The result is the integer n for which 2^n <= x < 2^(n+1).
+// If x == 0, the result is -1.
+func log2(x Word) int {
+ return bitLen(x) - 1
+}
+
+// Number of leading zeros in x.
+func leadingZeros(x Word) uint {
+ return uint(_W - bitLen(x))
+}
+
+// q = (u1<<_W + u0 - r)/y
+// Adapted from Warren, Hacker's Delight, p. 152.
+func divWW_g(u1, u0, v Word) (q, r Word) {
+ if u1 >= v {
+ return 1<<_W - 1, 1<<_W - 1
+ }
+
+ s := leadingZeros(v)
+ v <<= s
+
+ vn1 := v >> _W2
+ vn0 := v & _M2
+ un32 := u1<<s | u0>>(_W-s)
+ un10 := u0 << s
+ un1 := un10 >> _W2
+ un0 := un10 & _M2
+ q1 := un32 / vn1
+ rhat := un32 - q1*vn1
+
+ for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
+ q1--
+ rhat += vn1
+ if rhat >= _B2 {
+ break
+ }
+ }
+
+ un21 := un32*_B2 + un1 - q1*v
+ q0 := un21 / vn1
+ rhat = un21 - q0*vn1
+
+ for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
+ q0--
+ rhat += vn1
+ if rhat >= _B2 {
+ break
+ }
+ }
+
+ return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
+}
+
+// Keep for performance debugging.
+// Using addWW_g is likely slower.
+const use_addWW_g = false
+
+// The resulting carry c is either 0 or 1.
+func addVV_g(z, x, y []Word) (c Word) {
+ if use_addWW_g {
+ for i := range z {
+ c, z[i] = addWW_g(x[i], y[i], c)
+ }
+ return
+ }
+
+ for i, xi := range x[:len(z)] {
+ yi := y[i]
+ zi := xi + yi + c
+ z[i] = zi
+ // see "Hacker's Delight", section 2-12 (overflow detection)
+ c = (xi&yi | (xi|yi)&^zi) >> (_W - 1)
+ }
+ return
+}
+
+// The resulting carry c is either 0 or 1.
+func subVV_g(z, x, y []Word) (c Word) {
+ if use_addWW_g {
+ for i := range z {
+ c, z[i] = subWW_g(x[i], y[i], c)
+ }
+ return
+ }
+
+ for i, xi := range x[:len(z)] {
+ yi := y[i]
+ zi := xi - yi - c
+ z[i] = zi
+ // see "Hacker's Delight", section 2-12 (overflow detection)
+ c = (yi&^xi | (yi|^xi)&zi) >> (_W - 1)
+ }
+ return
+}
+
+// Argument y must be either 0 or 1.
+// The resulting carry c is either 0 or 1.
+func addVW_g(z, x []Word, y Word) (c Word) {
+ if use_addWW_g {
+ c = y
+ for i := range z {
+ c, z[i] = addWW_g(x[i], c, 0)
+ }
+ return
+ }
+
+ c = y
+ for i, xi := range x[:len(z)] {
+ zi := xi + c
+ z[i] = zi
+ c = xi &^ zi >> (_W - 1)
+ }
+ return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+ if use_addWW_g {
+ c = y
+ for i := range z {
+ c, z[i] = subWW_g(x[i], c, 0)
+ }
+ return
+ }
+
+ c = y
+ for i, xi := range x[:len(z)] {
+ zi := xi - c
+ z[i] = zi
+ c = (zi &^ xi) >> (_W - 1)
+ }
+ return
+}
+
+func shlVU_g(z, x []Word, s uint) (c Word) {
+ if n := len(z); n > 0 {
+ ŝ := _W - s
+ w1 := x[n-1]
+ c = w1 >> ŝ
+ for i := n - 1; i > 0; i-- {
+ w := w1
+ w1 = x[i-1]
+ z[i] = w<<s | w1>>ŝ
+ }
+ z[0] = w1 << s
+ }
+ return
+}
+
+func shrVU_g(z, x []Word, s uint) (c Word) {
+ if n := len(z); n > 0 {
+ ŝ := _W - s
+ w1 := x[0]
+ c = w1 << ŝ
+ for i := 0; i < n-1; i++ {
+ w := w1
+ w1 = x[i+1]
+ z[i] = w>>s | w1<<ŝ
+ }
+ z[n-1] = w1 >> s
+ }
+ return
+}
+
+func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
+ c = r
+ for i := range z {
+ c, z[i] = mulAddWWW_g(x[i], y, c)
+ }
+ return
+}
+
+// TODO(gri) Remove use of addWW_g here and then we can remove addWW_g and subWW_g.
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+ for i := range z {
+ z1, z0 := mulAddWWW_g(x[i], y, z[i])
+ c, z[i] = addWW_g(z0, c, 0)
+ c += z1
+ }
+ return
+}
+
+func divWVW_g(z []Word, xn Word, x []Word, y Word) (r Word) {
+ r = xn
+ for i := len(z) - 1; i >= 0; i-- {
+ z[i], r = divWW_g(r, x[i], y)
+ }
+ return
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+
+
+package big
+
+func mulWW(x, y Word) (z1, z0 Word) {
+ return mulWW_g(x, y)
+}
+
+func divWW(x1, x0, y Word) (q, r Word) {
+ return divWW_g(x1, x0, y)
+}
+
+func addVV(z, x, y []Word) (c Word) {
+ return addVV_g(z, x, y)
+}
+
+func subVV(z, x, y []Word) (c Word) {
+ return subVV_g(z, x, y)
+}
+
+func addVW(z, x []Word, y Word) (c Word) {
+ return addVW_g(z, x, y)
+}
+
+func subVW(z, x []Word, y Word) (c Word) {
+ return subVW_g(z, x, y)
+}
+
+func shlVU(z, x []Word, s uint) (c Word) {
+ return shlVU_g(z, x, s)
+}
+
+func shrVU(z, x []Word, s uint) (c Word) {
+ return shrVU_g(z, x, s)
+}
+
+func mulAddVWW(z, x []Word, y, r Word) (c Word) {
+ return mulAddVWW_g(z, x, y, r)
+}
+
+func addMulVVW(z, x []Word, y Word) (c Word) {
+ return addMulVVW_g(z, x, y)
+}
+
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
+ return divWVW_g(z, xn, x, y)
+}
+
+func bitLen(x Word) (n int) {
+ return bitLen_g(x)
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "math/rand"
+ "testing"
+)
+
+type funWW func(x, y, c Word) (z1, z0 Word)
+type argWW struct {
+ x, y, c, z1, z0 Word
+}
+
+var sumWW = []argWW{
+ {0, 0, 0, 0, 0},
+ {0, 1, 0, 0, 1},
+ {0, 0, 1, 0, 1},
+ {0, 1, 1, 0, 2},
+ {12345, 67890, 0, 0, 80235},
+ {12345, 67890, 1, 0, 80236},
+ {_M, 1, 0, 1, 0},
+ {_M, 0, 1, 1, 0},
+ {_M, 1, 1, 1, 1},
+ {_M, _M, 0, 1, _M - 1},
+ {_M, _M, 1, 1, _M},
+}
+
+func testFunWW(t *testing.T, msg string, f funWW, a argWW) {
+ z1, z0 := f(a.x, a.y, a.c)
+ if z1 != a.z1 || z0 != a.z0 {
+ t.Errorf("%s%+v\n\tgot z1:z0 = %#x:%#x; want %#x:%#x", msg, a, z1, z0, a.z1, a.z0)
+ }
+}
+
+func TestFunWW(t *testing.T) {
+ for _, a := range sumWW {
+ arg := a
+ testFunWW(t, "addWW_g", addWW_g, arg)
+
+ arg = argWW{a.y, a.x, a.c, a.z1, a.z0}
+ testFunWW(t, "addWW_g symmetric", addWW_g, arg)
+
+ arg = argWW{a.z0, a.x, a.c, a.z1, a.y}
+ testFunWW(t, "subWW_g", subWW_g, arg)
+
+ arg = argWW{a.z0, a.y, a.c, a.z1, a.x}
+ testFunWW(t, "subWW_g symmetric", subWW_g, arg)
+ }
+}
+
+type funVV func(z, x, y []Word) (c Word)
+type argVV struct {
+ z, x, y nat
+ c Word
+}
+
+var sumVV = []argVV{
+ {},
+ {nat{0}, nat{0}, nat{0}, 0},
+ {nat{1}, nat{1}, nat{0}, 0},
+ {nat{0}, nat{_M}, nat{1}, 1},
+ {nat{80235}, nat{12345}, nat{67890}, 0},
+ {nat{_M - 1}, nat{_M}, nat{_M}, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, nat{1, 0, 0, 0}, 1},
+ {nat{0, 0, 0, _M}, nat{_M, _M, _M, _M - 1}, nat{1, 0, 0, 0}, 0},
+ {nat{0, 0, 0, 0}, nat{_M, 0, _M, 0}, nat{1, _M, 0, _M}, 1},
+}
+
+func testFunVV(t *testing.T, msg string, f funVV, a argVV) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func TestFunVV(t *testing.T) {
+ for _, a := range sumVV {
+ arg := a
+ testFunVV(t, "addVV_g", addVV_g, arg)
+ testFunVV(t, "addVV", addVV, arg)
+
+ arg = argVV{a.z, a.y, a.x, a.c}
+ testFunVV(t, "addVV_g symmetric", addVV_g, arg)
+ testFunVV(t, "addVV symmetric", addVV, arg)
+
+ arg = argVV{a.x, a.z, a.y, a.c}
+ testFunVV(t, "subVV_g", subVV_g, arg)
+ testFunVV(t, "subVV", subVV, arg)
+
+ arg = argVV{a.y, a.z, a.x, a.c}
+ testFunVV(t, "subVV_g symmetric", subVV_g, arg)
+ testFunVV(t, "subVV symmetric", subVV, arg)
+ }
+}
+
+// Always the same seed for reproducible results.
+var rnd = rand.New(rand.NewSource(0))
+
+func rndW() Word {
+ return Word(rnd.Int63()<<1 | rnd.Int63n(2))
+}
+
+func rndV(n int) []Word {
+ v := make([]Word, n)
+ for i := range v {
+ v[i] = rndW()
+ }
+ return v
+}
+
+func benchmarkFunVV(b *testing.B, f funVV, n int) {
+ x := rndV(n)
+ y := rndV(n)
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _W))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ f(z, x, y)
+ }
+}
+
+func BenchmarkAddVV_1(b *testing.B) { benchmarkFunVV(b, addVV, 1) }
+func BenchmarkAddVV_2(b *testing.B) { benchmarkFunVV(b, addVV, 2) }
+func BenchmarkAddVV_3(b *testing.B) { benchmarkFunVV(b, addVV, 3) }
+func BenchmarkAddVV_4(b *testing.B) { benchmarkFunVV(b, addVV, 4) }
+func BenchmarkAddVV_5(b *testing.B) { benchmarkFunVV(b, addVV, 5) }
+func BenchmarkAddVV_1e1(b *testing.B) { benchmarkFunVV(b, addVV, 1e1) }
+func BenchmarkAddVV_1e2(b *testing.B) { benchmarkFunVV(b, addVV, 1e2) }
+func BenchmarkAddVV_1e3(b *testing.B) { benchmarkFunVV(b, addVV, 1e3) }
+func BenchmarkAddVV_1e4(b *testing.B) { benchmarkFunVV(b, addVV, 1e4) }
+func BenchmarkAddVV_1e5(b *testing.B) { benchmarkFunVV(b, addVV, 1e5) }
+
+type funVW func(z, x []Word, y Word) (c Word)
+type argVW struct {
+ z, x nat
+ y Word
+ c Word
+}
+
+var sumVW = []argVW{
+ {},
+ {nil, nil, 2, 2},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{1}, nat{0}, 1, 0},
+ {nat{1}, nat{1}, 0, 0},
+ {nat{0}, nat{_M}, 1, 1},
+ {nat{0, 0, 0, 0}, nat{_M, _M, _M, _M}, 1, 1},
+}
+
+var prodVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{_M}, 0, 0},
+ {nat{0}, nat{0}, _M, 0},
+ {nat{1}, nat{1}, 1, 0},
+ {nat{22793}, nat{991}, 23, 0},
+ {nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0},
+ {nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0},
+ {nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1 << 1, _M >> (_W - 1)},
+ {nat{_M << 7 & _M}, nat{_M}, 1 << 7, _M >> (_W - 7)},
+ {nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, _M >> (_W - 7)},
+}
+
+var lshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1, 1},
+ {nat{_M << 20 & _M}, nat{_M}, 20, _M >> (_W - 20)},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M << 1 & _M, _M, _M}, nat{_M, _M, _M}, 1, 1},
+ {nat{_M << 20 & _M, _M, _M}, nat{_M, _M, _M}, 20, _M >> (_W - 20)},
+}
+
+var rshVW = []argVW{
+ {},
+ {nat{0}, nat{0}, 0, 0},
+ {nat{0}, nat{0}, 1, 0},
+ {nat{0}, nat{0}, 20, 0},
+
+ {nat{_M}, nat{_M}, 0, 0},
+ {nat{_M >> 1}, nat{_M}, 1, _M << (_W - 1) & _M},
+ {nat{_M >> 20}, nat{_M}, 20, _M << (_W - 20) & _M},
+
+ {nat{_M, _M, _M}, nat{_M, _M, _M}, 0, 0},
+ {nat{_M, _M, _M >> 1}, nat{_M, _M, _M}, 1, _M << (_W - 1) & _M},
+ {nat{_M, _M, _M >> 20}, nat{_M, _M, _M}, 20, _M << (_W - 20) & _M},
+}
+
+func testFunVW(t *testing.T, msg string, f funVW, a argVW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+func makeFunVW(f func(z, x []Word, s uint) (c Word)) funVW {
+ return func(z, x []Word, s Word) (c Word) {
+ return f(z, x, uint(s))
+ }
+}
+
+func TestFunVW(t *testing.T) {
+ for _, a := range sumVW {
+ arg := a
+ testFunVW(t, "addVW_g", addVW_g, arg)
+ testFunVW(t, "addVW", addVW, arg)
+
+ arg = argVW{a.x, a.z, a.y, a.c}
+ testFunVW(t, "subVW_g", subVW_g, arg)
+ testFunVW(t, "subVW", subVW, arg)
+ }
+
+ shlVW_g := makeFunVW(shlVU_g)
+ shlVW := makeFunVW(shlVU)
+ for _, a := range lshVW {
+ arg := a
+ testFunVW(t, "shlVU_g", shlVW_g, arg)
+ testFunVW(t, "shlVU", shlVW, arg)
+ }
+
+ shrVW_g := makeFunVW(shrVU_g)
+ shrVW := makeFunVW(shrVU)
+ for _, a := range rshVW {
+ arg := a
+ testFunVW(t, "shrVU_g", shrVW_g, arg)
+ testFunVW(t, "shrVU", shrVW, arg)
+ }
+}
+
+func benchmarkFunVW(b *testing.B, f funVW, n int) {
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _S))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ f(z, x, y)
+ }
+}
+
+func BenchmarkAddVW_1(b *testing.B) { benchmarkFunVW(b, addVW, 1) }
+func BenchmarkAddVW_2(b *testing.B) { benchmarkFunVW(b, addVW, 2) }
+func BenchmarkAddVW_3(b *testing.B) { benchmarkFunVW(b, addVW, 3) }
+func BenchmarkAddVW_4(b *testing.B) { benchmarkFunVW(b, addVW, 4) }
+func BenchmarkAddVW_5(b *testing.B) { benchmarkFunVW(b, addVW, 5) }
+func BenchmarkAddVW_1e1(b *testing.B) { benchmarkFunVW(b, addVW, 1e1) }
+func BenchmarkAddVW_1e2(b *testing.B) { benchmarkFunVW(b, addVW, 1e2) }
+func BenchmarkAddVW_1e3(b *testing.B) { benchmarkFunVW(b, addVW, 1e3) }
+func BenchmarkAddVW_1e4(b *testing.B) { benchmarkFunVW(b, addVW, 1e4) }
+func BenchmarkAddVW_1e5(b *testing.B) { benchmarkFunVW(b, addVW, 1e5) }
+
+type funVWW func(z, x []Word, y, r Word) (c Word)
+type argVWW struct {
+ z, x nat
+ y, r Word
+ c Word
+}
+
+var prodVWW = []argVWW{
+ {},
+ {nat{0}, nat{0}, 0, 0, 0},
+ {nat{991}, nat{0}, 0, 991, 0},
+ {nat{0}, nat{_M}, 0, 0, 0},
+ {nat{991}, nat{_M}, 0, 991, 0},
+ {nat{0}, nat{0}, _M, 0, 0},
+ {nat{991}, nat{0}, _M, 991, 0},
+ {nat{1}, nat{1}, 1, 0, 0},
+ {nat{992}, nat{1}, 1, 991, 0},
+ {nat{22793}, nat{991}, 23, 0, 0},
+ {nat{22800}, nat{991}, 23, 7, 0},
+ {nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0, 0},
+ {nat{7, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 7, 0},
+ {nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0, 0},
+ {nat{991, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 991, 0},
+ {nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0, 0},
+ {nat{991, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 991, 0},
+ {nat{_M << 1 & _M}, nat{_M}, 1 << 1, 0, _M >> (_W - 1)},
+ {nat{_M<<1&_M + 1}, nat{_M}, 1 << 1, 1, _M >> (_W - 1)},
+ {nat{_M << 7 & _M}, nat{_M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6}, nat{_M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+ {nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 0, _M >> (_W - 7)},
+ {nat{_M<<7&_M + 1<<6, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, 1 << 6, _M >> (_W - 7)},
+}
+
+func testFunVWW(t *testing.T, msg string, f funVWW, a argVWW) {
+ z := make(nat, len(a.z))
+ c := f(z, a.x, a.y, a.r)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if c != a.c {
+ t.Errorf("%s%+v\n\tgot c = %#x; want %#x", msg, a, c, a.c)
+ }
+}
+
+// TODO(gri) mulAddVWW and divWVW are symmetric operations but
+// their signature is not symmetric. Try to unify.
+
+type funWVW func(z []Word, xn Word, x []Word, y Word) (r Word)
+type argWVW struct {
+ z nat
+ xn Word
+ x nat
+ y Word
+ r Word
+}
+
+func testFunWVW(t *testing.T, msg string, f funWVW, a argWVW) {
+ z := make(nat, len(a.z))
+ r := f(z, a.xn, a.x, a.y)
+ for i, zi := range z {
+ if zi != a.z[i] {
+ t.Errorf("%s%+v\n\tgot z[%d] = %#x; want %#x", msg, a, i, zi, a.z[i])
+ break
+ }
+ }
+ if r != a.r {
+ t.Errorf("%s%+v\n\tgot r = %#x; want %#x", msg, a, r, a.r)
+ }
+}
+
+func TestFunVWW(t *testing.T) {
+ for _, a := range prodVWW {
+ arg := a
+ testFunVWW(t, "mulAddVWW_g", mulAddVWW_g, arg)
+ testFunVWW(t, "mulAddVWW", mulAddVWW, arg)
+
+ if a.y != 0 && a.r < a.y {
+ arg := argWVW{a.x, a.c, a.z, a.y, a.r}
+ testFunWVW(t, "divWVW_g", divWVW_g, arg)
+ testFunWVW(t, "divWVW", divWVW, arg)
+ }
+ }
+}
+
+var mulWWTests = []struct {
+ x, y Word
+ q, r Word
+}{
+ {_M, _M, _M - 1, 1},
+ // 32 bit only: {0xc47dfa8c, 50911, 0x98a4, 0x998587f4},
+}
+
+func TestMulWW(t *testing.T) {
+ for i, test := range mulWWTests {
+ q, r := mulWW_g(test.x, test.y)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+var mulAddWWWTests = []struct {
+ x, y, c Word
+ q, r Word
+}{
+ // TODO(agl): These will only work on 64-bit platforms.
+ // {15064310297182388543, 0xe7df04d2d35d5d80, 13537600649892366549, 13644450054494335067, 10832252001440893781},
+ // {15064310297182388543, 0xdab2f18048baa68d, 13644450054494335067, 12869334219691522700, 14233854684711418382},
+ {_M, _M, 0, _M - 1, 1},
+ {_M, _M, _M, _M, 0},
+}
+
+func TestMulAddWWW(t *testing.T) {
+ for i, test := range mulAddWWWTests {
+ q, r := mulAddWWW_g(test.x, test.y, test.c)
+ if q != test.q || r != test.r {
+ t.Errorf("#%d got (%x, %x) want (%x, %x)", i, q, r, test.q, test.r)
+ }
+ }
+}
+
+func benchmarkAddMulVVW(b *testing.B, n int) {
+ x := rndV(n)
+ y := rndW()
+ z := make([]Word, n)
+ b.SetBytes(int64(n * _W))
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ addMulVVW(z, x, y)
+ }
+}
+
+func BenchmarkAddMulVVW_1(b *testing.B) { benchmarkAddMulVVW(b, 1) }
+func BenchmarkAddMulVVW_2(b *testing.B) { benchmarkAddMulVVW(b, 2) }
+func BenchmarkAddMulVVW_3(b *testing.B) { benchmarkAddMulVVW(b, 3) }
+func BenchmarkAddMulVVW_4(b *testing.B) { benchmarkAddMulVVW(b, 4) }
+func BenchmarkAddMulVVW_5(b *testing.B) { benchmarkAddMulVVW(b, 5) }
+func BenchmarkAddMulVVW_1e1(b *testing.B) { benchmarkAddMulVVW(b, 1e1) }
+func BenchmarkAddMulVVW_1e2(b *testing.B) { benchmarkAddMulVVW(b, 1e2) }
+func BenchmarkAddMulVVW_1e3(b *testing.B) { benchmarkAddMulVVW(b, 1e3) }
+func BenchmarkAddMulVVW_1e4(b *testing.B) { benchmarkAddMulVVW(b, 1e4) }
+func BenchmarkAddMulVVW_1e5(b *testing.B) { benchmarkAddMulVVW(b, 1e5) }
+
+func testWordBitLen(t *testing.T, fname string, f func(Word) int) {
+ for i := 0; i <= _W; i++ {
+ x := Word(1) << uint(i-1) // i == 0 => x == 0
+ n := f(x)
+ if n != i {
+ t.Errorf("got %d; want %d for %s(%#x)", n, i, fname, x)
+ }
+ }
+}
+
+func TestWordBitLen(t *testing.T) {
+ testWordBitLen(t, "bitLen", bitLen)
+ testWordBitLen(t, "bitLen_g", bitLen_g)
+}
+
+// runs b.N iterations of bitLen called on a Word containing (1 << nbits)-1.
+func benchmarkBitLenN(b *testing.B, nbits uint) {
+ testword := Word((uint64(1) << nbits) - 1)
+ for i := 0; i < b.N; i++ {
+ bitLen(testword)
+ }
+}
+
+// Individual bitLen tests. Numbers chosen to examine both sides
+// of powers-of-two boundaries.
+func BenchmarkBitLen0(b *testing.B) { benchmarkBitLenN(b, 0) }
+func BenchmarkBitLen1(b *testing.B) { benchmarkBitLenN(b, 1) }
+func BenchmarkBitLen2(b *testing.B) { benchmarkBitLenN(b, 2) }
+func BenchmarkBitLen3(b *testing.B) { benchmarkBitLenN(b, 3) }
+func BenchmarkBitLen4(b *testing.B) { benchmarkBitLenN(b, 4) }
+func BenchmarkBitLen5(b *testing.B) { benchmarkBitLenN(b, 5) }
+func BenchmarkBitLen8(b *testing.B) { benchmarkBitLenN(b, 8) }
+func BenchmarkBitLen9(b *testing.B) { benchmarkBitLenN(b, 9) }
+func BenchmarkBitLen16(b *testing.B) { benchmarkBitLenN(b, 16) }
+func BenchmarkBitLen17(b *testing.B) { benchmarkBitLenN(b, 17) }
+func BenchmarkBitLen31(b *testing.B) { benchmarkBitLenN(b, 31) }
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements the Bits type used for testing Float operations
+// via an independent (albeit slower) representations for floating-point
+// numbers.
+
+package big
+
+import (
+ "fmt"
+ "sort"
+ "testing"
+)
+
+// A Bits value b represents a finite floating-point number x of the form
+//
+// x = 2**b[0] + 2**b[1] + ... 2**b[len(b)-1]
+//
+// The order of slice elements is not significant. Negative elements may be
+// used to form fractions. A Bits value is normalized if each b[i] occurs at
+// most once. For instance Bits{0, 0, 1} is not normalized but represents the
+// same floating-point number as Bits{2}, which is normalized. The zero (nil)
+// value of Bits is a ready to use Bits value and represents the value 0.
+type Bits []int
+
+func (x Bits) add(y Bits) Bits {
+ return append(x, y...)
+}
+
+func (x Bits) mul(y Bits) Bits {
+ var p Bits
+ for _, x := range x {
+ for _, y := range y {
+ p = append(p, x+y)
+ }
+ }
+ return p
+}
+
+func TestMulBits(t *testing.T) {
+ for _, test := range []struct {
+ x, y, want Bits
+ }{
+ {nil, nil, nil},
+ {Bits{}, Bits{}, nil},
+ {Bits{0}, Bits{0}, Bits{0}},
+ {Bits{0}, Bits{1}, Bits{1}},
+ {Bits{1}, Bits{1, 2, 3}, Bits{2, 3, 4}},
+ {Bits{-1}, Bits{1}, Bits{0}},
+ {Bits{-10, -1, 0, 1, 10}, Bits{1, 2, 3}, Bits{-9, -8, -7, 0, 1, 2, 1, 2, 3, 2, 3, 4, 11, 12, 13}},
+ } {
+ got := fmt.Sprintf("%v", test.x.mul(test.y))
+ want := fmt.Sprintf("%v", test.want)
+ if got != want {
+ t.Errorf("%v * %v = %s; want %s", test.x, test.y, got, want)
+ }
+
+ }
+}
+
+// norm returns the normalized bits for x: It removes multiple equal entries
+// by treating them as an addition (e.g., Bits{5, 5} => Bits{6}), and it sorts
+// the result list for reproducible results.
+func (x Bits) norm() Bits {
+ m := make(map[int]bool)
+ for _, b := range x {
+ for m[b] {
+ m[b] = false
+ b++
+ }
+ m[b] = true
+ }
+ var z Bits
+ for b, set := range m {
+ if set {
+ z = append(z, b)
+ }
+ }
+ sort.Ints([]int(z))
+ return z
+}
+
+func TestNormBits(t *testing.T) {
+ for _, test := range []struct {
+ x, want Bits
+ }{
+ {nil, nil},
+ {Bits{}, Bits{}},
+ {Bits{0}, Bits{0}},
+ {Bits{0, 0}, Bits{1}},
+ {Bits{3, 1, 1}, Bits{2, 3}},
+ {Bits{10, 9, 8, 7, 6, 6}, Bits{11}},
+ } {
+ got := fmt.Sprintf("%v", test.x.norm())
+ want := fmt.Sprintf("%v", test.want)
+ if got != want {
+ t.Errorf("normBits(%v) = %s; want %s", test.x, got, want)
+ }
+
+ }
+}
+
+// round returns the Float value corresponding to x after rounding x
+// to prec bits according to mode.
+func (x Bits) round(prec uint, mode RoundingMode) *Float {
+ x = x.norm()
+
+ // determine range
+ var min, max int
+ for i, b := range x {
+ if i == 0 || b < min {
+ min = b
+ }
+ if i == 0 || b > max {
+ max = b
+ }
+ }
+ prec0 := uint(max + 1 - min)
+ if prec >= prec0 {
+ return x.Float()
+ }
+ // prec < prec0
+
+ // determine bit 0, rounding, and sticky bit, and result bits z
+ var bit0, rbit, sbit uint
+ var z Bits
+ r := max - int(prec)
+ for _, b := range x {
+ switch {
+ case b == r:
+ rbit = 1
+ case b < r:
+ sbit = 1
+ default:
+ // b > r
+ if b == r+1 {
+ bit0 = 1
+ }
+ z = append(z, b)
+ }
+ }
+
+ // round
+ f := z.Float() // rounded to zero
+ if mode == ToNearestAway {
+ panic("not yet implemented")
+ }
+ if mode == ToNearestEven && rbit == 1 && (sbit == 1 || sbit == 0 && bit0 != 0) || mode == AwayFromZero {
+ // round away from zero
+ f.SetMode(ToZero).SetPrec(prec)
+ f.Add(f, Bits{int(r) + 1}.Float())
+ }
+ return f
+}
+
+// Float returns the *Float z of the smallest possible precision such that
+// z = sum(2**bits[i]), with i = range bits. If multiple bits[i] are equal,
+// they are added: Bits{0, 1, 0}.Float() == 2**0 + 2**1 + 2**0 = 4.
+func (bits Bits) Float() *Float {
+ // handle 0
+ if len(bits) == 0 {
+ return new(Float)
+ }
+ // len(bits) > 0
+
+ // determine lsb exponent
+ var min int
+ for i, b := range bits {
+ if i == 0 || b < min {
+ min = b
+ }
+ }
+
+ // create bit pattern
+ x := NewInt(0)
+ for _, b := range bits {
+ badj := b - min
+ // propagate carry if necessary
+ for x.Bit(badj) != 0 {
+ x.SetBit(x, badj, 0)
+ badj++
+ }
+ x.SetBit(x, badj, 1)
+ }
+
+ // create corresponding float
+ z := new(Float).SetInt(x) // normalized
+ if e := int64(z.exp) + int64(min); MinExp <= e && e <= MaxExp {
+ z.exp = int32(e)
+ } else {
+ // this should never happen for our test cases
+ panic("exponent out of range")
+ }
+ return z
+}
+
+func TestFromBits(t *testing.T) {
+ for _, test := range []struct {
+ bits Bits
+ want string
+ }{
+ // all different bit numbers
+ {nil, "0"},
+ {Bits{0}, "0x.8p1"},
+ {Bits{1}, "0x.8p2"},
+ {Bits{-1}, "0x.8p0"},
+ {Bits{63}, "0x.8p64"},
+ {Bits{33, -30}, "0x.8000000000000001p34"},
+ {Bits{255, 0}, "0x.8000000000000000000000000000000000000000000000000000000000000001p256"},
+
+ // multiple equal bit numbers
+ {Bits{0, 0}, "0x.8p2"},
+ {Bits{0, 0, 0, 0}, "0x.8p3"},
+ {Bits{0, 1, 0}, "0x.8p3"},
+ {append(Bits{2, 1, 0} /* 7 */, Bits{3, 1} /* 10 */ ...), "0x.88p5" /* 17 */},
+ } {
+ f := test.bits.Float()
+ if got := f.Format('p', 0); got != test.want {
+ t.Errorf("setBits(%v) = %s; want %s", test.bits, got, test.want)
+ }
+ }
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file prints execution times for the Mul benchmark
+// given different Karatsuba thresholds. The result may be
+// used to manually fine-tune the threshold constant. The
+// results are somewhat fragile; use repeated runs to get
+// a clear picture.
+
+// Usage: go test -run=TestCalibrate -calibrate
+
+package big
+
+import (
+ "flag"
+ "fmt"
+ "testing"
+ "time"
+)
+
+var calibrate = flag.Bool("calibrate", false, "run calibration test")
+
+func karatsubaLoad(b *testing.B) {
+ BenchmarkMul(b)
+}
+
+// measureKaratsuba returns the time to run a Karatsuba-relevant benchmark
+// given Karatsuba threshold th.
+func measureKaratsuba(th int) time.Duration {
+ th, karatsubaThreshold = karatsubaThreshold, th
+ res := testing.Benchmark(karatsubaLoad)
+ karatsubaThreshold = th
+ return time.Duration(res.NsPerOp())
+}
+
+func computeThresholds() {
+ fmt.Printf("Multiplication times for varying Karatsuba thresholds\n")
+ fmt.Printf("(run repeatedly for good results)\n")
+
+ // determine Tk, the work load execution time using basic multiplication
+ Tb := measureKaratsuba(1e9) // th == 1e9 => Karatsuba multiplication disabled
+ fmt.Printf("Tb = %10s\n", Tb)
+
+ // thresholds
+ th := 4
+ th1 := -1
+ th2 := -1
+
+ var deltaOld time.Duration
+ for count := -1; count != 0 && th < 128; count-- {
+ // determine Tk, the work load execution time using Karatsuba multiplication
+ Tk := measureKaratsuba(th)
+
+ // improvement over Tb
+ delta := (Tb - Tk) * 100 / Tb
+
+ fmt.Printf("th = %3d Tk = %10s %4d%%", th, Tk, delta)
+
+ // determine break-even point
+ if Tk < Tb && th1 < 0 {
+ th1 = th
+ fmt.Print(" break-even point")
+ }
+
+ // determine diminishing return
+ if 0 < delta && delta < deltaOld && th2 < 0 {
+ th2 = th
+ fmt.Print(" diminishing return")
+ }
+ deltaOld = delta
+
+ fmt.Println()
+
+ // trigger counter
+ if th1 >= 0 && th2 >= 0 && count < 0 {
+ count = 10 // this many extra measurements after we got both thresholds
+ }
+
+ th++
+ }
+}
+
+func TestCalibrate(t *testing.T) {
+ if *calibrate {
+ computeThresholds()
+ }
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision decimal numbers.
+// The implementation is for float to decimal conversion only;
+// not general purpose use.
+// The only operations are precise conversion from binary to
+// decimal and rounding.
+//
+// The key observation and some code (shr) is borrowed from
+// strconv/decimal.go: conversion of binary fractional values can be done
+// precisely in multi-precision decimal because 2 divides 10 (required for
+// >> of mantissa); but conversion of decimal floating-point values cannot
+// be done precisely in binary representation.
+//
+// In contrast to strconv/decimal.go, only right shift is implemented in
+// decimal format - left shift can be done precisely in binary format.
+
+package big
+
+// A decimal represents a floating-point number in decimal representation.
+// The value of a decimal x is x.mant * 10 ** x.exp with 0.5 <= x.mant < 1,
+// with the most-significant mantissa digit at index 0.
+type decimal struct {
+ mant []byte // mantissa ASCII digits, big-endian
+ exp int // exponent, valid if len(mant) > 0
+}
+
+// Maximum shift amount that can be done in one pass without overflow.
+// A Word has _W bits and (1<<maxShift - 1)*10 + 9 must fit into Word.
+const maxShift = _W - 4
+
+// TODO(gri) Since we know the desired decimal precision when converting
+// a floating-point number, we may be able to limit the number of decimal
+// digits that need to be computed by init by providing an additional
+// precision argument and keeping track of when a number was truncated early
+// (equivalent of "sticky bit" in binary rounding).
+
+// TODO(gri) Along the same lines, enforce some limit to shift magnitudes
+// to avoid "infinitely" long running conversions (until we run out of space).
+
+// Init initializes x to the decimal representation of m << shift (for
+// shift >= 0), or m >> -shift (for shift < 0).
+func (x *decimal) init(m nat, shift int) {
+ // special case 0
+ if len(m) == 0 {
+ x.mant = x.mant[:0]
+ return
+ }
+
+ // Optimization: If we need to shift right, first remove any trailing
+ // zero bits from m to reduce shift amount that needs to be done in
+ // decimal format (since that is likely slower).
+ if shift < 0 {
+ ntz := m.trailingZeroBits()
+ s := uint(-shift)
+ if s >= ntz {
+ s = ntz // shift at most ntz bits
+ }
+ m = nat(nil).shr(m, s)
+ shift += int(s)
+ }
+
+ // Do any shift left in binary representation.
+ if shift > 0 {
+ m = nat(nil).shl(m, uint(shift))
+ shift = 0
+ }
+
+ // Convert mantissa into decimal representation.
+ s := m.decimalString() // TODO(gri) avoid string conversion here
+ n := len(s)
+ x.exp = n
+ // Trim trailing zeros; instead the exponent is tracking
+ // the decimal point independent of the number of digits.
+ for n > 0 && s[n-1] == '0' {
+ n--
+ }
+ x.mant = append(x.mant[:0], s[:n]...)
+
+ // Do any (remaining) shift right in decimal representation.
+ if shift < 0 {
+ for shift < -maxShift {
+ shr(x, maxShift)
+ shift += maxShift
+ }
+ shr(x, uint(-shift))
+ }
+}
+
+// Possibly optimization: The current implementation of nat.string takes
+// a charset argument. When a right shift is needed, we could provide
+// "\x00\x01...\x09" instead of "012..9" (as in nat.decimalString) and
+// avoid the repeated +'0' and -'0' operations in decimal.shr (and do a
+// single +'0' pass at the end).
+
+// shr implements x >> s, for s <= maxShift.
+func shr(x *decimal, s uint) {
+ // Division by 1<<s using shift-and-subtract algorithm.
+
+ // pick up enough leading digits to cover first shift
+ r := 0 // read index
+ var n Word
+ for n>>s == 0 && r < len(x.mant) {
+ ch := Word(x.mant[r])
+ r++
+ n = n*10 + ch - '0'
+ }
+ if n == 0 {
+ // x == 0; shouldn't get here, but handle anyway
+ x.mant = x.mant[:0]
+ return
+ }
+ for n>>s == 0 {
+ r++
+ n *= 10
+ }
+ x.exp += 1 - r
+
+ // read a digit, write a digit
+ w := 0 // write index
+ for r < len(x.mant) {
+ ch := Word(x.mant[r])
+ r++
+ d := n >> s
+ n -= d << s
+ x.mant[w] = byte(d + '0')
+ w++
+ n = n*10 + ch - '0'
+ }
+
+ // write extra digits that still fit
+ for n > 0 && w < len(x.mant) {
+ d := n >> s
+ n -= d << s
+ x.mant[w] = byte(d + '0')
+ w++
+ n = n * 10
+ }
+ x.mant = x.mant[:w] // the number may be shorter (e.g. 1024 >> 10)
+
+ // append additional digits that didn't fit
+ for n > 0 {
+ d := n >> s
+ n -= d << s
+ x.mant = append(x.mant, byte(d+'0'))
+ n = n * 10
+ }
+
+ trim(x)
+}
+
+func (x *decimal) String() string {
+ if len(x.mant) == 0 {
+ return "0"
+ }
+
+ var buf []byte
+ switch {
+ case x.exp <= 0:
+ // 0.00ddd
+ buf = append(buf, "0."...)
+ buf = appendZeros(buf, -x.exp)
+ buf = append(buf, x.mant...)
+
+ case /* 0 < */ x.exp < len(x.mant):
+ // dd.ddd
+ buf = append(buf, x.mant[:x.exp]...)
+ buf = append(buf, '.')
+ buf = append(buf, x.mant[x.exp:]...)
+
+ default: // len(x.mant) <= x.exp
+ // ddd00
+ buf = append(buf, x.mant...)
+ buf = appendZeros(buf, x.exp-len(x.mant))
+ }
+
+ return string(buf)
+}
+
+// appendZeros appends n 0 digits to buf and returns buf.
+func appendZeros(buf []byte, n int) []byte {
+ for ; n > 0; n-- {
+ buf = append(buf, '0')
+ }
+ return buf
+}
+
+// shouldRoundUp reports if x should be rounded up
+// if shortened to n digits. n must be a valid index
+// for x.mant.
+func shouldRoundUp(x *decimal, n int) bool {
+ if x.mant[n] == '5' && n+1 == len(x.mant) {
+ // exactly halfway - round to even
+ return n > 0 && (x.mant[n-1]-'0')&1 != 0
+ }
+ // not halfway - digit tells all (x.mant has no trailing zeros)
+ return x.mant[n] >= '5'
+}
+
+// round sets x to (at most) n mantissa digits by rounding it
+// to the nearest even value with n (or fever) mantissa digits.
+// If n < 0, x remains unchanged.
+func (x *decimal) round(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+
+ if shouldRoundUp(x, n) {
+ x.roundUp(n)
+ } else {
+ x.roundDown(n)
+ }
+}
+
+func (x *decimal) roundUp(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+ // 0 <= n < len(x.mant)
+
+ // find first digit < '9'
+ for n > 0 && x.mant[n-1] >= '9' {
+ n--
+ }
+
+ if n == 0 {
+ // all digits are '9's => round up to '1' and update exponent
+ x.mant[0] = '1' // ok since len(x.mant) > n
+ x.mant = x.mant[:1]
+ x.exp++
+ return
+ }
+
+ // n > 0 && x.mant[n-1] < '9'
+ x.mant[n-1]++
+ x.mant = x.mant[:n]
+ // x already trimmed
+}
+
+func (x *decimal) roundDown(n int) {
+ if n < 0 || n >= len(x.mant) {
+ return // nothing to do
+ }
+ x.mant = x.mant[:n]
+ trim(x)
+}
+
+// trim cuts off any trailing zeros from x's mantissa;
+// they are meaningless for the value of x.
+func trim(x *decimal) {
+ i := len(x.mant)
+ for i > 0 && x.mant[i-1] == '0' {
+ i--
+ }
+ x.mant = x.mant[:i]
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import "testing"
+
+func TestDecimalString(t *testing.T) {
+ for _, test := range []struct {
+ x decimal
+ want string
+ }{
+ {want: "0"},
+ {decimal{nil, 1000}, "0"}, // exponent of 0 is ignored
+ {decimal{[]byte("12345"), 0}, "0.12345"},
+ {decimal{[]byte("12345"), -3}, "0.00012345"},
+ {decimal{[]byte("12345"), +3}, "123.45"},
+ {decimal{[]byte("12345"), +10}, "1234500000"},
+ } {
+ if got := test.x.String(); got != test.want {
+ t.Errorf("%v == %s; want %s", test.x, got, test.want)
+ }
+ }
+}
+
+func TestDecimalInit(t *testing.T) {
+ for _, test := range []struct {
+ x Word
+ shift int
+ want string
+ }{
+ {0, 0, "0"},
+ {0, -100, "0"},
+ {0, 100, "0"},
+ {1, 0, "1"},
+ {1, 10, "1024"},
+ {1, 100, "1267650600228229401496703205376"},
+ {1, -100, "0.0000000000000000000000000000007888609052210118054117285652827862296732064351090230047702789306640625"},
+ {12345678, 8, "3160493568"},
+ {12345678, -8, "48225.3046875"},
+ {195312, 9, "99999744"},
+ {1953125, 9, "1000000000"},
+ } {
+ var d decimal
+ d.init(nat{test.x}.norm(), test.shift)
+ if got := d.String(); got != test.want {
+ t.Errorf("%d << %d == %s; want %s", test.x, test.shift, got, test.want)
+ }
+ }
+}
+
+func TestDecimalRounding(t *testing.T) {
+ for _, test := range []struct {
+ x uint64
+ n int
+ down, even, up string
+ }{
+ {0, 0, "0", "0", "0"},
+ {0, 1, "0", "0", "0"},
+
+ {1, 0, "0", "0", "10"},
+ {5, 0, "0", "0", "10"},
+ {9, 0, "0", "10", "10"},
+
+ {15, 1, "10", "20", "20"},
+ {45, 1, "40", "40", "50"},
+ {95, 1, "90", "100", "100"},
+
+ {12344999, 4, "12340000", "12340000", "12350000"},
+ {12345000, 4, "12340000", "12340000", "12350000"},
+ {12345001, 4, "12340000", "12350000", "12350000"},
+ {23454999, 4, "23450000", "23450000", "23460000"},
+ {23455000, 4, "23450000", "23460000", "23460000"},
+ {23455001, 4, "23450000", "23460000", "23460000"},
+
+ {99994999, 4, "99990000", "99990000", "100000000"},
+ {99995000, 4, "99990000", "100000000", "100000000"},
+ {99999999, 4, "99990000", "100000000", "100000000"},
+
+ {12994999, 4, "12990000", "12990000", "13000000"},
+ {12995000, 4, "12990000", "13000000", "13000000"},
+ {12999999, 4, "12990000", "13000000", "13000000"},
+ } {
+ x := nat(nil).setUint64(test.x)
+
+ var d decimal
+ d.init(x, 0)
+ d.roundDown(test.n)
+ if got := d.String(); got != test.down {
+ t.Errorf("roundDown(%d, %d) = %s; want %s", test.x, test.n, got, test.down)
+ }
+
+ d.init(x, 0)
+ d.round(test.n)
+ if got := d.String(); got != test.even {
+ t.Errorf("round(%d, %d) = %s; want %s", test.x, test.n, got, test.even)
+ }
+
+ d.init(x, 0)
+ d.roundUp(test.n)
+ if got := d.String(); got != test.up {
+ t.Errorf("roundUp(%d, %d) = %s; want %s", test.x, test.n, got, test.up)
+ }
+ }
+}
--- /dev/null
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "log"
+ "math/big"
+)
+
+func ExampleRat_SetString() {
+ r := new(big.Rat)
+ r.SetString("355/113")
+ fmt.Println(r.FloatString(3))
+ // Output: 3.142
+}
+
+func ExampleInt_SetString() {
+ i := new(big.Int)
+ i.SetString("644", 8) // octal
+ fmt.Println(i)
+ // Output: 420
+}
+
+func ExampleRat_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ r := new(big.Rat)
+ _, err := fmt.Sscan("1.5000", r)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(r)
+ }
+ // Output: 3/2
+}
+
+func ExampleInt_Scan() {
+ // The Scan function is rarely used directly;
+ // the fmt package recognizes it as an implementation of fmt.Scanner.
+ i := new(big.Int)
+ _, err := fmt.Sscan("18446744073709551617", i)
+ if err != nil {
+ log.Println("error scanning value:", err)
+ } else {
+ fmt.Println(i)
+ }
+ // Output: 18446744073709551617
+}
--- /dev/null
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision floating-point numbers.
+// Like in the GNU MPFR library (http://www.mpfr.org/), operands
+// can be of mixed precision. Unlike MPFR, the rounding mode is
+// not specified with each operation, but with each operand. The
+// rounding mode of the result operand determines the rounding
+// mode of an operation. This is a from-scratch implementation.
+
+package big
+
+import (
+ "fmt"
+ "math"
+)
+
+const debugFloat = true // enable for debugging
+
+// A nonzero finite Float represents a multi-precision floating point number
+//
+// sign × mantissa × 2**exponent
+//
+// with 0.5 <= mantissa < 1.0, and MinExp <= exponent <= MaxExp.
+// A Float may also be zero (+0, -0), infinite (+Inf, -Inf) or
+// not-a-number (NaN). Except for NaNs, all Floats are ordered,
+// and the ordering of two Floats x and y is defined by x.Cmp(y).
+// NaNs are always different from any other Float value.
+//
+// Each Float value also has a precision, rounding mode, and accuracy.
+// The precision is the maximum number of mantissa bits available to
+// represent the value. The rounding mode specifies how a result should
+// be rounded to fit into the mantissa bits, and accuracy describes the
+// rounding error with respect to the exact result.
+//
+// All operations, including setters, that specify a *Float variable for
+// the result (usually via the receiver with the exception of MantExp),
+// round the numeric result according to the precision and rounding mode
+// of the result variable, unless specified otherwise.
+//
+// If the provided result precision is 0 (see below), it is set to the
+// precision of the argument with the largest precision value before any
+// rounding takes place, and the rounding mode remains unchanged. Thus,
+// uninitialized Floats provided as result arguments will have their
+// precision set to a reasonable value determined by the operands and
+// their mode is the zero value for RoundingMode (ToNearestEven).
+//
+// By setting the desired precision to 24 or 53 and using matching rounding
+// mode (typically ToNearestEven), Float operations produce the same results
+// as the corresponding float32 or float64 IEEE-754 arithmetic. Exponent
+// underflow and overflow lead to a 0 or an Infinity for different values
+// than IEEE-754 because Float exponents have a much larger range.
+//
+// The zero (uninitialized) value for a Float is ready to use and represents
+// the number +0.0 exactly, with precision 0 and rounding mode ToNearestEven.
+//
+type Float struct {
+ prec uint32
+ mode RoundingMode
+ acc Accuracy
+ form form
+ neg bool
+ mant nat
+ exp int32
+}
+
+// NewFloat allocates and returns a new Float set to x,
+// with precision 53 and rounding mode ToNearestEven.
+func NewFloat(x float64) *Float {
+ return new(Float).SetFloat64(x)
+}
+
+// Exponent and precision limits.
+const (
+ MaxExp = math.MaxInt32 // largest supported exponent
+ MinExp = math.MinInt32 // smallest supported exponent
+ MaxPrec = math.MaxUint32 // largest (theoretically) supported precision; likely memory-limited
+)
+
+// Internal representation: The mantissa bits x.mant of a nonzero finite
+// Float x are stored in a nat slice long enough to hold up to x.prec bits;
+// the slice may (but doesn't have to) be shorter if the mantissa contains
+// trailing 0 bits. x.mant is normalized if the msb of x.mant == 1 (i.e.,
+// the msb is shifted all the way "to the left"). Thus, if the mantissa has
+// trailing 0 bits or x.prec is not a multiple of the the Word size _W,
+// x.mant[0] has trailing zero bits. The msb of the mantissa corresponds
+// to the value 0.5; the exponent x.exp shifts the binary point as needed.
+//
+// A zero or non-finite Float x ignores x.mant and x.exp. A NaN x ignores
+// the sign x.neg.
+//
+// x form neg mant exp
+// ----------------------------------------------------------
+// ±0 zero sign - -
+// 0 < |x| < +Inf finite sign mantissa exponent
+// ±Inf inf sign - -
+// NaN nan - - -
+
+// A form value describes the internal representation.
+type form byte
+
+// The form value order is relevant - do not change!
+const (
+ zero form = iota
+ finite
+ inf
+ nan
+)
+
+// RoundingMode determines how a Float value is rounded to the
+// desired precision. Rounding may change the Float value; the
+// rounding error is described by the Float's Accuracy.
+type RoundingMode byte
+
+// The following rounding modes are supported.
+const (
+ ToNearestEven RoundingMode = iota // == IEEE 754-2008 roundTiesToEven
+ ToNearestAway // == IEEE 754-2008 roundTiesToAway
+ ToZero // == IEEE 754-2008 roundTowardZero
+ AwayFromZero // no IEEE 754-2008 equivalent
+ ToNegativeInf // == IEEE 754-2008 roundTowardNegative
+ ToPositiveInf // == IEEE 754-2008 roundTowardPositive
+)
+
+//go:generate stringer -type=RoundingMode
+
+// Accuracy describes the rounding error produced by the most recent
+// operation that generated a Float value, relative to the exact value.
+// The accuracy is Undef for operations on and resulting in NaNs since
+// they are neither Below nor Above any other value.
+type Accuracy byte
+
+// Constants describing the Accuracy of a Float.
+const (
+ Exact Accuracy = 0
+ Below Accuracy = 1 << 0
+ Above Accuracy = 1 << 1
+ Undef Accuracy = Below | Above
+)
+
+//go:generate stringer -type=Accuracy
+
+// SetPrec sets z's precision to prec and returns the (possibly) rounded
+// value of z. Rounding occurs according to z's rounding mode if the mantissa
+// cannot be represented in prec bits without loss of precision.
+// SetPrec(0) maps all finite values to ±0; infinite and NaN values remain
+// unchanged. If prec > MaxPrec, it is set to MaxPrec.
+func (z *Float) SetPrec(prec uint) *Float {
+ z.acc = Exact // optimistically assume no rounding is needed
+
+ // special case
+ if prec == 0 {
+ z.prec = 0
+ if z.form == finite {
+ // truncate z to 0
+ z.acc = makeAcc(z.neg)
+ z.form = zero
+ }
+ return z
+ }
+
+ // general case
+ if prec > MaxPrec {
+ prec = MaxPrec
+ }
+ old := z.prec
+ z.prec = uint32(prec)
+ if z.prec < old {
+ z.round(0)
+ }
+ return z
+}
+
+func makeAcc(above bool) Accuracy {
+ if above {
+ return Above
+ }
+ return Below
+}
+
+// SetMode sets z's rounding mode to mode and returns an exact z.
+// z remains unchanged otherwise.
+// z.SetMode(z.Mode()) is a cheap way to set z's accuracy to Exact.
+func (z *Float) SetMode(mode RoundingMode) *Float {
+ z.mode = mode
+ z.acc = Exact
+ return z
+}
+
+// Prec returns the mantissa precision of x in bits.
+// The result may be 0 for |x| == 0, |x| == Inf, or NaN.
+func (x *Float) Prec() uint {
+ return uint(x.prec)
+}
+
+// MinPrec returns the minimum precision required to represent x exactly
+// (i.e., the smallest prec before x.SetPrec(prec) would start rounding x).
+// The result is 0 if x is 0 or not finite.
+func (x *Float) MinPrec() uint {
+ if x.form != finite {
+ return 0
+ }
+ return uint(len(x.mant))*_W - x.mant.trailingZeroBits()
+}
+
+// Mode returns the rounding mode of x.
+func (x *Float) Mode() RoundingMode {
+ return x.mode
+}
+
+// Acc returns the accuracy of x produced by the most recent operation.
+func (x *Float) Acc() Accuracy {
+ return x.acc
+}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x is ±0 or NaN
+// +1 if x > 0
+//
+func (x *Float) Sign() int {
+ if debugFloat {
+ x.validate()
+ }
+ if x.form == zero || x.form == nan {
+ return 0
+ }
+ if x.neg {
+ return -1
+ }
+ return 1
+}
+
+// MantExp breaks x into its mantissa and exponent components
+// and returns the exponent. If a non-nil mant argument is
+// provided its value is set to the mantissa of x, with the
+// same precision and rounding mode as x. The components
+// satisfy x == mant × 2**exp, with 0.5 <= |mant| < 1.0.
+// Calling MantExp with a nil argument is an efficient way to
+// get the exponent of the receiver.
+//
+// Special cases are:
+//
+// ( ±0).MantExp(mant) = 0, with mant set to ±0
+// (±Inf).MantExp(mant) = 0, with mant set to ±Inf
+// ( NaN).MantExp(mant) = 0, with mant set to NaN
+//
+// x and mant may be the same in which case x is set to its
+// mantissa value.
+func (x *Float) MantExp(mant *Float) (exp int) {
+ if debugFloat {
+ x.validate()
+ }
+ if x.form == finite {
+ exp = int(x.exp)
+ }
+ if mant != nil {
+ mant.Copy(x)
+ if mant.form == finite {
+ mant.exp = 0
+ }
+ }
+ return
+}
+
+func (z *Float) setExpAndRound(exp int64, sbit uint) {
+ if exp < MinExp {
+ // underflow
+ z.acc = makeAcc(z.neg)
+ z.form = zero
+ return
+ }
+
+ if exp > MaxExp {
+ // overflow
+ z.acc = makeAcc(!z.neg)
+ z.form = inf
+ return
+ }
+
+ z.form = finite
+ z.exp = int32(exp)
+ z.round(sbit)
+}
+
+// SetMantExp sets z to mant × 2**exp and and returns z.
+// The result z has the same precision and rounding mode
+// as mant. SetMantExp is an inverse of MantExp but does
+// not require 0.5 <= |mant| < 1.0. Specifically:
+//
+// mant := new(Float)
+// new(Float).SetMantExp(mant, x.SetMantExp(mant)).Cmp(x).Eql() is true
+//
+// Special cases are:
+//
+// z.SetMantExp( ±0, exp) = ±0
+// z.SetMantExp(±Inf, exp) = ±Inf
+// z.SetMantExp( NaN, exp) = NaN
+//
+// z and mant may be the same in which case z's exponent
+// is set to exp.
+func (z *Float) SetMantExp(mant *Float, exp int) *Float {
+ if debugFloat {
+ z.validate()
+ mant.validate()
+ }
+ z.Copy(mant)
+ if z.form != finite {
+ return z
+ }
+ z.setExpAndRound(int64(z.exp)+int64(exp), 0)
+ return z
+}
+
+// IsNeg reports whether x is negative.
+// A NaN value is not negative.
+func (x *Float) IsNeg() bool {
+ return x.neg && x.form != nan
+}
+
+// IsZero reports whether x is +0 or -0.
+func (x *Float) IsZero() bool {
+ return x.form == zero
+}
+
+// IsFinite reports whether -Inf < x < Inf.
+// A NaN value is not finite.
+func (x *Float) IsFinite() bool {
+ return x.form <= finite
+}
+
+// IsInf reports whether x is +Inf or -Inf.
+func (x *Float) IsInf() bool {
+ return x.form == inf
+}
+
+// IsNaN reports whether x is a NaN value.
+func (x *Float) IsNaN() bool {
+ return x.form == nan
+}
+
+// IsInt reports whether x is an integer.
+// ±Inf and NaN values are not integers.
+func (x *Float) IsInt() bool {
+ if debugFloat {
+ x.validate()
+ }
+ // special cases
+ if x.form != finite {
+ return x.form == zero
+ }
+ // x.form == finite
+ if x.exp <= 0 {
+ return false
+ }
+ // x.exp > 0
+ return x.prec <= uint32(x.exp) || x.MinPrec() <= uint(x.exp) // not enough bits for fractional mantissa
+}
+
+// debugging support
+func (x *Float) validate() {
+ if !debugFloat {
+ // avoid performance bugs
+ panic("validate called but debugFloat is not set")
+ }
+ if x.form != finite {
+ return
+ }
+ m := len(x.mant)
+ if m == 0 {
+ panic("nonzero finite number with empty mantissa")
+ }
+ const msb = 1 << (_W - 1)
+ if x.mant[m-1]&msb == 0 {
+ panic(fmt.Sprintf("msb not set in last word %#x of %s", x.mant[m-1], x.Format('p', 0)))
+ }
+ if x.prec == 0 {
+ panic("zero precision finite number")
+ }
+}
+
+// round rounds z according to z.mode to z.prec bits and sets z.acc accordingly.
+// sbit must be 0 or 1 and summarizes any "sticky bit" information one might
+// have before calling round. z's mantissa must be normalized (with the msb set)
+// or empty.
+//
+// CAUTION: The rounding modes ToNegativeInf, ToPositiveInf are affected by the
+// sign of z. For correct rounding, the sign of z must be set correctly before
+// calling round.
+func (z *Float) round(sbit uint) {
+ if debugFloat {
+ z.validate()
+ if z.form > finite {
+ panic(fmt.Sprintf("round called for non-finite value %s", z))
+ }
+ }
+ // z.form <= finite
+
+ z.acc = Exact
+ if z.form == zero {
+ return
+ }
+ // z.form == finite && len(z.mant) > 0
+ // m > 0 implies z.prec > 0 (checked by validate)
+
+ m := uint32(len(z.mant)) // present mantissa length in words
+ bits := m * _W // present mantissa bits
+ if bits <= z.prec {
+ // mantissa fits => nothing to do
+ return
+ }
+ // bits > z.prec
+
+ n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
+
+ // Rounding is based on two bits: the rounding bit (rbit) and the
+ // sticky bit (sbit). The rbit is the bit immediately before the
+ // z.prec leading mantissa bits (the "0.5"). The sbit is set if any
+ // of the bits before the rbit are set (the "0.25", "0.125", etc.):
+ //
+ // rbit sbit => "fractional part"
+ //
+ // 0 0 == 0
+ // 0 1 > 0 , < 0.5
+ // 1 0 == 0.5
+ // 1 1 > 0.5, < 1.0
+
+ // bits > z.prec: mantissa too large => round
+ r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
+ rbit := z.mant.bit(r) // rounding bit
+ if sbit == 0 {
+ sbit = z.mant.sticky(r)
+ }
+ if debugFloat && sbit&^1 != 0 {
+ panic(fmt.Sprintf("invalid sbit %#x", sbit))
+ }
+
+ // convert ToXInf rounding modes
+ mode := z.mode
+ switch mode {
+ case ToNegativeInf:
+ mode = ToZero
+ if z.neg {
+ mode = AwayFromZero
+ }
+ case ToPositiveInf:
+ mode = AwayFromZero
+ if z.neg {
+ mode = ToZero
+ }
+ }
+
+ // cut off extra words
+ if m > n {
+ copy(z.mant, z.mant[m-n:]) // move n last words to front
+ z.mant = z.mant[:n]
+ }
+
+ // determine number of trailing zero bits t
+ t := n*_W - z.prec // 0 <= t < _W
+ lsb := Word(1) << t
+
+ // make rounding decision
+ // TODO(gri) This can be simplified (see Bits.round in bits_test.go).
+ switch mode {
+ case ToZero:
+ // nothing to do
+ case ToNearestEven, ToNearestAway:
+ if rbit == 0 {
+ // rounding bits == 0b0x
+ mode = ToZero
+ } else if sbit == 1 {
+ // rounding bits == 0b11
+ mode = AwayFromZero
+ }
+ case AwayFromZero:
+ if rbit|sbit == 0 {
+ mode = ToZero
+ }
+ default:
+ // ToXInf modes have been converted to ToZero or AwayFromZero
+ panic("unreachable")
+ }
+
+ // round and determine accuracy
+ switch mode {
+ case ToZero:
+ if rbit|sbit != 0 {
+ z.acc = Below
+ }
+
+ case ToNearestEven, ToNearestAway:
+ if debugFloat && rbit != 1 {
+ panic("internal error in rounding")
+ }
+ if mode == ToNearestEven && sbit == 0 && z.mant[0]&lsb == 0 {
+ z.acc = Below
+ break
+ }
+ // mode == ToNearestAway || sbit == 1 || z.mant[0]&lsb != 0
+ fallthrough
+
+ case AwayFromZero:
+ // add 1 to mantissa
+ if addVW(z.mant, z.mant, lsb) != 0 {
+ // overflow => shift mantissa right by 1 and add msb
+ shrVU(z.mant, z.mant, 1)
+ z.mant[n-1] |= 1 << (_W - 1)
+ // adjust exponent
+ if z.exp < MaxExp {
+ z.exp++
+ } else {
+ // exponent overflow
+ z.acc = makeAcc(!z.neg)
+ z.form = inf
+ return
+ }
+ }
+ z.acc = Above
+ }
+
+ // zero out trailing bits in least-significant word
+ z.mant[0] &^= lsb - 1
+
+ // update accuracy
+ if z.acc != Exact && z.neg {
+ z.acc ^= Below | Above
+ }
+
+ if debugFloat {
+ z.validate()
+ }
+
+ return
+}
+
+// nlz returns the number of leading zero bits in x.
+func nlz(x Word) uint {
+ return _W - uint(bitLen(x))
+}
+
+func nlz64(x uint64) uint {
+ // TODO(gri) this can be done more nicely
+ if _W == 32 {
+ if x>>32 == 0 {
+ return 32 + nlz(Word(x))
+ }
+ return nlz(Word(x >> 32))
+ }
+ if _W == 64 {
+ return nlz(Word(x))
+ }
+ panic("unreachable")
+}
+
+func (z *Float) setBits64(neg bool, x uint64) *Float {
+ if z.prec == 0 {
+ z.prec = 64
+ }
+ z.acc = Exact
+ z.neg = neg
+ if x == 0 {
+ z.form = zero
+ return z
+ }
+ // x != 0
+ z.form = finite
+ s := nlz64(x)
+ z.mant = z.mant.setUint64(x << s)
+ z.exp = int32(64 - s) // always fits
+ if z.prec < 64 {
+ z.round(0)
+ }
+ return z
+}
+
+// SetUint64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 64 (and rounding will have
+// no effect).
+func (z *Float) SetUint64(x uint64) *Float {
+ return z.setBits64(false, x)
+}
+
+// SetInt64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 64 (and rounding will have
+// no effect).
+func (z *Float) SetInt64(x int64) *Float {
+ u := x
+ if u < 0 {
+ u = -u
+ }
+ // We cannot simply call z.SetUint64(uint64(u)) and change
+ // the sign afterwards because the sign affects rounding.
+ return z.setBits64(x < 0, uint64(u))
+}
+
+// SetFloat64 sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to 53 (and rounding will have
+// no effect).
+func (z *Float) SetFloat64(x float64) *Float {
+ if z.prec == 0 {
+ z.prec = 53
+ }
+ if math.IsNaN(x) {
+ return z.SetNaN()
+ }
+ z.acc = Exact
+ z.neg = math.Signbit(x) // handle -0, -Inf correctly
+ if x == 0 {
+ z.form = zero
+ return z
+ }
+ if math.IsInf(x, 0) {
+ z.form = inf
+ return z
+ }
+ // normalized x != 0
+ z.form = finite
+ fmant, exp := math.Frexp(x) // get normalized mantissa
+ z.mant = z.mant.setUint64(1<<63 | math.Float64bits(fmant)<<11)
+ z.exp = int32(exp) // always fits
+ if z.prec < 53 {
+ z.round(0)
+ }
+ return z
+}
+
+// fnorm normalizes mantissa m by shifting it to the left
+// such that the msb of the most-significant word (msw) is 1.
+// It returns the shift amount. It assumes that len(m) != 0.
+func fnorm(m nat) int64 {
+ if debugFloat && (len(m) == 0 || m[len(m)-1] == 0) {
+ panic("msw of mantissa is 0")
+ }
+ s := nlz(m[len(m)-1])
+ if s > 0 {
+ c := shlVU(m, m, s)
+ if debugFloat && c != 0 {
+ panic("nlz or shlVU incorrect")
+ }
+ }
+ return int64(s)
+}
+
+// SetInt sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the larger of x.BitLen()
+// or 64 (and rounding will have no effect).
+func (z *Float) SetInt(x *Int) *Float {
+ // TODO(gri) can be more efficient if z.prec > 0
+ // but small compared to the size of x, or if there
+ // are many trailing 0's.
+ bits := uint32(x.BitLen())
+ if z.prec == 0 {
+ z.prec = umax32(bits, 64)
+ }
+ z.acc = Exact
+ z.neg = x.neg
+ if len(x.abs) == 0 {
+ z.form = zero
+ return z
+ }
+ // x != 0
+ z.mant = z.mant.set(x.abs)
+ fnorm(z.mant)
+ z.setExpAndRound(int64(bits), 0)
+ return z
+}
+
+// SetRat sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the largest of a.BitLen(),
+// b.BitLen(), or 64; with x = a/b.
+func (z *Float) SetRat(x *Rat) *Float {
+ if x.IsInt() {
+ return z.SetInt(x.Num())
+ }
+ var a, b Float
+ a.SetInt(x.Num())
+ b.SetInt(x.Denom())
+ if z.prec == 0 {
+ z.prec = umax32(a.prec, b.prec)
+ }
+ return z.Quo(&a, &b)
+}
+
+// SetInf sets z to the infinite Float +Inf for sign >= 0,
+// or -Inf for sign < 0, and returns z. The precision of
+// z is unchanged and the result is always Exact.
+func (z *Float) SetInf(sign int) *Float {
+ z.acc = Exact
+ z.form = inf
+ z.neg = sign < 0
+ return z
+}
+
+// SetNaN sets z to a NaN value, and returns z.
+// The precision of z is unchanged and the result accuracy is always Undef.
+func (z *Float) SetNaN() *Float {
+ z.acc = Undef
+ z.form = nan
+ return z
+}
+
+// Set sets z to the (possibly rounded) value of x and returns z.
+// If z's precision is 0, it is changed to the precision of x
+// before setting z (and rounding will have no effect).
+// Rounding is performed according to z's precision and rounding
+// mode; and z's accuracy reports the result error relative to the
+// exact (not rounded) result.
+func (z *Float) Set(x *Float) *Float {
+ if debugFloat {
+ x.validate()
+ }
+ z.acc = Exact
+ if z != x {
+ z.form = x.form
+ z.neg = x.neg
+ if x.form == finite {
+ z.exp = x.exp
+ z.mant = z.mant.set(x.mant)
+ }
+ if z.prec == 0 {
+ z.prec = x.prec
+ } else if z.prec < x.prec {
+ z.round(0)
+ }
+ }
+ return z
+}
+
+// Copy sets z to x, with the same precision, rounding mode, and
+// accuracy as x, and returns z. x is not changed even if z and
+// x are the same.
+func (z *Float) Copy(x *Float) *Float {
+ if debugFloat {
+ x.validate()
+ }
+ if z != x {
+ z.prec = x.prec
+ z.mode = x.mode
+ z.acc = x.acc
+ z.form = x.form
+ z.neg = x.neg
+ if z.form == finite {
+ z.mant = z.mant.set(x.mant)
+ z.exp = x.exp
+ }
+ }
+ return z
+}
+
+func high32(x nat) uint32 {
+ // TODO(gri) This can be done more efficiently on 32bit platforms.
+ return uint32(high64(x) >> 32)
+}
+
+func high64(x nat) uint64 {
+ i := len(x)
+ if i == 0 {
+ return 0
+ }
+ // i > 0
+ v := uint64(x[i-1])
+ if _W == 32 {
+ v <<= 32
+ if i > 1 {
+ v |= uint64(x[i-2])
+ }
+ }
+ return v
+}
+
+// Uint64 returns the unsigned integer resulting from truncating x
+// towards zero. If 0 <= x <= math.MaxUint64, the result is Exact
+// if x is an integer and Below otherwise.
+// The result is (0, Above) for x < 0, (math.MaxUint64, Below)
+// for x > math.MaxUint64, and (0, Undef) for NaNs.
+func (x *Float) Uint64() (uint64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ if x.neg {
+ return 0, Above
+ }
+ // 0 < x < +Inf
+ if x.exp <= 0 {
+ // 0 < x < 1
+ return 0, Below
+ }
+ // 1 <= x < Inf
+ if x.exp <= 64 {
+ // u = trunc(x) fits into a uint64
+ u := high64(x.mant) >> (64 - uint32(x.exp))
+ if x.MinPrec() <= 64 {
+ return u, Exact
+ }
+ return u, Below // x truncated
+ }
+ // x too large
+ return math.MaxUint64, Below
+
+ case zero:
+ return 0, Exact
+
+ case inf:
+ if x.neg {
+ return 0, Above
+ }
+ return math.MaxUint64, Below
+
+ case nan:
+ return 0, Undef
+ }
+
+ panic("unreachable")
+}
+
+// Int64 returns the integer resulting from truncating x towards zero.
+// If math.MinInt64 <= x <= math.MaxInt64, the result is Exact if x is
+// an integer, and Above (x < 0) or Below (x > 0) otherwise.
+// The result is (math.MinInt64, Above) for x < math.MinInt64,
+// (math.MaxInt64, Below) for x > math.MaxInt64, and (0, Undef) for NaNs.
+func (x *Float) Int64() (int64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ acc := makeAcc(x.neg)
+ if x.exp <= 0 {
+ // 0 < |x| < 1
+ return 0, acc
+ }
+ // x.exp > 0
+
+ // 1 <= |x| < +Inf
+ if x.exp <= 63 {
+ // i = trunc(x) fits into an int64 (excluding math.MinInt64)
+ i := int64(high64(x.mant) >> (64 - uint32(x.exp)))
+ if x.neg {
+ i = -i
+ }
+ if x.MinPrec() <= uint(x.exp) {
+ return i, Exact
+ }
+ return i, acc // x truncated
+ }
+ if x.neg {
+ // check for special case x == math.MinInt64 (i.e., x == -(0.5 << 64))
+ if x.exp == 64 && x.MinPrec() == 1 {
+ acc = Exact
+ }
+ return math.MinInt64, acc
+ }
+ // x too large
+ return math.MaxInt64, Below
+
+ case zero:
+ return 0, Exact
+
+ case inf:
+ if x.neg {
+ return math.MinInt64, Above
+ }
+ return math.MaxInt64, Below
+
+ case nan:
+ return 0, Undef
+ }
+
+ panic("unreachable")
+}
+
+// TODO(gri) Float32 and Float64 are very similar internally but for the
+// floatxx parameters and some conversions. Should factor out shared code.
+
+// Float32 returns the float32 value nearest to x. If x is too small to be
+// represented by a float32 (|x| < math.SmallestNonzeroFloat32), the result
+// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
+// If x is too large to be represented by a float32 (|x| > math.MaxFloat32),
+// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
+// The result is (NaN, Undef) for NaNs.
+func (x *Float) Float32() (float32, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+
+ const (
+ fbits = 32 // float size
+ mbits = 23 // mantissa size (excluding implicit msb)
+ ebits = fbits - mbits - 1 // 8 exponent size
+ bias = 1<<(ebits-1) - 1 // 127 exponent bias
+ dmin = 1 - bias - mbits // -149 smallest unbiased exponent (denormal)
+ emin = 1 - bias // -126 smallest unbiased exponent (normal)
+ emax = bias // 127 largest unbiased exponent (normal)
+ )
+
+ // Float mantissae m have an explicit msb and are in the range 0.5 <= m < 1.0.
+ // floatxx mantissae have an implicit msb and are in the range 1.0 <= m < 2.0.
+ // For a given mantissa m, we need to add 1 to a floatxx exponent to get the
+ // corresponding Float exponent.
+ // (see also implementation of math.Ldexp for similar code)
+
+ if x.exp < dmin+1 {
+ // underflow
+ if x.neg {
+ var z float32
+ return -z, Above
+ }
+ return 0.0, Below
+ }
+ // x.exp >= dmin+1
+
+ var r Float
+ r.prec = mbits + 1 // +1 for implicit msb
+ if x.exp < emin+1 {
+ // denormal number - round to fewer bits
+ r.prec = uint32(x.exp - dmin)
+ }
+ r.Set(x)
+
+ // Rounding may have caused r to overflow to ±Inf
+ // (rounding never causes underflows to 0).
+ if r.form == inf {
+ r.exp = emax + 2 // cause overflow below
+ }
+
+ if r.exp > emax+1 {
+ // overflow
+ if x.neg {
+ return float32(math.Inf(-1)), Below
+ }
+ return float32(math.Inf(+1)), Above
+ }
+ // dmin+1 <= r.exp <= emax+1
+
+ var s uint32
+ if r.neg {
+ s = 1 << (fbits - 1)
+ }
+
+ m := high32(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
+
+ // Rounding may have caused a denormal number to
+ // become normal. Check again.
+ c := float32(1.0)
+ if r.exp < emin+1 {
+ // denormal number
+ r.exp += mbits
+ c = 1.0 / (1 << mbits) // 2**-mbits
+ }
+ // emin+1 <= r.exp <= emax+1
+ e := uint32(r.exp-emin) << mbits
+
+ return c * math.Float32frombits(s|e|m), r.acc
+
+ case zero:
+ if x.neg {
+ var z float32
+ return -z, Exact
+ }
+ return 0.0, Exact
+
+ case inf:
+ if x.neg {
+ return float32(math.Inf(-1)), Exact
+ }
+ return float32(math.Inf(+1)), Exact
+
+ case nan:
+ return float32(math.NaN()), Undef
+ }
+
+ panic("unreachable")
+}
+
+// Float64 returns the float64 value nearest to x. If x is too small to be
+// represented by a float64 (|x| < math.SmallestNonzeroFloat64), the result
+// is (0, Below) or (-0, Above), respectively, depending on the sign of x.
+// If x is too large to be represented by a float64 (|x| > math.MaxFloat64),
+// the result is (+Inf, Above) or (-Inf, Below), depending on the sign of x.
+// The result is (NaN, Undef) for NaNs.
+func (x *Float) Float64() (float64, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+
+ const (
+ fbits = 64 // float size
+ mbits = 52 // mantissa size (excluding implicit msb)
+ ebits = fbits - mbits - 1 // 11 exponent size
+ bias = 1<<(ebits-1) - 1 // 1023 exponent bias
+ dmin = 1 - bias - mbits // -1074 smallest unbiased exponent (denormal)
+ emin = 1 - bias // -1022 smallest unbiased exponent (normal)
+ emax = bias // 1023 largest unbiased exponent (normal)
+ )
+
+ // Float mantissae m have an explicit msb and are in the range 0.5 <= m < 1.0.
+ // floatxx mantissae have an implicit msb and are in the range 1.0 <= m < 2.0.
+ // For a given mantissa m, we need to add 1 to a floatxx exponent to get the
+ // corresponding Float exponent.
+ // (see also implementation of math.Ldexp for similar code)
+
+ if x.exp < dmin+1 {
+ // underflow
+ if x.neg {
+ var z float64
+ return -z, Above
+ }
+ return 0.0, Below
+ }
+ // x.exp >= dmin+1
+
+ var r Float
+ r.prec = mbits + 1 // +1 for implicit msb
+ if x.exp < emin+1 {
+ // denormal number - round to fewer bits
+ r.prec = uint32(x.exp - dmin)
+ }
+ r.Set(x)
+
+ // Rounding may have caused r to overflow to ±Inf
+ // (rounding never causes underflows to 0).
+ if r.form == inf {
+ r.exp = emax + 2 // cause overflow below
+ }
+
+ if r.exp > emax+1 {
+ // overflow
+ if x.neg {
+ return math.Inf(-1), Below
+ }
+ return math.Inf(+1), Above
+ }
+ // dmin+1 <= r.exp <= emax+1
+
+ var s uint64
+ if r.neg {
+ s = 1 << (fbits - 1)
+ }
+
+ m := high64(r.mant) >> ebits & (1<<mbits - 1) // cut off msb (implicit 1 bit)
+
+ // Rounding may have caused a denormal number to
+ // become normal. Check again.
+ c := 1.0
+ if r.exp < emin+1 {
+ // denormal number
+ r.exp += mbits
+ c = 1.0 / (1 << mbits) // 2**-mbits
+ }
+ // emin+1 <= r.exp <= emax+1
+ e := uint64(r.exp-emin) << mbits
+
+ return c * math.Float64frombits(s|e|m), r.acc
+
+ case zero:
+ if x.neg {
+ var z float64
+ return -z, Exact
+ }
+ return 0.0, Exact
+
+ case inf:
+ if x.neg {
+ return math.Inf(-1), Exact
+ }
+ return math.Inf(+1), Exact
+
+ case nan:
+ return math.NaN(), Undef
+ }
+
+ panic("unreachable")
+}
+
+// Int returns the result of truncating x towards zero;
+// or nil if x is an infinity or NaN.
+// The result is Exact if x.IsInt(); otherwise it is Below
+// for x > 0, and Above for x < 0.
+// If a non-nil *Int argument z is provided, Int stores
+// the result in z instead of allocating a new Int.
+func (x *Float) Int(z *Int) (*Int, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ if z == nil && x.form <= finite {
+ z = new(Int)
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ acc := makeAcc(x.neg)
+ if x.exp <= 0 {
+ // 0 < |x| < 1
+ return z.SetInt64(0), acc
+ }
+ // x.exp > 0
+
+ // 1 <= |x| < +Inf
+ // determine minimum required precision for x
+ allBits := uint(len(x.mant)) * _W
+ exp := uint(x.exp)
+ if x.MinPrec() <= exp {
+ acc = Exact
+ }
+ // shift mantissa as needed
+ if z == nil {
+ z = new(Int)
+ }
+ z.neg = x.neg
+ switch {
+ case exp > allBits:
+ z.abs = z.abs.shl(x.mant, exp-allBits)
+ default:
+ z.abs = z.abs.set(x.mant)
+ case exp < allBits:
+ z.abs = z.abs.shr(x.mant, allBits-exp)
+ }
+ return z, acc
+
+ case zero:
+ return z.SetInt64(0), Exact
+
+ case inf:
+ return nil, makeAcc(x.neg)
+
+ case nan:
+ return nil, Undef
+ }
+
+ panic("unreachable")
+}
+
+// Rat returns the rational number corresponding to x;
+// or nil if x is an infinity or NaN.
+// The result is Exact is x is not an Inf or NaN.
+// If a non-nil *Rat argument z is provided, Rat stores
+// the result in z instead of allocating a new Rat.
+func (x *Float) Rat(z *Rat) (*Rat, Accuracy) {
+ if debugFloat {
+ x.validate()
+ }
+
+ if z == nil && x.form <= finite {
+ z = new(Rat)
+ }
+
+ switch x.form {
+ case finite:
+ // 0 < |x| < +Inf
+ allBits := int32(len(x.mant)) * _W
+ // build up numerator and denominator
+ z.a.neg = x.neg
+ switch {
+ case x.exp > allBits:
+ z.a.abs = z.a.abs.shl(x.mant, uint(x.exp-allBits))
+ z.b.abs = z.b.abs[:0] // == 1 (see Rat)
+ // z already in normal form
+ default:
+ z.a.abs = z.a.abs.set(x.mant)
+ z.b.abs = z.b.abs[:0] // == 1 (see Rat)
+ // z already in normal form
+ case x.exp < allBits:
+ z.a.abs = z.a.abs.set(x.mant)
+ t := z.b.abs.setUint64(1)
+ z.b.abs = t.shl(t, uint(allBits-x.exp))
+ z.norm()
+ }
+ return z, Exact
+
+ case zero:
+ return z.SetInt64(0), Exact
+
+ case inf:
+ return nil, makeAcc(x.neg)
+
+ case nan:
+ return nil, Undef
+ }
+
+ panic("unreachable")
+}
+
+// Abs sets z to the (possibly rounded) value |x| (the absolute value of x)
+// and returns z.
+func (z *Float) Abs(x *Float) *Float {
+ z.Set(x)
+ z.neg = false
+ return z
+}
+
+// Neg sets z to the (possibly rounded) value of x with its sign negated,
+// and returns z.
+func (z *Float) Neg(x *Float) *Float {
+ z.Set(x)
+ z.neg = !z.neg
+ return z
+}
+
+func validateBinaryOperands(x, y *Float) {
+ if !debugFloat {
+ // avoid performance bugs
+ panic("validateBinaryOperands called but debugFloat is not set")
+ }
+ if len(x.mant) == 0 {
+ panic("empty mantissa for x")
+ }
+ if len(y.mant) == 0 {
+ panic("empty mantissa for y")
+ }
+}
+
+// z = x + y, ignoring signs of x and y for the addition
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) uadd(x, y *Float) {
+ // Note: This implementation requires 2 shifts most of the
+ // time. It is also inefficient if exponents or precisions
+ // differ by wide margins. The following article describes
+ // an efficient (but much more complicated) implementation
+ // compatible with the internal representation used here:
+ //
+ // Vincent Lefèvre: "The Generic Multiple-Precision Floating-
+ // Point Addition With Exact Rounding (as in the MPFR Library)"
+ // http://www.vinc17.net/research/papers/rnc6.pdf
+
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // compute exponents ex, ey for mantissa with "binary point"
+ // on the right (mantissa.0) - use int64 to avoid overflow
+ ex := int64(x.exp) - int64(len(x.mant))*_W
+ ey := int64(y.exp) - int64(len(y.mant))*_W
+
+ // TODO(gri) having a combined add-and-shift primitive
+ // could make this code significantly faster
+ switch {
+ case ex < ey:
+ // cannot re-use z.mant w/o testing for aliasing
+ t := nat(nil).shl(y.mant, uint(ey-ex))
+ z.mant = z.mant.add(x.mant, t)
+ default:
+ // ex == ey, no shift needed
+ z.mant = z.mant.add(x.mant, y.mant)
+ case ex > ey:
+ // cannot re-use z.mant w/o testing for aliasing
+ t := nat(nil).shl(x.mant, uint(ex-ey))
+ z.mant = z.mant.add(t, y.mant)
+ ex = ey
+ }
+ // len(z.mant) > 0
+
+ z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
+}
+
+// z = x - y for |x| > |y|, ignoring signs of x and y for the subtraction
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) usub(x, y *Float) {
+ // This code is symmetric to uadd.
+ // We have not factored the common code out because
+ // eventually uadd (and usub) should be optimized
+ // by special-casing, and the code will diverge.
+
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ ex := int64(x.exp) - int64(len(x.mant))*_W
+ ey := int64(y.exp) - int64(len(y.mant))*_W
+
+ switch {
+ case ex < ey:
+ // cannot re-use z.mant w/o testing for aliasing
+ t := nat(nil).shl(y.mant, uint(ey-ex))
+ z.mant = t.sub(x.mant, t)
+ default:
+ // ex == ey, no shift needed
+ z.mant = z.mant.sub(x.mant, y.mant)
+ case ex > ey:
+ // cannot re-use z.mant w/o testing for aliasing
+ t := nat(nil).shl(x.mant, uint(ex-ey))
+ z.mant = t.sub(t, y.mant)
+ ex = ey
+ }
+
+ // operands may have cancelled each other out
+ if len(z.mant) == 0 {
+ z.acc = Exact
+ z.form = zero
+ z.neg = false
+ return
+ }
+ // len(z.mant) > 0
+
+ z.setExpAndRound(ex+int64(len(z.mant))*_W-fnorm(z.mant), 0)
+}
+
+// z = x * y, ignoring signs of x and y for the multiplication
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) umul(x, y *Float) {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // Note: This is doing too much work if the precision
+ // of z is less than the sum of the precisions of x
+ // and y which is often the case (e.g., if all floats
+ // have the same precision).
+ // TODO(gri) Optimize this for the common case.
+
+ e := int64(x.exp) + int64(y.exp)
+ z.mant = z.mant.mul(x.mant, y.mant)
+
+ z.setExpAndRound(e-fnorm(z.mant), 0)
+}
+
+// z = x / y, ignoring signs of x and y for the division
+// but using the sign of z for rounding the result.
+// x and y must have a non-empty mantissa and valid exponent.
+func (z *Float) uquo(x, y *Float) {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ // mantissa length in words for desired result precision + 1
+ // (at least one extra bit so we get the rounding bit after
+ // the division)
+ n := int(z.prec/_W) + 1
+
+ // compute adjusted x.mant such that we get enough result precision
+ xadj := x.mant
+ if d := n - len(x.mant) + len(y.mant); d > 0 {
+ // d extra words needed => add d "0 digits" to x
+ xadj = make(nat, len(x.mant)+d)
+ copy(xadj[d:], x.mant)
+ }
+ // TODO(gri): If we have too many digits (d < 0), we should be able
+ // to shorten x for faster division. But we must be extra careful
+ // with rounding in that case.
+
+ // Compute d before division since there may be aliasing of x.mant
+ // (via xadj) or y.mant with z.mant.
+ d := len(xadj) - len(y.mant)
+
+ // divide
+ var r nat
+ z.mant, r = z.mant.div(nil, xadj, y.mant)
+ e := int64(x.exp) - int64(y.exp) - int64(d-len(z.mant))*_W
+
+ // The result is long enough to include (at least) the rounding bit.
+ // If there's a non-zero remainder, the corresponding fractional part
+ // (if it were computed), would have a non-zero sticky bit (if it were
+ // zero, it couldn't have a non-zero remainder).
+ var sbit uint
+ if len(r) > 0 {
+ sbit = 1
+ }
+
+ z.setExpAndRound(e-fnorm(z.mant), sbit)
+}
+
+// ucmp returns Below, Exact, or Above, depending
+// on whether |x| < |y|, |x| == |y|, or |x| > |y|.
+// x and y must have a non-empty mantissa and valid exponent.
+func (x *Float) ucmp(y *Float) Accuracy {
+ if debugFloat {
+ validateBinaryOperands(x, y)
+ }
+
+ switch {
+ case x.exp < y.exp:
+ return Below
+ case x.exp > y.exp:
+ return Above
+ }
+ // x.exp == y.exp
+
+ // compare mantissas
+ i := len(x.mant)
+ j := len(y.mant)
+ for i > 0 || j > 0 {
+ var xm, ym Word
+ if i > 0 {
+ i--
+ xm = x.mant[i]
+ }
+ if j > 0 {
+ j--
+ ym = y.mant[j]
+ }
+ switch {
+ case xm < ym:
+ return Below
+ case xm > ym:
+ return Above
+ }
+ }
+
+ return Exact
+}
+
+// Handling of sign bit as defined by IEEE 754-2008, section 6.3:
+//
+// When neither the inputs nor result are NaN, the sign of a product or
+// quotient is the exclusive OR of the operands’ signs; the sign of a sum,
+// or of a difference x−y regarded as a sum x+(−y), differs from at most
+// one of the addends’ signs; and the sign of the result of conversions,
+// the quantize operation, the roundToIntegral operations, and the
+// roundToIntegralExact (see 5.3.1) is the sign of the first or only operand.
+// These rules shall apply even when operands or results are zero or infinite.
+//
+// When the sum of two operands with opposite signs (or the difference of
+// two operands with like signs) is exactly zero, the sign of that sum (or
+// difference) shall be +0 in all rounding-direction attributes except
+// roundTowardNegative; under that attribute, the sign of an exact zero
+// sum (or difference) shall be −0. However, x+x = x−(−x) retains the same
+// sign as x even when x is zero.
+//
+// See also: http://play.golang.org/p/RtH3UCt5IH
+
+// Add sets z to the rounded sum x+y and returns z. If z's precision is 0,
+// it is changed to the larger of x's or y's precision before the operation.
+// Rounding is performed according to z's precision and rounding mode; and
+// z's accuracy reports the result error relative to the exact (not rounded)
+// result.
+// BUG(gri) Float.Add returns NaN if an operand is Inf.
+// BUG(gri) When rounding ToNegativeInf, the sign of Float values rounded to 0 is incorrect.
+func (z *Float) Add(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ // special cases
+ if x.form != finite || y.form != finite {
+ if x.form > finite || y.form > finite {
+ // TODO(gri) handle Inf separately
+ return z.SetNaN()
+ }
+ if x.form == zero {
+ z.Set(y)
+ if z.form == zero {
+ z.neg = x.neg && y.neg // -0 + -0 == -0
+ }
+ return z
+ }
+ // y == ±0
+ return z.Set(x)
+ }
+
+ // x, y != 0
+ z.neg = x.neg
+ if x.neg == y.neg {
+ // x + y == x + y
+ // (-x) + (-y) == -(x + y)
+ z.uadd(x, y)
+ } else {
+ // x + (-y) == x - y == -(y - x)
+ // (-x) + y == y - x == -(x - y)
+ if x.ucmp(y) == Above {
+ z.usub(x, y)
+ } else {
+ z.neg = !z.neg
+ z.usub(y, x)
+ }
+ }
+
+ return z
+}
+
+// Sub sets z to the rounded difference x-y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// BUG(gri) Float.Sub returns NaN if an operand is Inf.
+func (z *Float) Sub(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ // special cases
+ if x.form != finite || y.form != finite {
+ if x.form > finite || y.form > finite {
+ // TODO(gri) handle Inf separately
+ return z.SetNaN()
+ }
+ if x.form == zero {
+ z.Neg(y)
+ if z.form == zero {
+ z.neg = x.neg && !y.neg // -0 - 0 == -0
+ }
+ return z
+ }
+ // y == ±0
+ return z.Set(x)
+ }
+
+ // x, y != 0
+ z.neg = x.neg
+ if x.neg != y.neg {
+ // x - (-y) == x + y
+ // (-x) - y == -(x + y)
+ z.uadd(x, y)
+ } else {
+ // x - y == x - y == -(y - x)
+ // (-x) - (-y) == y - x == -(x - y)
+ if x.ucmp(y) == Above {
+ z.usub(x, y)
+ } else {
+ z.neg = !z.neg
+ z.usub(y, x)
+ }
+ }
+
+ return z
+}
+
+// Mul sets z to the rounded product x*y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// BUG(gri) Float.Mul returns NaN if an operand is Inf.
+func (z *Float) Mul(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ z.neg = x.neg != y.neg
+
+ // special cases
+ if x.form != finite || y.form != finite {
+ if x.form > finite || y.form > finite {
+ // TODO(gri) handle Inf separately
+ return z.SetNaN()
+ }
+ // x == ±0 || y == ±0
+ z.acc = Exact
+ z.form = zero
+ return z
+ }
+
+ // x, y != 0
+ z.umul(x, y)
+
+ return z
+}
+
+// Quo sets z to the rounded quotient x/y and returns z.
+// Precision, rounding, and accuracy reporting are as for Add.
+// BUG(gri) Float.Quo returns NaN if an operand is Inf.
+func (z *Float) Quo(x, y *Float) *Float {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if z.prec == 0 {
+ z.prec = umax32(x.prec, y.prec)
+ }
+
+ z.neg = x.neg != y.neg
+
+ // special cases
+ z.acc = Exact
+ if x.form != finite || y.form != finite {
+ if x.form > finite || y.form > finite {
+ // TODO(gri) handle Inf separately
+ return z.SetNaN()
+ }
+ // x == ±0 || y == ±0
+ if x.form == zero {
+ if y.form == zero {
+ return z.SetNaN()
+ }
+ z.form = zero
+ return z
+ }
+ // y == ±0
+ z.form = inf
+ return z
+ }
+
+ // x, y != 0
+ z.uquo(x, y)
+
+ return z
+}
+
+type cmpResult struct {
+ acc Accuracy
+}
+
+// Cmp compares x and y and returns:
+//
+// Below if x < y
+// Exact if x == y (incl. -0 == 0, -Inf == -Inf, and +Inf == +Inf)
+// Above if x > y
+// Undef if any of x, y is NaN
+//
+func (x *Float) Cmp(y *Float) cmpResult {
+ if debugFloat {
+ x.validate()
+ y.validate()
+ }
+
+ if x.form == nan || y.form == nan {
+ return cmpResult{Undef}
+ }
+
+ mx := x.ord()
+ my := y.ord()
+ switch {
+ case mx < my:
+ return cmpResult{Below}
+ case mx > my:
+ return cmpResult{Above}
+ }
+ // mx == my
+
+ // only if |mx| == 1 we have to compare the mantissae
+ switch mx {
+ case -1:
+ return cmpResult{y.ucmp(x)}
+ case +1:
+ return cmpResult{x.ucmp(y)}
+ }
+
+ return cmpResult{Exact}
+}
+
+// The following accessors simplify testing of Cmp results.
+func (res cmpResult) Acc() Accuracy { return res.acc }
+func (res cmpResult) Eql() bool { return res.acc == Exact }
+func (res cmpResult) Neq() bool { return res.acc != Exact }
+func (res cmpResult) Lss() bool { return res.acc == Below }
+func (res cmpResult) Leq() bool { return res.acc&Above == 0 }
+func (res cmpResult) Gtr() bool { return res.acc == Above }
+func (res cmpResult) Geq() bool { return res.acc&Below == 0 }
+
+// ord classifies x and returns:
+//
+// -2 if -Inf == x
+// -1 if -Inf < x < 0
+// 0 if x == 0 (signed or unsigned)
+// +1 if 0 < x < +Inf
+// +2 if x == +Inf
+//
+// x must not be NaN.
+func (x *Float) ord() int {
+ var m int
+ switch x.form {
+ case finite:
+ m = 1
+ case zero:
+ return 0
+ case inf:
+ m = 2
+ default:
+ panic("unreachable")
+ }
+ if x.neg {
+ m = -m
+ }
+ return m
+}
+
+func umax32(x, y uint32) uint32 {
+ if x > y {
+ return x
+ }
+ return y
+}
--- /dev/null
+// Copyright 2014 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "fmt"
+ "math"
+ "strconv"
+ "strings"
+ "testing"
+)
+
+func (x *Float) uint64() uint64 {
+ u, acc := x.Uint64()
+ if acc != Exact {
+ panic(fmt.Sprintf("%s is not a uint64", x.Format('g', 10)))
+ }
+ return u
+}
+
+func (x *Float) int64() int64 {
+ i, acc := x.Int64()
+ if acc != Exact {
+ panic(fmt.Sprintf("%s is not an int64", x.Format('g', 10)))
+ }
+ return i
+}
+
+func TestFloatZeroValue(t *testing.T) {
+ // zero (uninitialized) value is a ready-to-use 0.0
+ var x Float
+ if s := x.Format('f', 1); s != "0.0" {
+ t.Errorf("zero value = %s; want 0.0", s)
+ }
+
+ // zero value has precision 0
+ if prec := x.Prec(); prec != 0 {
+ t.Errorf("prec = %d; want 0", prec)
+ }
+
+ // zero value can be used in any and all positions of binary operations
+ make := func(x int) *Float {
+ var f Float
+ if x != 0 {
+ f.SetInt64(int64(x))
+ }
+ // x == 0 translates into the zero value
+ return &f
+ }
+ for _, test := range []struct {
+ z, x, y, want int
+ opname rune
+ op func(z, x, y *Float) *Float
+ }{
+ {0, 0, 0, 0, '+', (*Float).Add},
+ {0, 1, 2, 3, '+', (*Float).Add},
+ {1, 2, 0, 2, '+', (*Float).Add},
+ {2, 0, 1, 1, '+', (*Float).Add},
+
+ {0, 0, 0, 0, '-', (*Float).Sub},
+ {0, 1, 2, -1, '-', (*Float).Sub},
+ {1, 2, 0, 2, '-', (*Float).Sub},
+ {2, 0, 1, -1, '-', (*Float).Sub},
+
+ {0, 0, 0, 0, '*', (*Float).Mul},
+ {0, 1, 2, 2, '*', (*Float).Mul},
+ {1, 2, 0, 0, '*', (*Float).Mul},
+ {2, 0, 1, 0, '*', (*Float).Mul},
+
+ {0, 0, 0, 0, '/', (*Float).Quo}, // = Nan
+ {0, 2, 1, 2, '/', (*Float).Quo},
+ {1, 2, 0, 0, '/', (*Float).Quo}, // = +Inf
+ {2, 0, 1, 0, '/', (*Float).Quo},
+ } {
+ z := make(test.z)
+ test.op(z, make(test.x), make(test.y))
+ got := 0
+ if z.IsFinite() {
+ got = int(z.int64())
+ }
+ if got != test.want {
+ t.Errorf("%d %c %d = %d; want %d", test.x, test.opname, test.y, got, test.want)
+ }
+ }
+
+ // TODO(gri) test how precision is set for zero value results
+}
+
+func makeFloat(s string) *Float {
+ var x Float
+
+ switch s {
+ case "0":
+ return &x
+ case "-0":
+ return x.Neg(&x)
+ case "Inf", "+Inf":
+ return x.SetInf(+1)
+ case "-Inf":
+ return x.SetInf(-1)
+ case "NaN", "-NaN":
+ return x.SetNaN()
+ }
+
+ x.SetPrec(1000)
+ if _, ok := x.SetString(s); !ok {
+ panic(fmt.Sprintf("%q is not a valid float", s))
+ }
+ return &x
+}
+
+func TestFloatSetPrec(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ prec uint
+ want string
+ acc Accuracy
+ }{
+ // prec 0
+ {"0", 0, "0", Exact},
+ {"-0", 0, "-0", Exact},
+ {"-Inf", 0, "-Inf", Exact},
+ {"+Inf", 0, "+Inf", Exact},
+ {"NaN", 0, "NaN", Exact},
+ {"123", 0, "0", Below},
+ {"-123", 0, "-0", Above},
+
+ // prec at upper limit
+ {"0", MaxPrec, "0", Exact},
+ {"-0", MaxPrec, "-0", Exact},
+ {"-Inf", MaxPrec, "-Inf", Exact},
+ {"+Inf", MaxPrec, "+Inf", Exact},
+ {"NaN", MaxPrec, "NaN", Exact},
+
+ // just a few regular cases - general rounding is tested elsewhere
+ {"1.5", 1, "2", Above},
+ {"-1.5", 1, "-2", Below},
+ {"123", 1e6, "123", Exact},
+ {"-123", 1e6, "-123", Exact},
+ } {
+ x := makeFloat(test.x).SetPrec(test.prec)
+ prec := test.prec
+ if prec > MaxPrec {
+ prec = MaxPrec
+ }
+ if got := x.Prec(); got != prec {
+ t.Errorf("%s.SetPrec(%d).Prec() == %d; want %d", test.x, test.prec, got, prec)
+ }
+ if got, acc := x.String(), x.Acc(); got != test.want || acc != test.acc {
+ t.Errorf("%s.SetPrec(%d) = %s (%s); want %s (%s)", test.x, test.prec, got, acc, test.want, test.acc)
+ }
+ }
+}
+
+func TestFloatMinPrec(t *testing.T) {
+ const max = 100
+ for _, test := range []struct {
+ x string
+ want uint
+ }{
+ {"0", 0},
+ {"-0", 0},
+ {"+Inf", 0},
+ {"-Inf", 0},
+ {"NaN", 0},
+ {"1", 1},
+ {"2", 1},
+ {"3", 2},
+ {"0x8001", 16},
+ {"0x8001p-1000", 16},
+ {"0x8001p+1000", 16},
+ {"0.1", max},
+ } {
+ x := makeFloat(test.x).SetPrec(max)
+ if got := x.MinPrec(); got != test.want {
+ t.Errorf("%s.MinPrec() = %d; want %d", test.x, got, test.want)
+ }
+ }
+}
+
+func TestFloatSign(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ s int
+ }{
+ {"-Inf", -1},
+ {"-1", -1},
+ {"-0", 0},
+ {"+0", 0},
+ {"+1", +1},
+ {"+Inf", +1},
+ {"NaN", 0},
+ } {
+ x := makeFloat(test.x)
+ s := x.Sign()
+ if s != test.s {
+ t.Errorf("%s.Sign() = %d; want %d", test.x, s, test.s)
+ }
+ }
+}
+
+// feq(x, y) is like x.Cmp(y) == 0 but it also considers the sign of 0 (0 != -0).
+// Caution: Two NaN's are equal with this function!
+func feq(x, y *Float) bool {
+ if x.IsNaN() || y.IsNaN() {
+ return x.IsNaN() && y.IsNaN()
+ }
+ return x.Cmp(y).Eql() && x.IsNeg() == y.IsNeg()
+}
+
+func TestFloatMantExp(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ mant string
+ exp int
+ }{
+ {"0", "0", 0},
+ {"+0", "0", 0},
+ {"-0", "-0", 0},
+ {"Inf", "+Inf", 0},
+ {"+Inf", "+Inf", 0},
+ {"-Inf", "-Inf", 0},
+ {"NaN", "NaN", 0},
+ {"1.5", "0.75", 1},
+ {"1.024e3", "0.5", 11},
+ {"-0.125", "-0.5", -2},
+ } {
+ x := makeFloat(test.x)
+ mant := makeFloat(test.mant)
+ m := new(Float)
+ e := x.MantExp(m)
+ if !feq(m, mant) || e != test.exp {
+ t.Errorf("%s.MantExp() = %s, %d; want %s, %d", test.x, m.Format('g', 10), e, test.mant, test.exp)
+ }
+ }
+}
+
+func TestFloatMantExpAliasing(t *testing.T) {
+ x := makeFloat("0.5p10")
+ if e := x.MantExp(x); e != 10 {
+ t.Fatalf("Float.MantExp aliasing error: got %d; want 10", e)
+ }
+ if want := makeFloat("0.5"); !feq(x, want) {
+ t.Fatalf("Float.MantExp aliasing error: got %s; want %s", x.Format('g', 10), want.Format('g', 10))
+ }
+}
+
+func TestFloatSetMantExp(t *testing.T) {
+ for _, test := range []struct {
+ frac string
+ exp int
+ z string
+ }{
+ {"0", 0, "0"},
+ {"+0", 0, "0"},
+ {"-0", 0, "-0"},
+ {"Inf", 1234, "+Inf"},
+ {"+Inf", -1234, "+Inf"},
+ {"-Inf", -1234, "-Inf"},
+ {"0", MinExp, "0"},
+ {"0.25", MinExp, "+0"}, // exponent underflow
+ {"-0.25", MinExp, "-0"}, // exponent underflow
+ {"1", MaxExp, "+Inf"}, // exponent overflow
+ {"2", MaxExp - 1, "+Inf"}, // exponent overflow
+ {"0.75", 1, "1.5"},
+ {"0.5", 11, "1024"},
+ {"-0.5", -2, "-0.125"},
+ {"32", 5, "1024"},
+ {"1024", -10, "1"},
+ } {
+ frac := makeFloat(test.frac)
+ want := makeFloat(test.z)
+ var z Float
+ z.SetMantExp(frac, test.exp)
+ if !feq(&z, want) {
+ t.Errorf("SetMantExp(%s, %d) = %s; want %s", test.frac, test.exp, z.Format('g', 10), test.z)
+ }
+ // test inverse property
+ mant := new(Float)
+ if z.SetMantExp(mant, want.MantExp(mant)).Cmp(want).Neq() {
+ t.Errorf("Inverse property not satisfied: got %s; want %s", z.Format('g', 10), test.z)
+ }
+ }
+}
+
+func TestFloatPredicates(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ neg, zero, finite, inf, nan bool
+ }{
+ {x: "-Inf", neg: true, inf: true},
+ {x: "-1", neg: true, finite: true},
+ {x: "-0", neg: true, zero: true, finite: true},
+ {x: "0", zero: true, finite: true},
+ {x: "1", finite: true},
+ {x: "+Inf", inf: true},
+ {x: "NaN", nan: true},
+ } {
+ x := makeFloat(test.x)
+ if got := x.IsNeg(); got != test.neg {
+ t.Errorf("(%s).IsNeg() = %v; want %v", test.x, got, test.neg)
+ }
+ if got := x.IsZero(); got != test.zero {
+ t.Errorf("(%s).IsZero() = %v; want %v", test.x, got, test.zero)
+ }
+ if got := x.IsFinite(); got != test.finite {
+ t.Errorf("(%s).IsFinite() = %v; want %v", test.x, got, test.finite)
+ }
+ if got := x.IsInf(); got != test.inf {
+ t.Errorf("(%s).IsInf() = %v; want %v", test.x, got, test.inf)
+ }
+ if got := x.IsNaN(); got != test.nan {
+ t.Errorf("(%s).IsNaN() = %v; want %v", test.x, got, test.nan)
+ }
+ }
+}
+
+func TestFloatIsInt(t *testing.T) {
+ for _, test := range []string{
+ "0 int",
+ "-0 int",
+ "1 int",
+ "-1 int",
+ "0.5",
+ "1.23",
+ "1.23e1",
+ "1.23e2 int",
+ "0.000000001e+8",
+ "0.000000001e+9 int",
+ "1.2345e200 int",
+ "Inf",
+ "+Inf",
+ "-Inf",
+ "NaN",
+ } {
+ s := strings.TrimSuffix(test, " int")
+ want := s != test
+ if got := makeFloat(s).IsInt(); got != want {
+ t.Errorf("%s.IsInt() == %t", s, got)
+ }
+ }
+}
+
+func fromBinary(s string) int64 {
+ x, err := strconv.ParseInt(s, 2, 64)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func toBinary(x int64) string {
+ return strconv.FormatInt(x, 2)
+}
+
+func testFloatRound(t *testing.T, x, r int64, prec uint, mode RoundingMode) {
+ // verify test data
+ var ok bool
+ switch mode {
+ case ToNearestEven, ToNearestAway:
+ ok = true // nothing to do for now
+ case ToZero:
+ if x < 0 {
+ ok = r >= x
+ } else {
+ ok = r <= x
+ }
+ case AwayFromZero:
+ if x < 0 {
+ ok = r <= x
+ } else {
+ ok = r >= x
+ }
+ case ToNegativeInf:
+ ok = r <= x
+ case ToPositiveInf:
+ ok = r >= x
+ default:
+ panic("unreachable")
+ }
+ if !ok {
+ t.Fatalf("incorrect test data for prec = %d, %s: x = %s, r = %s", prec, mode, toBinary(x), toBinary(r))
+ }
+
+ // compute expected accuracy
+ a := Exact
+ switch {
+ case r < x:
+ a = Below
+ case r > x:
+ a = Above
+ }
+
+ // round
+ f := new(Float).SetMode(mode).SetInt64(x).SetPrec(prec)
+
+ // check result
+ r1 := f.int64()
+ p1 := f.Prec()
+ a1 := f.Acc()
+ if r1 != r || p1 != prec || a1 != a {
+ t.Errorf("round %s (%d bits, %s) incorrect: got %s (%d bits, %s); want %s (%d bits, %s)",
+ toBinary(x), prec, mode,
+ toBinary(r1), p1, a1,
+ toBinary(r), prec, a)
+ return
+ }
+
+ // g and f should be the same
+ // (rounding by SetPrec after SetInt64 using default precision
+ // should be the same as rounding by SetInt64 after setting the
+ // precision)
+ g := new(Float).SetMode(mode).SetPrec(prec).SetInt64(x)
+ if !feq(g, f) {
+ t.Errorf("round %s (%d bits, %s) not symmetric: got %s and %s; want %s",
+ toBinary(x), prec, mode,
+ toBinary(g.int64()),
+ toBinary(r1),
+ toBinary(r),
+ )
+ return
+ }
+
+ // h and f should be the same
+ // (repeated rounding should be idempotent)
+ h := new(Float).SetMode(mode).SetPrec(prec).Set(f)
+ if !feq(h, f) {
+ t.Errorf("round %s (%d bits, %s) not idempotent: got %s and %s; want %s",
+ toBinary(x), prec, mode,
+ toBinary(h.int64()),
+ toBinary(r1),
+ toBinary(r),
+ )
+ return
+ }
+}
+
+// TestFloatRound tests basic rounding.
+func TestFloatRound(t *testing.T) {
+ for _, test := range []struct {
+ prec uint
+ x, zero, neven, naway, away string // input, results rounded to prec bits
+ }{
+ {5, "1000", "1000", "1000", "1000", "1000"},
+ {5, "1001", "1001", "1001", "1001", "1001"},
+ {5, "1010", "1010", "1010", "1010", "1010"},
+ {5, "1011", "1011", "1011", "1011", "1011"},
+ {5, "1100", "1100", "1100", "1100", "1100"},
+ {5, "1101", "1101", "1101", "1101", "1101"},
+ {5, "1110", "1110", "1110", "1110", "1110"},
+ {5, "1111", "1111", "1111", "1111", "1111"},
+
+ {4, "1000", "1000", "1000", "1000", "1000"},
+ {4, "1001", "1001", "1001", "1001", "1001"},
+ {4, "1010", "1010", "1010", "1010", "1010"},
+ {4, "1011", "1011", "1011", "1011", "1011"},
+ {4, "1100", "1100", "1100", "1100", "1100"},
+ {4, "1101", "1101", "1101", "1101", "1101"},
+ {4, "1110", "1110", "1110", "1110", "1110"},
+ {4, "1111", "1111", "1111", "1111", "1111"},
+
+ {3, "1000", "1000", "1000", "1000", "1000"},
+ {3, "1001", "1000", "1000", "1010", "1010"},
+ {3, "1010", "1010", "1010", "1010", "1010"},
+ {3, "1011", "1010", "1100", "1100", "1100"},
+ {3, "1100", "1100", "1100", "1100", "1100"},
+ {3, "1101", "1100", "1100", "1110", "1110"},
+ {3, "1110", "1110", "1110", "1110", "1110"},
+ {3, "1111", "1110", "10000", "10000", "10000"},
+
+ {3, "1000001", "1000000", "1000000", "1000000", "1010000"},
+ {3, "1001001", "1000000", "1010000", "1010000", "1010000"},
+ {3, "1010001", "1010000", "1010000", "1010000", "1100000"},
+ {3, "1011001", "1010000", "1100000", "1100000", "1100000"},
+ {3, "1100001", "1100000", "1100000", "1100000", "1110000"},
+ {3, "1101001", "1100000", "1110000", "1110000", "1110000"},
+ {3, "1110001", "1110000", "1110000", "1110000", "10000000"},
+ {3, "1111001", "1110000", "10000000", "10000000", "10000000"},
+
+ {2, "1000", "1000", "1000", "1000", "1000"},
+ {2, "1001", "1000", "1000", "1000", "1100"},
+ {2, "1010", "1000", "1000", "1100", "1100"},
+ {2, "1011", "1000", "1100", "1100", "1100"},
+ {2, "1100", "1100", "1100", "1100", "1100"},
+ {2, "1101", "1100", "1100", "1100", "10000"},
+ {2, "1110", "1100", "10000", "10000", "10000"},
+ {2, "1111", "1100", "10000", "10000", "10000"},
+
+ {2, "1000001", "1000000", "1000000", "1000000", "1100000"},
+ {2, "1001001", "1000000", "1000000", "1000000", "1100000"},
+ {2, "1010001", "1000000", "1100000", "1100000", "1100000"},
+ {2, "1011001", "1000000", "1100000", "1100000", "1100000"},
+ {2, "1100001", "1100000", "1100000", "1100000", "10000000"},
+ {2, "1101001", "1100000", "1100000", "1100000", "10000000"},
+ {2, "1110001", "1100000", "10000000", "10000000", "10000000"},
+ {2, "1111001", "1100000", "10000000", "10000000", "10000000"},
+
+ {1, "1000", "1000", "1000", "1000", "1000"},
+ {1, "1001", "1000", "1000", "1000", "10000"},
+ {1, "1010", "1000", "1000", "1000", "10000"},
+ {1, "1011", "1000", "1000", "1000", "10000"},
+ {1, "1100", "1000", "10000", "10000", "10000"},
+ {1, "1101", "1000", "10000", "10000", "10000"},
+ {1, "1110", "1000", "10000", "10000", "10000"},
+ {1, "1111", "1000", "10000", "10000", "10000"},
+
+ {1, "1000001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1001001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1010001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1011001", "1000000", "1000000", "1000000", "10000000"},
+ {1, "1100001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1101001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1110001", "1000000", "10000000", "10000000", "10000000"},
+ {1, "1111001", "1000000", "10000000", "10000000", "10000000"},
+ } {
+ x := fromBinary(test.x)
+ z := fromBinary(test.zero)
+ e := fromBinary(test.neven)
+ n := fromBinary(test.naway)
+ a := fromBinary(test.away)
+ prec := test.prec
+
+ testFloatRound(t, x, z, prec, ToZero)
+ testFloatRound(t, x, e, prec, ToNearestEven)
+ testFloatRound(t, x, n, prec, ToNearestAway)
+ testFloatRound(t, x, a, prec, AwayFromZero)
+
+ testFloatRound(t, x, z, prec, ToNegativeInf)
+ testFloatRound(t, x, a, prec, ToPositiveInf)
+
+ testFloatRound(t, -x, -a, prec, ToNegativeInf)
+ testFloatRound(t, -x, -z, prec, ToPositiveInf)
+ }
+}
+
+// TestFloatRound24 tests that rounding a float64 to 24 bits
+// matches IEEE-754 rounding to nearest when converting a
+// float64 to a float32 (excluding denormal numbers).
+func TestFloatRound24(t *testing.T) {
+ const x0 = 1<<26 - 0x10 // 11...110000 (26 bits)
+ for d := 0; d <= 0x10; d++ {
+ x := float64(x0 + d)
+ f := new(Float).SetPrec(24).SetFloat64(x)
+ got, _ := f.Float32()
+ want := float32(x)
+ if got != want {
+ t.Errorf("Round(%g, 24) = %g; want %g", x, got, want)
+ }
+ }
+}
+
+func TestFloatSetUint64(t *testing.T) {
+ for _, want := range []uint64{
+ 0,
+ 1,
+ 2,
+ 10,
+ 100,
+ 1<<32 - 1,
+ 1 << 32,
+ 1<<64 - 1,
+ } {
+ var f Float
+ f.SetUint64(want)
+ if got := f.uint64(); got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Format('p', 0), want)
+ }
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x uint64 = 0x8765432187654321 // 64 bits needed
+ for prec := uint(1); prec <= 64; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetUint64(x)
+ got := f.uint64()
+ want := x &^ (1<<(64-prec) - 1) // cut off (round to zero) low 64-prec bits
+ if got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Format('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetInt64(t *testing.T) {
+ for _, want := range []int64{
+ 0,
+ 1,
+ 2,
+ 10,
+ 100,
+ 1<<32 - 1,
+ 1 << 32,
+ 1<<63 - 1,
+ } {
+ for i := range [2]int{} {
+ if i&1 != 0 {
+ want = -want
+ }
+ var f Float
+ f.SetInt64(want)
+ if got := f.int64(); got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Format('p', 0), want)
+ }
+ }
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x int64 = 0x7654321076543210 // 63 bits needed
+ for prec := uint(1); prec <= 63; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetInt64(x)
+ got := f.int64()
+ want := x &^ (1<<(63-prec) - 1) // cut off (round to zero) low 63-prec bits
+ if got != want {
+ t.Errorf("got %#x (%s); want %#x", got, f.Format('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetFloat64(t *testing.T) {
+ for _, want := range []float64{
+ 0,
+ 1,
+ 2,
+ 12345,
+ 1e10,
+ 1e100,
+ 3.14159265e10,
+ 2.718281828e-123,
+ 1.0 / 3,
+ math.MaxFloat32,
+ math.MaxFloat64,
+ math.SmallestNonzeroFloat32,
+ math.SmallestNonzeroFloat64,
+ math.Inf(-1),
+ math.Inf(0),
+ -math.Inf(1),
+ } {
+ for i := range [2]int{} {
+ if i&1 != 0 {
+ want = -want
+ }
+ var f Float
+ f.SetFloat64(want)
+ if got, acc := f.Float64(); got != want || acc != Exact {
+ t.Errorf("got %g (%s, %s); want %g (Exact)", got, f.Format('p', 0), acc, want)
+ }
+ }
+ }
+
+ // test NaN
+ var f Float
+ f.SetFloat64(math.NaN())
+ if got, acc := f.Float64(); !math.IsNaN(got) || acc != Undef {
+ t.Errorf("got %g (%s, %s); want %g (undef)", got, f.Format('p', 0), acc, math.NaN())
+ }
+
+ // test basic rounding behavior (exhaustive rounding testing is done elsewhere)
+ const x uint64 = 0x8765432143218 // 53 bits needed
+ for prec := uint(1); prec <= 52; prec++ {
+ f := new(Float).SetPrec(prec).SetMode(ToZero).SetFloat64(float64(x))
+ got, _ := f.Float64()
+ want := float64(x &^ (1<<(52-prec) - 1)) // cut off (round to zero) low 53-prec bits
+ if got != want {
+ t.Errorf("got %g (%s); want %g", got, f.Format('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetInt(t *testing.T) {
+ for _, want := range []string{
+ "0",
+ "1",
+ "-1",
+ "1234567890",
+ "123456789012345678901234567890",
+ "123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
+ } {
+ var x Int
+ _, ok := x.SetString(want, 0)
+ if !ok {
+ t.Errorf("invalid integer %s", want)
+ continue
+ }
+ n := x.BitLen()
+
+ var f Float
+ f.SetInt(&x)
+
+ // check precision
+ if n < 64 {
+ n = 64
+ }
+ if prec := f.Prec(); prec != uint(n) {
+ t.Errorf("got prec = %d; want %d", prec, n)
+ }
+
+ // check value
+ got := f.Format('g', 100)
+ if got != want {
+ t.Errorf("got %s (%s); want %s", got, f.Format('p', 0), want)
+ }
+ }
+
+ // TODO(gri) test basic rounding behavior
+}
+
+func TestFloatSetRat(t *testing.T) {
+ for _, want := range []string{
+ "0",
+ "1",
+ "-1",
+ "1234567890",
+ "123456789012345678901234567890",
+ "123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890",
+ "1.2",
+ "3.14159265",
+ // TODO(gri) expand
+ } {
+ var x Rat
+ _, ok := x.SetString(want)
+ if !ok {
+ t.Errorf("invalid fraction %s", want)
+ continue
+ }
+ n := max(x.Num().BitLen(), x.Denom().BitLen())
+
+ var f1, f2 Float
+ f2.SetPrec(1000)
+ f1.SetRat(&x)
+ f2.SetRat(&x)
+
+ // check precision when set automatically
+ if n < 64 {
+ n = 64
+ }
+ if prec := f1.Prec(); prec != uint(n) {
+ t.Errorf("got prec = %d; want %d", prec, n)
+ }
+
+ got := f2.Format('g', 100)
+ if got != want {
+ t.Errorf("got %s (%s); want %s", got, f2.Format('p', 0), want)
+ }
+ }
+}
+
+func TestFloatSetInf(t *testing.T) {
+ var f Float
+ for _, test := range []struct {
+ sign int
+ prec uint
+ want string
+ }{
+ {0, 0, "+Inf"},
+ {100, 0, "+Inf"},
+ {-1, 0, "-Inf"},
+ {0, 10, "+Inf"},
+ {100, 20, "+Inf"},
+ {-1, 30, "-Inf"},
+ } {
+ x := f.SetPrec(test.prec).SetInf(test.sign)
+ if got := x.String(); got != test.want || x.Prec() != test.prec {
+ t.Errorf("SetInf(%d) = %s (prec = %d); want %s (prec = %d)", test.sign, got, x.Prec(), test.want, test.prec)
+ }
+ }
+}
+
+func TestFloatUint64(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out uint64
+ acc Accuracy
+ }{
+ {"-Inf", 0, Above},
+ {"-1", 0, Above},
+ {"-1e-1000", 0, Above},
+ {"-0", 0, Exact},
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"18446744073709551615", 18446744073709551615, Exact},
+ {"18446744073709551615.000000000000000000001", math.MaxUint64, Below},
+ {"18446744073709551616", math.MaxUint64, Below},
+ {"1e10000", math.MaxUint64, Below},
+ {"+Inf", math.MaxUint64, Below},
+ {"NaN", 0, Undef},
+ } {
+ x := makeFloat(test.x)
+ out, acc := x.Uint64()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
+ }
+ }
+}
+
+func TestFloatInt64(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out int64
+ acc Accuracy
+ }{
+ {"-Inf", math.MinInt64, Above},
+ {"-1e10000", math.MinInt64, Above},
+ {"-9223372036854775809", math.MinInt64, Above},
+ {"-9223372036854775808.000000000000000000001", math.MinInt64, Above},
+ {"-9223372036854775808", -9223372036854775808, Exact},
+ {"-9223372036854775807.000000000000000000001", -9223372036854775807, Above},
+ {"-9223372036854775807", -9223372036854775807, Exact},
+ {"-12345.000000000000000000001", -12345, Above},
+ {"-12345.0", -12345, Exact},
+ {"-1.000000000000000000001", -1, Above},
+ {"-1.5", -1, Above},
+ {"-1", -1, Exact},
+ {"-1e-1000", 0, Above},
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"1.5", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"9223372036854775807", 9223372036854775807, Exact},
+ {"9223372036854775807.000000000000000000001", math.MaxInt64, Below},
+ {"9223372036854775808", math.MaxInt64, Below},
+ {"1e10000", math.MaxInt64, Below},
+ {"+Inf", math.MaxInt64, Below},
+ {"NaN", 0, Undef},
+ } {
+ x := makeFloat(test.x)
+ out, acc := x.Int64()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %d (%s); want %d (%s)", test.x, out, acc, test.out, test.acc)
+ }
+ }
+}
+
+func TestFloatFloat32(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ out float32
+ acc Accuracy
+ }{
+ {"-Inf", float32(math.Inf(-1)), Exact},
+ {"-0x1.ffffff0p2147483646", float32(-math.Inf(+1)), Below}, // overflow in rounding
+ {"-1e10000", float32(math.Inf(-1)), Below}, // overflow
+ {"-0x1p128", float32(math.Inf(-1)), Below}, // overflow
+ {"-0x1.ffffff0p127", float32(-math.Inf(+1)), Below}, // overflow
+ {"-0x1.fffffe8p127", -math.MaxFloat32, Above},
+ {"-0x1.fffffe0p127", -math.MaxFloat32, Exact},
+ {"-12345.000000000000000000001", -12345, Above},
+ {"-12345.0", -12345, Exact},
+ {"-1.000000000000000000001", -1, Above},
+ {"-1", -1, Exact},
+ {"-0x0.000002p-126", -math.SmallestNonzeroFloat32, Exact},
+ {"-0x0.000002p-127", -0, Above}, // underflow
+ {"-1e-1000", -0, Above}, // underflow
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below}, // underflow
+ {"0x0.000002p-127", 0, Below}, // underflow
+ {"0x0.000002p-126", math.SmallestNonzeroFloat32, Exact},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"0x1.fffffe0p127", math.MaxFloat32, Exact},
+ {"0x1.fffffe8p127", math.MaxFloat32, Below},
+ {"0x1.ffffff0p127", float32(math.Inf(+1)), Above}, // overflow
+ {"0x1p128", float32(math.Inf(+1)), Above}, // overflow
+ {"1e10000", float32(math.Inf(+1)), Above}, // overflow
+ {"0x1.ffffff0p2147483646", float32(math.Inf(+1)), Above}, // overflow in rounding
+ {"+Inf", float32(math.Inf(+1)), Exact},
+ } {
+ // conversion should match strconv where syntax is agreeable
+ if f, err := strconv.ParseFloat(test.x, 32); err == nil && float32(f) != test.out {
+ t.Errorf("%s: got %g; want %g (incorrect test data)", test.x, f, test.out)
+ }
+
+ x := makeFloat(test.x)
+ out, acc := x.Float32()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %g (%#x, %s); want %g (%#x, %s)", test.x, out, math.Float32bits(out), acc, test.out, math.Float32bits(test.out), test.acc)
+ }
+
+ // test that x.SetFloat64(float64(f)).Float32() == f
+ var x2 Float
+ out2, acc2 := x2.SetFloat64(float64(out)).Float32()
+ if out2 != out || acc2 != Exact {
+ t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
+ }
+ }
+
+ // test NaN
+ x := makeFloat("NaN")
+ if out, acc := x.Float32(); out == out || acc != Undef {
+ t.Errorf("NaN: got %g (%s); want NaN (Undef)", out, acc)
+ }
+}
+
+func TestFloatFloat64(t *testing.T) {
+ const smallestNormalFloat64 = 2.2250738585072014e-308 // 1p-1022
+ for _, test := range []struct {
+ x string
+ out float64
+ acc Accuracy
+ }{
+ {"-Inf", math.Inf(-1), Exact},
+ {"-0x1.fffffffffffff8p2147483646", -math.Inf(+1), Below}, // overflow in rounding
+ {"-1e10000", math.Inf(-1), Below}, // overflow
+ {"-0x1p1024", math.Inf(-1), Below}, // overflow
+ {"-0x1.fffffffffffff8p1023", -math.Inf(+1), Below}, // overflow
+ {"-0x1.fffffffffffff4p1023", -math.MaxFloat64, Above},
+ {"-0x1.fffffffffffff0p1023", -math.MaxFloat64, Exact},
+ {"-12345.000000000000000000001", -12345, Above},
+ {"-12345.0", -12345, Exact},
+ {"-1.000000000000000000001", -1, Above},
+ {"-1", -1, Exact},
+ {"-0x0.0000000000001p-1022", -math.SmallestNonzeroFloat64, Exact},
+ {"-0x0.0000000000001p-1023", -0, Above}, // underflow
+ {"-1e-1000", -0, Above}, // underflow
+ {"0", 0, Exact},
+ {"1e-1000", 0, Below}, // underflow
+ {"0x0.0000000000001p-1023", 0, Below}, // underflow
+ {"0x0.0000000000001p-1022", math.SmallestNonzeroFloat64, Exact},
+ {"1", 1, Exact},
+ {"1.000000000000000000001", 1, Below},
+ {"12345.0", 12345, Exact},
+ {"12345.000000000000000000001", 12345, Below},
+ {"0x1.fffffffffffff0p1023", math.MaxFloat64, Exact},
+ {"0x1.fffffffffffff4p1023", math.MaxFloat64, Below},
+ {"0x1.fffffffffffff8p1023", math.Inf(+1), Above}, // overflow
+ {"0x1p1024", math.Inf(+1), Above}, // overflow
+ {"1e10000", math.Inf(+1), Above}, // overflow
+ {"0x1.fffffffffffff8p2147483646", math.Inf(+1), Above}, // overflow in rounding
+ {"+Inf", math.Inf(+1), Exact},
+
+ // selected denormalized values that were handled incorrectly in the past
+ {"0x.fffffffffffffp-1022", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},
+ {"4503599627370495p-1074", smallestNormalFloat64 - math.SmallestNonzeroFloat64, Exact},
+
+ // http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ {"2.2250738585072011e-308", 2.225073858507201e-308, Below},
+ // http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ {"2.2250738585072012e-308", 2.2250738585072014e-308, Above},
+ } {
+ // conversion should match strconv where syntax is agreeable
+ if f, err := strconv.ParseFloat(test.x, 64); err == nil && f != test.out {
+ t.Errorf("%s: got %g; want %g (incorrect test data)", test.x, f, test.out)
+ }
+
+ x := makeFloat(test.x)
+ out, acc := x.Float64()
+ if out != test.out || acc != test.acc {
+ t.Errorf("%s: got %g (%#x, %s); want %g (%#x, %s)", test.x, out, math.Float64bits(out), acc, test.out, math.Float64bits(test.out), test.acc)
+ }
+
+ // test that x.SetFloat64(f).Float64() == f
+ var x2 Float
+ out2, acc2 := x2.SetFloat64(out).Float64()
+ if out2 != out || acc2 != Exact {
+ t.Errorf("idempotency test: got %g (%s); want %g (Exact)", out2, acc2, out)
+ }
+ }
+
+ // test NaN
+ x := makeFloat("NaN")
+ if out, acc := x.Float64(); out == out || acc != Undef {
+ t.Errorf("NaN: got %g (%s); want NaN (Undef)", out, acc)
+ }
+}
+
+func TestFloatInt(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ want string
+ acc Accuracy
+ }{
+ {"0", "0", Exact},
+ {"+0", "0", Exact},
+ {"-0", "0", Exact},
+ {"Inf", "nil", Below},
+ {"+Inf", "nil", Below},
+ {"-Inf", "nil", Above},
+ {"NaN", "nil", Undef},
+ {"1", "1", Exact},
+ {"-1", "-1", Exact},
+ {"1.23", "1", Below},
+ {"-1.23", "-1", Above},
+ {"123e-2", "1", Below},
+ {"123e-3", "0", Below},
+ {"123e-4", "0", Below},
+ {"1e-1000", "0", Below},
+ {"-1e-1000", "0", Above},
+ {"1e+10", "10000000000", Exact},
+ {"1e+100", "10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", Exact},
+ } {
+ x := makeFloat(test.x)
+ res, acc := x.Int(nil)
+ got := "nil"
+ if res != nil {
+ got = res.String()
+ }
+ if got != test.want || acc != test.acc {
+ t.Errorf("%s: got %s (%s); want %s (%s)", test.x, got, acc, test.want, test.acc)
+ }
+ }
+
+ // check that supplied *Int is used
+ for _, f := range []string{"0", "1", "-1", "1234"} {
+ x := makeFloat(f)
+ i := new(Int)
+ if res, _ := x.Int(i); res != i {
+ t.Errorf("(%s).Int is not using supplied *Int", f)
+ }
+ }
+}
+
+func TestFloatRat(t *testing.T) {
+ for _, test := range []struct {
+ x, want string
+ acc Accuracy
+ }{
+ {"0", "0/1", Exact},
+ {"+0", "0/1", Exact},
+ {"-0", "0/1", Exact},
+ {"Inf", "nil", Below},
+ {"+Inf", "nil", Below},
+ {"-Inf", "nil", Above},
+ {"NaN", "nil", Undef},
+ {"1", "1/1", Exact},
+ {"-1", "-1/1", Exact},
+ {"1.25", "5/4", Exact},
+ {"-1.25", "-5/4", Exact},
+ {"1e10", "10000000000/1", Exact},
+ {"1p10", "1024/1", Exact},
+ {"-1p-10", "-1/1024", Exact},
+ {"3.14159265", "7244019449799623199/2305843009213693952", Exact},
+ } {
+ x := makeFloat(test.x).SetPrec(64)
+ res, acc := x.Rat(nil)
+ got := "nil"
+ if res != nil {
+ got = res.String()
+ }
+ if got != test.want {
+ t.Errorf("%s: got %s; want %s", test.x, got, test.want)
+ continue
+ }
+ if acc != test.acc {
+ t.Errorf("%s: got %s; want %s", test.x, acc, test.acc)
+ continue
+ }
+
+ // inverse conversion
+ if res != nil {
+ got := new(Float).SetPrec(64).SetRat(res)
+ if got.Cmp(x).Neq() {
+ t.Errorf("%s: got %s; want %s", test.x, got, x)
+ }
+ }
+ }
+
+ // check that supplied *Rat is used
+ for _, f := range []string{"0", "1", "-1", "1234"} {
+ x := makeFloat(f)
+ r := new(Rat)
+ if res, _ := x.Rat(r); res != r {
+ t.Errorf("(%s).Rat is not using supplied *Rat", f)
+ }
+ }
+}
+
+func TestFloatAbs(t *testing.T) {
+ for _, test := range []string{
+ "0",
+ "1",
+ "1234",
+ "1.23e-2",
+ "1e-1000",
+ "1e1000",
+ "Inf",
+ "NaN",
+ } {
+ p := makeFloat(test)
+ a := new(Float).Abs(p)
+ if !feq(a, p) {
+ t.Errorf("%s: got %s; want %s", test, a.Format('g', 10), test)
+ }
+
+ n := makeFloat("-" + test)
+ a.Abs(n)
+ if !feq(a, p) {
+ t.Errorf("-%s: got %s; want %s", test, a.Format('g', 10), test)
+ }
+ }
+}
+
+func TestFloatNeg(t *testing.T) {
+ for _, test := range []string{
+ "0",
+ "1",
+ "1234",
+ "1.23e-2",
+ "1e-1000",
+ "1e1000",
+ "Inf",
+ "NaN",
+ } {
+ p1 := makeFloat(test)
+ n1 := makeFloat("-" + test)
+ n2 := new(Float).Neg(p1)
+ p2 := new(Float).Neg(n2)
+ if !feq(n2, n1) {
+ t.Errorf("%s: got %s; want %s", test, n2.Format('g', 10), n1.Format('g', 10))
+ }
+ if !feq(p2, p1) {
+ t.Errorf("%s: got %s; want %s", test, p2.Format('g', 10), p1.Format('g', 10))
+ }
+ }
+}
+
+func TestFloatInc(t *testing.T) {
+ const n = 10
+ for _, prec := range precList {
+ if 1<<prec < n {
+ continue // prec must be large enough to hold all numbers from 0 to n
+ }
+ var x, one Float
+ x.SetPrec(prec)
+ one.SetInt64(1)
+ for i := 0; i < n; i++ {
+ x.Add(&x, &one)
+ }
+ if x.Cmp(new(Float).SetInt64(n)).Neq() {
+ t.Errorf("prec = %d: got %s; want %d", prec, &x, n)
+ }
+ }
+}
+
+// Selected precisions with which to run various tests.
+var precList = [...]uint{1, 2, 5, 8, 10, 16, 23, 24, 32, 50, 53, 64, 100, 128, 500, 511, 512, 513, 1000, 10000}
+
+// Selected bits with which to run various tests.
+// Each entry is a list of bits representing a floating-point number (see fromBits).
+var bitsList = [...]Bits{
+ {}, // = 0
+ {0}, // = 1
+ {1}, // = 2
+ {-1}, // = 1/2
+ {10}, // = 2**10 == 1024
+ {-10}, // = 2**-10 == 1/1024
+ {100, 10, 1}, // = 2**100 + 2**10 + 2**1
+ {0, -1, -2, -10},
+ // TODO(gri) add more test cases
+}
+
+// TestFloatAdd tests Float.Add/Sub by comparing the result of a "manual"
+// addition/subtraction of arguments represented by Bits values with the
+// respective Float addition/subtraction for a variety of precisions
+// and rounding modes.
+func TestFloatAdd(t *testing.T) {
+ for _, xbits := range bitsList {
+ for _, ybits := range bitsList {
+ // exact values
+ x := xbits.Float()
+ y := ybits.Float()
+ zbits := xbits.add(ybits)
+ z := zbits.Float()
+
+ for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for _, prec := range precList {
+ got := new(Float).SetPrec(prec).SetMode(mode)
+ got.Add(x, y)
+ want := zbits.round(prec, mode)
+ if got.Cmp(want).Neq() {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t+ %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, x, xbits, y, ybits, got, want)
+ }
+
+ got.Sub(z, x)
+ want = ybits.round(prec, mode)
+ if got.Cmp(want).Neq() {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t- %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, z, zbits, x, xbits, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatAdd32 tests that Float.Add/Sub of numbers with
+// 24bit mantissa behaves like float32 addition/subtraction
+// (excluding denormal numbers).
+func TestFloatAdd32(t *testing.T) {
+ // chose base such that we cross the mantissa precision limit
+ const base = 1<<26 - 0x10 // 11...110000 (26 bits)
+ for d := 0; d <= 0x10; d++ {
+ for i := range [2]int{} {
+ x0, y0 := float64(base), float64(d)
+ if i&1 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(24)
+
+ z.Add(x, y)
+ got, acc := z.Float32()
+ want := float32(y0) + float32(x0)
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
+ }
+
+ z.Sub(z, y)
+ got, acc = z.Float32()
+ want = float32(want) - float32(y0)
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
+ }
+ }
+ }
+}
+
+// TestFloatAdd64 tests that Float.Add/Sub of numbers with
+// 53bit mantissa behaves like float64 addition/subtraction.
+func TestFloatAdd64(t *testing.T) {
+ // chose base such that we cross the mantissa precision limit
+ const base = 1<<55 - 0x10 // 11...110000 (55 bits)
+ for d := 0; d <= 0x10; d++ {
+ for i := range [2]int{} {
+ x0, y0 := float64(base), float64(d)
+ if i&1 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(53)
+
+ z.Add(x, y)
+ got, acc := z.Float64()
+ want := x0 + y0
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g + %g = %g (%s); want %g (Exact)", d, x0, y0, got, acc, want)
+ }
+
+ z.Sub(z, y)
+ got, acc = z.Float64()
+ want -= y0
+ if got != want || acc != Exact {
+ t.Errorf("d = %d: %g - %g = %g (%s); want %g (Exact)", d, x0+y0, y0, got, acc, want)
+ }
+ }
+ }
+}
+
+// TestFloatMul tests Float.Mul/Quo by comparing the result of a "manual"
+// multiplication/division of arguments represented by Bits values with the
+// respective Float multiplication/division for a variety of precisions
+// and rounding modes.
+func TestFloatMul(t *testing.T) {
+ for _, xbits := range bitsList {
+ for _, ybits := range bitsList {
+ // exact values
+ x := xbits.Float()
+ y := ybits.Float()
+ zbits := xbits.mul(ybits)
+ z := zbits.Float()
+
+ for i, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for _, prec := range precList {
+ got := new(Float).SetPrec(prec).SetMode(mode)
+ got.Mul(x, y)
+ want := zbits.round(prec, mode)
+ if got.Cmp(want).Neq() {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t* %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, x, xbits, y, ybits, got, want)
+ }
+
+ if x.IsZero() {
+ continue // ignore div-0 case (not invertable)
+ }
+ got.Quo(z, x)
+ want = ybits.round(prec, mode)
+ if got.Cmp(want).Neq() {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s %v\n\t/ %s %v\n\t= %s\n\twant %s",
+ i, prec, mode, z, zbits, x, xbits, got, want)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatMul64 tests that Float.Mul/Quo of numbers with
+// 53bit mantissa behaves like float64 multiplication/division.
+func TestFloatMul64(t *testing.T) {
+ for _, test := range []struct {
+ x, y float64
+ }{
+ {0, 0},
+ {0, 1},
+ {1, 1},
+ {1, 1.5},
+ {1.234, 0.5678},
+ {2.718281828, 3.14159265358979},
+ {2.718281828e10, 3.14159265358979e-32},
+ {1.0 / 3, 1e200},
+ } {
+ for i := range [8]int{} {
+ x0, y0 := test.x, test.y
+ if i&1 != 0 {
+ x0 = -x0
+ }
+ if i&2 != 0 {
+ y0 = -y0
+ }
+ if i&4 != 0 {
+ x0, y0 = y0, x0
+ }
+
+ x := NewFloat(x0)
+ y := NewFloat(y0)
+ z := new(Float).SetPrec(53)
+
+ z.Mul(x, y)
+ got, _ := z.Float64()
+ want := x0 * y0
+ if got != want {
+ t.Errorf("%g * %g = %g; want %g", x0, y0, got, want)
+ }
+
+ if y0 == 0 {
+ continue // avoid division-by-zero
+ }
+ z.Quo(z, y)
+ got, _ = z.Float64()
+ want /= y0
+ if got != want {
+ t.Errorf("%g / %g = %g; want %g", x0*y0, y0, got, want)
+ }
+ }
+ }
+}
+
+func TestIssue6866(t *testing.T) {
+ for _, prec := range precList {
+ two := new(Float).SetPrec(prec).SetInt64(2)
+ one := new(Float).SetPrec(prec).SetInt64(1)
+ three := new(Float).SetPrec(prec).SetInt64(3)
+ msix := new(Float).SetPrec(prec).SetInt64(-6)
+ psix := new(Float).SetPrec(prec).SetInt64(+6)
+
+ p := new(Float).SetPrec(prec)
+ z1 := new(Float).SetPrec(prec)
+ z2 := new(Float).SetPrec(prec)
+
+ // z1 = 2 + 1.0/3*-6
+ p.Quo(one, three)
+ p.Mul(p, msix)
+ z1.Add(two, p)
+
+ // z2 = 2 - 1.0/3*+6
+ p.Quo(one, three)
+ p.Mul(p, psix)
+ z2.Sub(two, p)
+
+ if z1.Cmp(z2).Neq() {
+ t.Fatalf("prec %d: got z1 = %s != z2 = %s; want z1 == z2\n", prec, z1, z2)
+ }
+ if z1.Sign() != 0 {
+ t.Errorf("prec %d: got z1 = %s; want 0", prec, z1)
+ }
+ if z2.Sign() != 0 {
+ t.Errorf("prec %d: got z2 = %s; want 0", prec, z2)
+ }
+ }
+}
+
+func TestFloatQuo(t *testing.T) {
+ // TODO(gri) make the test vary these precisions
+ preci := 200 // precision of integer part
+ precf := 20 // precision of fractional part
+
+ for i := 0; i < 8; i++ {
+ // compute accurate (not rounded) result z
+ bits := Bits{preci - 1}
+ if i&3 != 0 {
+ bits = append(bits, 0)
+ }
+ if i&2 != 0 {
+ bits = append(bits, -1)
+ }
+ if i&1 != 0 {
+ bits = append(bits, -precf)
+ }
+ z := bits.Float()
+
+ // compute accurate x as z*y
+ y := NewFloat(3.14159265358979323e123)
+
+ x := new(Float).SetPrec(z.Prec() + y.Prec()).SetMode(ToZero)
+ x.Mul(z, y)
+
+ // leave for debugging
+ // fmt.Printf("x = %s\ny = %s\nz = %s\n", x, y, z)
+
+ if got := x.Acc(); got != Exact {
+ t.Errorf("got acc = %s; want exact", got)
+ }
+
+ // round accurate z for a variety of precisions and
+ // modes and compare against result of x / y.
+ for _, mode := range [...]RoundingMode{ToZero, ToNearestEven, AwayFromZero} {
+ for d := -5; d < 5; d++ {
+ prec := uint(preci + d)
+ got := new(Float).SetPrec(prec).SetMode(mode).Quo(x, y)
+ want := bits.round(prec, mode)
+ if got.Cmp(want).Neq() {
+ t.Errorf("i = %d, prec = %d, %s:\n\t %s\n\t/ %s\n\t= %s\n\twant %s",
+ i, prec, mode, x, y, got, want)
+ }
+ }
+ }
+ }
+}
+
+// TestFloatQuoSmoke tests all divisions x/y for values x, y in the range [-n, +n];
+// it serves as a smoke test for basic correctness of division.
+func TestFloatQuoSmoke(t *testing.T) {
+ n := 1000
+ if testing.Short() {
+ n = 10
+ }
+
+ const dprec = 3 // max. precision variation
+ const prec = 10 + dprec // enough bits to hold n precisely
+ for x := -n; x <= n; x++ {
+ for y := -n; y < n; y++ {
+ if y == 0 {
+ continue
+ }
+
+ a := float64(x)
+ b := float64(y)
+ c := a / b
+
+ // vary operand precision (only ok as long as a, b can be represented correctly)
+ for ad := -dprec; ad <= dprec; ad++ {
+ for bd := -dprec; bd <= dprec; bd++ {
+ A := new(Float).SetPrec(uint(prec + ad)).SetFloat64(a)
+ B := new(Float).SetPrec(uint(prec + bd)).SetFloat64(b)
+ C := new(Float).SetPrec(53).Quo(A, B) // C has float64 mantissa width
+
+ cc, acc := C.Float64()
+ if cc != c {
+ t.Errorf("%g/%g = %s; want %.5g\n", a, b, C.Format('g', 5), c)
+ continue
+ }
+ if acc != Exact {
+ t.Errorf("%g/%g got %s result; want exact result", a, b, acc)
+ }
+ }
+ }
+ }
+ }
+}
+
+// TestFloatArithmeticSpecialValues tests that Float operations produce the
+// correct results for combinations of zero (±0), finite (±1 and ±2.71828),
+// and non-finite (±Inf, NaN) operands.
+func TestFloatArithmeticSpecialValues(t *testing.T) {
+ zero := 0.0
+ args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1), math.NaN()}
+ xx := new(Float)
+ yy := new(Float)
+ got := new(Float)
+ want := new(Float)
+ for i := 0; i < 4; i++ {
+ for _, x := range args {
+ xx.SetFloat64(x)
+ // check conversion is correct
+ // (no need to do this for y, since we see exactly the
+ // same values there)
+ if got, acc := xx.Float64(); !math.IsNaN(x) && (got != x || acc != Exact) {
+ t.Errorf("Float(%g) == %g (%s)", x, got, acc)
+ }
+ for _, y := range args {
+ yy.SetFloat64(y)
+ var op string
+ var z float64
+ switch i {
+ case 0:
+ op = "+"
+ z = x + y
+ got.Add(xx, yy)
+ case 1:
+ op = "-"
+ z = x - y
+ got.Sub(xx, yy)
+ case 2:
+ op = "*"
+ z = x * y
+ got.Mul(xx, yy)
+ case 3:
+ op = "/"
+ z = x / y
+ got.Quo(xx, yy)
+ default:
+ panic("unreachable")
+ }
+ // At the moment an Inf operand always leads to a NaN result (known bug).
+ // TODO(gri) remove this once the bug is fixed.
+ if math.IsInf(x, 0) || math.IsInf(y, 0) {
+ want.SetNaN()
+ } else {
+ want.SetFloat64(z)
+ }
+ if !feq(got, want) {
+ t.Errorf("%5g %s %5g = %5s; want %5s", x, op, y, got, want)
+ }
+ }
+ }
+ }
+}
+
+func TestFloatArithmeticOverflow(t *testing.T) {
+ for _, test := range []struct {
+ prec uint
+ mode RoundingMode
+ op byte
+ x, y, want string
+ acc Accuracy
+ }{
+ {4, ToNearestEven, '+', "0", "0", "0", Exact}, // smoke test
+ {4, ToNearestEven, '+', "0x.8p0", "0x.8p0", "0x.8p1", Exact}, // smoke test
+
+ {4, ToNearestEven, '+', "0", "0x.8p2147483647", "0x.8p2147483647", Exact},
+ {4, ToNearestEven, '+', "0x.8p2147483500", "0x.8p2147483647", "0x.8p2147483647", Below}, // rounded to zero
+ {4, ToNearestEven, '+', "0x.8p2147483647", "0x.8p2147483647", "+Inf", Above}, // exponent overflow in +
+ {4, ToNearestEven, '+', "-0x.8p2147483647", "-0x.8p2147483647", "-Inf", Below}, // exponent overflow in +
+ {4, ToNearestEven, '-', "-0x.8p2147483647", "0x.8p2147483647", "-Inf", Below}, // exponent overflow in -
+
+ {4, ToZero, '+', "0x.fp2147483647", "0x.8p2147483643", "0x.fp2147483647", Below}, // rounded to zero
+ {4, ToNearestEven, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above}, // exponent overflow in rounding
+ {4, AwayFromZero, '+', "0x.fp2147483647", "0x.8p2147483643", "+Inf", Above}, // exponent overflow in rounding
+
+ {4, AwayFromZero, '-', "-0x.fp2147483647", "0x.8p2147483644", "-Inf", Below}, // exponent overflow in rounding
+ {4, ToNearestEven, '-', "-0x.fp2147483647", "0x.8p2147483643", "-Inf", Below}, // exponent overflow in rounding
+ {4, ToZero, '-', "-0x.fp2147483647", "0x.8p2147483643", "-0x.fp2147483647", Above}, // rounded to zero
+
+ {4, ToNearestEven, '+', "0", "0x.8p-2147483648", "0x.8p-2147483648", Exact},
+ {4, ToNearestEven, '+', "0x.8p-2147483648", "0x.8p-2147483648", "0x.8p-2147483647", Exact},
+
+ {4, ToNearestEven, '*', "1", "0x.8p2147483647", "0x.8p2147483647", Exact},
+ {4, ToNearestEven, '*', "2", "0x.8p2147483647", "+Inf", Above}, // exponent overflow in *
+ {4, ToNearestEven, '*', "-2", "0x.8p2147483647", "-Inf", Below}, // exponent overflow in *
+
+ {4, ToNearestEven, '/', "0.5", "0x.8p2147483647", "0x.8p-2147483646", Exact},
+ {4, ToNearestEven, '/', "0x.8p0", "0x.8p2147483647", "0x.8p-2147483646", Exact},
+ {4, ToNearestEven, '/', "0x.8p-1", "0x.8p2147483647", "0x.8p-2147483647", Exact},
+ {4, ToNearestEven, '/', "0x.8p-2", "0x.8p2147483647", "0x.8p-2147483648", Exact},
+ {4, ToNearestEven, '/', "0x.8p-3", "0x.8p2147483647", "0", Below}, // exponent underflow in /
+ } {
+ x := makeFloat(test.x)
+ y := makeFloat(test.y)
+ z := new(Float).SetPrec(test.prec).SetMode(test.mode)
+ switch test.op {
+ case '+':
+ z.Add(x, y)
+ case '-':
+ z.Sub(x, y)
+ case '*':
+ z.Mul(x, y)
+ case '/':
+ z.Quo(x, y)
+ default:
+ panic("unreachable")
+ }
+ if got := z.Format('p', 0); got != test.want || z.Acc() != test.acc {
+ t.Errorf(
+ "prec = %d (%s): %s %c %s = %s (%s); want %s (%s)",
+ test.prec, test.mode, x.Format('p', 0), test.op, y.Format('p', 0), got, z.Acc(), test.want, test.acc,
+ )
+ }
+ }
+}
+
+// TODO(gri) Add tests that check correctness in the presence of aliasing.
+
+// For rounding modes ToNegativeInf and ToPositiveInf, rounding is affected
+// by the sign of the value to be rounded. Test that rounding happens after
+// the sign of a result has been set.
+// This test uses specific values that are known to fail if rounding is
+// "factored" out before setting the result sign.
+func TestFloatArithmeticRounding(t *testing.T) {
+ for _, test := range []struct {
+ mode RoundingMode
+ prec uint
+ x, y, want int64
+ op byte
+ }{
+ {ToZero, 3, -0x8, -0x1, -0x8, '+'},
+ {AwayFromZero, 3, -0x8, -0x1, -0xa, '+'},
+ {ToNegativeInf, 3, -0x8, -0x1, -0xa, '+'},
+
+ {ToZero, 3, -0x8, 0x1, -0x8, '-'},
+ {AwayFromZero, 3, -0x8, 0x1, -0xa, '-'},
+ {ToNegativeInf, 3, -0x8, 0x1, -0xa, '-'},
+
+ {ToZero, 3, -0x9, 0x1, -0x8, '*'},
+ {AwayFromZero, 3, -0x9, 0x1, -0xa, '*'},
+ {ToNegativeInf, 3, -0x9, 0x1, -0xa, '*'},
+
+ {ToZero, 3, -0x9, 0x1, -0x8, '/'},
+ {AwayFromZero, 3, -0x9, 0x1, -0xa, '/'},
+ {ToNegativeInf, 3, -0x9, 0x1, -0xa, '/'},
+ } {
+ var x, y, z Float
+ x.SetInt64(test.x)
+ y.SetInt64(test.y)
+ z.SetPrec(test.prec).SetMode(test.mode)
+ switch test.op {
+ case '+':
+ z.Add(&x, &y)
+ case '-':
+ z.Sub(&x, &y)
+ case '*':
+ z.Mul(&x, &y)
+ case '/':
+ z.Quo(&x, &y)
+ default:
+ panic("unreachable")
+ }
+ if got, acc := z.Int64(); got != test.want || acc != Exact {
+ t.Errorf("%s, %d bits: %d %c %d = %d (%s); want %d (Exact)",
+ test.mode, test.prec, test.x, test.op, test.y, got, acc, test.want,
+ )
+ }
+ }
+}
+
+// TestFloatCmpSpecialValues tests that Cmp produces the correct results for
+// combinations of zero (±0), finite (±1 and ±2.71828), and non-finite (±Inf,
+// NaN) operands.
+func TestFloatCmpSpecialValues(t *testing.T) {
+ zero := 0.0
+ args := []float64{math.Inf(-1), -2.71828, -1, -zero, zero, 1, 2.71828, math.Inf(1), math.NaN()}
+ xx := new(Float)
+ yy := new(Float)
+ for i := 0; i < 4; i++ {
+ for _, x := range args {
+ xx.SetFloat64(x)
+ // check conversion is correct
+ // (no need to do this for y, since we see exactly the
+ // same values there)
+ if got, acc := xx.Float64(); !math.IsNaN(x) && (got != x || acc != Exact) {
+ t.Errorf("Float(%g) == %g (%s)", x, got, acc)
+ }
+ for _, y := range args {
+ yy.SetFloat64(y)
+ got := xx.Cmp(yy).Acc()
+ want := Undef
+ switch {
+ case x < y:
+ want = Below
+ case x == y:
+ want = Exact
+ case x > y:
+ want = Above
+ }
+ if got != want {
+ t.Errorf("(%g).Cmp(%g) = %s; want %s", x, y, got, want)
+ }
+ }
+ }
+ }
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements float-to-string conversion functions.
+
+package big
+
+import (
+ "fmt"
+ "io"
+ "strconv"
+ "strings"
+)
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s must be a floating-point number of the same format as accepted
+// by Scan, with number prefixes permitted.
+func (z *Float) SetString(s string) (*Float, bool) {
+ r := strings.NewReader(s)
+
+ f, _, err := z.Scan(r, 0)
+ if err != nil {
+ return nil, false
+ }
+
+ // there should be no unread characters left
+ if _, err = r.ReadByte(); err != io.EOF {
+ return nil, false
+ }
+
+ return f, true
+}
+
+// Scan scans the number corresponding to the longest possible prefix
+// of r representing a floating-point number with a mantissa in the
+// given conversion base (the exponent is always a decimal number).
+// It sets z to the (possibly rounded) value of the corresponding
+// floating-point number, and returns z, the actual base b, and an
+// error err, if any. If z's precision is 0, it is changed to 64
+// before rounding takes effect. The number must be of the form:
+//
+// number = [ sign ] [ prefix ] mantissa [ exponent ] .
+// sign = "+" | "-" .
+// prefix = "0" ( "x" | "X" | "b" | "B" ) .
+// mantissa = digits | digits "." [ digits ] | "." digits .
+// exponent = ( "E" | "e" | "p" ) [ sign ] digits .
+// digits = digit { digit } .
+// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
+//
+// The base argument must be 0, 2, 10, or 16. Providing an invalid base
+// argument will lead to a run-time panic.
+//
+// For base 0, the number prefix determines the actual base: A prefix of
+// "0x" or "0X" selects base 16, and a "0b" or "0B" prefix selects
+// base 2; otherwise, the actual base is 10 and no prefix is accepted.
+// The octal prefix "0" is not supported (a leading "0" is simply
+// considered a "0").
+//
+// A "p" exponent indicates a binary (rather then decimal) exponent;
+// for instance "0x1.fffffffffffffp1023" (using base 0) represents the
+// maximum float64 value. For hexadecimal mantissae, the exponent must
+// be binary, if present (an "e" or "E" exponent indicator cannot be
+// distinguished from a mantissa digit).
+//
+// The returned *Float f is nil and the value of z is valid but not
+// defined if an error is reported.
+//
+// BUG(gri) The Float.Scan signature conflicts with Scan(s fmt.ScanState, ch rune) error.
+func (z *Float) Scan(r io.ByteScanner, base int) (f *Float, b int, err error) {
+ prec := z.prec
+ if prec == 0 {
+ prec = 64
+ }
+
+ // NaNs ignore sign, mantissa, and exponent so we can set
+ // them below while having a valid value for z in case of
+ // errors.
+ z.SetNaN()
+
+ // sign
+ z.neg, err = scanSign(r)
+ if err != nil {
+ return
+ }
+
+ // mantissa
+ var fcount int // fractional digit count; valid if <= 0
+ z.mant, b, fcount, err = z.mant.scan(r, base, true)
+ if err != nil {
+ return
+ }
+
+ // exponent
+ var exp int64
+ var ebase int
+ exp, ebase, err = scanExponent(r, true)
+ if err != nil {
+ return
+ }
+
+ // special-case 0
+ if len(z.mant) == 0 {
+ z.prec = prec
+ z.acc = Exact
+ z.form = zero
+ f = z
+ return
+ }
+ // len(z.mant) > 0
+
+ // The mantissa may have a decimal point (fcount <= 0) and there
+ // may be a nonzero exponent exp. The decimal point amounts to a
+ // division by b**(-fcount). An exponent means multiplication by
+ // ebase**exp. Finally, mantissa normalization (shift left) requires
+ // a correcting multiplication by 2**(-shiftcount). Multiplications
+ // are commutative, so we can apply them in any order as long as there
+ // is no loss of precision. We only have powers of 2 and 10; keep
+ // track via separate exponents exp2 and exp10.
+
+ // normalize mantissa and get initial binary exponent
+ var exp2 = int64(len(z.mant))*_W - fnorm(z.mant)
+
+ // determine binary or decimal exponent contribution of decimal point
+ var exp10 int64
+ if fcount < 0 {
+ // The mantissa has a "decimal" point ddd.dddd; and
+ // -fcount is the number of digits to the right of '.'.
+ // Adjust relevant exponent accodingly.
+ switch b {
+ case 16:
+ fcount *= 4 // hexadecimal digits are 4 bits each
+ fallthrough
+ case 2:
+ exp2 += int64(fcount)
+ default: // b == 10
+ exp10 = int64(fcount)
+ }
+ // we don't need fcount anymore
+ }
+
+ // take actual exponent into account
+ if ebase == 2 {
+ exp2 += exp
+ } else { // ebase == 10
+ exp10 += exp
+ }
+ // we don't need exp anymore
+
+ // apply 2**exp2
+ if MinExp <= exp2 && exp2 <= MaxExp {
+ z.prec = prec
+ z.form = finite
+ z.exp = int32(exp2)
+ f = z
+ } else {
+ err = fmt.Errorf("exponent overflow")
+ return
+ }
+
+ if exp10 == 0 {
+ // no decimal exponent to consider
+ z.round(0)
+ return
+ }
+ // exp10 != 0
+
+ // compute decimal exponent power
+ expabs := exp10
+ if expabs < 0 {
+ expabs = -expabs
+ }
+ powTen := nat(nil).expNN(natTen, nat(nil).setUint64(uint64(expabs)), nil)
+ fpowTen := new(Float).SetInt(new(Int).SetBits(powTen))
+
+ // apply 10**exp10
+ if exp10 < 0 {
+ z.uquo(z, fpowTen)
+ } else {
+ z.umul(z, fpowTen)
+ }
+
+ return
+}
+
+// Parse is like z.Scan(r, base), but instead of reading from an
+// io.ByteScanner, it parses the string s. An error is also returned
+// if the string contains invalid or trailing bytes not belonging to
+// the number.
+func (z *Float) Parse(s string, base int) (f *Float, b int, err error) {
+ r := strings.NewReader(s)
+
+ if f, b, err = z.Scan(r, base); err != nil {
+ return
+ }
+
+ // entire string must have been consumed
+ if ch, err2 := r.ReadByte(); err2 == nil {
+ err = fmt.Errorf("expected end of string, found %q", ch)
+ } else if err2 != io.EOF {
+ err = err2
+ }
+
+ return
+}
+
+// ScanFloat is like f.Scan(r, base) with f set to the given precision
+// and rounding mode.
+func ScanFloat(r io.ByteScanner, base int, prec uint, mode RoundingMode) (f *Float, b int, err error) {
+ return new(Float).SetPrec(prec).SetMode(mode).Scan(r, base)
+}
+
+// ParseFloat is like f.Parse(s, base) with f set to the given precision
+// and rounding mode.
+func ParseFloat(s string, base int, prec uint, mode RoundingMode) (f *Float, b int, err error) {
+ return new(Float).SetPrec(prec).SetMode(mode).Parse(s, base)
+}
+
+// Format converts the floating-point number x to a string according
+// to the given format and precision prec. The format is one of:
+//
+// 'e' -d.dddde±dd, decimal exponent, at least two (possibly 0) exponent digits
+// 'E' -d.ddddE±dd, decimal exponent, at least two (possibly 0) exponent digits
+// 'f' -ddddd.dddd, no exponent
+// 'g' like 'e' for large exponents, like 'f' otherwise
+// 'G' like 'E' for large exponents, like 'f' otherwise
+// 'b' -ddddddp±dd, binary exponent
+// 'p' -0x.dddp±dd, binary exponent, hexadecimal mantissa
+//
+// For the binary exponent formats, the mantissa is printed in normalized form:
+//
+// 'b' decimal integer mantissa using x.Prec() bits, or -0
+// 'p' hexadecimal fraction with 0.5 <= 0.mantissa < 1.0, or -0
+//
+// The precision prec controls the number of digits (excluding the exponent)
+// printed by the 'e', 'E', 'f', 'g', and 'G' formats. For 'e', 'E', and 'f'
+// it is the number of digits after the decimal point. For 'g' and 'G' it is
+// the total number of digits. A negative precision selects the smallest
+// number of digits necessary such that ParseFloat will return f exactly.
+// The prec value is ignored for the 'b' or 'p' format.
+//
+// BUG(gri) Float.Format does not accept negative precisions.
+func (x *Float) Format(format byte, prec int) string {
+ const extra = 10 // TODO(gri) determine a good/better value here
+ return string(x.Append(make([]byte, 0, prec+extra), format, prec))
+}
+
+// Append appends the string form of the floating-point number x,
+// as generated by x.Format, to buf and returns the extended buffer.
+func (x *Float) Append(buf []byte, format byte, prec int) []byte {
+ // TODO(gri) factor out handling of sign?
+
+ // Inf
+ if x.IsInf() {
+ var ch byte = '+'
+ if x.neg {
+ ch = '-'
+ }
+ buf = append(buf, ch)
+ return append(buf, "Inf"...)
+ }
+
+ // NaN
+ if x.IsNaN() {
+ return append(buf, "NaN"...)
+ }
+
+ // easy formats
+ switch format {
+ case 'b':
+ return x.bstring(buf)
+ case 'p':
+ return x.pstring(buf)
+ }
+
+ return x.bigFtoa(buf, format, prec)
+}
+
+// BUG(gri): Float.String uses x.Format('g', 10) rather than x.Format('g', -1).
+func (x *Float) String() string {
+ return x.Format('g', 10)
+}
+
+// bstring appends the string of x in the format ["-"] mantissa "p" exponent
+// with a decimal mantissa and a binary exponent, or ["-"] "0" if x is zero,
+// and returns the extended buffer.
+// The mantissa is normalized such that is uses x.Prec() bits in binary
+// representation.
+func (x *Float) bstring(buf []byte) []byte {
+ if x.neg {
+ buf = append(buf, '-')
+ }
+ if x.form == zero {
+ return append(buf, '0')
+ }
+
+ if debugFloat && x.form != finite {
+ panic("non-finite float")
+ }
+ // x != 0
+
+ // adjust mantissa to use exactly x.prec bits
+ m := x.mant
+ switch w := uint32(len(x.mant)) * _W; {
+ case w < x.prec:
+ m = nat(nil).shl(m, uint(x.prec-w))
+ case w > x.prec:
+ m = nat(nil).shr(m, uint(w-x.prec))
+ }
+
+ buf = append(buf, m.decimalString()...)
+ buf = append(buf, 'p')
+ e := int64(x.exp) - int64(x.prec)
+ if e >= 0 {
+ buf = append(buf, '+')
+ }
+ return strconv.AppendInt(buf, e, 10)
+}
+
+// pstring appends the string of x in the format ["-"] "0x." mantissa "p" exponent
+// with a hexadecimal mantissa and a binary exponent, or ["-"] "0" if x is zero,
+// ad returns the extended buffer.
+// The mantissa is normalized such that 0.5 <= 0.mantissa < 1.0.
+func (x *Float) pstring(buf []byte) []byte {
+ if x.neg {
+ buf = append(buf, '-')
+ }
+ if x.form == zero {
+ return append(buf, '0')
+ }
+
+ if debugFloat && x.form != finite {
+ panic("non-finite float")
+ }
+ // x != 0
+
+ // remove trailing 0 words early
+ // (no need to convert to hex 0's and trim later)
+ m := x.mant
+ i := 0
+ for i < len(m) && m[i] == 0 {
+ i++
+ }
+ m = m[i:]
+
+ buf = append(buf, "0x."...)
+ buf = append(buf, strings.TrimRight(x.mant.hexString(), "0")...)
+ buf = append(buf, 'p')
+ return strconv.AppendInt(buf, int64(x.exp), 10)
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "math"
+ "strconv"
+ "testing"
+)
+
+func TestFloatSetFloat64String(t *testing.T) {
+ for _, test := range []struct {
+ s string
+ x float64
+ }{
+ // basics
+ {"0", 0},
+ {"-0", -0},
+ {"+0", 0},
+ {"1", 1},
+ {"-1", -1},
+ {"+1", 1},
+ {"1.234", 1.234},
+ {"-1.234", -1.234},
+ {"+1.234", 1.234},
+ {".1", 0.1},
+ {"1.", 1},
+ {"+1.", 1},
+
+ // various zeros
+ {"0e100", 0},
+ {"-0e+100", 0},
+ {"+0e-100", 0},
+ {"0E100", 0},
+ {"-0E+100", 0},
+ {"+0E-100", 0},
+
+ // various decimal exponent formats
+ {"1.e10", 1e10},
+ {"1e+10", 1e10},
+ {"+1e-10", 1e-10},
+ {"1E10", 1e10},
+ {"1.E+10", 1e10},
+ {"+1E-10", 1e-10},
+
+ // misc decimal values
+ {"3.14159265", 3.14159265},
+ {"-687436.79457e-245", -687436.79457e-245},
+ {"-687436.79457E245", -687436.79457e245},
+ {".0000000000000000000000000000000000000001", 1e-40},
+ {"+10000000000000000000000000000000000000000e-0", 1e40},
+
+ // decimal mantissa, binary exponent
+ {"0p0", 0},
+ {"-0p0", -0},
+ {"1p10", 1 << 10},
+ {"1p+10", 1 << 10},
+ {"+1p-10", 1.0 / (1 << 10)},
+ {"1024p-12", 0.25},
+ {"-1p10", -1024},
+ {"1.5p1", 3},
+
+ // binary mantissa, decimal exponent
+ {"0b0", 0},
+ {"-0b0", -0},
+ {"0b0e+10", 0},
+ {"-0b0e-10", -0},
+ {"0b1010", 10},
+ {"0B1010E2", 1000},
+ {"0b.1", 0.5},
+ {"0b.001", 0.125},
+ {"0b.001e3", 125},
+
+ // binary mantissa, binary exponent
+ {"0b0p+10", 0},
+ {"-0b0p-10", -0},
+ {"0b.1010p4", 10},
+ {"0b1p-1", 0.5},
+ {"0b001p-3", 0.125},
+ {"0b.001p3", 1},
+ {"0b0.01p2", 1},
+
+ // hexadecimal mantissa and exponent
+ {"0x0", 0},
+ {"-0x0", -0},
+ {"0x0p+10", 0},
+ {"-0x0p-10", -0},
+ {"0xff", 255},
+ {"0X.8p1", 1},
+ {"-0X0.00008p16", -0.5},
+ {"0x0.0000000000001p-1022", math.SmallestNonzeroFloat64},
+ {"0x1.fffffffffffffp1023", math.MaxFloat64},
+ } {
+ var x Float
+ x.SetPrec(53)
+ _, ok := x.SetString(test.s)
+ if !ok {
+ t.Errorf("%s: parse error", test.s)
+ continue
+ }
+ f, _ := x.Float64()
+ want := new(Float).SetFloat64(test.x)
+ if x.Cmp(want).Neq() {
+ t.Errorf("%s: got %s (%v); want %v", test.s, &x, f, test.x)
+ }
+ }
+}
+
+const (
+ below1e23 = 99999999999999974834176
+ above1e23 = 100000000000000008388608
+)
+
+func TestFloat64Format(t *testing.T) {
+ for _, test := range []struct {
+ x float64
+ format byte
+ prec int
+ want string
+ }{
+ {0, 'f', 0, "0"},
+ {math.Copysign(0, -1), 'f', 0, "-0"},
+ {1, 'f', 0, "1"},
+ {-1, 'f', 0, "-1"},
+
+ {1.459, 'e', 0, "1e+00"},
+ {2.459, 'e', 1, "2.5e+00"},
+ {3.459, 'e', 2, "3.46e+00"},
+ {4.459, 'e', 3, "4.459e+00"},
+ {5.459, 'e', 4, "5.4590e+00"},
+
+ {1.459, 'f', 0, "1"},
+ {2.459, 'f', 1, "2.5"},
+ {3.459, 'f', 2, "3.46"},
+ {4.459, 'f', 3, "4.459"},
+ {5.459, 'f', 4, "5.4590"},
+
+ {0, 'b', 0, "0"},
+ {math.Copysign(0, -1), 'b', 0, "-0"},
+ {1.0, 'b', 0, "4503599627370496p-52"},
+ {-1.0, 'b', 0, "-4503599627370496p-52"},
+ {4503599627370496, 'b', 0, "4503599627370496p+0"},
+
+ {0, 'p', 0, "0"},
+ {math.Copysign(0, -1), 'p', 0, "-0"},
+ {1024.0, 'p', 0, "0x.8p11"},
+ {-1024.0, 'p', 0, "-0x.8p11"},
+
+ // all test cases below from strconv/ftoa_test.go
+ {1, 'e', 5, "1.00000e+00"},
+ {1, 'f', 5, "1.00000"},
+ {1, 'g', 5, "1"},
+ // {1, 'g', -1, "1"},
+ // {20, 'g', -1, "20"},
+ // {1234567.8, 'g', -1, "1.2345678e+06"},
+ // {200000, 'g', -1, "200000"},
+ // {2000000, 'g', -1, "2e+06"},
+
+ // g conversion and zero suppression
+ {400, 'g', 2, "4e+02"},
+ {40, 'g', 2, "40"},
+ {4, 'g', 2, "4"},
+ {.4, 'g', 2, "0.4"},
+ {.04, 'g', 2, "0.04"},
+ {.004, 'g', 2, "0.004"},
+ {.0004, 'g', 2, "0.0004"},
+ {.00004, 'g', 2, "4e-05"},
+ {.000004, 'g', 2, "4e-06"},
+
+ {0, 'e', 5, "0.00000e+00"},
+ {0, 'f', 5, "0.00000"},
+ {0, 'g', 5, "0"},
+ // {0, 'g', -1, "0"},
+
+ {-1, 'e', 5, "-1.00000e+00"},
+ {-1, 'f', 5, "-1.00000"},
+ {-1, 'g', 5, "-1"},
+ // {-1, 'g', -1, "-1"},
+
+ {12, 'e', 5, "1.20000e+01"},
+ {12, 'f', 5, "12.00000"},
+ {12, 'g', 5, "12"},
+ // {12, 'g', -1, "12"},
+
+ {123456700, 'e', 5, "1.23457e+08"},
+ {123456700, 'f', 5, "123456700.00000"},
+ {123456700, 'g', 5, "1.2346e+08"},
+ // {123456700, 'g', -1, "1.234567e+08"},
+
+ {1.2345e6, 'e', 5, "1.23450e+06"},
+ {1.2345e6, 'f', 5, "1234500.00000"},
+ {1.2345e6, 'g', 5, "1.2345e+06"},
+
+ {1e23, 'e', 17, "9.99999999999999916e+22"},
+ {1e23, 'f', 17, "99999999999999991611392.00000000000000000"},
+ {1e23, 'g', 17, "9.9999999999999992e+22"},
+
+ // {1e23, 'e', -1, "1e+23"},
+ // {1e23, 'f', -1, "100000000000000000000000"},
+ // {1e23, 'g', -1, "1e+23"},
+
+ {below1e23, 'e', 17, "9.99999999999999748e+22"},
+ {below1e23, 'f', 17, "99999999999999974834176.00000000000000000"},
+ {below1e23, 'g', 17, "9.9999999999999975e+22"},
+
+ // {below1e23, 'e', -1, "9.999999999999997e+22"},
+ // {below1e23, 'f', -1, "99999999999999970000000"},
+ // {below1e23, 'g', -1, "9.999999999999997e+22"},
+
+ {above1e23, 'e', 17, "1.00000000000000008e+23"},
+ {above1e23, 'f', 17, "100000000000000008388608.00000000000000000"},
+ // {above1e23, 'g', 17, "1.0000000000000001e+23"},
+
+ // {above1e23, 'e', -1, "1.0000000000000001e+23"},
+ // {above1e23, 'f', -1, "100000000000000010000000"},
+ // {above1e23, 'g', -1, "1.0000000000000001e+23"},
+
+ // {fdiv(5e-304, 1e20), 'g', -1, "5e-324"},
+ // {fdiv(-5e-304, 1e20), 'g', -1, "-5e-324"},
+
+ // {32, 'g', -1, "32"},
+ // {32, 'g', 0, "3e+01"},
+
+ // {100, 'x', -1, "%x"},
+
+ // {math.NaN(), 'g', -1, "NaN"},
+ // {-math.NaN(), 'g', -1, "NaN"},
+ {math.Inf(0), 'g', -1, "+Inf"},
+ {math.Inf(-1), 'g', -1, "-Inf"},
+ {-math.Inf(0), 'g', -1, "-Inf"},
+
+ {-1, 'b', -1, "-4503599627370496p-52"},
+
+ // fixed bugs
+ {0.9, 'f', 1, "0.9"},
+ {0.09, 'f', 1, "0.1"},
+ {0.0999, 'f', 1, "0.1"},
+ {0.05, 'f', 1, "0.1"},
+ {0.05, 'f', 0, "0"},
+ {0.5, 'f', 1, "0.5"},
+ {0.5, 'f', 0, "0"},
+ {1.5, 'f', 0, "2"},
+
+ // http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ // {2.2250738585072012e-308, 'g', -1, "2.2250738585072014e-308"},
+ // http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ // {2.2250738585072011e-308, 'g', -1, "2.225073858507201e-308"},
+
+ // Issue 2625.
+ {383260575764816448, 'f', 0, "383260575764816448"},
+ // {383260575764816448, 'g', -1, "3.8326057576481645e+17"},
+ } {
+ f := new(Float).SetFloat64(test.x)
+ got := f.Format(test.format, test.prec)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ }
+
+ if test.format == 'b' && test.x == 0 {
+ continue // 'b' format in strconv.Float requires knowledge of bias for 0.0
+ }
+ if test.format == 'p' {
+ continue // 'p' format not supported in strconv.Format
+ }
+
+ // verify that Float format matches strconv format
+ want := strconv.FormatFloat(test.x, test.format, test.prec, 64)
+ if got != want {
+ t.Errorf("%v: got %s; want %s (strconv)", test, got, want)
+ }
+ }
+}
+
+func TestFloatFormat(t *testing.T) {
+ for _, test := range []struct {
+ x string
+ prec uint
+ format byte
+ digits int
+ want string
+ }{
+ {"0", 10, 'f', 0, "0"},
+ {"-0", 10, 'f', 0, "-0"},
+ {"1", 10, 'f', 0, "1"},
+ {"-1", 10, 'f', 0, "-1"},
+
+ {"1.459", 100, 'e', 0, "1e+00"},
+ {"2.459", 100, 'e', 1, "2.5e+00"},
+ {"3.459", 100, 'e', 2, "3.46e+00"},
+ {"4.459", 100, 'e', 3, "4.459e+00"},
+ {"5.459", 100, 'e', 4, "5.4590e+00"},
+
+ {"1.459", 100, 'E', 0, "1E+00"},
+ {"2.459", 100, 'E', 1, "2.5E+00"},
+ {"3.459", 100, 'E', 2, "3.46E+00"},
+ {"4.459", 100, 'E', 3, "4.459E+00"},
+ {"5.459", 100, 'E', 4, "5.4590E+00"},
+
+ {"1.459", 100, 'f', 0, "1"},
+ {"2.459", 100, 'f', 1, "2.5"},
+ {"3.459", 100, 'f', 2, "3.46"},
+ {"4.459", 100, 'f', 3, "4.459"},
+ {"5.459", 100, 'f', 4, "5.4590"},
+
+ {"1.459", 100, 'g', 0, "1"},
+ {"2.459", 100, 'g', 1, "2"},
+ {"3.459", 100, 'g', 2, "3.5"},
+ {"4.459", 100, 'g', 3, "4.46"},
+ {"5.459", 100, 'g', 4, "5.459"},
+
+ {"1459", 53, 'g', 0, "1e+03"},
+ {"2459", 53, 'g', 1, "2e+03"},
+ {"3459", 53, 'g', 2, "3.5e+03"},
+ {"4459", 53, 'g', 3, "4.46e+03"},
+ {"5459", 53, 'g', 4, "5459"},
+
+ {"1459", 53, 'G', 0, "1E+03"},
+ {"2459", 53, 'G', 1, "2E+03"},
+ {"3459", 53, 'G', 2, "3.5E+03"},
+ {"4459", 53, 'G', 3, "4.46E+03"},
+ {"5459", 53, 'G', 4, "5459"},
+
+ {"3", 10, 'e', 40, "3.0000000000000000000000000000000000000000e+00"},
+ {"3", 10, 'f', 40, "3.0000000000000000000000000000000000000000"},
+ {"3", 10, 'g', 40, "3"},
+
+ {"3e40", 100, 'e', 40, "3.0000000000000000000000000000000000000000e+40"},
+ {"3e40", 100, 'f', 4, "30000000000000000000000000000000000000000.0000"},
+ {"3e40", 100, 'g', 40, "3e+40"},
+
+ // TODO(gri) need tests for actual large Floats
+
+ {"0", 53, 'b', 0, "0"},
+ {"-0", 53, 'b', 0, "-0"},
+ {"1.0", 53, 'b', 0, "4503599627370496p-52"},
+ {"-1.0", 53, 'b', 0, "-4503599627370496p-52"},
+ {"4503599627370496", 53, 'b', 0, "4503599627370496p+0"},
+
+ // issue 9939
+ {"3", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"03", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.0", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.00", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+ {"3.000", 350, 'b', 0, "1720123961992553633708115671476565205597423741876210842803191629540192157066363606052513914832594264915968p-348"},
+
+ {"3", 350, 'p', 0, "0x.cp2"},
+ {"03", 350, 'p', 0, "0x.cp2"},
+ {"3.", 350, 'p', 0, "0x.cp2"},
+ {"3.0", 350, 'p', 0, "0x.cp2"},
+ {"3.00", 350, 'p', 0, "0x.cp2"},
+ {"3.000", 350, 'p', 0, "0x.cp2"},
+
+ {"0", 64, 'p', 0, "0"},
+ {"-0", 64, 'p', 0, "-0"},
+ {"1024.0", 64, 'p', 0, "0x.8p11"},
+ {"-1024.0", 64, 'p', 0, "-0x.8p11"},
+
+ // unsupported format
+ {"3.14", 64, 'x', 0, "%x"},
+ } {
+ f, _, err := ParseFloat(test.x, 0, test.prec, ToNearestEven)
+ if err != nil {
+ t.Errorf("%v: %s", test, err)
+ continue
+ }
+
+ got := f.Format(test.format, test.digits)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ }
+
+ // compare with strconv.FormatFloat output if possible
+ // ('p' format is not supported by strconv.FormatFloat,
+ // and its output for 0.0 prints a biased exponent value
+ // as in 0p-1074 which makes no sense to emulate here)
+ if test.prec == 53 && test.format != 'p' && f.Sign() != 0 {
+ f64, acc := f.Float64()
+ if acc != Exact {
+ t.Errorf("%v: expected exact conversion to float64", test)
+ continue
+ }
+ got := strconv.FormatFloat(f64, test.format, test.digits, 64)
+ if got != test.want {
+ t.Errorf("%v: got %s; want %s", test, got, test.want)
+ }
+ }
+ }
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big_test
+
+import (
+ "fmt"
+ "math"
+ "math/big"
+)
+
+func ExampleFloat_Add() {
+ // Operating on numbers of different precision.
+ var x, y, z big.Float
+ x.SetInt64(1000) // x is automatically set to 64bit precision
+ y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
+ z.SetPrec(32)
+ z.Add(&x, &y)
+ fmt.Printf("x = %s (%s, prec = %d, acc = %s)\n", &x, x.Format('p', 0), x.Prec(), x.Acc())
+ fmt.Printf("y = %s (%s, prec = %d, acc = %s)\n", &y, y.Format('p', 0), y.Prec(), y.Acc())
+ fmt.Printf("z = %s (%s, prec = %d, acc = %s)\n", &z, z.Format('p', 0), z.Prec(), z.Acc())
+ // Output:
+ // x = 1000 (0x.fap10, prec = 64, acc = Exact)
+ // y = 2.718281828 (0x.adf85458248cd8p2, prec = 53, acc = Exact)
+ // z = 1002.718282 (0x.faadf854p10, prec = 32, acc = Below)
+}
+
+func Example_Shift() {
+ // Implementing Float "shift" by modifying the (binary) exponents directly.
+ for s := -5; s <= 5; s++ {
+ x := big.NewFloat(0.5)
+ x.SetMantExp(x, x.MantExp(nil)+s) // shift x by s
+ fmt.Println(x)
+ }
+ // Output:
+ // 0.015625
+ // 0.03125
+ // 0.0625
+ // 0.125
+ // 0.25
+ // 0.5
+ // 1
+ // 2
+ // 4
+ // 8
+ // 16
+}
+
+func ExampleFloat_Cmp() {
+ inf := math.Inf(1)
+ zero := 0.0
+ nan := math.NaN()
+
+ operands := []float64{-inf, -1.2, -zero, 0, +1.2, +inf, nan}
+
+ fmt.Println(" x y cmp eql neq lss leq gtr geq")
+ fmt.Println("-----------------------------------------------")
+ for _, x64 := range operands {
+ x := big.NewFloat(x64)
+ for _, y64 := range operands {
+ y := big.NewFloat(y64)
+ t := x.Cmp(y)
+ fmt.Printf(
+ "%4s %4s %5s %c %c %c %c %c %c\n",
+ x, y, t.Acc(),
+ mark(t.Eql()), mark(t.Neq()), mark(t.Lss()), mark(t.Leq()), mark(t.Gtr()), mark(t.Geq()))
+ }
+ fmt.Println()
+ }
+
+ // Output:
+ // x y cmp eql neq lss leq gtr geq
+ // -----------------------------------------------
+ // -Inf -Inf Exact ● ○ ○ ● ○ ●
+ // -Inf -1.2 Below ○ ● ● ● ○ ○
+ // -Inf -0 Below ○ ● ● ● ○ ○
+ // -Inf 0 Below ○ ● ● ● ○ ○
+ // -Inf 1.2 Below ○ ● ● ● ○ ○
+ // -Inf +Inf Below ○ ● ● ● ○ ○
+ // -Inf NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // -1.2 -Inf Above ○ ● ○ ○ ● ●
+ // -1.2 -1.2 Exact ● ○ ○ ● ○ ●
+ // -1.2 -0 Below ○ ● ● ● ○ ○
+ // -1.2 0 Below ○ ● ● ● ○ ○
+ // -1.2 1.2 Below ○ ● ● ● ○ ○
+ // -1.2 +Inf Below ○ ● ● ● ○ ○
+ // -1.2 NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // -0 -Inf Above ○ ● ○ ○ ● ●
+ // -0 -1.2 Above ○ ● ○ ○ ● ●
+ // -0 -0 Exact ● ○ ○ ● ○ ●
+ // -0 0 Exact ● ○ ○ ● ○ ●
+ // -0 1.2 Below ○ ● ● ● ○ ○
+ // -0 +Inf Below ○ ● ● ● ○ ○
+ // -0 NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // 0 -Inf Above ○ ● ○ ○ ● ●
+ // 0 -1.2 Above ○ ● ○ ○ ● ●
+ // 0 -0 Exact ● ○ ○ ● ○ ●
+ // 0 0 Exact ● ○ ○ ● ○ ●
+ // 0 1.2 Below ○ ● ● ● ○ ○
+ // 0 +Inf Below ○ ● ● ● ○ ○
+ // 0 NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // 1.2 -Inf Above ○ ● ○ ○ ● ●
+ // 1.2 -1.2 Above ○ ● ○ ○ ● ●
+ // 1.2 -0 Above ○ ● ○ ○ ● ●
+ // 1.2 0 Above ○ ● ○ ○ ● ●
+ // 1.2 1.2 Exact ● ○ ○ ● ○ ●
+ // 1.2 +Inf Below ○ ● ● ● ○ ○
+ // 1.2 NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // +Inf -Inf Above ○ ● ○ ○ ● ●
+ // +Inf -1.2 Above ○ ● ○ ○ ● ●
+ // +Inf -0 Above ○ ● ○ ○ ● ●
+ // +Inf 0 Above ○ ● ○ ○ ● ●
+ // +Inf 1.2 Above ○ ● ○ ○ ● ●
+ // +Inf +Inf Exact ● ○ ○ ● ○ ●
+ // +Inf NaN Undef ○ ● ○ ○ ○ ○
+ //
+ // NaN -Inf Undef ○ ● ○ ○ ○ ○
+ // NaN -1.2 Undef ○ ● ○ ○ ○ ○
+ // NaN -0 Undef ○ ● ○ ○ ○ ○
+ // NaN 0 Undef ○ ● ○ ○ ○ ○
+ // NaN 1.2 Undef ○ ● ○ ○ ○ ○
+ // NaN +Inf Undef ○ ● ○ ○ ○ ○
+ // NaN NaN Undef ○ ● ○ ○ ○ ○
+}
+
+func mark(p bool) rune {
+ if p {
+ return '●'
+ }
+ return '○'
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements the 'e', 'f', 'g' floating-point formats.
+// It is closely following the corresponding implementation in
+// strconv/ftoa.go, but modified and simplified for big.Float.
+
+// Algorithm:
+// 1) convert Float to multiprecision decimal
+// 2) round to desired precision
+// 3) read digits out and format
+
+package big
+
+import "strconv"
+
+// TODO(gri) Consider moving sign into decimal - could make the signatures below cleaner.
+
+// bigFtoa formats a float for the %e, %E, %f, %g, and %G formats.
+func (f *Float) bigFtoa(buf []byte, fmt byte, prec int) []byte {
+ if debugFloat && !f.IsFinite() {
+ panic("non-finite float")
+ }
+
+ // 1) convert Float to multiprecision decimal
+ var mant nat
+ if f.form == finite {
+ mant = f.mant
+ }
+ var d decimal
+ d.init(mant, int(f.exp)-f.mant.bitLen())
+
+ // 2) round to desired precision
+ shortest := false
+ if prec < 0 {
+ shortest = true
+ panic("unimplemented")
+ // TODO(gri) complete this
+ // roundShortest(&d, f.mant, int(f.exp))
+ // Precision for shortest representation mode.
+ switch fmt {
+ case 'e', 'E':
+ prec = len(d.mant) - 1
+ case 'f':
+ prec = max(len(d.mant)-d.exp, 0)
+ case 'g', 'G':
+ prec = len(d.mant)
+ }
+ } else {
+ // round appropriately
+ switch fmt {
+ case 'e', 'E':
+ // one digit before and number of digits after decimal point
+ d.round(1 + prec)
+ case 'f':
+ // number of digits before and after decimal point
+ d.round(d.exp + prec)
+ case 'g', 'G':
+ if prec == 0 {
+ prec = 1
+ }
+ d.round(prec)
+ }
+ }
+
+ // 3) read digits out and format
+ switch fmt {
+ case 'e', 'E':
+ return fmtE(buf, fmt, prec, f.neg, d)
+ case 'f':
+ return fmtF(buf, prec, f.neg, d)
+ case 'g', 'G':
+ // trim trailing fractional zeros in %e format
+ eprec := prec
+ if eprec > len(d.mant) && len(d.mant) >= d.exp {
+ eprec = len(d.mant)
+ }
+ // %e is used if the exponent from the conversion
+ // is less than -4 or greater than or equal to the precision.
+ // If precision was the shortest possible, use eprec = 6 for
+ // this decision.
+ if shortest {
+ eprec = 6
+ }
+ exp := d.exp - 1
+ if exp < -4 || exp >= eprec {
+ if prec > len(d.mant) {
+ prec = len(d.mant)
+ }
+ return fmtE(buf, fmt+'e'-'g', prec-1, f.neg, d)
+ }
+ if prec > d.exp {
+ prec = len(d.mant)
+ }
+ return fmtF(buf, max(prec-d.exp, 0), f.neg, d)
+ }
+
+ // unknown format
+ return append(buf, '%', fmt)
+}
+
+// %e: -d.ddddde±dd
+func fmtE(buf []byte, fmt byte, prec int, neg bool, d decimal) []byte {
+ // sign
+ if neg {
+ buf = append(buf, '-')
+ }
+
+ // first digit
+ ch := byte('0')
+ if len(d.mant) > 0 {
+ ch = d.mant[0]
+ }
+ buf = append(buf, ch)
+
+ // .moredigits
+ if prec > 0 {
+ buf = append(buf, '.')
+ i := 1
+ m := min(len(d.mant), prec+1)
+ if i < m {
+ buf = append(buf, d.mant[i:m]...)
+ i = m
+ }
+ for ; i <= prec; i++ {
+ buf = append(buf, '0')
+ }
+ }
+
+ // e±
+ buf = append(buf, fmt)
+ var exp int64
+ if len(d.mant) > 0 {
+ exp = int64(d.exp) - 1 // -1 because first digit was printed before '.'
+ }
+ if exp < 0 {
+ ch = '-'
+ exp = -exp
+ } else {
+ ch = '+'
+ }
+ buf = append(buf, ch)
+
+ // dd...d
+ if exp < 10 {
+ buf = append(buf, '0') // at least 2 exponent digits
+ }
+ return strconv.AppendInt(buf, exp, 10)
+}
+
+// %f: -ddddddd.ddddd
+func fmtF(buf []byte, prec int, neg bool, d decimal) []byte {
+ // sign
+ if neg {
+ buf = append(buf, '-')
+ }
+
+ // integer, padded with zeros as needed
+ if d.exp > 0 {
+ m := min(len(d.mant), d.exp)
+ buf = append(buf, d.mant[:m]...)
+ for ; m < d.exp; m++ {
+ buf = append(buf, '0')
+ }
+ } else {
+ buf = append(buf, '0')
+ }
+
+ // fraction
+ if prec > 0 {
+ buf = append(buf, '.')
+ for i := 0; i < prec; i++ {
+ ch := byte('0')
+ if j := d.exp + i; 0 <= j && j < len(d.mant) {
+ ch = d.mant[j]
+ }
+ buf = append(buf, ch)
+ }
+ }
+
+ return buf
+}
+
+func min(x, y int) int {
+ if x < y {
+ return x
+ }
+ return y
+}
--- /dev/null
+// Copyright 2012 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements a GCD benchmark.
+// Usage: go test math/big -test.bench GCD
+
+package big
+
+import (
+ "math/rand"
+ "testing"
+)
+
+// randInt returns a pseudo-random Int in the range [1<<(size-1), (1<<size) - 1]
+func randInt(r *rand.Rand, size uint) *Int {
+ n := new(Int).Lsh(intOne, size-1)
+ x := new(Int).Rand(r, n)
+ return x.Add(x, n) // make sure result > 1<<(size-1)
+}
+
+func runGCD(b *testing.B, aSize, bSize uint) {
+ b.StopTimer()
+ var r = rand.New(rand.NewSource(1234))
+ aa := randInt(r, aSize)
+ bb := randInt(r, bSize)
+ b.StartTimer()
+ for i := 0; i < b.N; i++ {
+ new(Int).GCD(nil, nil, aa, bb)
+ }
+}
+
+func BenchmarkGCD10x10(b *testing.B) { runGCD(b, 10, 10) }
+func BenchmarkGCD10x100(b *testing.B) { runGCD(b, 10, 100) }
+func BenchmarkGCD10x1000(b *testing.B) { runGCD(b, 10, 1000) }
+func BenchmarkGCD10x10000(b *testing.B) { runGCD(b, 10, 10000) }
+func BenchmarkGCD10x100000(b *testing.B) { runGCD(b, 10, 100000) }
+func BenchmarkGCD100x100(b *testing.B) { runGCD(b, 100, 100) }
+func BenchmarkGCD100x1000(b *testing.B) { runGCD(b, 100, 1000) }
+func BenchmarkGCD100x10000(b *testing.B) { runGCD(b, 100, 10000) }
+func BenchmarkGCD100x100000(b *testing.B) { runGCD(b, 100, 100000) }
+func BenchmarkGCD1000x1000(b *testing.B) { runGCD(b, 1000, 1000) }
+func BenchmarkGCD1000x10000(b *testing.B) { runGCD(b, 1000, 10000) }
+func BenchmarkGCD1000x100000(b *testing.B) { runGCD(b, 1000, 100000) }
+func BenchmarkGCD10000x10000(b *testing.B) { runGCD(b, 10000, 10000) }
+func BenchmarkGCD10000x100000(b *testing.B) { runGCD(b, 10000, 100000) }
+func BenchmarkGCD100000x100000(b *testing.B) { runGCD(b, 100000, 100000) }
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// A little test program and benchmark for rational arithmetics.
+// Computes a Hilbert matrix, its inverse, multiplies them
+// and verifies that the product is the identity matrix.
+
+package big
+
+import (
+ "fmt"
+ "testing"
+)
+
+type matrix struct {
+ n, m int
+ a []*Rat
+}
+
+func (a *matrix) at(i, j int) *Rat {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ return a.a[i*a.m+j]
+}
+
+func (a *matrix) set(i, j int, x *Rat) {
+ if !(0 <= i && i < a.n && 0 <= j && j < a.m) {
+ panic("index out of range")
+ }
+ a.a[i*a.m+j] = x
+}
+
+func newMatrix(n, m int) *matrix {
+ if !(0 <= n && 0 <= m) {
+ panic("illegal matrix")
+ }
+ a := new(matrix)
+ a.n = n
+ a.m = m
+ a.a = make([]*Rat, n*m)
+ return a
+}
+
+func newUnit(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x := NewRat(0, 1)
+ if i == j {
+ x.SetInt64(1)
+ }
+ a.set(i, j, x)
+ }
+ }
+ return a
+}
+
+func newHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ a.set(i, j, NewRat(1, int64(i+j+1)))
+ }
+ }
+ return a
+}
+
+func newInverseHilbert(n int) *matrix {
+ a := newMatrix(n, n)
+ for i := 0; i < n; i++ {
+ for j := 0; j < n; j++ {
+ x1 := new(Rat).SetInt64(int64(i + j + 1))
+ x2 := new(Rat).SetInt(new(Int).Binomial(int64(n+i), int64(n-j-1)))
+ x3 := new(Rat).SetInt(new(Int).Binomial(int64(n+j), int64(n-i-1)))
+ x4 := new(Rat).SetInt(new(Int).Binomial(int64(i+j), int64(i)))
+
+ x1.Mul(x1, x2)
+ x1.Mul(x1, x3)
+ x1.Mul(x1, x4)
+ x1.Mul(x1, x4)
+
+ if (i+j)&1 != 0 {
+ x1.Neg(x1)
+ }
+
+ a.set(i, j, x1)
+ }
+ }
+ return a
+}
+
+func (a *matrix) mul(b *matrix) *matrix {
+ if a.m != b.n {
+ panic("illegal matrix multiply")
+ }
+ c := newMatrix(a.n, b.m)
+ for i := 0; i < c.n; i++ {
+ for j := 0; j < c.m; j++ {
+ x := NewRat(0, 1)
+ for k := 0; k < a.m; k++ {
+ x.Add(x, new(Rat).Mul(a.at(i, k), b.at(k, j)))
+ }
+ c.set(i, j, x)
+ }
+ }
+ return c
+}
+
+func (a *matrix) eql(b *matrix) bool {
+ if a.n != b.n || a.m != b.m {
+ return false
+ }
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ if a.at(i, j).Cmp(b.at(i, j)) != 0 {
+ return false
+ }
+ }
+ }
+ return true
+}
+
+func (a *matrix) String() string {
+ s := ""
+ for i := 0; i < a.n; i++ {
+ for j := 0; j < a.m; j++ {
+ s += fmt.Sprintf("\t%s", a.at(i, j))
+ }
+ s += "\n"
+ }
+ return s
+}
+
+func doHilbert(t *testing.T, n int) {
+ a := newHilbert(n)
+ b := newInverseHilbert(n)
+ I := newUnit(n)
+ ab := a.mul(b)
+ if !ab.eql(I) {
+ if t == nil {
+ panic("Hilbert failed")
+ }
+ t.Errorf("a = %s\n", a)
+ t.Errorf("b = %s\n", b)
+ t.Errorf("a*b = %s\n", ab)
+ t.Errorf("I = %s\n", I)
+ }
+}
+
+func TestHilbert(t *testing.T) {
+ doHilbert(t, 10)
+}
+
+func BenchmarkHilbert(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ doHilbert(nil, 10)
+ }
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements signed multi-precision integers.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "math/rand"
+ "strings"
+)
+
+// An Int represents a signed multi-precision integer.
+// The zero value for an Int represents the value 0.
+type Int struct {
+ neg bool // sign
+ abs nat // absolute value of the integer
+}
+
+var intOne = &Int{false, natOne}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Int) Sign() int {
+ if len(x.abs) == 0 {
+ return 0
+ }
+ if x.neg {
+ return -1
+ }
+ return 1
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Int) SetInt64(x int64) *Int {
+ neg := false
+ if x < 0 {
+ neg = true
+ x = -x
+ }
+ z.abs = z.abs.setUint64(uint64(x))
+ z.neg = neg
+ return z
+}
+
+// SetUint64 sets z to x and returns z.
+func (z *Int) SetUint64(x uint64) *Int {
+ z.abs = z.abs.setUint64(x)
+ z.neg = false
+ return z
+}
+
+// NewInt allocates and returns a new Int set to x.
+func NewInt(x int64) *Int {
+ return new(Int).SetInt64(x)
+}
+
+// Set sets z to x and returns z.
+func (z *Int) Set(x *Int) *Int {
+ if z != x {
+ z.abs = z.abs.set(x.abs)
+ z.neg = x.neg
+ }
+ return z
+}
+
+// Bits provides raw (unchecked but fast) access to x by returning its
+// absolute value as a little-endian Word slice. The result and x share
+// the same underlying array.
+// Bits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (x *Int) Bits() []Word {
+ return x.abs
+}
+
+// SetBits provides raw (unchecked but fast) access to z by setting its
+// value to abs, interpreted as a little-endian Word slice, and returning
+// z. The result and abs share the same underlying array.
+// SetBits is intended to support implementation of missing low-level Int
+// functionality outside this package; it should be avoided otherwise.
+func (z *Int) SetBits(abs []Word) *Int {
+ z.abs = nat(abs).norm()
+ z.neg = false
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Int) Abs(x *Int) *Int {
+ z.Set(x)
+ z.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Int) Neg(x *Int) *Int {
+ z.Set(x)
+ z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
+ return z
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Int) Add(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg == y.neg {
+ // x + y == x + y
+ // (-x) + (-y) == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x + (-y) == x - y == -(y - x)
+ // (-x) + y == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Int) Sub(x, y *Int) *Int {
+ neg := x.neg
+ if x.neg != y.neg {
+ // x - (-y) == x + y
+ // (-x) - y == -(x + y)
+ z.abs = z.abs.add(x.abs, y.abs)
+ } else {
+ // x - y == x - y == -(y - x)
+ // (-x) - (-y) == y - x == -(x - y)
+ if x.abs.cmp(y.abs) >= 0 {
+ z.abs = z.abs.sub(x.abs, y.abs)
+ } else {
+ neg = !neg
+ z.abs = z.abs.sub(y.abs, x.abs)
+ }
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+ return z
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Int) Mul(x, y *Int) *Int {
+ // x * y == x * y
+ // x * (-y) == -(x * y)
+ // (-x) * y == -(x * y)
+ // (-x) * (-y) == x * y
+ z.abs = z.abs.mul(x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// MulRange sets z to the product of all integers
+// in the range [a, b] inclusively and returns z.
+// If a > b (empty range), the result is 1.
+func (z *Int) MulRange(a, b int64) *Int {
+ switch {
+ case a > b:
+ return z.SetInt64(1) // empty range
+ case a <= 0 && b >= 0:
+ return z.SetInt64(0) // range includes 0
+ }
+ // a <= b && (b < 0 || a > 0)
+
+ neg := false
+ if a < 0 {
+ neg = (b-a)&1 == 0
+ a, b = -b, -a
+ }
+
+ z.abs = z.abs.mulRange(uint64(a), uint64(b))
+ z.neg = neg
+ return z
+}
+
+// Binomial sets z to the binomial coefficient of (n, k) and returns z.
+func (z *Int) Binomial(n, k int64) *Int {
+ var a, b Int
+ a.MulRange(n-k+1, n)
+ b.MulRange(1, k)
+ return z.Quo(&a, &b)
+}
+
+// Quo sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Quo implements truncated division (like Go); see QuoRem for more details.
+func (z *Int) Quo(x, y *Int) *Int {
+ z.abs, _ = z.abs.div(nil, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
+ return z
+}
+
+// Rem sets z to the remainder x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Rem implements truncated modulus (like Go); see QuoRem for more details.
+func (z *Int) Rem(x, y *Int) *Int {
+ _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
+ z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
+ return z
+}
+
+// QuoRem sets z to the quotient x/y and r to the remainder x%y
+// and returns the pair (z, r) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// QuoRem implements T-division and modulus (like Go):
+//
+// q = x/y with the result truncated to zero
+// r = x - y*q
+//
+// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
+// See DivMod for Euclidean division and modulus (unlike Go).
+//
+func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
+ z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
+ z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
+ return z, r
+}
+
+// Div sets z to the quotient x/y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Div implements Euclidean division (unlike Go); see DivMod for more details.
+func (z *Int) Div(x, y *Int) *Int {
+ y_neg := y.neg // z may be an alias for y
+ var r Int
+ z.QuoRem(x, y, &r)
+ if r.neg {
+ if y_neg {
+ z.Add(z, intOne)
+ } else {
+ z.Sub(z, intOne)
+ }
+ }
+ return z
+}
+
+// Mod sets z to the modulus x%y for y != 0 and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
+func (z *Int) Mod(x, y *Int) *Int {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ var q Int
+ q.QuoRem(x, y, z)
+ if z.neg {
+ if y0.neg {
+ z.Sub(z, y0)
+ } else {
+ z.Add(z, y0)
+ }
+ }
+ return z
+}
+
+// DivMod sets z to the quotient x div y and m to the modulus x mod y
+// and returns the pair (z, m) for y != 0.
+// If y == 0, a division-by-zero run-time panic occurs.
+//
+// DivMod implements Euclidean division and modulus (unlike Go):
+//
+// q = x div y such that
+// m = x - y*q with 0 <= m < |q|
+//
+// (See Raymond T. Boute, ``The Euclidean definition of the functions
+// div and mod''. ACM Transactions on Programming Languages and
+// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
+// ACM press.)
+// See QuoRem for T-division and modulus (like Go).
+//
+func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
+ y0 := y // save y
+ if z == y || alias(z.abs, y.abs) {
+ y0 = new(Int).Set(y)
+ }
+ z.QuoRem(x, y, m)
+ if m.neg {
+ if y0.neg {
+ z.Add(z, intOne)
+ m.Sub(m, y0)
+ } else {
+ z.Sub(z, intOne)
+ m.Add(m, y0)
+ }
+ }
+ return z, m
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Int) Cmp(y *Int) (r int) {
+ // x cmp y == x cmp y
+ // x cmp (-y) == x
+ // (-x) cmp y == y
+ // (-x) cmp (-y) == -(x cmp y)
+ switch {
+ case x.neg == y.neg:
+ r = x.abs.cmp(y.abs)
+ if x.neg {
+ r = -r
+ }
+ case x.neg:
+ r = -1
+ default:
+ r = 1
+ }
+ return
+}
+
+// low32 returns the least significant 32 bits of z.
+func low32(z nat) uint32 {
+ if len(z) == 0 {
+ return 0
+ }
+ return uint32(z[0])
+}
+
+// low64 returns the least significant 64 bits of z.
+func low64(z nat) uint64 {
+ if len(z) == 0 {
+ return 0
+ }
+ v := uint64(z[0])
+ if _W == 32 && len(z) > 1 {
+ v |= uint64(z[1]) << 32
+ }
+ return v
+}
+
+// Int64 returns the int64 representation of x.
+// If x cannot be represented in an int64, the result is undefined.
+func (x *Int) Int64() int64 {
+ v := int64(low64(x.abs))
+ if x.neg {
+ v = -v
+ }
+ return v
+}
+
+// Uint64 returns the uint64 representation of x.
+// If x cannot be represented in a uint64, the result is undefined.
+func (x *Int) Uint64() uint64 {
+ return low64(x.abs)
+}
+
+// SetString sets z to the value of s, interpreted in the given base,
+// and returns z and a boolean indicating success. If SetString fails,
+// the value of z is undefined but the returned value is nil.
+//
+// The base argument must be 0 or a value between 2 and MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) SetString(s string, base int) (*Int, bool) {
+ r := strings.NewReader(s)
+ _, _, err := z.scan(r, base)
+ if err != nil {
+ return nil, false
+ }
+ _, err = r.ReadByte()
+ if err != io.EOF {
+ return nil, false
+ }
+ return z, true // err == io.EOF => scan consumed all of s
+}
+
+// SetBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z *Int) SetBytes(buf []byte) *Int {
+ z.abs = z.abs.setBytes(buf)
+ z.neg = false
+ return z
+}
+
+// Bytes returns the absolute value of x as a big-endian byte slice.
+func (x *Int) Bytes() []byte {
+ buf := make([]byte, len(x.abs)*_S)
+ return buf[x.abs.bytes(buf):]
+}
+
+// BitLen returns the length of the absolute value of x in bits.
+// The bit length of 0 is 0.
+func (x *Int) BitLen() int {
+ return x.abs.bitLen()
+}
+
+// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
+// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
+// See Knuth, volume 2, section 4.6.3.
+func (z *Int) Exp(x, y, m *Int) *Int {
+ var yWords nat
+ if !y.neg {
+ yWords = y.abs
+ }
+ // y >= 0
+
+ var mWords nat
+ if m != nil {
+ mWords = m.abs // m.abs may be nil for m == 0
+ }
+
+ z.abs = z.abs.expNN(x.abs, yWords, mWords)
+ z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
+ if z.neg && len(mWords) > 0 {
+ // make modulus result positive
+ z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
+ z.neg = false
+ }
+
+ return z
+}
+
+// GCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
+// If either a or b is <= 0, GCD sets z = x = y = 0.
+func (z *Int) GCD(x, y, a, b *Int) *Int {
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ z.SetInt64(0)
+ if x != nil {
+ x.SetInt64(0)
+ }
+ if y != nil {
+ y.SetInt64(0)
+ }
+ return z
+ }
+ if x == nil && y == nil {
+ return z.binaryGCD(a, b)
+ }
+
+ A := new(Int).Set(a)
+ B := new(Int).Set(b)
+
+ X := new(Int)
+ Y := new(Int).SetInt64(1)
+
+ lastX := new(Int).SetInt64(1)
+ lastY := new(Int)
+
+ q := new(Int)
+ temp := new(Int)
+
+ for len(B.abs) > 0 {
+ r := new(Int)
+ q, r = q.QuoRem(A, B, r)
+
+ A, B = B, r
+
+ temp.Set(X)
+ X.Mul(X, q)
+ X.neg = !X.neg
+ X.Add(X, lastX)
+ lastX.Set(temp)
+
+ temp.Set(Y)
+ Y.Mul(Y, q)
+ Y.neg = !Y.neg
+ Y.Add(Y, lastY)
+ lastY.Set(temp)
+ }
+
+ if x != nil {
+ *x = *lastX
+ }
+
+ if y != nil {
+ *y = *lastY
+ }
+
+ *z = *A
+ return z
+}
+
+// binaryGCD sets z to the greatest common divisor of a and b, which both must
+// be > 0, and returns z.
+// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
+func (z *Int) binaryGCD(a, b *Int) *Int {
+ u := z
+ v := new(Int)
+
+ // use one Euclidean iteration to ensure that u and v are approx. the same size
+ switch {
+ case len(a.abs) > len(b.abs):
+ u.Set(b)
+ v.Rem(a, b)
+ case len(a.abs) < len(b.abs):
+ u.Set(a)
+ v.Rem(b, a)
+ default:
+ u.Set(a)
+ v.Set(b)
+ }
+
+ // v might be 0 now
+ if len(v.abs) == 0 {
+ return u
+ }
+ // u > 0 && v > 0
+
+ // determine largest k such that u = u' << k, v = v' << k
+ k := u.abs.trailingZeroBits()
+ if vk := v.abs.trailingZeroBits(); vk < k {
+ k = vk
+ }
+ u.Rsh(u, k)
+ v.Rsh(v, k)
+
+ // determine t (we know that u > 0)
+ t := new(Int)
+ if u.abs[0]&1 != 0 {
+ // u is odd
+ t.Neg(v)
+ } else {
+ t.Set(u)
+ }
+
+ for len(t.abs) > 0 {
+ // reduce t
+ t.Rsh(t, t.abs.trailingZeroBits())
+ if t.neg {
+ v, t = t, v
+ v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
+ } else {
+ u, t = t, u
+ }
+ t.Sub(u, v)
+ }
+
+ return z.Lsh(u, k)
+}
+
+// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
+// If it returns true, x is prime with probability 1 - 1/4^n.
+// If it returns false, x is not prime. n must be > 0.
+func (x *Int) ProbablyPrime(n int) bool {
+ if n <= 0 {
+ panic("non-positive n for ProbablyPrime")
+ }
+ return !x.neg && x.abs.probablyPrime(n)
+}
+
+// Rand sets z to a pseudo-random number in [0, n) and returns z.
+func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
+ z.neg = false
+ if n.neg == true || len(n.abs) == 0 {
+ z.abs = nil
+ return z
+ }
+ z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
+ return z
+}
+
+// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
+// and returns z. If g and n are not relatively prime, the result is undefined.
+func (z *Int) ModInverse(g, n *Int) *Int {
+ var d Int
+ d.GCD(z, nil, g, n)
+ // x and y are such that g*x + n*y = d. Since g and n are
+ // relatively prime, d = 1. Taking that modulo n results in
+ // g*x = 1, therefore x is the inverse element.
+ if z.neg {
+ z.Add(z, n)
+ }
+ return z
+}
+
+// Lsh sets z = x << n and returns z.
+func (z *Int) Lsh(x *Int, n uint) *Int {
+ z.abs = z.abs.shl(x.abs, n)
+ z.neg = x.neg
+ return z
+}
+
+// Rsh sets z = x >> n and returns z.
+func (z *Int) Rsh(x *Int, n uint) *Int {
+ if x.neg {
+ // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
+ t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
+ t = t.shr(t, n)
+ z.abs = t.add(t, natOne)
+ z.neg = true // z cannot be zero if x is negative
+ return z
+ }
+
+ z.abs = z.abs.shr(x.abs, n)
+ z.neg = false
+ return z
+}
+
+// Bit returns the value of the i'th bit of x. That is, it
+// returns (x>>i)&1. The bit index i must be >= 0.
+func (x *Int) Bit(i int) uint {
+ if i == 0 {
+ // optimization for common case: odd/even test of x
+ if len(x.abs) > 0 {
+ return uint(x.abs[0] & 1) // bit 0 is same for -x
+ }
+ return 0
+ }
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := nat(nil).sub(x.abs, natOne)
+ return t.bit(uint(i)) ^ 1
+ }
+
+ return x.abs.bit(uint(i))
+}
+
+// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
+// That is, if b is 1 SetBit sets z = x | (1 << i);
+// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
+// SetBit will panic.
+func (z *Int) SetBit(x *Int, i int, b uint) *Int {
+ if i < 0 {
+ panic("negative bit index")
+ }
+ if x.neg {
+ t := z.abs.sub(x.abs, natOne)
+ t = t.setBit(t, uint(i), b^1)
+ z.abs = t.add(t, natOne)
+ z.neg = len(z.abs) > 0
+ return z
+ }
+ z.abs = z.abs.setBit(x.abs, uint(i), b)
+ z.neg = false
+ return z
+}
+
+// And sets z = x & y and returns z.
+func (z *Int) And(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x & y == x & y
+ z.abs = z.abs.and(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // & is symmetric
+ }
+
+ // x & (-y) == x & ^(y-1) == x &^ (y-1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// AndNot sets z = x &^ y and returns z.
+func (z *Int) AndNot(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.andNot(y1, x1)
+ z.neg = false
+ return z
+ }
+
+ // x &^ y == x &^ y
+ z.abs = z.abs.andNot(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ if x.neg {
+ // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
+ z.neg = true // z cannot be zero if x is negative and y is positive
+ return z
+ }
+
+ // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.and(x.abs, y1)
+ z.neg = false
+ return z
+}
+
+// Or sets z = x | y and returns z.
+func (z *Int) Or(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
+ z.neg = true // z cannot be zero if x and y are negative
+ return z
+ }
+
+ // x | y == x | y
+ z.abs = z.abs.or(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // | is symmetric
+ }
+
+ // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
+ z.neg = true // z cannot be zero if one of x or y is negative
+ return z
+}
+
+// Xor sets z = x ^ y and returns z.
+func (z *Int) Xor(x, y *Int) *Int {
+ if x.neg == y.neg {
+ if x.neg {
+ // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
+ x1 := nat(nil).sub(x.abs, natOne)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.xor(x1, y1)
+ z.neg = false
+ return z
+ }
+
+ // x ^ y == x ^ y
+ z.abs = z.abs.xor(x.abs, y.abs)
+ z.neg = false
+ return z
+ }
+
+ // x.neg != y.neg
+ if x.neg {
+ x, y = y, x // ^ is symmetric
+ }
+
+ // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
+ y1 := nat(nil).sub(y.abs, natOne)
+ z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
+ z.neg = true // z cannot be zero if only one of x or y is negative
+ return z
+}
+
+// Not sets z = ^x and returns z.
+func (z *Int) Not(x *Int) *Int {
+ if x.neg {
+ // ^(-x) == ^(^(x-1)) == x-1
+ z.abs = z.abs.sub(x.abs, natOne)
+ z.neg = false
+ return z
+ }
+
+ // ^x == -x-1 == -(x+1)
+ z.abs = z.abs.add(x.abs, natOne)
+ z.neg = true // z cannot be zero if x is positive
+ return z
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const intGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Int) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
+ i := x.abs.bytes(buf) - 1 // i >= 0
+ b := intGobVersion << 1 // make space for sign bit
+ if x.neg {
+ b |= 1
+ }
+ buf[i] = b
+ return buf[i:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Int) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Int{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != intGobVersion {
+ return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
+ }
+ z.neg = b&1 != 0
+ z.abs = z.abs.setBytes(buf[1:])
+ return nil
+}
+
+// MarshalJSON implements the json.Marshaler interface.
+func (z *Int) MarshalJSON() ([]byte, error) {
+ // TODO(gri): get rid of the []byte/string conversions
+ return []byte(z.String()), nil
+}
+
+// UnmarshalJSON implements the json.Unmarshaler interface.
+func (z *Int) UnmarshalJSON(text []byte) error {
+ // TODO(gri): get rid of the []byte/string conversions
+ if _, ok := z.SetString(string(text), 0); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+ }
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (z *Int) MarshalText() (text []byte, err error) {
+ return []byte(z.String()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (z *Int) UnmarshalText(text []byte) error {
+ if _, ok := z.SetString(string(text), 0); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
+ }
+ return nil
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/hex"
+ "encoding/json"
+ "encoding/xml"
+ "fmt"
+ "math/rand"
+ "testing"
+ "testing/quick"
+)
+
+func isNormalized(x *Int) bool {
+ if len(x.abs) == 0 {
+ return !x.neg
+ }
+ // len(x.abs) > 0
+ return x.abs[len(x.abs)-1] != 0
+}
+
+type funZZ func(z, x, y *Int) *Int
+type argZZ struct {
+ z, x, y *Int
+}
+
+var sumZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(0)},
+ {NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+ {NewInt(-1), NewInt(-1), NewInt(0)},
+ {NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+ {NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
+}
+
+var prodZZ = []argZZ{
+ {NewInt(0), NewInt(0), NewInt(0)},
+ {NewInt(0), NewInt(1), NewInt(0)},
+ {NewInt(1), NewInt(1), NewInt(1)},
+ {NewInt(-991 * 991), NewInt(991), NewInt(-991)},
+ // TODO(gri) add larger products
+}
+
+func TestSignZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ s := a.z.Sign()
+ e := a.z.Cmp(&zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, a.z)
+ }
+ }
+}
+
+func TestSetZ(t *testing.T) {
+ for _, a := range sumZZ {
+ var z Int
+ z.Set(a.z)
+ if !isNormalized(&z) {
+ t.Errorf("%v is not normalized", z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func TestAbsZ(t *testing.T) {
+ var zero Int
+ for _, a := range sumZZ {
+ var z Int
+ z.Abs(a.z)
+ var e Int
+ e.Set(a.z)
+ if e.Cmp(&zero) < 0 {
+ e.Sub(&zero, &e)
+ }
+ if z.Cmp(&e) != 0 {
+ t.Errorf("got z = %v; want %v", z, e)
+ }
+ }
+}
+
+func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
+ var z Int
+ f(&z, a.x, a.y)
+ if !isNormalized(&z) {
+ t.Errorf("%s%v is not normalized", msg, z)
+ }
+ if (&z).Cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
+ }
+}
+
+func TestSumZZ(t *testing.T) {
+ AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
+ SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
+ for _, a := range sumZZ {
+ arg := a
+ testFunZZ(t, "AddZZ", AddZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
+
+ arg = argZZ{a.x, a.z, a.y}
+ testFunZZ(t, "SubZZ", SubZZ, arg)
+
+ arg = argZZ{a.y, a.z, a.x}
+ testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
+ }
+}
+
+func TestProdZZ(t *testing.T) {
+ MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
+ for _, a := range prodZZ {
+ arg := a
+ testFunZZ(t, "MulZZ", MulZZ, arg)
+
+ arg = argZZ{a.z, a.y, a.x}
+ testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
+ }
+}
+
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+ z := make([]byte, len(x)+len(y))
+
+ // multiply
+ k0 := len(z) - 1
+ for j := len(y) - 1; j >= 0; j-- {
+ d := int(y[j])
+ if d != 0 {
+ k := k0
+ carry := 0
+ for i := len(x) - 1; i >= 0; i-- {
+ t := int(z[k]) + int(x[i])*d + carry
+ z[k], carry = byte(t), t>>8
+ k--
+ }
+ z[k] = byte(carry)
+ }
+ k0--
+ }
+
+ // normalize (remove leading 0's)
+ i := 0
+ for i < len(z) && z[i] == 0 {
+ i++
+ }
+
+ return z[i:]
+}
+
+func checkMul(a, b []byte) bool {
+ var x, y, z1 Int
+ x.SetBytes(a)
+ y.SetBytes(b)
+ z1.Mul(&x, &y)
+
+ var z2 Int
+ z2.SetBytes(mulBytes(a, b))
+
+ return z1.Cmp(&z2) == 0
+}
+
+func TestMul(t *testing.T) {
+ if err := quick.Check(checkMul, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+var mulRangesZ = []struct {
+ a, b int64
+ prod string
+}{
+ // entirely positive ranges are covered by mulRangesN
+ {-1, 1, "0"},
+ {-2, -1, "2"},
+ {-3, -2, "6"},
+ {-3, -1, "-6"},
+ {1, 3, "6"},
+ {-10, -10, "-10"},
+ {0, -1, "1"}, // empty range
+ {-1, -100, "1"}, // empty range
+ {-1, 1, "0"}, // range includes 0
+ {-1e9, 0, "0"}, // range includes 0
+ {-1e9, 1e9, "0"}, // range includes 0
+ {-10, -1, "3628800"}, // 10!
+ {-20, -2, "-2432902008176640000"}, // -20!
+ {-99, -1,
+ "-933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "511852109168640000000000000000000000", // -99!
+ },
+}
+
+func TestMulRangeZ(t *testing.T) {
+ var tmp Int
+ // test entirely positive ranges
+ for i, r := range mulRangesN {
+ prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
+ if prod != r.prod {
+ t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
+ }
+ }
+ // test other ranges
+ for i, r := range mulRangesZ {
+ prod := tmp.MulRange(r.a, r.b).String()
+ if prod != r.prod {
+ t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []struct {
+ x, y int64
+ q, r int64 // T-division
+ d, m int64 // Euclidian division
+}{
+ {5, 3, 1, 2, 1, 2},
+ {-5, 3, -1, -2, -2, 1},
+ {5, -3, -1, 2, -1, 2},
+ {-5, -3, 1, -2, 2, 1},
+ {1, 2, 0, 1, 0, 1},
+ {8, 4, 2, 0, 2, 0},
+}
+
+func TestDivisionSigns(t *testing.T) {
+ for i, test := range divisionSignsTests {
+ x := NewInt(test.x)
+ y := NewInt(test.y)
+ q := NewInt(test.q)
+ r := NewInt(test.r)
+ d := NewInt(test.d)
+ m := NewInt(test.m)
+
+ q1 := new(Int).Quo(x, y)
+ r1 := new(Int).Rem(x, y)
+ if !isNormalized(q1) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+ }
+ if !isNormalized(r1) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+ }
+ if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+ }
+
+ q2, r2 := new(Int).QuoRem(x, y, new(Int))
+ if !isNormalized(q2) {
+ t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+ }
+ if !isNormalized(r2) {
+ t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+ }
+ if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+ t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+ }
+
+ d1 := new(Int).Div(x, y)
+ m1 := new(Int).Mod(x, y)
+ if !isNormalized(d1) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d1)
+ }
+ if !isNormalized(m1) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+ }
+ if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+ }
+
+ d2, m2 := new(Int).DivMod(x, y, new(Int))
+ if !isNormalized(d2) {
+ t.Errorf("#%d Div: %v is not normalized", i, *d2)
+ }
+ if !isNormalized(m2) {
+ t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+ }
+ if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+ t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
+ }
+ }
+}
+
+func norm(x nat) nat {
+ i := len(x)
+ for i > 0 && x[i-1] == 0 {
+ i--
+ }
+ return x[:i]
+}
+
+func TestBits(t *testing.T) {
+ for _, test := range []nat{
+ nil,
+ {0},
+ {1},
+ {0, 1, 2, 3, 4},
+ {4, 3, 2, 1, 0},
+ {4, 3, 2, 1, 0, 0, 0, 0},
+ } {
+ var z Int
+ z.neg = true
+ got := z.SetBits(test)
+ want := norm(test)
+ if got.abs.cmp(want) != 0 {
+ t.Errorf("SetBits(%v) = %v; want %v", test, got.abs, want)
+ }
+
+ if got.neg {
+ t.Errorf("SetBits(%v): got negative result", test)
+ }
+
+ bits := nat(z.Bits())
+ if bits.cmp(want) != 0 {
+ t.Errorf("%v.Bits() = %v; want %v", z.abs, bits, want)
+ }
+ }
+}
+
+func checkSetBytes(b []byte) bool {
+ hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
+ hex2 := hex.EncodeToString(b)
+
+ for len(hex1) < len(hex2) {
+ hex1 = "0" + hex1
+ }
+
+ for len(hex1) > len(hex2) {
+ hex2 = "0" + hex2
+ }
+
+ return hex1 == hex2
+}
+
+func TestSetBytes(t *testing.T) {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkBytes(b []byte) bool {
+ b2 := new(Int).SetBytes(b).Bytes()
+ return bytes.Equal(b, b2)
+}
+
+func TestBytes(t *testing.T) {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
+ t.Error(err)
+ }
+}
+
+func checkQuo(x, y []byte) bool {
+ u := new(Int).SetBytes(x)
+ v := new(Int).SetBytes(y)
+
+ if len(v.abs) == 0 {
+ return true
+ }
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+
+ if r.Cmp(v) >= 0 {
+ return false
+ }
+
+ uprime := new(Int).Set(q)
+ uprime.Mul(uprime, v)
+ uprime.Add(uprime, r)
+
+ return uprime.Cmp(u) == 0
+}
+
+var quoTests = []struct {
+ x, y string
+ q, r string
+}{
+ {
+ "476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
+ "9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
+ "50911",
+ "1",
+ },
+ {
+ "11510768301994997771168",
+ "1328165573307167369775",
+ "8",
+ "885443715537658812968",
+ },
+}
+
+func TestQuo(t *testing.T) {
+ if err := quick.Check(checkQuo, nil); err != nil {
+ t.Error(err)
+ }
+
+ for i, test := range quoTests {
+ x, _ := new(Int).SetString(test.x, 10)
+ y, _ := new(Int).SetString(test.y, 10)
+ expectedQ, _ := new(Int).SetString(test.q, 10)
+ expectedR, _ := new(Int).SetString(test.r, 10)
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(x, y, r)
+
+ if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
+ t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+ }
+ }
+}
+
+func TestQuoStepD6(t *testing.T) {
+ // See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
+ // a code path which only triggers 1 in 10^{-19} cases.
+
+ u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
+ v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
+
+ r := new(Int)
+ q, r := new(Int).QuoRem(u, v, r)
+ const expectedQ64 = "18446744073709551613"
+ const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
+ const expectedQ32 = "4294967293"
+ const expectedR32 = "39614081266355540837921718287"
+ if q.String() != expectedQ64 && q.String() != expectedQ32 ||
+ r.String() != expectedR64 && r.String() != expectedR32 {
+ t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
+ }
+}
+
+var bitLenTests = []struct {
+ in string
+ out int
+}{
+ {"-1", 1},
+ {"0", 0},
+ {"1", 1},
+ {"2", 2},
+ {"4", 3},
+ {"0xabc", 12},
+ {"0x8000", 16},
+ {"0x80000000", 32},
+ {"0x800000000000", 48},
+ {"0x8000000000000000", 64},
+ {"0x80000000000000000000", 80},
+ {"-0x4000000000000000000000", 87},
+}
+
+func TestBitLen(t *testing.T) {
+ for i, test := range bitLenTests {
+ x, ok := new(Int).SetString(test.in, 0)
+ if !ok {
+ t.Errorf("#%d test input invalid: %s", i, test.in)
+ continue
+ }
+
+ if n := x.BitLen(); n != test.out {
+ t.Errorf("#%d got %d want %d", i, n, test.out)
+ }
+ }
+}
+
+var expTests = []struct {
+ x, y, m string
+ out string
+}{
+ // y <= 0
+ {"0", "0", "", "1"},
+ {"1", "0", "", "1"},
+ {"-10", "0", "", "1"},
+ {"1234", "-1", "", "1"},
+
+ // m == 1
+ {"0", "0", "1", "0"},
+ {"1", "0", "1", "0"},
+ {"-10", "0", "1", "0"},
+ {"1234", "-1", "1", "0"},
+
+ // misc
+ {"5", "-7", "", "1"},
+ {"-5", "-7", "", "1"},
+ {"5", "0", "", "1"},
+ {"-5", "0", "", "1"},
+ {"5", "1", "", "5"},
+ {"-5", "1", "", "-5"},
+ {"-5", "1", "7", "2"},
+ {"-2", "3", "2", "0"},
+ {"5", "2", "", "25"},
+ {"1", "65537", "2", "1"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {"0x8000000000000000", "-1000000", "6719", "1"},
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+ // test case for issue 8822
+ {
+ "-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
+ "0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+ "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
+ "21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+ },
+}
+
+func TestExp(t *testing.T) {
+ for i, test := range expTests {
+ x, ok1 := new(Int).SetString(test.x, 0)
+ y, ok2 := new(Int).SetString(test.y, 0)
+ out, ok3 := new(Int).SetString(test.out, 0)
+
+ var ok4 bool
+ var m *Int
+
+ if len(test.m) == 0 {
+ m, ok4 = nil, true
+ } else {
+ m, ok4 = new(Int).SetString(test.m, 0)
+ }
+
+ if !ok1 || !ok2 || !ok3 || !ok4 {
+ t.Errorf("#%d: error in input", i)
+ continue
+ }
+
+ z1 := new(Int).Exp(x, y, m)
+ if !isNormalized(z1) {
+ t.Errorf("#%d: %v is not normalized", i, *z1)
+ }
+ if z1.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z1, out)
+ }
+
+ if m == nil {
+ // The result should be the same as for m == 0;
+ // specifically, there should be no div-zero panic.
+ m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+ z2 := new(Int).Exp(x, y, m)
+ if z2.Cmp(z1) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z2, z1)
+ }
+ }
+ }
+}
+
+func checkGcd(aBytes, bBytes []byte) bool {
+ x := new(Int)
+ y := new(Int)
+ a := new(Int).SetBytes(aBytes)
+ b := new(Int).SetBytes(bBytes)
+
+ d := new(Int).GCD(x, y, a, b)
+ x.Mul(x, a)
+ y.Mul(y, b)
+ x.Add(x, y)
+
+ return x.Cmp(d) == 0
+}
+
+var gcdTests = []struct {
+ d, x, y, a, b string
+}{
+ // a <= 0 || b <= 0
+ {"0", "0", "0", "0", "0"},
+ {"0", "0", "0", "0", "7"},
+ {"0", "0", "0", "11", "0"},
+ {"0", "0", "0", "-77", "35"},
+ {"0", "0", "0", "64515", "-24310"},
+ {"0", "0", "0", "-64515", "-24310"},
+
+ {"1", "-9", "47", "120", "23"},
+ {"7", "1", "-2", "77", "35"},
+ {"935", "-3", "8", "64515", "24310"},
+ {"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
+ {"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
+
+ // test early exit (after one Euclidean iteration) in binaryGCD
+ {"1", "", "", "1", "98920366548084643601728869055592650835572950932266967461790948584315647051443"},
+}
+
+func testGcd(t *testing.T, d, x, y, a, b *Int) {
+ var X *Int
+ if x != nil {
+ X = new(Int)
+ }
+ var Y *Int
+ if y != nil {
+ Y = new(Int)
+ }
+
+ D := new(Int).GCD(X, Y, a, b)
+ if D.Cmp(d) != 0 {
+ t.Errorf("GCD(%s, %s): got d = %s, want %s", a, b, D, d)
+ }
+ if x != nil && X.Cmp(x) != 0 {
+ t.Errorf("GCD(%s, %s): got x = %s, want %s", a, b, X, x)
+ }
+ if y != nil && Y.Cmp(y) != 0 {
+ t.Errorf("GCD(%s, %s): got y = %s, want %s", a, b, Y, y)
+ }
+
+ // binaryGCD requires a > 0 && b > 0
+ if a.Sign() <= 0 || b.Sign() <= 0 {
+ return
+ }
+
+ D.binaryGCD(a, b)
+ if D.Cmp(d) != 0 {
+ t.Errorf("binaryGcd(%s, %s): got d = %s, want %s", a, b, D, d)
+ }
+}
+
+func TestGcd(t *testing.T) {
+ for _, test := range gcdTests {
+ d, _ := new(Int).SetString(test.d, 0)
+ x, _ := new(Int).SetString(test.x, 0)
+ y, _ := new(Int).SetString(test.y, 0)
+ a, _ := new(Int).SetString(test.a, 0)
+ b, _ := new(Int).SetString(test.b, 0)
+
+ testGcd(t, d, nil, nil, a, b)
+ testGcd(t, d, x, nil, a, b)
+ testGcd(t, d, nil, y, a, b)
+ testGcd(t, d, x, y, a, b)
+ }
+
+ quick.Check(checkGcd, nil)
+}
+
+var primes = []string{
+ "2",
+ "3",
+ "5",
+ "7",
+ "11",
+
+ "13756265695458089029",
+ "13496181268022124907",
+ "10953742525620032441",
+ "17908251027575790097",
+
+ // http://golang.org/issue/638
+ "18699199384836356663",
+
+ "98920366548084643601728869055592650835572950932266967461790948584315647051443",
+ "94560208308847015747498523884063394671606671904944666360068158221458669711639",
+
+ // http://primes.utm.edu/lists/small/small3.html
+ "449417999055441493994709297093108513015373787049558499205492347871729927573118262811508386655998299074566974373711472560655026288668094291699357843464363003144674940345912431129144354948751003607115263071543163",
+ "230975859993204150666423538988557839555560243929065415434980904258310530753006723857139742334640122533598517597674807096648905501653461687601339782814316124971547968912893214002992086353183070342498989426570593",
+ "5521712099665906221540423207019333379125265462121169655563495403888449493493629943498064604536961775110765377745550377067893607246020694972959780839151452457728855382113555867743022746090187341871655890805971735385789993",
+ "203956878356401977405765866929034577280193993314348263094772646453283062722701277632936616063144088173312372882677123879538709400158306567338328279154499698366071906766440037074217117805690872792848149112022286332144876183376326512083574821647933992961249917319836219304274280243803104015000563790123",
+}
+
+var composites = []string{
+ "0",
+ "1",
+ "21284175091214687912771199898307297748211672914763848041968395774954376176754",
+ "6084766654921918907427900243509372380954290099172559290432744450051395395951",
+ "84594350493221918389213352992032324280367711247940675652888030554255915464401",
+ "82793403787388584738507275144194252681",
+}
+
+func TestProbablyPrime(t *testing.T) {
+ nreps := 20
+ if testing.Short() {
+ nreps = 1
+ }
+ for i, s := range primes {
+ p, _ := new(Int).SetString(s, 10)
+ if !p.ProbablyPrime(nreps) {
+ t.Errorf("#%d prime found to be non-prime (%s)", i, s)
+ }
+ }
+
+ for i, s := range composites {
+ c, _ := new(Int).SetString(s, 10)
+ if c.ProbablyPrime(nreps) {
+ t.Errorf("#%d composite found to be prime (%s)", i, s)
+ }
+ if testing.Short() {
+ break
+ }
+ }
+
+ // check that ProbablyPrime panics if n <= 0
+ c := NewInt(11) // a prime
+ for _, n := range []int{-1, 0, 1} {
+ func() {
+ defer func() {
+ if n <= 0 && recover() == nil {
+ t.Fatalf("expected panic from ProbablyPrime(%d)", n)
+ }
+ }()
+ if !c.ProbablyPrime(n) {
+ t.Fatalf("%v should be a prime", c)
+ }
+ }()
+ }
+}
+
+type intShiftTest struct {
+ in string
+ shift uint
+ out string
+}
+
+var rshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"-0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "0"},
+ {"1", 2, "0"},
+ {"2", 0, "2"},
+ {"2", 1, "1"},
+ {"-1", 0, "-1"},
+ {"-1", 1, "-1"},
+ {"-1", 10, "-1"},
+ {"-100", 2, "-25"},
+ {"-100", 3, "-13"},
+ {"-100", 100, "-1"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "2147483648"},
+ {"4294967296", 2, "1073741824"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"18446744073709551616", 1, "9223372036854775808"},
+ {"18446744073709551616", 2, "4611686018427387904"},
+ {"18446744073709551616", 64, "1"},
+ {"340282366920938463463374607431768211456", 64, "18446744073709551616"},
+ {"340282366920938463463374607431768211456", 128, "1"},
+}
+
+func TestRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Rsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestRshSelf(t *testing.T) {
+ for i, test := range rshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Rsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+var lshTests = []intShiftTest{
+ {"0", 0, "0"},
+ {"0", 1, "0"},
+ {"0", 2, "0"},
+ {"1", 0, "1"},
+ {"1", 1, "2"},
+ {"1", 2, "4"},
+ {"2", 0, "2"},
+ {"2", 1, "4"},
+ {"2", 2, "8"},
+ {"-87", 1, "-174"},
+ {"4294967296", 0, "4294967296"},
+ {"4294967296", 1, "8589934592"},
+ {"4294967296", 2, "17179869184"},
+ {"18446744073709551616", 0, "18446744073709551616"},
+ {"9223372036854775808", 1, "18446744073709551616"},
+ {"4611686018427387904", 2, "18446744073709551616"},
+ {"1", 64, "18446744073709551616"},
+ {"18446744073709551616", 64, "340282366920938463463374607431768211456"},
+ {"1", 128, "340282366920938463463374607431768211456"},
+}
+
+func TestLsh(t *testing.T) {
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ out := new(Int).Lsh(in, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ }
+}
+
+func TestLshSelf(t *testing.T) {
+ for i, test := range lshTests {
+ z, _ := new(Int).SetString(test.in, 10)
+ expected, _ := new(Int).SetString(test.out, 10)
+ z.Lsh(z, test.shift)
+
+ if !isNormalized(z) {
+ t.Errorf("#%d: %v is not normalized", i, *z)
+ }
+ if z.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, z, expected)
+ }
+ }
+}
+
+func TestLshRsh(t *testing.T) {
+ for i, test := range rshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out = out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+ for i, test := range lshTests {
+ in, _ := new(Int).SetString(test.in, 10)
+ out := new(Int).Lsh(in, test.shift)
+ out.Rsh(out, test.shift)
+
+ if !isNormalized(out) {
+ t.Errorf("#%d: %v is not normalized", i, *out)
+ }
+ if in.Cmp(out) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+var int64Tests = []int64{
+ 0,
+ 1,
+ -1,
+ 4294967295,
+ -4294967295,
+ 4294967296,
+ -4294967296,
+ 9223372036854775807,
+ -9223372036854775807,
+ -9223372036854775808,
+}
+
+func TestInt64(t *testing.T) {
+ for i, testVal := range int64Tests {
+ in := NewInt(testVal)
+ out := in.Int64()
+
+ if out != testVal {
+ t.Errorf("#%d got %d want %d", i, out, testVal)
+ }
+ }
+}
+
+var uint64Tests = []uint64{
+ 0,
+ 1,
+ 4294967295,
+ 4294967296,
+ 8589934591,
+ 8589934592,
+ 9223372036854775807,
+ 9223372036854775808,
+ 18446744073709551615, // 1<<64 - 1
+}
+
+func TestUint64(t *testing.T) {
+ in := new(Int)
+ for i, testVal := range uint64Tests {
+ in.SetUint64(testVal)
+ out := in.Uint64()
+
+ if out != testVal {
+ t.Errorf("#%d got %d want %d", i, out, testVal)
+ }
+
+ str := fmt.Sprint(testVal)
+ strOut := in.String()
+ if strOut != str {
+ t.Errorf("#%d.String got %s want %s", i, strOut, str)
+ }
+ }
+}
+
+var bitwiseTests = []struct {
+ x, y string
+ and, or, xor, andNot string
+}{
+ {"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
+ {"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
+ {"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
+ {"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+ {"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
+ {"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
+ {"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
+ {"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
+ {"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
+ {"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
+ {"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
+ {"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
+ {"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
+ {"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0x8c40c2d8822caa04120b8321400",
+ },
+ {
+ "0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "0x8c40c2d8822caa04120b8321401",
+ "-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
+ "-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
+ "0x1000001186210100001000009048c2000",
+ },
+ {
+ "-0x1000009dc6e3d9822cba04129bcbe3401",
+ "-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+ "-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+ "-0x1000001186210100001000009048c2001",
+ "0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+ "0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
+ },
+}
+
+type bitFun func(z, x, y *Int) *Int
+
+func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ out := f(new(Int), x, y)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, out, expected)
+ }
+}
+
+func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+ self := new(Int)
+ self.Set(x)
+ expected := new(Int)
+ expected.SetString(exp, 0)
+
+ self = f(self, self, y)
+ if self.Cmp(expected) != 0 {
+ t.Errorf("%s: got %s want %s", msg, self, expected)
+ }
+}
+
+func altBit(x *Int, i int) uint {
+ z := new(Int).Rsh(x, uint(i))
+ z = z.And(z, NewInt(1))
+ if z.Cmp(new(Int)) != 0 {
+ return 1
+ }
+ return 0
+}
+
+func altSetBit(z *Int, x *Int, i int, b uint) *Int {
+ one := NewInt(1)
+ m := one.Lsh(one, uint(i))
+ switch b {
+ case 1:
+ return z.Or(x, m)
+ case 0:
+ return z.AndNot(x, m)
+ }
+ panic("set bit is not 0 or 1")
+}
+
+func testBitset(t *testing.T, x *Int) {
+ n := x.BitLen()
+ z := new(Int).Set(x)
+ z1 := new(Int).Set(x)
+ for i := 0; i < n+10; i++ {
+ old := z.Bit(i)
+ old1 := altBit(z1, i)
+ if old != old1 {
+ t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
+ }
+ z := new(Int).SetBit(z, i, 1)
+ z1 := altSetBit(new(Int), z1, i, 1)
+ if z.Bit(i) == 0 {
+ t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
+ }
+ z.SetBit(z, i, 0)
+ altSetBit(z1, z1, i, 0)
+ if z.Bit(i) != 0 {
+ t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
+ }
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
+ }
+ altSetBit(z1, z1, i, old)
+ z.SetBit(z, i, old)
+ if z.Cmp(z1) != 0 {
+ t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
+ }
+ }
+ if z.Cmp(x) != 0 {
+ t.Errorf("bitset: got %s want %s", z, x)
+ }
+}
+
+var bitsetTests = []struct {
+ x string
+ i int
+ b uint
+}{
+ {"0", 0, 0},
+ {"0", 200, 0},
+ {"1", 0, 1},
+ {"1", 1, 0},
+ {"-1", 0, 1},
+ {"-1", 200, 1},
+ {"0x2000000000000000000000000000", 108, 0},
+ {"0x2000000000000000000000000000", 109, 1},
+ {"0x2000000000000000000000000000", 110, 0},
+ {"-0x2000000000000000000000000001", 108, 1},
+ {"-0x2000000000000000000000000001", 109, 0},
+ {"-0x2000000000000000000000000001", 110, 1},
+}
+
+func TestBitSet(t *testing.T) {
+ for _, test := range bitwiseTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ testBitset(t, x)
+ x = new(Int)
+ x.SetString(test.y, 0)
+ testBitset(t, x)
+ }
+ for i, test := range bitsetTests {
+ x := new(Int)
+ x.SetString(test.x, 0)
+ b := x.Bit(test.i)
+ if b != test.b {
+ t.Errorf("#%d got %v want %v", i, b, test.b)
+ }
+ }
+ z := NewInt(1)
+ z.SetBit(NewInt(0), 2, 1)
+ if z.Cmp(NewInt(4)) != 0 {
+ t.Errorf("destination leaked into result; got %s want 4", z)
+ }
+}
+
+func BenchmarkBitset(b *testing.B) {
+ z := new(Int)
+ z.SetBit(z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNeg(b *testing.B) {
+ z := NewInt(-1)
+ z.SetBit(z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ z.SetBit(z, i&512, 0)
+ }
+}
+
+func BenchmarkBitsetOrig(b *testing.B) {
+ z := new(Int)
+ altSetBit(z, z, 512, 1)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 1)
+ }
+}
+
+func BenchmarkBitsetNegOrig(b *testing.B) {
+ z := NewInt(-1)
+ altSetBit(z, z, 512, 0)
+ b.ResetTimer()
+ b.StartTimer()
+ for i := b.N - 1; i >= 0; i-- {
+ altSetBit(z, z, i&512, 0)
+ }
+}
+
+func TestBitwise(t *testing.T) {
+ x := new(Int)
+ y := new(Int)
+ for _, test := range bitwiseTests {
+ x.SetString(test.x, 0)
+ y.SetString(test.y, 0)
+
+ testBitFun(t, "and", (*Int).And, x, y, test.and)
+ testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
+ testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+ testBitFun(t, "or", (*Int).Or, x, y, test.or)
+ testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
+ testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
+ testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
+ }
+}
+
+var notTests = []struct {
+ in string
+ out string
+}{
+ {"0", "-1"},
+ {"1", "-2"},
+ {"7", "-8"},
+ {"0", "-1"},
+ {"-81910", "81909"},
+ {
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "-298472983472983471903246121093472394872319615612417471234712062",
+ },
+}
+
+func TestNot(t *testing.T) {
+ in := new(Int)
+ out := new(Int)
+ expected := new(Int)
+ for i, test := range notTests {
+ in.SetString(test.in, 10)
+ expected.SetString(test.out, 10)
+ out = out.Not(in)
+ if out.Cmp(expected) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, expected)
+ }
+ out = out.Not(out)
+ if out.Cmp(in) != 0 {
+ t.Errorf("#%d: got %s want %s", i, out, in)
+ }
+ }
+}
+
+var modInverseTests = []struct {
+ element string
+ modulus string
+}{
+ {"1234567", "458948883992"},
+ {"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
+}
+
+func TestModInverse(t *testing.T) {
+ var element, modulus, gcd, inverse Int
+ one := NewInt(1)
+ for i, test := range modInverseTests {
+ (&element).SetString(test.element, 10)
+ (&modulus).SetString(test.modulus, 10)
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("#%d: failed (e·e^(-1)=%s)", i, &inverse)
+ }
+ }
+ // exhaustive test for small values
+ for n := 2; n < 100; n++ {
+ (&modulus).SetInt64(int64(n))
+ for x := 1; x < n; x++ {
+ (&element).SetInt64(int64(x))
+ (&gcd).GCD(nil, nil, &element, &modulus)
+ if (&gcd).Cmp(one) != 0 {
+ continue
+ }
+ (&inverse).ModInverse(&element, &modulus)
+ (&inverse).Mul(&inverse, &element)
+ (&inverse).Mod(&inverse, &modulus)
+ if (&inverse).Cmp(one) != 0 {
+ t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+ }
+ }
+ }
+}
+
+var encodingTests = []string{
+ "-539345864568634858364538753846587364875430589374589",
+ "-678645873",
+ "-100",
+ "-2",
+ "-1",
+ "0",
+ "1",
+ "2",
+ "10",
+ "42",
+ "1234567890",
+ "298472983472983471903246121093472394872319615612417471234712061",
+}
+
+func TestIntGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Int
+ tx.SetString(test, 10)
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ }
+ var rx Int
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+// Sending a nil Int pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilIntInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Int, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Int
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Int
+ if out[0].Cmp(&zero) != 0 {
+ t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+ }
+}
+
+func TestIntJSONEncoding(t *testing.T) {
+ for _, test := range encodingTests {
+ var tx Int
+ tx.SetString(test, 10)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ }
+ var rx Int
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+var intVals = []string{
+ "-141592653589793238462643383279502884197169399375105820974944592307816406286",
+ "-1415926535897932384626433832795028841971",
+ "-141592653589793",
+ "-1",
+ "0",
+ "1",
+ "141592653589793",
+ "1415926535897932384626433832795028841971",
+ "141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+func TestIntJSONEncodingTextMarshaller(t *testing.T) {
+ for _, num := range intVals {
+ var tx Int
+ tx.SetString(num, 0)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+func TestIntXMLEncodingTextMarshaller(t *testing.T) {
+ for _, num := range intVals {
+ var tx Int
+ tx.SetString(num, 0)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Int
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+func TestIssue2607(t *testing.T) {
+ // This code sequence used to hang.
+ n := NewInt(10)
+ n.Rand(rand.New(rand.NewSource(9)), n)
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements int-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+)
+
+func (x *Int) String() string {
+ switch {
+ case x == nil:
+ return "<nil>"
+ case x.neg:
+ return "-" + x.abs.decimalString()
+ }
+ return x.abs.decimalString()
+}
+
+func charset(ch rune) string {
+ switch ch {
+ case 'b':
+ return lowercaseDigits[0:2]
+ case 'o':
+ return lowercaseDigits[0:8]
+ case 'd', 's', 'v':
+ return lowercaseDigits[0:10]
+ case 'x':
+ return lowercaseDigits[0:16]
+ case 'X':
+ return uppercaseDigits[0:16]
+ }
+ return "" // unknown format
+}
+
+// write count copies of text to s
+func writeMultiple(s fmt.State, text string, count int) {
+ if len(text) > 0 {
+ b := []byte(text)
+ for ; count > 0; count-- {
+ s.Write(b)
+ }
+ }
+}
+
+// Format is a support routine for fmt.Formatter. It accepts
+// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
+// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+// Also supported are the full suite of package fmt's format
+// verbs for integral types, including '+', '-', and ' '
+// for sign control, '#' for leading zero in octal and for
+// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
+// respectively, specification of minimum digits precision,
+// output field width, space or zero padding, and left or
+// right justification.
+//
+func (x *Int) Format(s fmt.State, ch rune) {
+ cs := charset(ch)
+
+ // special cases
+ switch {
+ case cs == "":
+ // unknown format
+ fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
+ return
+ case x == nil:
+ fmt.Fprint(s, "<nil>")
+ return
+ }
+
+ // determine sign character
+ sign := ""
+ switch {
+ case x.neg:
+ sign = "-"
+ case s.Flag('+'): // supersedes ' ' when both specified
+ sign = "+"
+ case s.Flag(' '):
+ sign = " "
+ }
+
+ // determine prefix characters for indicating output base
+ prefix := ""
+ if s.Flag('#') {
+ switch ch {
+ case 'o': // octal
+ prefix = "0"
+ case 'x': // hexadecimal
+ prefix = "0x"
+ case 'X':
+ prefix = "0X"
+ }
+ }
+
+ // determine digits with base set by len(cs) and digit characters from cs
+ digits := x.abs.string(cs)
+
+ // number of characters for the three classes of number padding
+ var left int // space characters to left of digits for right justification ("%8d")
+ var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
+ var right int // space characters to right of digits for left justification ("%-8d")
+
+ // determine number padding from precision: the least number of digits to output
+ precision, precisionSet := s.Precision()
+ if precisionSet {
+ switch {
+ case len(digits) < precision:
+ zeroes = precision - len(digits) // count of zero padding
+ case digits == "0" && precision == 0:
+ return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
+ }
+ }
+
+ // determine field pad from width: the least number of characters to output
+ length := len(sign) + len(prefix) + zeroes + len(digits)
+ if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
+ switch d := width - length; {
+ case s.Flag('-'):
+ // pad on the right with spaces; supersedes '0' when both specified
+ right = d
+ case s.Flag('0') && !precisionSet:
+ // pad with zeroes unless precision also specified
+ zeroes = d
+ default:
+ // pad on the left with spaces
+ left = d
+ }
+ }
+
+ // print number as [left pad][sign][prefix][zero pad][digits][right pad]
+ writeMultiple(s, " ", left)
+ writeMultiple(s, sign, 1)
+ writeMultiple(s, prefix, 1)
+ writeMultiple(s, "0", zeroes)
+ writeMultiple(s, digits, 1)
+ writeMultiple(s, " ", right)
+}
+
+// scan sets z to the integer value corresponding to the longest possible prefix
+// read from r representing a signed integer number in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined but the returned value is nil. The
+// syntax follows the syntax of integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z *Int) scan(r io.ByteScanner, base int) (*Int, int, error) {
+ // determine sign
+ neg, err := scanSign(r)
+ if err != nil {
+ return nil, 0, err
+ }
+
+ // determine mantissa
+ z.abs, base, _, err = z.abs.scan(r, base, false)
+ if err != nil {
+ return nil, base, err
+ }
+ z.neg = len(z.abs) > 0 && neg // 0 has no sign
+
+ return z, base, nil
+}
+
+func scanSign(r io.ByteScanner) (neg bool, err error) {
+ var ch byte
+ if ch, err = r.ReadByte(); err != nil {
+ return false, err
+ }
+ switch ch {
+ case '-':
+ neg = true
+ case '+':
+ // nothing to do
+ default:
+ r.UnreadByte()
+ }
+ return
+}
+
+// byteReader is a local wrapper around fmt.ScanState;
+// it implements the ByteReader interface.
+type byteReader struct {
+ fmt.ScanState
+}
+
+func (r byteReader) ReadByte() (byte, error) {
+ ch, size, err := r.ReadRune()
+ if size != 1 && err == nil {
+ err = fmt.Errorf("invalid rune %#U", ch)
+ }
+ return byte(ch), err
+}
+
+func (r byteReader) UnreadByte() error {
+ return r.UnreadRune()
+}
+
+// Scan is a support routine for fmt.Scanner; it sets z to the value of
+// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
+// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+func (z *Int) Scan(s fmt.ScanState, ch rune) error {
+ s.SkipSpace() // skip leading space characters
+ base := 0
+ switch ch {
+ case 'b':
+ base = 2
+ case 'o':
+ base = 8
+ case 'd':
+ base = 10
+ case 'x', 'X':
+ base = 16
+ case 's', 'v':
+ // let scan determine the base
+ default:
+ return errors.New("Int.Scan: invalid verb")
+ }
+ _, _, err := z.scan(byteReader{s}, base)
+ return err
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "testing"
+)
+
+var stringTests = []struct {
+ in string
+ out string
+ base int
+ val int64
+ ok bool
+}{
+ {in: "", ok: false},
+ {in: "a", ok: false},
+ {in: "z", ok: false},
+ {in: "+", ok: false},
+ {in: "-", ok: false},
+ {in: "0b", ok: false},
+ {in: "0x", ok: false},
+ {in: "2", base: 2, ok: false},
+ {in: "0b2", base: 0, ok: false},
+ {in: "08", ok: false},
+ {in: "8", base: 8, ok: false},
+ {in: "0xg", base: 0, ok: false},
+ {in: "g", base: 16, ok: false},
+ {"0", "0", 0, 0, true},
+ {"0", "0", 10, 0, true},
+ {"0", "0", 16, 0, true},
+ {"+0", "0", 0, 0, true},
+ {"-0", "0", 0, 0, true},
+ {"10", "10", 0, 10, true},
+ {"10", "10", 10, 10, true},
+ {"10", "10", 16, 16, true},
+ {"-10", "-10", 16, -16, true},
+ {"+10", "10", 16, 16, true},
+ {"0x10", "16", 0, 16, true},
+ {in: "0x10", base: 16, ok: false},
+ {"-0x10", "-16", 0, -16, true},
+ {"+0x10", "16", 0, 16, true},
+ {"00", "0", 0, 0, true},
+ {"0", "0", 8, 0, true},
+ {"07", "7", 0, 7, true},
+ {"7", "7", 8, 7, true},
+ {"023", "19", 0, 19, true},
+ {"23", "23", 8, 19, true},
+ {"cafebabe", "cafebabe", 16, 0xcafebabe, true},
+ {"0b0", "0", 0, 0, true},
+ {"-111", "-111", 2, -7, true},
+ {"-0b111", "-7", 0, -7, true},
+ {"0b1001010111", "599", 0, 0x257, true},
+ {"1001010111", "1001010111", 2, 0x257, true},
+}
+
+func format(base int) string {
+ switch base {
+ case 2:
+ return "%b"
+ case 8:
+ return "%o"
+ case 16:
+ return "%x"
+ }
+ return "%d"
+}
+
+func TestGetString(t *testing.T) {
+ z := new(Int)
+ for i, test := range stringTests {
+ if !test.ok {
+ continue
+ }
+ z.SetInt64(test.val)
+
+ if test.base == 10 {
+ s := z.String()
+ if s != test.out {
+ t.Errorf("#%da got %s; want %s", i, s, test.out)
+ }
+ }
+
+ s := fmt.Sprintf(format(test.base), z)
+ if s != test.out {
+ t.Errorf("#%db got %s; want %s", i, s, test.out)
+ }
+ }
+}
+
+func TestSetString(t *testing.T) {
+ tmp := new(Int)
+ for i, test := range stringTests {
+ // initialize to a non-zero value so that issues with parsing
+ // 0 are detected
+ tmp.SetInt64(1234567890)
+ n1, ok1 := new(Int).SetString(test.in, test.base)
+ n2, ok2 := tmp.SetString(test.in, test.base)
+ expected := NewInt(test.val)
+ if ok1 != test.ok || ok2 != test.ok {
+ t.Errorf("#%d (input '%s') ok incorrect (should be %t)", i, test.in, test.ok)
+ continue
+ }
+ if !ok1 {
+ if n1 != nil {
+ t.Errorf("#%d (input '%s') n1 != nil", i, test.in)
+ }
+ continue
+ }
+ if !ok2 {
+ if n2 != nil {
+ t.Errorf("#%d (input '%s') n2 != nil", i, test.in)
+ }
+ continue
+ }
+
+ if ok1 && !isNormalized(n1) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n1)
+ }
+ if ok2 && !isNormalized(n2) {
+ t.Errorf("#%d (input '%s'): %v is not normalized", i, test.in, *n2)
+ }
+
+ if n1.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n1, test.val)
+ }
+ if n2.Cmp(expected) != 0 {
+ t.Errorf("#%d (input '%s') got: %s want: %d", i, test.in, n2, test.val)
+ }
+ }
+}
+
+var formatTests = []struct {
+ input string
+ format string
+ output string
+}{
+ {"<nil>", "%x", "<nil>"},
+ {"<nil>", "%#x", "<nil>"},
+ {"<nil>", "%#y", "%!y(big.Int=<nil>)"},
+
+ {"10", "%b", "1010"},
+ {"10", "%o", "12"},
+ {"10", "%d", "10"},
+ {"10", "%v", "10"},
+ {"10", "%x", "a"},
+ {"10", "%X", "A"},
+ {"-10", "%X", "-A"},
+ {"10", "%y", "%!y(big.Int=10)"},
+ {"-10", "%y", "%!y(big.Int=-10)"},
+
+ {"10", "%#b", "1010"},
+ {"10", "%#o", "012"},
+ {"10", "%#d", "10"},
+ {"10", "%#v", "10"},
+ {"10", "%#x", "0xa"},
+ {"10", "%#X", "0XA"},
+ {"-10", "%#X", "-0XA"},
+ {"10", "%#y", "%!y(big.Int=10)"},
+ {"-10", "%#y", "%!y(big.Int=-10)"},
+
+ {"1234", "%d", "1234"},
+ {"1234", "%3d", "1234"},
+ {"1234", "%4d", "1234"},
+ {"-1234", "%d", "-1234"},
+ {"1234", "% 5d", " 1234"},
+ {"1234", "%+5d", "+1234"},
+ {"1234", "%-5d", "1234 "},
+ {"1234", "%x", "4d2"},
+ {"1234", "%X", "4D2"},
+ {"-1234", "%3x", "-4d2"},
+ {"-1234", "%4x", "-4d2"},
+ {"-1234", "%5x", " -4d2"},
+ {"-1234", "%-5x", "-4d2 "},
+ {"1234", "%03d", "1234"},
+ {"1234", "%04d", "1234"},
+ {"1234", "%05d", "01234"},
+ {"1234", "%06d", "001234"},
+ {"-1234", "%06d", "-01234"},
+ {"1234", "%+06d", "+01234"},
+ {"1234", "% 06d", " 01234"},
+ {"1234", "%-6d", "1234 "},
+ {"1234", "%-06d", "1234 "},
+ {"-1234", "%-06d", "-1234 "},
+
+ {"1234", "%.3d", "1234"},
+ {"1234", "%.4d", "1234"},
+ {"1234", "%.5d", "01234"},
+ {"1234", "%.6d", "001234"},
+ {"-1234", "%.3d", "-1234"},
+ {"-1234", "%.4d", "-1234"},
+ {"-1234", "%.5d", "-01234"},
+ {"-1234", "%.6d", "-001234"},
+
+ {"1234", "%8.3d", " 1234"},
+ {"1234", "%8.4d", " 1234"},
+ {"1234", "%8.5d", " 01234"},
+ {"1234", "%8.6d", " 001234"},
+ {"-1234", "%8.3d", " -1234"},
+ {"-1234", "%8.4d", " -1234"},
+ {"-1234", "%8.5d", " -01234"},
+ {"-1234", "%8.6d", " -001234"},
+
+ {"1234", "%+8.3d", " +1234"},
+ {"1234", "%+8.4d", " +1234"},
+ {"1234", "%+8.5d", " +01234"},
+ {"1234", "%+8.6d", " +001234"},
+ {"-1234", "%+8.3d", " -1234"},
+ {"-1234", "%+8.4d", " -1234"},
+ {"-1234", "%+8.5d", " -01234"},
+ {"-1234", "%+8.6d", " -001234"},
+
+ {"1234", "% 8.3d", " 1234"},
+ {"1234", "% 8.4d", " 1234"},
+ {"1234", "% 8.5d", " 01234"},
+ {"1234", "% 8.6d", " 001234"},
+ {"-1234", "% 8.3d", " -1234"},
+ {"-1234", "% 8.4d", " -1234"},
+ {"-1234", "% 8.5d", " -01234"},
+ {"-1234", "% 8.6d", " -001234"},
+
+ {"1234", "%.3x", "4d2"},
+ {"1234", "%.4x", "04d2"},
+ {"1234", "%.5x", "004d2"},
+ {"1234", "%.6x", "0004d2"},
+ {"-1234", "%.3x", "-4d2"},
+ {"-1234", "%.4x", "-04d2"},
+ {"-1234", "%.5x", "-004d2"},
+ {"-1234", "%.6x", "-0004d2"},
+
+ {"1234", "%8.3x", " 4d2"},
+ {"1234", "%8.4x", " 04d2"},
+ {"1234", "%8.5x", " 004d2"},
+ {"1234", "%8.6x", " 0004d2"},
+ {"-1234", "%8.3x", " -4d2"},
+ {"-1234", "%8.4x", " -04d2"},
+ {"-1234", "%8.5x", " -004d2"},
+ {"-1234", "%8.6x", " -0004d2"},
+
+ {"1234", "%+8.3x", " +4d2"},
+ {"1234", "%+8.4x", " +04d2"},
+ {"1234", "%+8.5x", " +004d2"},
+ {"1234", "%+8.6x", " +0004d2"},
+ {"-1234", "%+8.3x", " -4d2"},
+ {"-1234", "%+8.4x", " -04d2"},
+ {"-1234", "%+8.5x", " -004d2"},
+ {"-1234", "%+8.6x", " -0004d2"},
+
+ {"1234", "% 8.3x", " 4d2"},
+ {"1234", "% 8.4x", " 04d2"},
+ {"1234", "% 8.5x", " 004d2"},
+ {"1234", "% 8.6x", " 0004d2"},
+ {"1234", "% 8.7x", " 00004d2"},
+ {"1234", "% 8.8x", " 000004d2"},
+ {"-1234", "% 8.3x", " -4d2"},
+ {"-1234", "% 8.4x", " -04d2"},
+ {"-1234", "% 8.5x", " -004d2"},
+ {"-1234", "% 8.6x", " -0004d2"},
+ {"-1234", "% 8.7x", "-00004d2"},
+ {"-1234", "% 8.8x", "-000004d2"},
+
+ {"1234", "%-8.3d", "1234 "},
+ {"1234", "%-8.4d", "1234 "},
+ {"1234", "%-8.5d", "01234 "},
+ {"1234", "%-8.6d", "001234 "},
+ {"1234", "%-8.7d", "0001234 "},
+ {"1234", "%-8.8d", "00001234"},
+ {"-1234", "%-8.3d", "-1234 "},
+ {"-1234", "%-8.4d", "-1234 "},
+ {"-1234", "%-8.5d", "-01234 "},
+ {"-1234", "%-8.6d", "-001234 "},
+ {"-1234", "%-8.7d", "-0001234"},
+ {"-1234", "%-8.8d", "-00001234"},
+
+ {"16777215", "%b", "111111111111111111111111"}, // 2**24 - 1
+
+ {"0", "%.d", ""},
+ {"0", "%.0d", ""},
+ {"0", "%3.d", ""},
+}
+
+func TestFormat(t *testing.T) {
+ for i, test := range formatTests {
+ var x *Int
+ if test.input != "<nil>" {
+ var ok bool
+ x, ok = new(Int).SetString(test.input, 0)
+ if !ok {
+ t.Errorf("#%d failed reading input %s", i, test.input)
+ }
+ }
+ output := fmt.Sprintf(test.format, x)
+ if output != test.output {
+ t.Errorf("#%d got %q; want %q, {%q, %q, %q}", i, output, test.output, test.input, test.format, test.output)
+ }
+ }
+}
+
+var scanTests = []struct {
+ input string
+ format string
+ output string
+ remaining int
+}{
+ {"1010", "%b", "10", 0},
+ {"0b1010", "%v", "10", 0},
+ {"12", "%o", "10", 0},
+ {"012", "%v", "10", 0},
+ {"10", "%d", "10", 0},
+ {"10", "%v", "10", 0},
+ {"a", "%x", "10", 0},
+ {"0xa", "%v", "10", 0},
+ {"A", "%X", "10", 0},
+ {"-A", "%X", "-10", 0},
+ {"+0b1011001", "%v", "89", 0},
+ {"0xA", "%v", "10", 0},
+ {"0 ", "%v", "0", 1},
+ {"2+3", "%v", "2", 2},
+ {"0XABC 12", "%v", "2748", 3},
+}
+
+func TestScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range scanTests {
+ x := new(Int)
+ buf.Reset()
+ buf.WriteString(test.input)
+ if _, err := fmt.Fscanf(&buf, test.format, x); err != nil {
+ t.Errorf("#%d error: %s", i, err)
+ }
+ if x.String() != test.output {
+ t.Errorf("#%d got %s; want %s", i, x.String(), test.output)
+ }
+ if buf.Len() != test.remaining {
+ t.Errorf("#%d got %d bytes remaining; want %d", i, buf.Len(), test.remaining)
+ }
+ }
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package big implements multi-precision arithmetic (big numbers).
+// The following numeric types are supported:
+//
+// Int signed integers
+// Rat rational numbers
+// Float floating-point numbers
+//
+// Methods are typically of the form:
+//
+// func (z *T) Unary(x *T) *T // z = op x
+// func (z *T) Binary(x, y *T) *T // z = x op y
+// func (x *T) M() T1 // v = x.M()
+//
+// with T one of Int, Rat, or Float. For unary and binary operations, the
+// result is the receiver (usually named z in that case); if it is one of
+// the operands x or y it may be overwritten (and its memory reused).
+// To enable chaining of operations, the result is also returned. Methods
+// returning a result other than *Int, *Rat, or *Float take an operand as
+// the receiver (usually named x in that case).
+//
+package big
+
+// This file contains operations on unsigned multi-precision integers.
+// These are the building blocks for the operations on signed integers
+// and rationals.
+
+import "math/rand"
+
+// An unsigned integer x of the form
+//
+// x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
+//
+// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
+// with the digits x[i] as the slice elements.
+//
+// A number is normalized if the slice contains no leading 0 digits.
+// During arithmetic operations, denormalized values may occur but are
+// always normalized before returning the final result. The normalized
+// representation of 0 is the empty or nil slice (length = 0).
+//
+type nat []Word
+
+var (
+ natOne = nat{1}
+ natTwo = nat{2}
+ natTen = nat{10}
+)
+
+func (z nat) clear() {
+ for i := range z {
+ z[i] = 0
+ }
+}
+
+func (z nat) norm() nat {
+ i := len(z)
+ for i > 0 && z[i-1] == 0 {
+ i--
+ }
+ return z[0:i]
+}
+
+func (z nat) make(n int) nat {
+ if n <= cap(z) {
+ return z[:n] // reuse z
+ }
+ // Choosing a good value for e has significant performance impact
+ // because it increases the chance that a value can be reused.
+ const e = 4 // extra capacity
+ return make(nat, n, n+e)
+}
+
+func (z nat) setWord(x Word) nat {
+ if x == 0 {
+ return z[:0]
+ }
+ z = z.make(1)
+ z[0] = x
+ return z
+}
+
+func (z nat) setUint64(x uint64) nat {
+ // single-digit values
+ if w := Word(x); uint64(w) == x {
+ return z.setWord(w)
+ }
+
+ // compute number of words n required to represent x
+ n := 0
+ for t := x; t > 0; t >>= _W {
+ n++
+ }
+
+ // split x into n words
+ z = z.make(n)
+ for i := range z {
+ z[i] = Word(x & _M)
+ x >>= _W
+ }
+
+ return z
+}
+
+func (z nat) set(x nat) nat {
+ z = z.make(len(x))
+ copy(z, x)
+ return z
+}
+
+func (z nat) add(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.add(y, x)
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z[:0]
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ c := addVV(z[0:n], x, y)
+ if m > n {
+ c = addVW(z[n:m], x[n:], c)
+ }
+ z[m] = c
+
+ return z.norm()
+}
+
+func (z nat) sub(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ panic("underflow")
+ case m == 0:
+ // n == 0 because m >= n; result is 0
+ return z[:0]
+ case n == 0:
+ // result is x
+ return z.set(x)
+ }
+ // m > 0
+
+ z = z.make(m)
+ c := subVV(z[0:n], x, y)
+ if m > n {
+ c = subVW(z[n:], x[n:], c)
+ }
+ if c != 0 {
+ panic("underflow")
+ }
+
+ return z.norm()
+}
+
+func (x nat) cmp(y nat) (r int) {
+ m := len(x)
+ n := len(y)
+ if m != n || m == 0 {
+ switch {
+ case m < n:
+ r = -1
+ case m > n:
+ r = 1
+ }
+ return
+ }
+
+ i := m - 1
+ for i > 0 && x[i] == y[i] {
+ i--
+ }
+
+ switch {
+ case x[i] < y[i]:
+ r = -1
+ case x[i] > y[i]:
+ r = 1
+ }
+ return
+}
+
+func (z nat) mulAddWW(x nat, y, r Word) nat {
+ m := len(x)
+ if m == 0 || y == 0 {
+ return z.setWord(r) // result is r
+ }
+ // m > 0
+
+ z = z.make(m + 1)
+ z[m] = mulAddVWW(z[0:m], x, y, r)
+
+ return z.norm()
+}
+
+// basicMul multiplies x and y and leaves the result in z.
+// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
+func basicMul(z, x, y nat) {
+ z[0 : len(x)+len(y)].clear() // initialize z
+ for i, d := range y {
+ if d != 0 {
+ z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
+ }
+ }
+}
+
+// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
+// Factored out for readability - do not use outside karatsuba.
+func karatsubaAdd(z, x nat, n int) {
+ if c := addVV(z[0:n], z, x); c != 0 {
+ addVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Like karatsubaAdd, but does subtract.
+func karatsubaSub(z, x nat, n int) {
+ if c := subVV(z[0:n], z, x); c != 0 {
+ subVW(z[n:n+n>>1], z[n:], c)
+ }
+}
+
+// Operands that are shorter than karatsubaThreshold are multiplied using
+// "grade school" multiplication; for longer operands the Karatsuba algorithm
+// is used.
+var karatsubaThreshold int = 40 // computed by calibrate.go
+
+// karatsuba multiplies x and y and leaves the result in z.
+// Both x and y must have the same length n and n must be a
+// power of 2. The result vector z must have len(z) >= 6*n.
+// The (non-normalized) result is placed in z[0 : 2*n].
+func karatsuba(z, x, y nat) {
+ n := len(y)
+
+ // Switch to basic multiplication if numbers are odd or small.
+ // (n is always even if karatsubaThreshold is even, but be
+ // conservative)
+ if n&1 != 0 || n < karatsubaThreshold || n < 2 {
+ basicMul(z, x, y)
+ return
+ }
+ // n&1 == 0 && n >= karatsubaThreshold && n >= 2
+
+ // Karatsuba multiplication is based on the observation that
+ // for two numbers x and y with:
+ //
+ // x = x1*b + x0
+ // y = y1*b + y0
+ //
+ // the product x*y can be obtained with 3 products z2, z1, z0
+ // instead of 4:
+ //
+ // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
+ // = z2*b*b + z1*b + z0
+ //
+ // with:
+ //
+ // xd = x1 - x0
+ // yd = y0 - y1
+ //
+ // z1 = xd*yd + z2 + z0
+ // = (x1-x0)*(y0 - y1) + z2 + z0
+ // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
+ // = x1*y0 - z2 - z0 + x0*y1 + z2 + z0
+ // = x1*y0 + x0*y1
+
+ // split x, y into "digits"
+ n2 := n >> 1 // n2 >= 1
+ x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
+ y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
+
+ // z is used for the result and temporary storage:
+ //
+ // 6*n 5*n 4*n 3*n 2*n 1*n 0*n
+ // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
+ //
+ // For each recursive call of karatsuba, an unused slice of
+ // z is passed in that has (at least) half the length of the
+ // caller's z.
+
+ // compute z0 and z2 with the result "in place" in z
+ karatsuba(z, x0, y0) // z0 = x0*y0
+ karatsuba(z[n:], x1, y1) // z2 = x1*y1
+
+ // compute xd (or the negative value if underflow occurs)
+ s := 1 // sign of product xd*yd
+ xd := z[2*n : 2*n+n2]
+ if subVV(xd, x1, x0) != 0 { // x1-x0
+ s = -s
+ subVV(xd, x0, x1) // x0-x1
+ }
+
+ // compute yd (or the negative value if underflow occurs)
+ yd := z[2*n+n2 : 3*n]
+ if subVV(yd, y0, y1) != 0 { // y0-y1
+ s = -s
+ subVV(yd, y1, y0) // y1-y0
+ }
+
+ // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
+ // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
+ p := z[n*3:]
+ karatsuba(p, xd, yd)
+
+ // save original z2:z0
+ // (ok to use upper half of z since we're done recursing)
+ r := z[n*4:]
+ copy(r, z[:n*2])
+
+ // add up all partial products
+ //
+ // 2*n n 0
+ // z = [ z2 | z0 ]
+ // + [ z0 ]
+ // + [ z2 ]
+ // + [ p ]
+ //
+ karatsubaAdd(z[n2:], r, n)
+ karatsubaAdd(z[n2:], r[n:], n)
+ if s > 0 {
+ karatsubaAdd(z[n2:], p, n)
+ } else {
+ karatsubaSub(z[n2:], p, n)
+ }
+}
+
+// alias reports whether x and y share the same base array.
+func alias(x, y nat) bool {
+ return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+}
+
+// addAt implements z += x<<(_W*i); z must be long enough.
+// (we don't use nat.add because we need z to stay the same
+// slice, and we don't need to normalize z after each addition)
+func addAt(z, x nat, i int) {
+ if n := len(x); n > 0 {
+ if c := addVV(z[i:i+n], z[i:], x); c != 0 {
+ j := i + n
+ if j < len(z) {
+ addVW(z[j:], z[j:], c)
+ }
+ }
+ }
+}
+
+func max(x, y int) int {
+ if x > y {
+ return x
+ }
+ return y
+}
+
+// karatsubaLen computes an approximation to the maximum k <= n such that
+// k = p<<i for a number p <= karatsubaThreshold and an i >= 0. Thus, the
+// result is the largest number that can be divided repeatedly by 2 before
+// becoming about the value of karatsubaThreshold.
+func karatsubaLen(n int) int {
+ i := uint(0)
+ for n > karatsubaThreshold {
+ n >>= 1
+ i++
+ }
+ return n << i
+}
+
+func (z nat) mul(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+
+ switch {
+ case m < n:
+ return z.mul(y, x)
+ case m == 0 || n == 0:
+ return z[:0]
+ case n == 1:
+ return z.mulAddWW(x, y[0], 0)
+ }
+ // m >= n > 1
+
+ // determine if z can be reused
+ if alias(z, x) || alias(z, y) {
+ z = nil // z is an alias for x or y - cannot reuse
+ }
+
+ // use basic multiplication if the numbers are small
+ if n < karatsubaThreshold {
+ z = z.make(m + n)
+ basicMul(z, x, y)
+ return z.norm()
+ }
+ // m >= n && n >= karatsubaThreshold && n >= 2
+
+ // determine Karatsuba length k such that
+ //
+ // x = xh*b + x0 (0 <= x0 < b)
+ // y = yh*b + y0 (0 <= y0 < b)
+ // b = 1<<(_W*k) ("base" of digits xi, yi)
+ //
+ k := karatsubaLen(n)
+ // k <= n
+
+ // multiply x0 and y0 via Karatsuba
+ x0 := x[0:k] // x0 is not normalized
+ y0 := y[0:k] // y0 is not normalized
+ z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
+ karatsuba(z, x0, y0)
+ z = z[0 : m+n] // z has final length but may be incomplete
+ z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
+
+ // If xh != 0 or yh != 0, add the missing terms to z. For
+ //
+ // xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
+ // yh = y1*b (0 <= y1 < b)
+ //
+ // the missing terms are
+ //
+ // x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
+ //
+ // since all the yi for i > 1 are 0 by choice of k: If any of them
+ // were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
+ // be a larger valid threshold contradicting the assumption about k.
+ //
+ if k < n || m != n {
+ var t nat
+
+ // add x0*y1*b
+ x0 := x0.norm()
+ y1 := y[k:] // y1 is normalized because y is
+ t = t.mul(x0, y1) // update t so we don't lose t's underlying array
+ addAt(z, t, k)
+
+ // add xi*y0<<i, xi*y1*b<<(i+k)
+ y0 := y0.norm()
+ for i := k; i < len(x); i += k {
+ xi := x[i:]
+ if len(xi) > k {
+ xi = xi[:k]
+ }
+ xi = xi.norm()
+ t = t.mul(xi, y0)
+ addAt(z, t, i)
+ t = t.mul(xi, y1)
+ addAt(z, t, i+k)
+ }
+ }
+
+ return z.norm()
+}
+
+// mulRange computes the product of all the unsigned integers in the
+// range [a, b] inclusively. If a > b (empty range), the result is 1.
+func (z nat) mulRange(a, b uint64) nat {
+ switch {
+ case a == 0:
+ // cut long ranges short (optimization)
+ return z.setUint64(0)
+ case a > b:
+ return z.setUint64(1)
+ case a == b:
+ return z.setUint64(a)
+ case a+1 == b:
+ return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
+ }
+ m := (a + b) / 2
+ return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
+}
+
+// q = (x-r)/y, with 0 <= r < y
+func (z nat) divW(x nat, y Word) (q nat, r Word) {
+ m := len(x)
+ switch {
+ case y == 0:
+ panic("division by zero")
+ case y == 1:
+ q = z.set(x) // result is x
+ return
+ case m == 0:
+ q = z[:0] // result is 0
+ return
+ }
+ // m > 0
+ z = z.make(m)
+ r = divWVW(z, 0, x, y)
+ q = z.norm()
+ return
+}
+
+func (z nat) div(z2, u, v nat) (q, r nat) {
+ if len(v) == 0 {
+ panic("division by zero")
+ }
+
+ if u.cmp(v) < 0 {
+ q = z[:0]
+ r = z2.set(u)
+ return
+ }
+
+ if len(v) == 1 {
+ var r2 Word
+ q, r2 = z.divW(u, v[0])
+ r = z2.setWord(r2)
+ return
+ }
+
+ q, r = z.divLarge(z2, u, v)
+ return
+}
+
+// q = (uIn-r)/v, with 0 <= r < y
+// Uses z as storage for q, and u as storage for r if possible.
+// See Knuth, Volume 2, section 4.3.1, Algorithm D.
+// Preconditions:
+// len(v) >= 2
+// len(uIn) >= len(v)
+func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
+ n := len(v)
+ m := len(uIn) - n
+
+ // determine if z can be reused
+ // TODO(gri) should find a better solution - this if statement
+ // is very costly (see e.g. time pidigits -s -n 10000)
+ if alias(z, uIn) || alias(z, v) {
+ z = nil // z is an alias for uIn or v - cannot reuse
+ }
+ q = z.make(m + 1)
+
+ qhatv := make(nat, n+1)
+ if alias(u, uIn) || alias(u, v) {
+ u = nil // u is an alias for uIn or v - cannot reuse
+ }
+ u = u.make(len(uIn) + 1)
+ u.clear() // TODO(gri) no need to clear if we allocated a new u
+
+ // D1.
+ shift := leadingZeros(v[n-1])
+ if shift > 0 {
+ // do not modify v, it may be used by another goroutine simultaneously
+ v1 := make(nat, n)
+ shlVU(v1, v, shift)
+ v = v1
+ }
+ u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+ // D2.
+ for j := m; j >= 0; j-- {
+ // D3.
+ qhat := Word(_M)
+ if u[j+n] != v[n-1] {
+ var rhat Word
+ qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1])
+
+ // x1 | x2 = q̂v_{n-2}
+ x1, x2 := mulWW(qhat, v[n-2])
+ // test if q̂v_{n-2} > br̂ + u_{j+n-2}
+ for greaterThan(x1, x2, rhat, u[j+n-2]) {
+ qhat--
+ prevRhat := rhat
+ rhat += v[n-1]
+ // v[n-1] >= 0, so this tests for overflow.
+ if rhat < prevRhat {
+ break
+ }
+ x1, x2 = mulWW(qhat, v[n-2])
+ }
+ }
+
+ // D4.
+ qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
+
+ c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
+ if c != 0 {
+ c := addVV(u[j:j+n], u[j:], v)
+ u[j+n] += c
+ qhat--
+ }
+
+ q[j] = qhat
+ }
+
+ q = q.norm()
+ shrVU(u, u, shift)
+ r = u.norm()
+
+ return q, r
+}
+
+// Length of x in bits. x must be normalized.
+func (x nat) bitLen() int {
+ if i := len(x) - 1; i >= 0 {
+ return i*_W + bitLen(x[i])
+ }
+ return 0
+}
+
+const deBruijn32 = 0x077CB531
+
+var deBruijn32Lookup = []byte{
+ 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
+ 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
+}
+
+const deBruijn64 = 0x03f79d71b4ca8b09
+
+var deBruijn64Lookup = []byte{
+ 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
+ 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
+ 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
+ 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func trailingZeroBits(x Word) uint {
+ // x & -x leaves only the right-most bit set in the word. Let k be the
+ // index of that bit. Since only a single bit is set, the value is two
+ // to the power of k. Multiplying by a power of two is equivalent to
+ // left shifting, in this case by k bits. The de Bruijn constant is
+ // such that all six bit, consecutive substrings are distinct.
+ // Therefore, if we have a left shifted version of this constant we can
+ // find by how many bits it was shifted by looking at which six bit
+ // substring ended up at the top of the word.
+ // (Knuth, volume 4, section 7.3.1)
+ switch _W {
+ case 32:
+ return uint(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
+ case 64:
+ return uint(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
+ default:
+ panic("unknown word size")
+ }
+}
+
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func (x nat) trailingZeroBits() uint {
+ if len(x) == 0 {
+ return 0
+ }
+ var i uint
+ for x[i] == 0 {
+ i++
+ }
+ // x[i] != 0
+ return i*_W + trailingZeroBits(x[i])
+}
+
+// z = x << s
+func (z nat) shl(x nat, s uint) nat {
+ m := len(x)
+ if m == 0 {
+ return z[:0]
+ }
+ // m > 0
+
+ n := m + int(s/_W)
+ z = z.make(n + 1)
+ z[n] = shlVU(z[n-m:n], x, s%_W)
+ z[0 : n-m].clear()
+
+ return z.norm()
+}
+
+// z = x >> s
+func (z nat) shr(x nat, s uint) nat {
+ m := len(x)
+ n := m - int(s/_W)
+ if n <= 0 {
+ return z[:0]
+ }
+ // n > 0
+
+ z = z.make(n)
+ shrVU(z, x[m-n:], s%_W)
+
+ return z.norm()
+}
+
+func (z nat) setBit(x nat, i uint, b uint) nat {
+ j := int(i / _W)
+ m := Word(1) << (i % _W)
+ n := len(x)
+ switch b {
+ case 0:
+ z = z.make(n)
+ copy(z, x)
+ if j >= n {
+ // no need to grow
+ return z
+ }
+ z[j] &^= m
+ return z.norm()
+ case 1:
+ if j >= n {
+ z = z.make(j + 1)
+ z[n:].clear()
+ } else {
+ z = z.make(n)
+ }
+ copy(z, x)
+ z[j] |= m
+ // no need to normalize
+ return z
+ }
+ panic("set bit is not 0 or 1")
+}
+
+// bit returns the value of the i'th bit, with lsb == bit 0.
+func (x nat) bit(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
+ return 0
+ }
+ // 0 <= j < len(x)
+ return uint(x[j] >> (i % _W) & 1)
+}
+
+// sticky returns 1 if there's a 1 bit within the
+// i least significant bits, otherwise it returns 0.
+func (x nat) sticky(i uint) uint {
+ j := i / _W
+ if j >= uint(len(x)) {
+ if len(x) == 0 {
+ return 0
+ }
+ return 1
+ }
+ // 0 <= j < len(x)
+ for _, x := range x[:j] {
+ if x != 0 {
+ return 1
+ }
+ }
+ if x[j]<<(_W-i%_W) != 0 {
+ return 1
+ }
+ return 0
+}
+
+func (z nat) and(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if m > n {
+ m = n
+ }
+ // m <= n
+
+ z = z.make(m)
+ for i := 0; i < m; i++ {
+ z[i] = x[i] & y[i]
+ }
+
+ return z.norm()
+}
+
+func (z nat) andNot(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ if n > m {
+ n = m
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] &^ y[i]
+ }
+ copy(z[n:m], x[n:m])
+
+ return z.norm()
+}
+
+func (z nat) or(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] | y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+func (z nat) xor(x, y nat) nat {
+ m := len(x)
+ n := len(y)
+ s := x
+ if m < n {
+ n, m = m, n
+ s = y
+ }
+ // m >= n
+
+ z = z.make(m)
+ for i := 0; i < n; i++ {
+ z[i] = x[i] ^ y[i]
+ }
+ copy(z[n:m], s[n:m])
+
+ return z.norm()
+}
+
+// greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
+func greaterThan(x1, x2, y1, y2 Word) bool {
+ return x1 > y1 || x1 == y1 && x2 > y2
+}
+
+// modW returns x % d.
+func (x nat) modW(d Word) (r Word) {
+ // TODO(agl): we don't actually need to store the q value.
+ var q nat
+ q = q.make(len(x))
+ return divWVW(q, 0, x, d)
+}
+
+// random creates a random integer in [0..limit), using the space in z if
+// possible. n is the bit length of limit.
+func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
+ if alias(z, limit) {
+ z = nil // z is an alias for limit - cannot reuse
+ }
+ z = z.make(len(limit))
+
+ bitLengthOfMSW := uint(n % _W)
+ if bitLengthOfMSW == 0 {
+ bitLengthOfMSW = _W
+ }
+ mask := Word((1 << bitLengthOfMSW) - 1)
+
+ for {
+ switch _W {
+ case 32:
+ for i := range z {
+ z[i] = Word(rand.Uint32())
+ }
+ case 64:
+ for i := range z {
+ z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
+ }
+ default:
+ panic("unknown word size")
+ }
+ z[len(limit)-1] &= mask
+ if z.cmp(limit) < 0 {
+ break
+ }
+ }
+
+ return z.norm()
+}
+
+// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
+// otherwise it sets z to x**y. The result is the value of z.
+func (z nat) expNN(x, y, m nat) nat {
+ if alias(z, x) || alias(z, y) {
+ // We cannot allow in-place modification of x or y.
+ z = nil
+ }
+
+ // x**y mod 1 == 0
+ if len(m) == 1 && m[0] == 1 {
+ return z.setWord(0)
+ }
+ // m == 0 || m > 1
+
+ // x**0 == 1
+ if len(y) == 0 {
+ return z.setWord(1)
+ }
+ // y > 0
+
+ if len(m) != 0 {
+ // We likely end up being as long as the modulus.
+ z = z.make(len(m))
+ }
+ z = z.set(x)
+
+ // If the base is non-trivial and the exponent is large, we use
+ // 4-bit, windowed exponentiation. This involves precomputing 14 values
+ // (x^2...x^15) but then reduces the number of multiply-reduces by a
+ // third. Even for a 32-bit exponent, this reduces the number of
+ // operations.
+ if len(x) > 1 && len(y) > 1 && len(m) > 0 {
+ return z.expNNWindowed(x, y, m)
+ }
+
+ v := y[len(y)-1] // v > 0 because y is normalized and y > 0
+ shift := leadingZeros(v) + 1
+ v <<= shift
+ var q nat
+
+ const mask = 1 << (_W - 1)
+
+ // We walk through the bits of the exponent one by one. Each time we
+ // see a bit, we square, thus doubling the power. If the bit is a one,
+ // we also multiply by x, thus adding one to the power.
+
+ w := _W - int(shift)
+ // zz and r are used to avoid allocating in mul and div as
+ // otherwise the arguments would alias.
+ var zz, r nat
+ for j := 0; j < w; j++ {
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+
+ for i := len(y) - 2; i >= 0; i-- {
+ v = y[i]
+
+ for j := 0; j < _W; j++ {
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+
+ if v&mask != 0 {
+ zz = zz.mul(z, x)
+ zz, z = z, zz
+ }
+
+ if len(m) != 0 {
+ zz, r = zz.div(r, z, m)
+ zz, r, q, z = q, z, zz, r
+ }
+
+ v <<= 1
+ }
+ }
+
+ return z.norm()
+}
+
+// expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
+func (z nat) expNNWindowed(x, y, m nat) nat {
+ // zz and r are used to avoid allocating in mul and div as otherwise
+ // the arguments would alias.
+ var zz, r nat
+
+ const n = 4
+ // powers[i] contains x^i.
+ var powers [1 << n]nat
+ powers[0] = natOne
+ powers[1] = x
+ for i := 2; i < 1<<n; i += 2 {
+ p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
+ *p = p.mul(*p2, *p2)
+ zz, r = zz.div(r, *p, m)
+ *p, r = r, *p
+ *p1 = p1.mul(*p, x)
+ zz, r = zz.div(r, *p1, m)
+ *p1, r = r, *p1
+ }
+
+ z = z.setWord(1)
+
+ for i := len(y) - 1; i >= 0; i-- {
+ yi := y[i]
+ for j := 0; j < _W; j += n {
+ if i != len(y)-1 || j != 0 {
+ // Unrolled loop for significant performance
+ // gain. Use go test -bench=".*" in crypto/rsa
+ // to check performance before making changes.
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ zz = zz.mul(z, z)
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+ }
+
+ zz = zz.mul(z, powers[yi>>(_W-n)])
+ zz, z = z, zz
+ zz, r = zz.div(r, z, m)
+ z, r = r, z
+
+ yi <<= n
+ }
+ }
+
+ return z.norm()
+}
+
+// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
+// If it returns true, n is prime with probability 1 - 1/4^reps.
+// If it returns false, n is not prime.
+func (n nat) probablyPrime(reps int) bool {
+ if len(n) == 0 {
+ return false
+ }
+
+ if len(n) == 1 {
+ if n[0] < 2 {
+ return false
+ }
+
+ if n[0]%2 == 0 {
+ return n[0] == 2
+ }
+
+ // We have to exclude these cases because we reject all
+ // multiples of these numbers below.
+ switch n[0] {
+ case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53:
+ return true
+ }
+ }
+
+ if n[0]&1 == 0 {
+ return false // n is even
+ }
+
+ const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
+ const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
+
+ var r Word
+ switch _W {
+ case 32:
+ r = n.modW(primesProduct32)
+ case 64:
+ r = n.modW(primesProduct64 & _M)
+ default:
+ panic("Unknown word size")
+ }
+
+ if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 ||
+ r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 {
+ return false
+ }
+
+ if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 ||
+ r%43 == 0 || r%47 == 0 || r%53 == 0) {
+ return false
+ }
+
+ nm1 := nat(nil).sub(n, natOne)
+ // determine q, k such that nm1 = q << k
+ k := nm1.trailingZeroBits()
+ q := nat(nil).shr(nm1, k)
+
+ nm3 := nat(nil).sub(nm1, natTwo)
+ rand := rand.New(rand.NewSource(int64(n[0])))
+
+ var x, y, quotient nat
+ nm3Len := nm3.bitLen()
+
+NextRandom:
+ for i := 0; i < reps; i++ {
+ x = x.random(rand, nm3, nm3Len)
+ x = x.add(x, natTwo)
+ y = y.expNN(x, q, n)
+ if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
+ continue
+ }
+ for j := uint(1); j < k; j++ {
+ y = y.mul(y, y)
+ quotient, y = quotient.div(y, y, n)
+ if y.cmp(nm1) == 0 {
+ continue NextRandom
+ }
+ if y.cmp(natOne) == 0 {
+ return false
+ }
+ }
+ return false
+ }
+
+ return true
+}
+
+// bytes writes the value of z into buf using big-endian encoding.
+// len(buf) must be >= len(z)*_S. The value of z is encoded in the
+// slice buf[i:]. The number i of unused bytes at the beginning of
+// buf is returned as result.
+func (z nat) bytes(buf []byte) (i int) {
+ i = len(buf)
+ for _, d := range z {
+ for j := 0; j < _S; j++ {
+ i--
+ buf[i] = byte(d)
+ d >>= 8
+ }
+ }
+
+ for i < len(buf) && buf[i] == 0 {
+ i++
+ }
+
+ return
+}
+
+// setBytes interprets buf as the bytes of a big-endian unsigned
+// integer, sets z to that value, and returns z.
+func (z nat) setBytes(buf []byte) nat {
+ z = z.make((len(buf) + _S - 1) / _S)
+
+ k := 0
+ s := uint(0)
+ var d Word
+ for i := len(buf); i > 0; i-- {
+ d |= Word(buf[i-1]) << s
+ if s += 8; s == _S*8 {
+ z[k] = d
+ k++
+ s = 0
+ d = 0
+ }
+ }
+ if k < len(z) {
+ z[k] = d
+ }
+
+ return z.norm()
+}
--- /dev/null
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "runtime"
+ "strings"
+ "testing"
+)
+
+var cmpTests = []struct {
+ x, y nat
+ r int
+}{
+ {nil, nil, 0},
+ {nil, nat(nil), 0},
+ {nat(nil), nil, 0},
+ {nat(nil), nat(nil), 0},
+ {nat{0}, nat{0}, 0},
+ {nat{0}, nat{1}, -1},
+ {nat{1}, nat{0}, 1},
+ {nat{1}, nat{1}, 0},
+ {nat{0, _M}, nat{1}, 1},
+ {nat{1}, nat{0, _M}, -1},
+ {nat{1, _M}, nat{0, _M}, 1},
+ {nat{0, _M}, nat{1, _M}, -1},
+ {nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
+ {nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
+}
+
+func TestCmp(t *testing.T) {
+ for i, a := range cmpTests {
+ r := a.x.cmp(a.y)
+ if r != a.r {
+ t.Errorf("#%d got r = %v; want %v", i, r, a.r)
+ }
+ }
+}
+
+type funNN func(z, x, y nat) nat
+type argNN struct {
+ z, x, y nat
+}
+
+var sumNN = []argNN{
+ {},
+ {nat{1}, nil, nat{1}},
+ {nat{1111111110}, nat{123456789}, nat{987654321}},
+ {nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
+ {nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
+ {nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
+}
+
+var prodNN = []argNN{
+ {},
+ {nil, nil, nil},
+ {nil, nat{991}, nil},
+ {nat{991}, nat{991}, nat{1}},
+ {nat{991 * 991}, nat{991}, nat{991}},
+ {nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
+ {nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
+ {nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
+ // 3^100 * 3^28 = 3^128
+ {
+ natFromString("11790184577738583171520872861412518665678211592275841109096961"),
+ natFromString("515377520732011331036461129765621272702107522001"),
+ natFromString("22876792454961"),
+ },
+ // z = 111....1 (70000 digits)
+ // x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
+ // y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
+ {
+ natFromString(strings.Repeat("1", 70000)),
+ natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
+ natFromString(strings.Repeat("1", 700)),
+ },
+ // z = 111....1 (20000 digits)
+ // x = 10^10000 + 1
+ // y = 111....1 (10000 digits)
+ {
+ natFromString(strings.Repeat("1", 20000)),
+ natFromString("1" + strings.Repeat("0", 9999) + "1"),
+ natFromString(strings.Repeat("1", 10000)),
+ },
+}
+
+func natFromString(s string) nat {
+ x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
+ if err != nil {
+ panic(err)
+ }
+ return x
+}
+
+func TestSet(t *testing.T) {
+ for _, a := range sumNN {
+ z := nat(nil).set(a.z)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("got z = %v; want %v", z, a.z)
+ }
+ }
+}
+
+func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
+ z := f(nil, a.x, a.y)
+ if z.cmp(a.z) != 0 {
+ t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
+ }
+}
+
+func TestFunNN(t *testing.T) {
+ for _, a := range sumNN {
+ arg := a
+ testFunNN(t, "add", nat.add, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "add symmetric", nat.add, arg)
+
+ arg = argNN{a.x, a.z, a.y}
+ testFunNN(t, "sub", nat.sub, arg)
+
+ arg = argNN{a.y, a.z, a.x}
+ testFunNN(t, "sub symmetric", nat.sub, arg)
+ }
+
+ for _, a := range prodNN {
+ arg := a
+ testFunNN(t, "mul", nat.mul, arg)
+
+ arg = argNN{a.z, a.y, a.x}
+ testFunNN(t, "mul symmetric", nat.mul, arg)
+ }
+}
+
+var mulRangesN = []struct {
+ a, b uint64
+ prod string
+}{
+ {0, 0, "0"},
+ {1, 1, "1"},
+ {1, 2, "2"},
+ {1, 3, "6"},
+ {10, 10, "10"},
+ {0, 100, "0"},
+ {0, 1e9, "0"},
+ {1, 0, "1"}, // empty range
+ {100, 1, "1"}, // empty range
+ {1, 10, "3628800"}, // 10!
+ {1, 20, "2432902008176640000"}, // 20!
+ {1, 100,
+ "933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "51185210916864000000000000000000000000", // 100!
+ },
+}
+
+func TestMulRangeN(t *testing.T) {
+ for i, r := range mulRangesN {
+ prod := nat(nil).mulRange(r.a, r.b).decimalString()
+ if prod != r.prod {
+ t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+// allocBytes returns the number of bytes allocated by invoking f.
+func allocBytes(f func()) uint64 {
+ var stats runtime.MemStats
+ runtime.ReadMemStats(&stats)
+ t := stats.TotalAlloc
+ f()
+ runtime.ReadMemStats(&stats)
+ return stats.TotalAlloc - t
+}
+
+// TestMulUnbalanced tests that multiplying numbers of different lengths
+// does not cause deep recursion and in turn allocate too much memory.
+// Test case for issue 3807.
+func TestMulUnbalanced(t *testing.T) {
+ defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
+ x := rndNat(50000)
+ y := rndNat(40)
+ allocSize := allocBytes(func() {
+ nat(nil).mul(x, y)
+ })
+ inputSize := uint64(len(x)+len(y)) * _S
+ if ratio := allocSize / uint64(inputSize); ratio > 10 {
+ t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
+ }
+}
+
+func rndNat(n int) nat {
+ return nat(rndV(n)).norm()
+}
+
+func BenchmarkMul(b *testing.B) {
+ mulx := rndNat(1e4)
+ muly := rndNat(1e4)
+ b.ResetTimer()
+ for i := 0; i < b.N; i++ {
+ var z nat
+ z.mul(mulx, muly)
+ }
+}
+
+func TestLeadingZeros(t *testing.T) {
+ var x Word = _B >> 1
+ for i := 0; i <= _W; i++ {
+ if int(leadingZeros(x)) != i {
+ t.Errorf("failed at %x: got %d want %d", x, leadingZeros(x), i)
+ }
+ x >>= 1
+ }
+}
+
+type shiftTest struct {
+ in nat
+ shift uint
+ out nat
+}
+
+var leftShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, natTwo},
+ {nat{1 << (_W - 1)}, 1, nat{0}},
+ {nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
+}
+
+func TestShiftLeft(t *testing.T) {
+ for i, test := range leftShiftTests {
+ var z nat
+ z = z.shl(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+var rightShiftTests = []shiftTest{
+ {nil, 0, nil},
+ {nil, 1, nil},
+ {natOne, 0, natOne},
+ {natOne, 1, nil},
+ {natTwo, 1, natOne},
+ {nat{0, 1}, 1, nat{1 << (_W - 1)}},
+ {nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
+}
+
+func TestShiftRight(t *testing.T) {
+ for i, test := range rightShiftTests {
+ var z nat
+ z = z.shr(test.in, test.shift)
+ for j, d := range test.out {
+ if j >= len(z) || z[j] != d {
+ t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+ break
+ }
+ }
+ }
+}
+
+type modWTest struct {
+ in string
+ dividend string
+ out string
+}
+
+var modWTests32 = []modWTest{
+ {"23492635982634928349238759823742", "252341", "220170"},
+}
+
+var modWTests64 = []modWTest{
+ {"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
+}
+
+func runModWTests(t *testing.T, tests []modWTest) {
+ for i, test := range tests {
+ in, _ := new(Int).SetString(test.in, 10)
+ d, _ := new(Int).SetString(test.dividend, 10)
+ out, _ := new(Int).SetString(test.out, 10)
+
+ r := in.abs.modW(d.abs[0])
+ if r != out.abs[0] {
+ t.Errorf("#%d failed: got %d want %s", i, r, out)
+ }
+ }
+}
+
+func TestModW(t *testing.T) {
+ if _W >= 32 {
+ runModWTests(t, modWTests32)
+ }
+ if _W >= 64 {
+ runModWTests(t, modWTests64)
+ }
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+ // test 0 case explicitly
+ if n := trailingZeroBits(0); n != 0 {
+ t.Errorf("got trailingZeroBits(0) = %d; want 0", n)
+ }
+
+ x := Word(1)
+ for i := uint(0); i < _W; i++ {
+ n := trailingZeroBits(x)
+ if n != i {
+ t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
+ }
+ x <<= 1
+ }
+
+ // test 0 case explicitly
+ if n := nat(nil).trailingZeroBits(); n != 0 {
+ t.Errorf("got nat(nil).trailingZeroBits() = %d; want 0", n)
+ }
+
+ y := nat(nil).set(natOne)
+ for i := uint(0); i <= 3*_W; i++ {
+ n := y.trailingZeroBits()
+ if n != i {
+ t.Errorf("got 0x%s.trailingZeroBits() = %d; want %d", y.hexString(), n, i)
+ }
+ y = y.shl(y, 1)
+ }
+}
+
+var expNNTests = []struct {
+ x, y, m string
+ out string
+}{
+ {"0", "0", "0", "1"},
+ {"0", "0", "1", "0"},
+ {"1", "1", "1", "0"},
+ {"2", "1", "1", "0"},
+ {"2", "2", "1", "0"},
+ {"10", "100000000000", "1", "0"},
+ {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+ {"0x8000000000000000", "2", "6719", "4944"},
+ {"0x8000000000000000", "3", "6719", "5447"},
+ {"0x8000000000000000", "1000", "6719", "1603"},
+ {"0x8000000000000000", "1000000", "6719", "3199"},
+ {
+ "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+ "298472983472983471903246121093472394872319615612417471234712061",
+ "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+ "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+ },
+}
+
+func TestExpNN(t *testing.T) {
+ for i, test := range expNNTests {
+ x := natFromString(test.x)
+ y := natFromString(test.y)
+ out := natFromString(test.out)
+
+ var m nat
+ if len(test.m) > 0 {
+ m = natFromString(test.m)
+ }
+
+ z := nat(nil).expNN(x, y, m)
+ if z.cmp(out) != 0 {
+ t.Errorf("#%d got %s want %s", i, z.decimalString(), out.decimalString())
+ }
+ }
+}
+
+func ExpHelper(b *testing.B, x, y Word) {
+ var z nat
+ for i := 0; i < b.N; i++ {
+ z.expWW(x, y)
+ }
+}
+
+func BenchmarkExp3Power0x10(b *testing.B) { ExpHelper(b, 3, 0x10) }
+func BenchmarkExp3Power0x40(b *testing.B) { ExpHelper(b, 3, 0x40) }
+func BenchmarkExp3Power0x100(b *testing.B) { ExpHelper(b, 3, 0x100) }
+func BenchmarkExp3Power0x400(b *testing.B) { ExpHelper(b, 3, 0x400) }
+func BenchmarkExp3Power0x1000(b *testing.B) { ExpHelper(b, 3, 0x1000) }
+func BenchmarkExp3Power0x4000(b *testing.B) { ExpHelper(b, 3, 0x4000) }
+func BenchmarkExp3Power0x10000(b *testing.B) { ExpHelper(b, 3, 0x10000) }
+func BenchmarkExp3Power0x40000(b *testing.B) { ExpHelper(b, 3, 0x40000) }
+func BenchmarkExp3Power0x100000(b *testing.B) { ExpHelper(b, 3, 0x100000) }
+func BenchmarkExp3Power0x400000(b *testing.B) { ExpHelper(b, 3, 0x400000) }
+
+func fibo(n int) nat {
+ switch n {
+ case 0:
+ return nil
+ case 1:
+ return nat{1}
+ }
+ f0 := fibo(0)
+ f1 := fibo(1)
+ var f2 nat
+ for i := 1; i < n; i++ {
+ f2 = f2.add(f0, f1)
+ f0, f1, f2 = f1, f2, f0
+ }
+ return f1
+}
+
+var fiboNums = []string{
+ "0",
+ "55",
+ "6765",
+ "832040",
+ "102334155",
+ "12586269025",
+ "1548008755920",
+ "190392490709135",
+ "23416728348467685",
+ "2880067194370816120",
+ "354224848179261915075",
+}
+
+func TestFibo(t *testing.T) {
+ for i, want := range fiboNums {
+ n := i * 10
+ got := fibo(n).decimalString()
+ if got != want {
+ t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
+ }
+ }
+}
+
+func BenchmarkFibo(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ fibo(1e0)
+ fibo(1e1)
+ fibo(1e2)
+ fibo(1e3)
+ fibo(1e4)
+ fibo(1e5)
+ }
+}
+
+var bitTests = []struct {
+ x string
+ i uint
+ want uint
+}{
+ {"0", 0, 0},
+ {"0", 1, 0},
+ {"0", 1000, 0},
+
+ {"0x1", 0, 1},
+ {"0x10", 0, 0},
+ {"0x10", 3, 0},
+ {"0x10", 4, 1},
+ {"0x10", 5, 0},
+
+ {"0x8000000000000000", 62, 0},
+ {"0x8000000000000000", 63, 1},
+ {"0x8000000000000000", 64, 0},
+
+ {"0x3" + strings.Repeat("0", 32), 127, 0},
+ {"0x3" + strings.Repeat("0", 32), 128, 1},
+ {"0x3" + strings.Repeat("0", 32), 129, 1},
+ {"0x3" + strings.Repeat("0", 32), 130, 0},
+}
+
+func TestBit(t *testing.T) {
+ for i, test := range bitTests {
+ x := natFromString(test.x)
+ if got := x.bit(test.i); got != test.want {
+ t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+ }
+ }
+}
+
+var stickyTests = []struct {
+ x string
+ i uint
+ want uint
+}{
+ {"0", 0, 0},
+ {"0", 1, 0},
+ {"0", 1000, 0},
+
+ {"0x1", 0, 0},
+ {"0x1", 1, 1},
+
+ {"0x1350", 0, 0},
+ {"0x1350", 4, 0},
+ {"0x1350", 5, 1},
+
+ {"0x8000000000000000", 63, 0},
+ {"0x8000000000000000", 64, 1},
+
+ {"0x1" + strings.Repeat("0", 100), 400, 0},
+ {"0x1" + strings.Repeat("0", 100), 401, 1},
+}
+
+func TestSticky(t *testing.T) {
+ for i, test := range stickyTests {
+ x := natFromString(test.x)
+ if got := x.sticky(test.i); got != test.want {
+ t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+ }
+ if test.want == 1 {
+ // all subsequent i's should also return 1
+ for d := uint(1); d <= 3; d++ {
+ if got := x.sticky(test.i + d); got != 1 {
+ t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
+ }
+ }
+ }
+ }
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements nat-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "math"
+ "sync"
+)
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 'z' - 'a' + 10 + 1
+
+// maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M.
+// For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word.
+// In other words, at most n digits in base b fit into a Word.
+// TODO(gri) replace this with a table, generated at build time.
+func maxPow(b Word) (p Word, n int) {
+ p, n = b, 1 // assuming b <= _M
+ for max := _M / b; p <= max; {
+ // p == b**n && p <= max
+ p *= b
+ n++
+ }
+ // p == b**n && p <= _M
+ return
+}
+
+// pow returns x**n for n > 0, and 1 otherwise.
+func pow(x Word, n int) (p Word) {
+ // n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1
+ // thus x**n == product of x**(2**i) for all i where bi == 1
+ // (Russian Peasant Method for exponentiation)
+ p = 1
+ for n > 0 {
+ if n&1 != 0 {
+ p *= x
+ }
+ x *= x
+ n >>= 1
+ }
+ return
+}
+
+// scan scans the number corresponding to the longest possible prefix
+// from r representing an unsigned number in a given conversion base.
+// It returns the corresponding natural number res, the actual base b,
+// a digit count, and a read or syntax error err, if any.
+//
+// number = [ prefix ] mantissa .
+// prefix = "0" [ "x" | "X" | "b" | "B" ] .
+// mantissa = digits | digits "." [ digits ] | "." digits .
+// digits = digit { digit } .
+// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
+//
+// Unless fracOk is set, the base argument must be 0 or a value between
+// 2 and MaxBase. If fracOk is set, the base argument must be one of
+// 0, 2, 10, or 16. Providing an invalid base argument leads to a run-
+// time panic.
+//
+// For base 0, the number prefix determines the actual base: A prefix of
+// ``0x'' or ``0X'' selects base 16; if fracOk is not set, the ``0'' prefix
+// selects base 8, and a ``0b'' or ``0B'' prefix selects base 2. Otherwise
+// the selected base is 10 and no prefix is accepted.
+//
+// If fracOk is set, an octal prefix is ignored (a leading ``0'' simply
+// stands for a zero digit), and a period followed by a fractional part
+// is permitted. The result value is computed as if there were no period
+// present; and the count value is used to determine the fractional part.
+//
+// A result digit count > 0 corresponds to the number of (non-prefix) digits
+// parsed. A digit count <= 0 indicates the presence of a period (if fracOk
+// is set, only), and -count is the number of fractional digits found.
+// In this case, the actual value of the scanned number is res * b**count.
+//
+func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) {
+ // reject illegal bases
+ baseOk := base == 0 ||
+ !fracOk && 2 <= base && base <= MaxBase ||
+ fracOk && (base == 2 || base == 10 || base == 16)
+ if !baseOk {
+ panic(fmt.Sprintf("illegal number base %d", base))
+ }
+
+ // one char look-ahead
+ ch, err := r.ReadByte()
+ if err != nil {
+ return
+ }
+
+ // determine actual base
+ b = base
+ if base == 0 {
+ // actual base is 10 unless there's a base prefix
+ b = 10
+ if ch == '0' {
+ count = 1
+ switch ch, err = r.ReadByte(); err {
+ case nil:
+ // possibly one of 0x, 0X, 0b, 0B
+ if !fracOk {
+ b = 8
+ }
+ switch ch {
+ case 'x', 'X':
+ b = 16
+ case 'b', 'B':
+ b = 2
+ }
+ switch b {
+ case 16, 2:
+ count = 0 // prefix is not counted
+ if ch, err = r.ReadByte(); err != nil {
+ // io.EOF is also an error in this case
+ return
+ }
+ case 8:
+ count = 0 // prefix is not counted
+ }
+ case io.EOF:
+ // input is "0"
+ res = z[:0]
+ err = nil
+ return
+ default:
+ // read error
+ return
+ }
+ }
+ }
+
+ // convert string
+ // Algorithm: Collect digits in groups of at most n digits in di
+ // and then use mulAddWW for every such group to add them to the
+ // result.
+ z = z[:0]
+ b1 := Word(b)
+ bn, n := maxPow(b1) // at most n digits in base b1 fit into Word
+ di := Word(0) // 0 <= di < b1**i < bn
+ i := 0 // 0 <= i < n
+ dp := -1 // position of decimal point
+ for {
+ if fracOk && ch == '.' {
+ fracOk = false
+ dp = count
+ // advance
+ if ch, err = r.ReadByte(); err != nil {
+ if err == io.EOF {
+ err = nil
+ break
+ }
+ return
+ }
+ }
+
+ // convert rune into digit value d1
+ var d1 Word
+ switch {
+ case '0' <= ch && ch <= '9':
+ d1 = Word(ch - '0')
+ case 'a' <= ch && ch <= 'z':
+ d1 = Word(ch - 'a' + 10)
+ case 'A' <= ch && ch <= 'Z':
+ d1 = Word(ch - 'A' + 10)
+ default:
+ d1 = MaxBase + 1
+ }
+ if d1 >= b1 {
+ r.UnreadByte() // ch does not belong to number anymore
+ break
+ }
+ count++
+
+ // collect d1 in di
+ di = di*b1 + d1
+ i++
+
+ // if di is "full", add it to the result
+ if i == n {
+ z = z.mulAddWW(z, bn, di)
+ di = 0
+ i = 0
+ }
+
+ // advance
+ if ch, err = r.ReadByte(); err != nil {
+ if err == io.EOF {
+ err = nil
+ break
+ }
+ return
+ }
+ }
+
+ if count == 0 {
+ // no digits found
+ switch {
+ case base == 0 && b == 8:
+ // there was only the octal prefix 0 (possibly followed by digits > 7);
+ // count as one digit and return base 10, not 8
+ count = 1
+ b = 10
+ case base != 0 || b != 8:
+ // there was neither a mantissa digit nor the octal prefix 0
+ err = errors.New("syntax error scanning number")
+ }
+ return
+ }
+ // count > 0
+
+ // add remaining digits to result
+ if i > 0 {
+ z = z.mulAddWW(z, pow(b1, i), di)
+ }
+ res = z.norm()
+
+ // adjust for fraction, if any
+ if dp >= 0 {
+ // 0 <= dp <= count > 0
+ count = dp - count
+ }
+
+ return
+}
+
+// Character sets for string conversion.
+const (
+ lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz"
+ uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+)
+
+// decimalString returns a decimal representation of x.
+// It calls x.string with the charset "0123456789".
+func (x nat) decimalString() string {
+ return x.string(lowercaseDigits[:10])
+}
+
+// hexString returns a hexadecimal representation of x.
+// It calls x.string with the charset "0123456789abcdef".
+func (x nat) hexString() string {
+ return x.string(lowercaseDigits[:16])
+}
+
+// string converts x to a string using digits from a charset; a digit with
+// value d is represented by charset[d]. The conversion base is determined
+// by len(charset), which must be >= 2 and <= 256.
+func (x nat) string(charset string) string {
+ b := Word(len(charset))
+
+ // special cases
+ switch {
+ case b < 2 || b > 256:
+ panic("invalid character set length")
+ case len(x) == 0:
+ return string(charset[0])
+ }
+
+ // allocate buffer for conversion
+ i := int(float64(x.bitLen())/math.Log2(float64(b))) + 1 // off by one at most
+ s := make([]byte, i)
+
+ // convert power of two and non power of two bases separately
+ if b == b&-b {
+ // shift is base-b digit size in bits
+ shift := trailingZeroBits(b) // shift > 0 because b >= 2
+ mask := Word(1)<<shift - 1
+ w := x[0]
+ nbits := uint(_W) // number of unprocessed bits in w
+
+ // convert less-significant words
+ for k := 1; k < len(x); k++ {
+ // convert full digits
+ for nbits >= shift {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ // convert any partial leading digit and advance to next word
+ if nbits == 0 {
+ // no partial digit remaining, just advance
+ w = x[k]
+ nbits = _W
+ } else {
+ // partial digit in current (k-1) and next (k) word
+ w |= x[k] << nbits
+ i--
+ s[i] = charset[w&mask]
+
+ // advance
+ w = x[k] >> (shift - nbits)
+ nbits = _W - (shift - nbits)
+ }
+ }
+
+ // convert digits of most-significant word (omit leading zeros)
+ for nbits >= 0 && w != 0 {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ } else {
+ bb, ndigits := maxPow(Word(b))
+
+ // construct table of successive squares of bb*leafSize to use in subdivisions
+ // result (table != nil) <=> (len(x) > leafSize > 0)
+ table := divisors(len(x), b, ndigits, bb)
+
+ // preserve x, create local copy for use by convertWords
+ q := nat(nil).set(x)
+
+ // convert q to string s in base b
+ q.convertWords(s, charset, b, ndigits, bb, table)
+
+ // strip leading zeros
+ // (x != 0; thus s must contain at least one non-zero digit
+ // and the loop will terminate)
+ i = 0
+ for zero := charset[0]; s[i] == zero; {
+ i++
+ }
+ }
+
+ return string(s[i:])
+}
+
+// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
+// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
+// repeated nat/Word division.
+//
+// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
+// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
+// Recursive conversion divides q by its approximate square root, yielding two parts, each half
+// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
+// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
+// is made better by splitting the subblocks recursively. Best is to split blocks until one more
+// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
+// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
+// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
+// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
+// specific hardware.
+//
+func (q nat) convertWords(s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) {
+ // split larger blocks recursively
+ if table != nil {
+ // len(q) > leafSize > 0
+ var r nat
+ index := len(table) - 1
+ for len(q) > leafSize {
+ // find divisor close to sqrt(q) if possible, but in any case < q
+ maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
+ minLength := maxLength >> 1 // ~= log2 sqrt(q)
+ for index > 0 && table[index-1].nbits > minLength {
+ index-- // desired
+ }
+ if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
+ index--
+ if index < 0 {
+ panic("internal inconsistency")
+ }
+ }
+
+ // split q into the two digit number (q'*bbb + r) to form independent subblocks
+ q, r = q.div(r, q, table[index].bbb)
+
+ // convert subblocks and collect results in s[:h] and s[h:]
+ h := len(s) - table[index].ndigits
+ r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
+ s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
+ }
+ }
+
+ // having split any large blocks now process the remaining (small) block iteratively
+ i := len(s)
+ var r Word
+ if b == 10 {
+ // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ // avoid % computation since r%10 == r - int(r/10)*10;
+ // this appears to be faster for BenchmarkString10000Base10
+ // and smaller strings (but a bit slower for larger ones)
+ t := r / 10
+ s[i] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
+ r = t
+ }
+ }
+ } else {
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ s[i] = charset[r%b]
+ r /= b
+ }
+ }
+ }
+
+ // prepend high-order zeroes
+ zero := charset[0]
+ for i > 0 { // while need more leading zeroes
+ i--
+ s[i] = zero
+ }
+}
+
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
+// Benchmark and configure leafSize using: go test -bench="Leaf"
+// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
+// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
+var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
+
+type divisor struct {
+ bbb nat // divisor
+ nbits int // bit length of divisor (discounting leading zeroes) ~= log2(bbb)
+ ndigits int // digit length of divisor in terms of output base digits
+}
+
+var cacheBase10 struct {
+ sync.Mutex
+ table [64]divisor // cached divisors for base 10
+}
+
+// expWW computes x**y
+func (z nat) expWW(x, y Word) nat {
+ return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
+}
+
+// construct table of powers of bb*leafSize to use in subdivisions
+func divisors(m int, b Word, ndigits int, bb Word) []divisor {
+ // only compute table when recursive conversion is enabled and x is large
+ if leafSize == 0 || m <= leafSize {
+ return nil
+ }
+
+ // determine k where (bb**leafSize)**(2**k) >= sqrt(x)
+ k := 1
+ for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
+ k++
+ }
+
+ // reuse and extend existing table of divisors or create new table as appropriate
+ var table []divisor // for b == 10, table overlaps with cacheBase10.table
+ if b == 10 {
+ cacheBase10.Lock()
+ table = cacheBase10.table[0:k] // reuse old table for this conversion
+ } else {
+ table = make([]divisor, k) // create new table for this conversion
+ }
+
+ // extend table
+ if table[k-1].ndigits == 0 {
+ // add new entries as needed
+ var larger nat
+ for i := 0; i < k; i++ {
+ if table[i].ndigits == 0 {
+ if i == 0 {
+ table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
+ table[0].ndigits = ndigits * leafSize
+ } else {
+ table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
+ table[i].ndigits = 2 * table[i-1].ndigits
+ }
+
+ // optimization: exploit aggregated extra bits in macro blocks
+ larger = nat(nil).set(table[i].bbb)
+ for mulAddVWW(larger, larger, b, 0) == 0 {
+ table[i].bbb = table[i].bbb.set(larger)
+ table[i].ndigits++
+ }
+
+ table[i].nbits = table[i].bbb.bitLen()
+ }
+ }
+ }
+
+ if b == 10 {
+ cacheBase10.Unlock()
+ }
+
+ return table
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "io"
+ "strings"
+ "testing"
+)
+
+func toString(x nat, charset string) string {
+ base := len(charset)
+
+ // special cases
+ switch {
+ case base < 2:
+ panic("illegal base")
+ case len(x) == 0:
+ return string(charset[0])
+ }
+
+ // allocate buffer for conversion
+ i := x.bitLen()/log2(Word(base)) + 1 // +1: round up
+ s := make([]byte, i)
+
+ // don't destroy x
+ q := nat(nil).set(x)
+
+ // convert
+ for len(q) > 0 {
+ i--
+ var r Word
+ q, r = q.divW(q, Word(base))
+ s[i] = charset[r]
+ }
+
+ return string(s[i:])
+}
+
+var strTests = []struct {
+ x nat // nat value to be converted
+ c string // conversion charset
+ s string // expected result
+}{
+ {nil, "01", "0"},
+ {nat{1}, "01", "1"},
+ {nat{0xc5}, "01", "11000101"},
+ {nat{03271}, lowercaseDigits[:8], "3271"},
+ {nat{10}, lowercaseDigits[:10], "10"},
+ {nat{1234567890}, uppercaseDigits[:10], "1234567890"},
+ {nat{0xdeadbeef}, lowercaseDigits[:16], "deadbeef"},
+ {nat{0xdeadbeef}, uppercaseDigits[:16], "DEADBEEF"},
+ {nat{0x229be7}, lowercaseDigits[:17], "1a2b3c"},
+ {nat{0x309663e6}, uppercaseDigits[:32], "O9COV6"},
+}
+
+func TestString(t *testing.T) {
+ // test invalid character set explicitly
+ var panicStr string
+ func() {
+ defer func() {
+ panicStr = recover().(string)
+ }()
+ natOne.string("0")
+ }()
+ if panicStr != "invalid character set length" {
+ t.Errorf("expected panic for invalid character set")
+ }
+
+ for _, a := range strTests {
+ s := a.x.string(a.c)
+ if s != a.s {
+ t.Errorf("string%+v\n\tgot s = %s; want %s", a, s, a.s)
+ }
+
+ x, b, _, err := nat(nil).scan(strings.NewReader(a.s), len(a.c), false)
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != len(a.c) {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, len(a.c))
+ }
+ if err != nil {
+ t.Errorf("scan%+v\n\tgot error = %s", a, err)
+ }
+ }
+}
+
+var natScanTests = []struct {
+ s string // string to be scanned
+ base int // input base
+ frac bool // fraction ok
+ x nat // expected nat
+ b int // expected base
+ count int // expected digit count
+ ok bool // expected success
+ next rune // next character (or 0, if at EOF)
+}{
+ // error: no mantissa
+ {},
+ {s: "?"},
+ {base: 10},
+ {base: 36},
+ {s: "?", base: 10},
+ {s: "0x"},
+ {s: "345", base: 2},
+
+ // error: incorrect use of decimal point
+ {s: ".0"},
+ {s: ".0", base: 10},
+ {s: ".", base: 0},
+ {s: "0x.0"},
+
+ // no errors
+ {"0", 0, false, nil, 10, 1, true, 0},
+ {"0", 10, false, nil, 10, 1, true, 0},
+ {"0", 36, false, nil, 36, 1, true, 0},
+ {"1", 0, false, nat{1}, 10, 1, true, 0},
+ {"1", 10, false, nat{1}, 10, 1, true, 0},
+ {"0 ", 0, false, nil, 10, 1, true, ' '},
+ {"08", 0, false, nil, 10, 1, true, '8'},
+ {"08", 10, false, nat{8}, 10, 2, true, 0},
+ {"018", 0, false, nat{1}, 8, 1, true, '8'},
+ {"0b1", 0, false, nat{1}, 2, 1, true, 0},
+ {"0b11000101", 0, false, nat{0xc5}, 2, 8, true, 0},
+ {"03271", 0, false, nat{03271}, 8, 4, true, 0},
+ {"10ab", 0, false, nat{10}, 10, 2, true, 'a'},
+ {"1234567890", 0, false, nat{1234567890}, 10, 10, true, 0},
+ {"xyz", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, true, 0},
+ {"xyz?", 36, false, nat{(33*36+34)*36 + 35}, 36, 3, true, '?'},
+ {"0x", 16, false, nil, 16, 1, true, 'x'},
+ {"0xdeadbeef", 0, false, nat{0xdeadbeef}, 16, 8, true, 0},
+ {"0XDEADBEEF", 0, false, nat{0xdeadbeef}, 16, 8, true, 0},
+
+ // no errors, decimal point
+ {"0.", 0, false, nil, 10, 1, true, '.'},
+ {"0.", 10, true, nil, 10, 0, true, 0},
+ {"0.1.2", 10, true, nat{1}, 10, -1, true, '.'},
+ {".000", 10, true, nil, 10, -3, true, 0},
+ {"12.3", 10, true, nat{123}, 10, -1, true, 0},
+ {"012.345", 10, true, nat{12345}, 10, -3, true, 0},
+}
+
+func TestScanBase(t *testing.T) {
+ for _, a := range natScanTests {
+ r := strings.NewReader(a.s)
+ x, b, count, err := nat(nil).scan(r, a.base, a.frac)
+ if err == nil && !a.ok {
+ t.Errorf("scan%+v\n\texpected error", a)
+ }
+ if err != nil {
+ if a.ok {
+ t.Errorf("scan%+v\n\tgot error = %s", a, err)
+ }
+ continue
+ }
+ if x.cmp(a.x) != 0 {
+ t.Errorf("scan%+v\n\tgot z = %v; want %v", a, x, a.x)
+ }
+ if b != a.b {
+ t.Errorf("scan%+v\n\tgot b = %d; want %d", a, b, a.base)
+ }
+ if count != a.count {
+ t.Errorf("scan%+v\n\tgot count = %d; want %d", a, count, a.count)
+ }
+ next, _, err := r.ReadRune()
+ if err == io.EOF {
+ next = 0
+ err = nil
+ }
+ if err == nil && next != a.next {
+ t.Errorf("scan%+v\n\tgot next = %q; want %q", a, next, a.next)
+ }
+ }
+}
+
+var pi = "3" +
+ "14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651" +
+ "32823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461" +
+ "28475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920" +
+ "96282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179" +
+ "31051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798" +
+ "60943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901" +
+ "22495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837" +
+ "29780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083" +
+ "81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909" +
+ "21642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151" +
+ "55748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035" +
+ "63707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104" +
+ "75216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992" +
+ "45863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818" +
+ "34797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548" +
+ "16136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179" +
+ "04946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886" +
+ "26945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645" +
+ "99581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745" +
+ "53050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382" +
+ "68683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244" +
+ "13654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767" +
+ "88952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288" +
+ "79710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821" +
+ "68299894872265880485756401427047755513237964145152374623436454285844479526586782105114135473573952311342716610" +
+ "21359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435" +
+ "06430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675" +
+ "14269123974894090718649423196156794520809514655022523160388193014209376213785595663893778708303906979207734672" +
+ "21825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539" +
+ "05796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007" +
+ "23055876317635942187312514712053292819182618612586732157919841484882916447060957527069572209175671167229109816" +
+ "90915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398" +
+ "31501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064" +
+ "20467525907091548141654985946163718027098199430992448895757128289059232332609729971208443357326548938239119325" +
+ "97463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100" +
+ "44929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915" +
+ "44110104468232527162010526522721116603966655730925471105578537634668206531098965269186205647693125705863566201" +
+ "85581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318" +
+ "58676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797" +
+ "27082668306343285878569830523580893306575740679545716377525420211495576158140025012622859413021647155097925923" +
+ "09907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111" +
+ "79042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043" +
+ "06332175182979866223717215916077166925474873898665494945011465406284336639379003976926567214638530673609657120" +
+ "91807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862" +
+ "94726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939" +
+ "78054193414473774418426312986080998886874132604721569516239658645730216315981931951673538129741677294786724229" +
+ "24654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001" +
+ "59377647165122893578601588161755782973523344604281512627203734314653197777416031990665541876397929334419521541" +
+ "34189948544473456738316249934191318148092777710386387734317720754565453220777092120190516609628049092636019759" +
+ "88281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267" +
+ "94561275318134078330336254232783944975382437205835311477119926063813346776879695970309833913077109870408591337"
+
+// Test case for BenchmarkScanPi.
+func TestScanPi(t *testing.T) {
+ var x nat
+ z, _, _, err := x.scan(strings.NewReader(pi), 10, false)
+ if err != nil {
+ t.Errorf("scanning pi: %s", err)
+ }
+ if s := z.decimalString(); s != pi {
+ t.Errorf("scanning pi: got %s", s)
+ }
+}
+
+func TestScanPiParallel(t *testing.T) {
+ const n = 2
+ c := make(chan int)
+ for i := 0; i < n; i++ {
+ go func() {
+ TestScanPi(t)
+ c <- 0
+ }()
+ }
+ for i := 0; i < n; i++ {
+ <-c
+ }
+}
+
+func BenchmarkScanPi(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var x nat
+ x.scan(strings.NewReader(pi), 10, false)
+ }
+}
+
+func BenchmarkStringPiParallel(b *testing.B) {
+ var x nat
+ x, _, _, _ = x.scan(strings.NewReader(pi), 0, false)
+ if x.decimalString() != pi {
+ panic("benchmark incorrect: conversion failed")
+ }
+ b.RunParallel(func(pb *testing.PB) {
+ for pb.Next() {
+ x.decimalString()
+ }
+ })
+}
+
+func BenchmarkScan10Base2(b *testing.B) { ScanHelper(b, 2, 10, 10) }
+func BenchmarkScan100Base2(b *testing.B) { ScanHelper(b, 2, 10, 100) }
+func BenchmarkScan1000Base2(b *testing.B) { ScanHelper(b, 2, 10, 1000) }
+func BenchmarkScan10000Base2(b *testing.B) { ScanHelper(b, 2, 10, 10000) }
+func BenchmarkScan100000Base2(b *testing.B) { ScanHelper(b, 2, 10, 100000) }
+
+func BenchmarkScan10Base8(b *testing.B) { ScanHelper(b, 8, 10, 10) }
+func BenchmarkScan100Base8(b *testing.B) { ScanHelper(b, 8, 10, 100) }
+func BenchmarkScan1000Base8(b *testing.B) { ScanHelper(b, 8, 10, 1000) }
+func BenchmarkScan10000Base8(b *testing.B) { ScanHelper(b, 8, 10, 10000) }
+func BenchmarkScan100000Base8(b *testing.B) { ScanHelper(b, 8, 10, 100000) }
+
+func BenchmarkScan10Base10(b *testing.B) { ScanHelper(b, 10, 10, 10) }
+func BenchmarkScan100Base10(b *testing.B) { ScanHelper(b, 10, 10, 100) }
+func BenchmarkScan1000Base10(b *testing.B) { ScanHelper(b, 10, 10, 1000) }
+func BenchmarkScan10000Base10(b *testing.B) { ScanHelper(b, 10, 10, 10000) }
+func BenchmarkScan100000Base10(b *testing.B) { ScanHelper(b, 10, 10, 100000) }
+
+func BenchmarkScan10Base16(b *testing.B) { ScanHelper(b, 16, 10, 10) }
+func BenchmarkScan100Base16(b *testing.B) { ScanHelper(b, 16, 10, 100) }
+func BenchmarkScan1000Base16(b *testing.B) { ScanHelper(b, 16, 10, 1000) }
+func BenchmarkScan10000Base16(b *testing.B) { ScanHelper(b, 16, 10, 10000) }
+func BenchmarkScan100000Base16(b *testing.B) { ScanHelper(b, 16, 10, 100000) }
+
+func ScanHelper(b *testing.B, base int, x, y Word) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+
+ var s string
+ s = z.string(lowercaseDigits[:base])
+ if t := toString(z, lowercaseDigits[:base]); t != s {
+ b.Fatalf("scanning: got %s; want %s", s, t)
+ }
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ z.scan(strings.NewReader(s), base, false)
+ }
+}
+
+func BenchmarkString10Base2(b *testing.B) { StringHelper(b, 2, 10, 10) }
+func BenchmarkString100Base2(b *testing.B) { StringHelper(b, 2, 10, 100) }
+func BenchmarkString1000Base2(b *testing.B) { StringHelper(b, 2, 10, 1000) }
+func BenchmarkString10000Base2(b *testing.B) { StringHelper(b, 2, 10, 10000) }
+func BenchmarkString100000Base2(b *testing.B) { StringHelper(b, 2, 10, 100000) }
+
+func BenchmarkString10Base8(b *testing.B) { StringHelper(b, 8, 10, 10) }
+func BenchmarkString100Base8(b *testing.B) { StringHelper(b, 8, 10, 100) }
+func BenchmarkString1000Base8(b *testing.B) { StringHelper(b, 8, 10, 1000) }
+func BenchmarkString10000Base8(b *testing.B) { StringHelper(b, 8, 10, 10000) }
+func BenchmarkString100000Base8(b *testing.B) { StringHelper(b, 8, 10, 100000) }
+
+func BenchmarkString10Base10(b *testing.B) { StringHelper(b, 10, 10, 10) }
+func BenchmarkString100Base10(b *testing.B) { StringHelper(b, 10, 10, 100) }
+func BenchmarkString1000Base10(b *testing.B) { StringHelper(b, 10, 10, 1000) }
+func BenchmarkString10000Base10(b *testing.B) { StringHelper(b, 10, 10, 10000) }
+func BenchmarkString100000Base10(b *testing.B) { StringHelper(b, 10, 10, 100000) }
+
+func BenchmarkString10Base16(b *testing.B) { StringHelper(b, 16, 10, 10) }
+func BenchmarkString100Base16(b *testing.B) { StringHelper(b, 16, 10, 100) }
+func BenchmarkString1000Base16(b *testing.B) { StringHelper(b, 16, 10, 1000) }
+func BenchmarkString10000Base16(b *testing.B) { StringHelper(b, 16, 10, 10000) }
+func BenchmarkString100000Base16(b *testing.B) { StringHelper(b, 16, 10, 100000) }
+
+func StringHelper(b *testing.B, base int, x, y Word) {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(x, y)
+ z.string(lowercaseDigits[:base]) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.string(lowercaseDigits[:base])
+ }
+}
+
+func BenchmarkLeafSize0(b *testing.B) { LeafSizeHelper(b, 10, 0) } // test without splitting
+func BenchmarkLeafSize1(b *testing.B) { LeafSizeHelper(b, 10, 1) }
+func BenchmarkLeafSize2(b *testing.B) { LeafSizeHelper(b, 10, 2) }
+func BenchmarkLeafSize3(b *testing.B) { LeafSizeHelper(b, 10, 3) }
+func BenchmarkLeafSize4(b *testing.B) { LeafSizeHelper(b, 10, 4) }
+func BenchmarkLeafSize5(b *testing.B) { LeafSizeHelper(b, 10, 5) }
+func BenchmarkLeafSize6(b *testing.B) { LeafSizeHelper(b, 10, 6) }
+func BenchmarkLeafSize7(b *testing.B) { LeafSizeHelper(b, 10, 7) }
+func BenchmarkLeafSize8(b *testing.B) { LeafSizeHelper(b, 10, 8) }
+func BenchmarkLeafSize9(b *testing.B) { LeafSizeHelper(b, 10, 9) }
+func BenchmarkLeafSize10(b *testing.B) { LeafSizeHelper(b, 10, 10) }
+func BenchmarkLeafSize11(b *testing.B) { LeafSizeHelper(b, 10, 11) }
+func BenchmarkLeafSize12(b *testing.B) { LeafSizeHelper(b, 10, 12) }
+func BenchmarkLeafSize13(b *testing.B) { LeafSizeHelper(b, 10, 13) }
+func BenchmarkLeafSize14(b *testing.B) { LeafSizeHelper(b, 10, 14) }
+func BenchmarkLeafSize15(b *testing.B) { LeafSizeHelper(b, 10, 15) }
+func BenchmarkLeafSize16(b *testing.B) { LeafSizeHelper(b, 10, 16) }
+func BenchmarkLeafSize32(b *testing.B) { LeafSizeHelper(b, 10, 32) } // try some large lengths
+func BenchmarkLeafSize64(b *testing.B) { LeafSizeHelper(b, 10, 64) }
+
+func LeafSizeHelper(b *testing.B, base Word, size int) {
+ b.StopTimer()
+ originalLeafSize := leafSize
+ resetTable(cacheBase10.table[:])
+ leafSize = size
+ b.StartTimer()
+
+ for d := 1; d <= 10000; d *= 10 {
+ b.StopTimer()
+ var z nat
+ z = z.expWW(base, Word(d)) // build target number
+ _ = z.string(lowercaseDigits[:base]) // warm divisor cache
+ b.StartTimer()
+
+ for i := 0; i < b.N; i++ {
+ _ = z.string(lowercaseDigits[:base])
+ }
+ }
+
+ b.StopTimer()
+ resetTable(cacheBase10.table[:])
+ leafSize = originalLeafSize
+ b.StartTimer()
+}
+
+func resetTable(table []divisor) {
+ if table != nil && table[0].bbb != nil {
+ for i := 0; i < len(table); i++ {
+ table[i].bbb = nil
+ table[i].nbits = 0
+ table[i].ndigits = 0
+ }
+ }
+}
+
+func TestStringPowers(t *testing.T) {
+ var b, p Word
+ for b = 2; b <= 16; b++ {
+ for p = 0; p <= 512; p++ {
+ x := nat(nil).expWW(b, p)
+ xs := x.string(lowercaseDigits[:b])
+ xs2 := toString(x, lowercaseDigits[:b])
+ if xs != xs2 {
+ t.Errorf("failed at %d ** %d in base %d: %s != %s", b, p, b, xs, xs2)
+ }
+ }
+ if b >= 3 && testing.Short() {
+ break
+ }
+ }
+}
--- /dev/null
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements multi-precision rational numbers.
+
+package big
+
+import (
+ "encoding/binary"
+ "errors"
+ "fmt"
+ "math"
+)
+
+// A Rat represents a quotient a/b of arbitrary precision.
+// The zero value for a Rat represents the value 0.
+type Rat struct {
+ // To make zero values for Rat work w/o initialization,
+ // a zero value of b (len(b) == 0) acts like b == 1.
+ // a.neg determines the sign of the Rat, b.neg is ignored.
+ a, b Int
+}
+
+// NewRat creates a new Rat with numerator a and denominator b.
+func NewRat(a, b int64) *Rat {
+ return new(Rat).SetFrac64(a, b)
+}
+
+// SetFloat64 sets z to exactly f and returns z.
+// If f is not finite, SetFloat returns nil.
+func (z *Rat) SetFloat64(f float64) *Rat {
+ const expMask = 1<<11 - 1
+ bits := math.Float64bits(f)
+ mantissa := bits & (1<<52 - 1)
+ exp := int((bits >> 52) & expMask)
+ switch exp {
+ case expMask: // non-finite
+ return nil
+ case 0: // denormal
+ exp -= 1022
+ default: // normal
+ mantissa |= 1 << 52
+ exp -= 1023
+ }
+
+ shift := 52 - exp
+
+ // Optimization (?): partially pre-normalise.
+ for mantissa&1 == 0 && shift > 0 {
+ mantissa >>= 1
+ shift--
+ }
+
+ z.a.SetUint64(mantissa)
+ z.a.neg = f < 0
+ z.b.Set(intOne)
+ if shift > 0 {
+ z.b.Lsh(&z.b, uint(shift))
+ } else {
+ z.a.Lsh(&z.a, uint(-shift))
+ }
+ return z.norm()
+}
+
+// quotToFloat32 returns the non-negative float32 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat32(a, b nat) (f float32, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 32
+
+ // mantissa
+ Msize = 23
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float32 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low32(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
+ if math.IsInf(float64(f), 0) {
+ exact = false
+ }
+ return
+}
+
+// quotToFloat64 returns the non-negative float64 value
+// nearest to the quotient a/b, using round-to-even in
+// halfway cases. It does not mutate its arguments.
+// Preconditions: b is non-zero; a and b have no common factors.
+func quotToFloat64(a, b nat) (f float64, exact bool) {
+ const (
+ // float size in bits
+ Fsize = 64
+
+ // mantissa
+ Msize = 52
+ Msize1 = Msize + 1 // incl. implicit 1
+ Msize2 = Msize1 + 1
+
+ // exponent
+ Esize = Fsize - Msize1
+ Ebias = 1<<(Esize-1) - 1
+ Emin = 1 - Ebias
+ Emax = Ebias
+ )
+
+ // TODO(adonovan): specialize common degenerate cases: 1.0, integers.
+ alen := a.bitLen()
+ if alen == 0 {
+ return 0, true
+ }
+ blen := b.bitLen()
+ if blen == 0 {
+ panic("division by zero")
+ }
+
+ // 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
+ // (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
+ // This is 2 or 3 more than the float64 mantissa field width of Msize:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in "normal" representation;
+ // - the low-order 1 will be used during rounding then discarded.
+ exp := alen - blen
+ var a2, b2 nat
+ a2 = a2.set(a)
+ b2 = b2.set(b)
+ if shift := Msize2 - exp; shift > 0 {
+ a2 = a2.shl(a2, uint(shift))
+ } else if shift < 0 {
+ b2 = b2.shl(b2, uint(-shift))
+ }
+
+ // 2. Compute quotient and remainder (q, r). NB: due to the
+ // extra shift, the low-order bit of q is logically the
+ // high-order bit of r.
+ var q nat
+ q, r := q.div(a2, a2, b2) // (recycle a2)
+ mantissa := low64(q)
+ haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
+
+ // 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>Msize2 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>Msize1 != 1 {
+ panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
+ }
+
+ // 4. Rounding.
+ if Emin-Msize <= exp && exp <= Emin {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint(Emin - (exp - 1)) // [1..Esize1)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = 2 - Ebias // == exp + shift
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<Msize2 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
+
+ f = math.Ldexp(float64(mantissa), exp-Msize1)
+ if math.IsInf(f, 0) {
+ exact = false
+ }
+ return
+}
+
+// Float32 returns the nearest float32 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float32, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float32() (f float32, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = b.set(natOne) // materialize denominator
+ }
+ f, exact = quotToFloat32(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// Float64 returns the nearest float64 value for x and a bool indicating
+// whether f represents x exactly. If the magnitude of x is too large to
+// be represented by a float64, f is an infinity and exact is false.
+// The sign of f always matches the sign of x, even if f == 0.
+func (x *Rat) Float64() (f float64, exact bool) {
+ b := x.b.abs
+ if len(b) == 0 {
+ b = b.set(natOne) // materialize denominator
+ }
+ f, exact = quotToFloat64(x.a.abs, b)
+ if x.a.neg {
+ f = -f
+ }
+ return
+}
+
+// SetFrac sets z to a/b and returns z.
+func (z *Rat) SetFrac(a, b *Int) *Rat {
+ z.a.neg = a.neg != b.neg
+ babs := b.abs
+ if len(babs) == 0 {
+ panic("division by zero")
+ }
+ if &z.a == b || alias(z.a.abs, babs) {
+ babs = nat(nil).set(babs) // make a copy
+ }
+ z.a.abs = z.a.abs.set(a.abs)
+ z.b.abs = z.b.abs.set(babs)
+ return z.norm()
+}
+
+// SetFrac64 sets z to a/b and returns z.
+func (z *Rat) SetFrac64(a, b int64) *Rat {
+ z.a.SetInt64(a)
+ if b == 0 {
+ panic("division by zero")
+ }
+ if b < 0 {
+ b = -b
+ z.a.neg = !z.a.neg
+ }
+ z.b.abs = z.b.abs.setUint64(uint64(b))
+ return z.norm()
+}
+
+// SetInt sets z to x (by making a copy of x) and returns z.
+func (z *Rat) SetInt(x *Int) *Rat {
+ z.a.Set(x)
+ z.b.abs = z.b.abs[:0]
+ return z
+}
+
+// SetInt64 sets z to x and returns z.
+func (z *Rat) SetInt64(x int64) *Rat {
+ z.a.SetInt64(x)
+ z.b.abs = z.b.abs[:0]
+ return z
+}
+
+// Set sets z to x (by making a copy of x) and returns z.
+func (z *Rat) Set(x *Rat) *Rat {
+ if z != x {
+ z.a.Set(&x.a)
+ z.b.Set(&x.b)
+ }
+ return z
+}
+
+// Abs sets z to |x| (the absolute value of x) and returns z.
+func (z *Rat) Abs(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = false
+ return z
+}
+
+// Neg sets z to -x and returns z.
+func (z *Rat) Neg(x *Rat) *Rat {
+ z.Set(x)
+ z.a.neg = len(z.a.abs) > 0 && !z.a.neg // 0 has no sign
+ return z
+}
+
+// Inv sets z to 1/x and returns z.
+func (z *Rat) Inv(x *Rat) *Rat {
+ if len(x.a.abs) == 0 {
+ panic("division by zero")
+ }
+ z.Set(x)
+ a := z.b.abs
+ if len(a) == 0 {
+ a = a.set(natOne) // materialize numerator
+ }
+ b := z.a.abs
+ if b.cmp(natOne) == 0 {
+ b = b[:0] // normalize denominator
+ }
+ z.a.abs, z.b.abs = a, b // sign doesn't change
+ return z
+}
+
+// Sign returns:
+//
+// -1 if x < 0
+// 0 if x == 0
+// +1 if x > 0
+//
+func (x *Rat) Sign() int {
+ return x.a.Sign()
+}
+
+// IsInt reports whether the denominator of x is 1.
+func (x *Rat) IsInt() bool {
+ return len(x.b.abs) == 0 || x.b.abs.cmp(natOne) == 0
+}
+
+// Num returns the numerator of x; it may be <= 0.
+// The result is a reference to x's numerator; it
+// may change if a new value is assigned to x, and vice versa.
+// The sign of the numerator corresponds to the sign of x.
+func (x *Rat) Num() *Int {
+ return &x.a
+}
+
+// Denom returns the denominator of x; it is always > 0.
+// The result is a reference to x's denominator; it
+// may change if a new value is assigned to x, and vice versa.
+func (x *Rat) Denom() *Int {
+ x.b.neg = false // the result is always >= 0
+ if len(x.b.abs) == 0 {
+ x.b.abs = x.b.abs.set(natOne) // materialize denominator
+ }
+ return &x.b
+}
+
+func (z *Rat) norm() *Rat {
+ switch {
+ case len(z.a.abs) == 0:
+ // z == 0 - normalize sign and denominator
+ z.a.neg = false
+ z.b.abs = z.b.abs[:0]
+ case len(z.b.abs) == 0:
+ // z is normalized int - nothing to do
+ case z.b.abs.cmp(natOne) == 0:
+ // z is int - normalize denominator
+ z.b.abs = z.b.abs[:0]
+ default:
+ neg := z.a.neg
+ z.a.neg = false
+ z.b.neg = false
+ if f := NewInt(0).binaryGCD(&z.a, &z.b); f.Cmp(intOne) != 0 {
+ z.a.abs, _ = z.a.abs.div(nil, z.a.abs, f.abs)
+ z.b.abs, _ = z.b.abs.div(nil, z.b.abs, f.abs)
+ if z.b.abs.cmp(natOne) == 0 {
+ // z is int - normalize denominator
+ z.b.abs = z.b.abs[:0]
+ }
+ }
+ z.a.neg = neg
+ }
+ return z
+}
+
+// mulDenom sets z to the denominator product x*y (by taking into
+// account that 0 values for x or y must be interpreted as 1) and
+// returns z.
+func mulDenom(z, x, y nat) nat {
+ switch {
+ case len(x) == 0:
+ return z.set(y)
+ case len(y) == 0:
+ return z.set(x)
+ }
+ return z.mul(x, y)
+}
+
+// scaleDenom computes x*f.
+// If f == 0 (zero value of denominator), the result is (a copy of) x.
+func scaleDenom(x *Int, f nat) *Int {
+ var z Int
+ if len(f) == 0 {
+ return z.Set(x)
+ }
+ z.abs = z.abs.mul(x.abs, f)
+ z.neg = x.neg
+ return &z
+}
+
+// Cmp compares x and y and returns:
+//
+// -1 if x < y
+// 0 if x == y
+// +1 if x > y
+//
+func (x *Rat) Cmp(y *Rat) int {
+ return scaleDenom(&x.a, y.b.abs).Cmp(scaleDenom(&y.a, x.b.abs))
+}
+
+// Add sets z to the sum x+y and returns z.
+func (z *Rat) Add(x, y *Rat) *Rat {
+ a1 := scaleDenom(&x.a, y.b.abs)
+ a2 := scaleDenom(&y.a, x.b.abs)
+ z.a.Add(a1, a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Sub sets z to the difference x-y and returns z.
+func (z *Rat) Sub(x, y *Rat) *Rat {
+ a1 := scaleDenom(&x.a, y.b.abs)
+ a2 := scaleDenom(&y.a, x.b.abs)
+ z.a.Sub(a1, a2)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Mul sets z to the product x*y and returns z.
+func (z *Rat) Mul(x, y *Rat) *Rat {
+ z.a.Mul(&x.a, &y.a)
+ z.b.abs = mulDenom(z.b.abs, x.b.abs, y.b.abs)
+ return z.norm()
+}
+
+// Quo sets z to the quotient x/y and returns z.
+// If y == 0, a division-by-zero run-time panic occurs.
+func (z *Rat) Quo(x, y *Rat) *Rat {
+ if len(y.a.abs) == 0 {
+ panic("division by zero")
+ }
+ a := scaleDenom(&x.a, y.b.abs)
+ b := scaleDenom(&y.a, x.b.abs)
+ z.a.abs = a.abs
+ z.b.abs = b.abs
+ z.a.neg = a.neg != b.neg
+ return z.norm()
+}
+
+// Gob codec version. Permits backward-compatible changes to the encoding.
+const ratGobVersion byte = 1
+
+// GobEncode implements the gob.GobEncoder interface.
+func (x *Rat) GobEncode() ([]byte, error) {
+ if x == nil {
+ return nil, nil
+ }
+ buf := make([]byte, 1+4+(len(x.a.abs)+len(x.b.abs))*_S) // extra bytes for version and sign bit (1), and numerator length (4)
+ i := x.b.abs.bytes(buf)
+ j := x.a.abs.bytes(buf[:i])
+ n := i - j
+ if int(uint32(n)) != n {
+ // this should never happen
+ return nil, errors.New("Rat.GobEncode: numerator too large")
+ }
+ binary.BigEndian.PutUint32(buf[j-4:j], uint32(n))
+ j -= 1 + 4
+ b := ratGobVersion << 1 // make space for sign bit
+ if x.a.neg {
+ b |= 1
+ }
+ buf[j] = b
+ return buf[j:], nil
+}
+
+// GobDecode implements the gob.GobDecoder interface.
+func (z *Rat) GobDecode(buf []byte) error {
+ if len(buf) == 0 {
+ // Other side sent a nil or default value.
+ *z = Rat{}
+ return nil
+ }
+ b := buf[0]
+ if b>>1 != ratGobVersion {
+ return errors.New(fmt.Sprintf("Rat.GobDecode: encoding version %d not supported", b>>1))
+ }
+ const j = 1 + 4
+ i := j + binary.BigEndian.Uint32(buf[j-4:j])
+ z.a.neg = b&1 != 0
+ z.a.abs = z.a.abs.setBytes(buf[j:i])
+ z.b.abs = z.b.abs.setBytes(buf[i:])
+ return nil
+}
+
+// MarshalText implements the encoding.TextMarshaler interface.
+func (r *Rat) MarshalText() (text []byte, err error) {
+ return []byte(r.RatString()), nil
+}
+
+// UnmarshalText implements the encoding.TextUnmarshaler interface.
+func (r *Rat) UnmarshalText(text []byte) error {
+ if _, ok := r.SetString(string(text)); !ok {
+ return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Rat", text)
+ }
+ return nil
+}
--- /dev/null
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "encoding/gob"
+ "encoding/json"
+ "encoding/xml"
+ "math"
+ "testing"
+)
+
+func TestZeroRat(t *testing.T) {
+ var x, y, z Rat
+ y.SetFrac64(0, 42)
+
+ if x.Cmp(&y) != 0 {
+ t.Errorf("x and y should be both equal and zero")
+ }
+
+ if s := x.String(); s != "0/1" {
+ t.Errorf("got x = %s, want 0/1", s)
+ }
+
+ if s := x.RatString(); s != "0" {
+ t.Errorf("got x = %s, want 0", s)
+ }
+
+ z.Add(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x+y = %s, want 0", s)
+ }
+
+ z.Sub(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x-y = %s, want 0", s)
+ }
+
+ z.Mul(&x, &y)
+ if s := z.RatString(); s != "0" {
+ t.Errorf("got x*y = %s, want 0", s)
+ }
+
+ // check for division by zero
+ defer func() {
+ if s := recover(); s == nil || s.(string) != "division by zero" {
+ panic(s)
+ }
+ }()
+ z.Quo(&x, &y)
+}
+
+func TestRatSign(t *testing.T) {
+ zero := NewRat(0, 1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ s := x.Sign()
+ e := x.Cmp(zero)
+ if s != e {
+ t.Errorf("got %d; want %d for z = %v", s, e, &x)
+ }
+ }
+}
+
+var ratCmpTests = []struct {
+ rat1, rat2 string
+ out int
+}{
+ {"0", "0/1", 0},
+ {"1/1", "1", 0},
+ {"-1", "-2/2", 0},
+ {"1", "0", 1},
+ {"0/1", "1/1", -1},
+ {"-5/1434770811533343057144", "-5/1434770811533343057145", -1},
+ {"49832350382626108453/8964749413", "49832350382626108454/8964749413", -1},
+ {"-37414950961700930/7204075375675961", "37414950961700930/7204075375675961", -1},
+ {"37414950961700930/7204075375675961", "74829901923401860/14408150751351922", 0},
+}
+
+func TestRatCmp(t *testing.T) {
+ for i, test := range ratCmpTests {
+ x, _ := new(Rat).SetString(test.rat1)
+ y, _ := new(Rat).SetString(test.rat2)
+
+ out := x.Cmp(y)
+ if out != test.out {
+ t.Errorf("#%d got out = %v; want %v", i, out, test.out)
+ }
+ }
+}
+
+func TestIsInt(t *testing.T) {
+ one := NewInt(1)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ i := x.IsInt()
+ e := x.Denom().Cmp(one) == 0
+ if i != e {
+ t.Errorf("got IsInt(%v) == %v; want %v", x, i, e)
+ }
+ }
+}
+
+func TestRatAbs(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Set(x)
+ if e.Cmp(zero) < 0 {
+ e.Sub(zero, e)
+ }
+ z := new(Rat).Abs(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Abs(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatNeg(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ e := new(Rat).Sub(zero, x)
+ z := new(Rat).Neg(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Neg(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+func TestRatInv(t *testing.T) {
+ zero := new(Rat)
+ for _, a := range setStringTests {
+ x, ok := new(Rat).SetString(a.in)
+ if !ok {
+ continue
+ }
+ if x.Cmp(zero) == 0 {
+ continue // avoid division by zero
+ }
+ e := new(Rat).SetFrac(x.Denom(), x.Num())
+ z := new(Rat).Inv(x)
+ if z.Cmp(e) != 0 {
+ t.Errorf("got Inv(%v) = %v; want %v", x, z, e)
+ }
+ }
+}
+
+type ratBinFun func(z, x, y *Rat) *Rat
+type ratBinArg struct {
+ x, y, z string
+}
+
+func testRatBin(t *testing.T, i int, name string, f ratBinFun, a ratBinArg) {
+ x, _ := new(Rat).SetString(a.x)
+ y, _ := new(Rat).SetString(a.y)
+ z, _ := new(Rat).SetString(a.z)
+ out := f(new(Rat), x, y)
+
+ if out.Cmp(z) != 0 {
+ t.Errorf("%s #%d got %s want %s", name, i, out, z)
+ }
+}
+
+var ratBinTests = []struct {
+ x, y string
+ sum, prod string
+}{
+ {"0", "0", "0", "0"},
+ {"0", "1", "1", "0"},
+ {"-1", "0", "-1", "0"},
+ {"-1", "1", "0", "-1"},
+ {"1", "1", "2", "1"},
+ {"1/2", "1/2", "1", "1/4"},
+ {"1/4", "1/3", "7/12", "1/12"},
+ {"2/5", "-14/3", "-64/15", "-28/15"},
+ {"4707/49292519774798173060", "-3367/70976135186689855734", "84058377121001851123459/1749296273614329067191168098769082663020", "-1760941/388732505247628681598037355282018369560"},
+ {"-61204110018146728334/3", "-31052192278051565633/2", "-215564796870448153567/6", "950260896245257153059642991192710872711/3"},
+ {"-854857841473707320655/4237645934602118692642972629634714039", "-18/31750379913563777419", "-27/133467566250814981", "15387441146526731771790/134546868362786310073779084329032722548987800600710485341"},
+ {"618575745270541348005638912139/19198433543745179392300736", "-19948846211000086/637313996471", "27674141753240653/30123979153216", "-6169936206128396568797607742807090270137721977/6117715203873571641674006593837351328"},
+ {"-3/26206484091896184128", "5/2848423294177090248", "15310893822118706237/9330894968229805033368778458685147968", "-5/24882386581946146755650075889827061248"},
+ {"26946729/330400702820", "41563965/225583428284", "1238218672302860271/4658307703098666660055", "224002580204097/14906584649915733312176"},
+ {"-8259900599013409474/7", "-84829337473700364773/56707961321161574960", "-468402123685491748914621885145127724451/396955729248131024720", "350340947706464153265156004876107029701/198477864624065512360"},
+ {"575775209696864/1320203974639986246357", "29/712593081308", "410331716733912717985762465/940768218243776489278275419794956", "808/45524274987585732633"},
+ {"1786597389946320496771/2066653520653241", "6269770/1992362624741777", "3559549865190272133656109052308126637/4117523232840525481453983149257", "8967230/3296219033"},
+ {"-36459180403360509753/32150500941194292113930", "9381566963714/9633539", "301622077145533298008420642898530153/309723104686531919656937098270", "-3784609207827/3426986245"},
+}
+
+func TestRatBin(t *testing.T) {
+ for i, test := range ratBinTests {
+ arg := ratBinArg{test.x, test.y, test.sum}
+ testRatBin(t, i, "Add", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.y, test.x, test.sum}
+ testRatBin(t, i, "Add symmetric", (*Rat).Add, arg)
+
+ arg = ratBinArg{test.sum, test.x, test.y}
+ testRatBin(t, i, "Sub", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.sum, test.y, test.x}
+ testRatBin(t, i, "Sub symmetric", (*Rat).Sub, arg)
+
+ arg = ratBinArg{test.x, test.y, test.prod}
+ testRatBin(t, i, "Mul", (*Rat).Mul, arg)
+
+ arg = ratBinArg{test.y, test.x, test.prod}
+ testRatBin(t, i, "Mul symmetric", (*Rat).Mul, arg)
+
+ if test.x != "0" {
+ arg = ratBinArg{test.prod, test.x, test.y}
+ testRatBin(t, i, "Quo", (*Rat).Quo, arg)
+ }
+
+ if test.y != "0" {
+ arg = ratBinArg{test.prod, test.y, test.x}
+ testRatBin(t, i, "Quo symmetric", (*Rat).Quo, arg)
+ }
+ }
+}
+
+func TestIssue820(t *testing.T) {
+ x := NewRat(3, 1)
+ y := NewRat(2, 1)
+ z := y.Quo(x, y)
+ q := NewRat(3, 2)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ y = NewRat(3, 1)
+ x = NewRat(2, 1)
+ z = y.Quo(x, y)
+ q = NewRat(2, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+
+ x = NewRat(3, 1)
+ z = x.Quo(x, x)
+ q = NewRat(3, 3)
+ if z.Cmp(q) != 0 {
+ t.Errorf("got %s want %s", z, q)
+ }
+}
+
+var setFrac64Tests = []struct {
+ a, b int64
+ out string
+}{
+ {0, 1, "0"},
+ {0, -1, "0"},
+ {1, 1, "1"},
+ {-1, 1, "-1"},
+ {1, -1, "-1"},
+ {-1, -1, "1"},
+ {-9223372036854775808, -9223372036854775808, "1"},
+}
+
+func TestRatSetFrac64Rat(t *testing.T) {
+ for i, test := range setFrac64Tests {
+ x := new(Rat).SetFrac64(test.a, test.b)
+ if x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+func TestRatGobEncoding(t *testing.T) {
+ var medium bytes.Buffer
+ enc := gob.NewEncoder(&medium)
+ dec := gob.NewDecoder(&medium)
+ for _, test := range encodingTests {
+ medium.Reset() // empty buffer for each test case (in case of failures)
+ var tx Rat
+ tx.SetString(test + ".14159265")
+ if err := enc.Encode(&tx); err != nil {
+ t.Errorf("encoding of %s failed: %s", &tx, err)
+ }
+ var rx Rat
+ if err := dec.Decode(&rx); err != nil {
+ t.Errorf("decoding of %s failed: %s", &tx, err)
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("transmission of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+}
+
+// Sending a nil Rat pointer (inside a slice) on a round trip through gob should yield a zero.
+// TODO: top-level nils.
+func TestGobEncodingNilRatInSlice(t *testing.T) {
+ buf := new(bytes.Buffer)
+ enc := gob.NewEncoder(buf)
+ dec := gob.NewDecoder(buf)
+
+ var in = make([]*Rat, 1)
+ err := enc.Encode(&in)
+ if err != nil {
+ t.Errorf("gob encode failed: %q", err)
+ }
+ var out []*Rat
+ err = dec.Decode(&out)
+ if err != nil {
+ t.Fatalf("gob decode failed: %q", err)
+ }
+ if len(out) != 1 {
+ t.Fatalf("wrong len; want 1 got %d", len(out))
+ }
+ var zero Rat
+ if out[0].Cmp(&zero) != 0 {
+ t.Errorf("transmission of (*Int)(nill) failed: got %s want 0", out)
+ }
+}
+
+var ratNums = []string{
+ "-141592653589793238462643383279502884197169399375105820974944592307816406286",
+ "-1415926535897932384626433832795028841971",
+ "-141592653589793",
+ "-1",
+ "0",
+ "1",
+ "141592653589793",
+ "1415926535897932384626433832795028841971",
+ "141592653589793238462643383279502884197169399375105820974944592307816406286",
+}
+
+var ratDenoms = []string{
+ "1",
+ "718281828459045",
+ "7182818284590452353602874713526624977572",
+ "718281828459045235360287471352662497757247093699959574966967627724076630353",
+}
+
+func TestRatJSONEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := json.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := json.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("JSON encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestRatXMLEncoding(t *testing.T) {
+ for _, num := range ratNums {
+ for _, denom := range ratDenoms {
+ var tx Rat
+ tx.SetString(num + "/" + denom)
+ b, err := xml.Marshal(&tx)
+ if err != nil {
+ t.Errorf("marshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ var rx Rat
+ if err := xml.Unmarshal(b, &rx); err != nil {
+ t.Errorf("unmarshaling of %s failed: %s", &tx, err)
+ continue
+ }
+ if rx.Cmp(&tx) != 0 {
+ t.Errorf("XML encoding of %s failed: got %s want %s", &tx, &rx, &tx)
+ }
+ }
+ }
+}
+
+func TestIssue2379(t *testing.T) {
+ // 1) no aliasing
+ q := NewRat(3, 2)
+ x := new(Rat)
+ x.SetFrac(NewInt(3), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("1) got %s want %s", x, q)
+ }
+
+ // 2) aliasing of numerator
+ x = NewRat(2, 3)
+ x.SetFrac(NewInt(3), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("2) got %s want %s", x, q)
+ }
+
+ // 3) aliasing of denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), NewInt(2))
+ if x.Cmp(q) != 0 {
+ t.Errorf("3) got %s want %s", x, q)
+ }
+
+ // 4) aliasing of numerator and denominator
+ x = NewRat(2, 3)
+ x.SetFrac(x.Denom(), x.Num())
+ if x.Cmp(q) != 0 {
+ t.Errorf("4) got %s want %s", x, q)
+ }
+
+ // 5) numerator and denominator are the same
+ q = NewRat(1, 1)
+ x = new(Rat)
+ n := NewInt(7)
+ x.SetFrac(n, n)
+ if x.Cmp(q) != 0 {
+ t.Errorf("5) got %s want %s", x, q)
+ }
+}
+
+func TestIssue3521(t *testing.T) {
+ a := new(Int)
+ b := new(Int)
+ a.SetString("64375784358435883458348587", 0)
+ b.SetString("4789759874531", 0)
+
+ // 0) a raw zero value has 1 as denominator
+ zero := new(Rat)
+ one := NewInt(1)
+ if zero.Denom().Cmp(one) != 0 {
+ t.Errorf("0) got %s want %s", zero.Denom(), one)
+ }
+
+ // 1a) a zero value remains zero independent of denominator
+ x := new(Rat)
+ x.Denom().Set(new(Int).Neg(b))
+ if x.Cmp(zero) != 0 {
+ t.Errorf("1a) got %s want %s", x, zero)
+ }
+
+ // 1b) a zero value may have a denominator != 0 and != 1
+ x.Num().Set(a)
+ qab := new(Rat).SetFrac(a, b)
+ if x.Cmp(qab) != 0 {
+ t.Errorf("1b) got %s want %s", x, qab)
+ }
+
+ // 2a) an integral value becomes a fraction depending on denominator
+ x.SetFrac64(10, 2)
+ x.Denom().SetInt64(3)
+ q53 := NewRat(5, 3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2a) got %s want %s", x, q53)
+ }
+
+ // 2b) an integral value becomes a fraction depending on denominator
+ x = NewRat(10, 2)
+ x.Denom().SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("2b) got %s want %s", x, q53)
+ }
+
+ // 3) changing the numerator/denominator of a Rat changes the Rat
+ x.SetFrac(a, b)
+ a = x.Num()
+ b = x.Denom()
+ a.SetInt64(5)
+ b.SetInt64(3)
+ if x.Cmp(q53) != 0 {
+ t.Errorf("3) got %s want %s", x, q53)
+ }
+}
+
+func TestFloat32Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float32 range, and check Rat.Float32()
+ // always picks the closest float32 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(1), 1 // soak test (~1.5s on x86-64)
+ if testing.Short() {
+ winc, einc = 5, 15 // quick test (~60ms on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 30; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -150; exp < 150; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float32()
+
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip32(t, f)
+ }
+ }
+ }
+ }
+}
+
+func TestFloat64Distribution(t *testing.T) {
+ // Generate a distribution of (sign, mantissa, exp) values
+ // broader than the float64 range, and check Rat.Float64()
+ // always picks the closest float64 approximation.
+ var add = []int64{
+ 0,
+ 1,
+ 3,
+ 5,
+ 7,
+ 9,
+ 11,
+ }
+ var winc, einc = uint64(1), 1 // soak test (~75s on x86-64)
+ if testing.Short() {
+ winc, einc = 10, 500 // quick test (~12ms on x86-64)
+ }
+
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 60; wid += winc {
+ b := 1<<wid + a
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -1100; exp < 1100; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float64()
+
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
+ b, exp, f, f, math.Ldexp(float64(b), exp), r)
+ }
+
+ checkNonLossyRoundtrip64(t, f)
+ }
+ }
+ }
+ }
+}
+
+// TestSetFloat64NonFinite checks that SetFloat64 of a non-finite value
+// returns nil.
+func TestSetFloat64NonFinite(t *testing.T) {
+ for _, f := range []float64{math.NaN(), math.Inf(+1), math.Inf(-1)} {
+ var r Rat
+ if r2 := r.SetFloat64(f); r2 != nil {
+ t.Errorf("SetFloat64(%g) was %v, want nil", f, r2)
+ }
+ }
+}
+
+// checkNonLossyRoundtrip32 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip32(t *testing.T, f float32) {
+ if !isFinite(float64(f)) {
+ return
+ }
+ r := new(Rat).SetFloat64(float64(f))
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(float64(%g) (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float32()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(float64(%g)).Float32() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// checkNonLossyRoundtrip64 checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip64(t *testing.T, f float64) {
+ if !isFinite(f) {
+ return
+ }
+ r := new(Rat).SetFloat64(f)
+ if r == nil {
+ t.Errorf("Rat.SetFloat64(%g (%b)) == nil", f, f)
+ return
+ }
+ f2, exact := r.Float64()
+ if f != f2 || !exact {
+ t.Errorf("Rat.SetFloat64(%g).Float64() = %g (%b), %v, want %g (%b), %v; delta = %b",
+ f, f2, f2, exact, f, f, true, f2-f)
+ }
+}
+
+// delta returns the absolute difference between r and f.
+func delta(r *Rat, f float64) *Rat {
+ d := new(Rat).Sub(r, new(Rat).SetFloat64(f))
+ return d.Abs(d)
+}
+
+// checkIsBestApprox32 checks that f is the best possible float32
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox32(t *testing.T, f float32, r *Rat) bool {
+ if math.Abs(float64(f)) >= math.MaxFloat32 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat32).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter32(f, float32(math.Inf(-1)))
+ f1 := math.Nextafter32(f, float32(math.Inf(+1)))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, float64(f))
+ df0 := delta(r, float64(f0))
+ df1 := delta(r, float64(f1))
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but previous float32 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float32() = %g (%b), but next float32 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven32(f) {
+ t.Errorf("Rat(%v).Float32() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+// checkIsBestApprox64 checks that f is the best possible float64
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox64(t *testing.T, f float64, r *Rat) bool {
+ if math.Abs(f) >= math.MaxFloat64 {
+ // Cannot check +Inf, -Inf, nor the float next to them (MaxFloat64).
+ // But we have tests for these special cases.
+ return true
+ }
+
+ // r must be strictly between f0 and f1, the floats bracketing f.
+ f0 := math.Nextafter(f, math.Inf(-1))
+ f1 := math.Nextafter(f, math.Inf(+1))
+
+ // For f to be correct, r must be closer to f than to f0 or f1.
+ df := delta(r, f)
+ df0 := delta(r, f0)
+ df1 := delta(r, f1)
+ if df.Cmp(df0) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but previous float64 %g (%b) is closer", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) > 0 {
+ t.Errorf("Rat(%v).Float64() = %g (%b), but next float64 %g (%b) is closer", r, f, f, f1, f1)
+ return false
+ }
+ if df.Cmp(df0) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f0, f0)
+ return false
+ }
+ if df.Cmp(df1) == 0 && !isEven64(f) {
+ t.Errorf("Rat(%v).Float64() = %g (%b); halfway should have rounded to %g (%b) instead", r, f, f, f1, f1)
+ return false
+ }
+ return true
+}
+
+func isEven32(f float32) bool { return math.Float32bits(f)&1 == 0 }
+func isEven64(f float64) bool { return math.Float64bits(f)&1 == 0 }
+
+func TestIsFinite(t *testing.T) {
+ finites := []float64{
+ 1.0 / 3,
+ 4891559871276714924261e+222,
+ math.MaxFloat64,
+ math.SmallestNonzeroFloat64,
+ -math.MaxFloat64,
+ -math.SmallestNonzeroFloat64,
+ }
+ for _, f := range finites {
+ if !isFinite(f) {
+ t.Errorf("!IsFinite(%g (%b))", f, f)
+ }
+ }
+ nonfinites := []float64{
+ math.NaN(),
+ math.Inf(-1),
+ math.Inf(+1),
+ }
+ for _, f := range nonfinites {
+ if isFinite(f) {
+ t.Errorf("IsFinite(%g, (%b))", f, f)
+ }
+ }
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file implements rat-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "strconv"
+ "strings"
+)
+
+func ratTok(ch rune) bool {
+ return strings.IndexRune("+-/0123456789.eE", ch) >= 0
+}
+
+// Scan is a support routine for fmt.Scanner. It accepts the formats
+// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
+func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
+ tok, err := s.Token(true, ratTok)
+ if err != nil {
+ return err
+ }
+ if strings.IndexRune("efgEFGv", ch) < 0 {
+ return errors.New("Rat.Scan: invalid verb")
+ }
+ if _, ok := z.SetString(string(tok)); !ok {
+ return errors.New("Rat.Scan: invalid syntax")
+ }
+ return nil
+}
+
+// SetString sets z to the value of s and returns z and a boolean indicating
+// success. s can be given as a fraction "a/b" or as a floating-point number
+// optionally followed by an exponent. If the operation failed, the value of
+// z is undefined but the returned value is nil.
+func (z *Rat) SetString(s string) (*Rat, bool) {
+ if len(s) == 0 {
+ return nil, false
+ }
+ // len(s) > 0
+
+ // parse fraction a/b, if any
+ if sep := strings.Index(s, "/"); sep >= 0 {
+ if _, ok := z.a.SetString(s[:sep], 0); !ok {
+ return nil, false
+ }
+ s = s[sep+1:]
+ var err error
+ if z.b.abs, _, _, err = z.b.abs.scan(strings.NewReader(s), 0, false); err != nil {
+ return nil, false
+ }
+ if len(z.b.abs) == 0 {
+ return nil, false
+ }
+ return z.norm(), true
+ }
+
+ // parse floating-point number
+ r := strings.NewReader(s)
+
+ // sign
+ neg, err := scanSign(r)
+ if err != nil {
+ return nil, false
+ }
+
+ // mantissa
+ var ecorr int
+ z.a.abs, _, ecorr, err = z.a.abs.scan(r, 10, true)
+ if err != nil {
+ return nil, false
+ }
+
+ // exponent
+ var exp int64
+ exp, _, err = scanExponent(r, false)
+ if err != nil {
+ return nil, false
+ }
+
+ // there should be no unread characters left
+ if _, err = r.ReadByte(); err != io.EOF {
+ return nil, false
+ }
+
+ // correct exponent
+ if ecorr < 0 {
+ exp += int64(ecorr)
+ }
+
+ // compute exponent power
+ expabs := exp
+ if expabs < 0 {
+ expabs = -expabs
+ }
+ powTen := nat(nil).expNN(natTen, nat(nil).setWord(Word(expabs)), nil)
+
+ // complete fraction
+ if exp < 0 {
+ z.b.abs = powTen
+ z.norm()
+ } else {
+ z.a.abs = z.a.abs.mul(z.a.abs, powTen)
+ z.b.abs = z.b.abs[:0]
+ }
+
+ z.a.neg = neg && len(z.a.abs) > 0 // 0 has no sign
+
+ return z, true
+}
+
+// scanExponent scans the longest possible prefix of r representing a decimal
+// ('e', 'E') or binary ('p') exponent, if any. It returns the exponent, the
+// exponent base (10 or 2), or a read or syntax error, if any.
+//
+// exponent = ( "E" | "e" | "p" ) [ sign ] digits .
+// sign = "+" | "-" .
+// digits = digit { digit } .
+// digit = "0" ... "9" .
+//
+// A binary exponent is only permitted if binExpOk is set.
+func scanExponent(r io.ByteScanner, binExpOk bool) (exp int64, base int, err error) {
+ base = 10
+
+ var ch byte
+ if ch, err = r.ReadByte(); err != nil {
+ if err == io.EOF {
+ err = nil // no exponent; same as e0
+ }
+ return
+ }
+
+ switch ch {
+ case 'e', 'E':
+ // ok
+ case 'p':
+ if binExpOk {
+ base = 2
+ break // ok
+ }
+ fallthrough // binary exponent not permitted
+ default:
+ r.UnreadByte()
+ return // no exponent; same as e0
+ }
+
+ var neg bool
+ if neg, err = scanSign(r); err != nil {
+ return
+ }
+
+ var digits []byte
+ if neg {
+ digits = append(digits, '-')
+ }
+
+ // no need to use nat.scan for exponent digits
+ // since we only care about int64 values - the
+ // from-scratch scan is easy enough and faster
+ for i := 0; ; i++ {
+ if ch, err = r.ReadByte(); err != nil {
+ if err != io.EOF || i == 0 {
+ return
+ }
+ err = nil
+ break // i > 0
+ }
+ if ch < '0' || '9' < ch {
+ if i == 0 {
+ r.UnreadByte()
+ err = fmt.Errorf("invalid exponent (missing digits)")
+ return
+ }
+ break // i > 0
+ }
+ digits = append(digits, byte(ch))
+ }
+ // i > 0 => we have at least one digit
+
+ exp, err = strconv.ParseInt(string(digits), 10, 64)
+ return
+}
+
+// String returns a string representation of x in the form "a/b" (even if b == 1).
+func (x *Rat) String() string {
+ s := "/1"
+ if len(x.b.abs) != 0 {
+ s = "/" + x.b.abs.decimalString()
+ }
+ return x.a.String() + s
+}
+
+// RatString returns a string representation of x in the form "a/b" if b != 1,
+// and in the form "a" if b == 1.
+func (x *Rat) RatString() string {
+ if x.IsInt() {
+ return x.a.String()
+ }
+ return x.String()
+}
+
+// FloatString returns a string representation of x in decimal form with prec
+// digits of precision after the decimal point and the last digit rounded.
+func (x *Rat) FloatString(prec int) string {
+ if x.IsInt() {
+ s := x.a.String()
+ if prec > 0 {
+ s += "." + strings.Repeat("0", prec)
+ }
+ return s
+ }
+ // x.b.abs != 0
+
+ q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)
+
+ p := natOne
+ if prec > 0 {
+ p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil)
+ }
+
+ r = r.mul(r, p)
+ r, r2 := r.div(nat(nil), r, x.b.abs)
+
+ // see if we need to round up
+ r2 = r2.add(r2, r2)
+ if x.b.abs.cmp(r2) <= 0 {
+ r = r.add(r, natOne)
+ if r.cmp(p) >= 0 {
+ q = nat(nil).add(q, natOne)
+ r = nat(nil).sub(r, p)
+ }
+ }
+
+ s := q.decimalString()
+ if x.a.neg {
+ s = "-" + s
+ }
+
+ if prec > 0 {
+ rs := r.decimalString()
+ leadingZeros := prec - len(rs)
+ s += "." + strings.Repeat("0", leadingZeros) + rs
+ }
+
+ return s
+}
--- /dev/null
+// Copyright 2015 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+ "bytes"
+ "fmt"
+ "math"
+ "strconv"
+ "strings"
+ "testing"
+)
+
+type StringTest struct {
+ in, out string
+ ok bool
+}
+
+var setStringTests = []StringTest{
+ {"0", "0", true},
+ {"-0", "0", true},
+ {"1", "1", true},
+ {"-1", "-1", true},
+ {"1.", "1", true},
+ {"1e0", "1", true},
+ {"1.e1", "10", true},
+ {in: "1e"},
+ {in: "1.e"},
+ {in: "1e+14e-5"},
+ {in: "1e4.5"},
+ {in: "r"},
+ {in: "a/b"},
+ {in: "a.b"},
+ {"-0.1", "-1/10", true},
+ {"-.1", "-1/10", true},
+ {"2/4", "1/2", true},
+ {".25", "1/4", true},
+ {"-1/5", "-1/5", true},
+ {"8129567.7690E14", "812956776900000000000", true},
+ {"78189e+4", "781890000", true},
+ {"553019.8935e+8", "55301989350000", true},
+ {"98765432109876543210987654321e-10", "98765432109876543210987654321/10000000000", true},
+ {"9877861857500000E-7", "3951144743/4", true},
+ {"2169378.417e-3", "2169378417/1000000", true},
+ {"884243222337379604041632732738665534", "884243222337379604041632732738665534", true},
+ {"53/70893980658822810696", "53/70893980658822810696", true},
+ {"106/141787961317645621392", "53/70893980658822810696", true},
+ {"204211327800791583.81095", "4084226556015831676219/20000", true},
+ {in: "1/0"},
+}
+
+// These are not supported by fmt.Fscanf.
+var setStringTests2 = []StringTest{
+ {"0x10", "16", true},
+ {"-010/1", "-8", true}, // TODO(gri) should we even permit octal here?
+ {"-010.", "-10", true},
+ {"0x10/0x20", "1/2", true},
+ {"0b1000/3", "8/3", true},
+ // TODO(gri) add more tests
+}
+
+func TestRatSetString(t *testing.T) {
+ var tests []StringTest
+ tests = append(tests, setStringTests...)
+ tests = append(tests, setStringTests2...)
+
+ for i, test := range tests {
+ x, ok := new(Rat).SetString(test.in)
+
+ if ok {
+ if !test.ok {
+ t.Errorf("#%d SetString(%q) expected failure", i, test.in)
+ } else if x.RatString() != test.out {
+ t.Errorf("#%d SetString(%q) got %s want %s", i, test.in, x.RatString(), test.out)
+ }
+ } else if x != nil {
+ t.Errorf("#%d SetString(%q) got %p want nil", i, test.in, x)
+ }
+ }
+}
+
+func TestRatScan(t *testing.T) {
+ var buf bytes.Buffer
+ for i, test := range setStringTests {
+ x := new(Rat)
+ buf.Reset()
+ buf.WriteString(test.in)
+
+ _, err := fmt.Fscanf(&buf, "%v", x)
+ if err == nil != test.ok {
+ if test.ok {
+ t.Errorf("#%d (%s) error: %s", i, test.in, err)
+ } else {
+ t.Errorf("#%d (%s) expected error", i, test.in)
+ }
+ continue
+ }
+ if err == nil && x.RatString() != test.out {
+ t.Errorf("#%d got %s want %s", i, x.RatString(), test.out)
+ }
+ }
+}
+
+var floatStringTests = []struct {
+ in string
+ prec int
+ out string
+}{
+ {"0", 0, "0"},
+ {"0", 4, "0.0000"},
+ {"1", 0, "1"},
+ {"1", 2, "1.00"},
+ {"-1", 0, "-1"},
+ {".25", 2, "0.25"},
+ {".25", 1, "0.3"},
+ {".25", 3, "0.250"},
+ {"-1/3", 3, "-0.333"},
+ {"-2/3", 4, "-0.6667"},
+ {"0.96", 1, "1.0"},
+ {"0.999", 2, "1.00"},
+ {"0.9", 0, "1"},
+ {".25", -1, "0"},
+ {".55", -1, "1"},
+}
+
+func TestFloatString(t *testing.T) {
+ for i, test := range floatStringTests {
+ x, _ := new(Rat).SetString(test.in)
+
+ if x.FloatString(test.prec) != test.out {
+ t.Errorf("#%d got %s want %s", i, x.FloatString(test.prec), test.out)
+ }
+ }
+}
+
+// Test inputs to Rat.SetString. The prefix "long:" causes the test
+// to be skipped in --test.short mode. (The threshold is about 500us.)
+var float64inputs = []string{
+ // Constants plundered from strconv/testfp.txt.
+
+ // Table 1: Stress Inputs for Conversion to 53-bit Binary, < 1/2 ULP
+ "5e+125",
+ "69e+267",
+ "999e-026",
+ "7861e-034",
+ "75569e-254",
+ "928609e-261",
+ "9210917e+080",
+ "84863171e+114",
+ "653777767e+273",
+ "5232604057e-298",
+ "27235667517e-109",
+ "653532977297e-123",
+ "3142213164987e-294",
+ "46202199371337e-072",
+ "231010996856685e-073",
+ "9324754620109615e+212",
+ "78459735791271921e+049",
+ "272104041512242479e+200",
+ "6802601037806061975e+198",
+ "20505426358836677347e-221",
+ "836168422905420598437e-234",
+ "4891559871276714924261e+222",
+
+ // Table 2: Stress Inputs for Conversion to 53-bit Binary, > 1/2 ULP
+ "9e-265",
+ "85e-037",
+ "623e+100",
+ "3571e+263",
+ "81661e+153",
+ "920657e-023",
+ "4603285e-024",
+ "87575437e-309",
+ "245540327e+122",
+ "6138508175e+120",
+ "83356057653e+193",
+ "619534293513e+124",
+ "2335141086879e+218",
+ "36167929443327e-159",
+ "609610927149051e-255",
+ "3743626360493413e-165",
+ "94080055902682397e-242",
+ "899810892172646163e+283",
+ "7120190517612959703e+120",
+ "25188282901709339043e-252",
+ "308984926168550152811e-052",
+ "6372891218502368041059e+064",
+
+ // Table 14: Stress Inputs for Conversion to 24-bit Binary, <1/2 ULP
+ "5e-20",
+ "67e+14",
+ "985e+15",
+ "7693e-42",
+ "55895e-16",
+ "996622e-44",
+ "7038531e-32",
+ "60419369e-46",
+ "702990899e-20",
+ "6930161142e-48",
+ "25933168707e+13",
+ "596428896559e+20",
+
+ // Table 15: Stress Inputs for Conversion to 24-bit Binary, >1/2 ULP
+ "3e-23",
+ "57e+18",
+ "789e-35",
+ "2539e-18",
+ "76173e+28",
+ "887745e-11",
+ "5382571e-37",
+ "82381273e-35",
+ "750486563e-38",
+ "3752432815e-39",
+ "75224575729e-45",
+ "459926601011e+15",
+
+ // Constants plundered from strconv/atof_test.go.
+
+ "0",
+ "1",
+ "+1",
+ "1e23",
+ "1E23",
+ "100000000000000000000000",
+ "1e-100",
+ "123456700",
+ "99999999999999974834176",
+ "100000000000000000000001",
+ "100000000000000008388608",
+ "100000000000000016777215",
+ "100000000000000016777216",
+ "-1",
+ "-0.1",
+ "-0", // NB: exception made for this input
+ "1e-20",
+ "625e-3",
+
+ // largest float64
+ "1.7976931348623157e308",
+ "-1.7976931348623157e308",
+ // next float64 - too large
+ "1.7976931348623159e308",
+ "-1.7976931348623159e308",
+ // the border is ...158079
+ // borderline - okay
+ "1.7976931348623158e308",
+ "-1.7976931348623158e308",
+ // borderline - too large
+ "1.797693134862315808e308",
+ "-1.797693134862315808e308",
+
+ // a little too large
+ "1e308",
+ "2e308",
+ "1e309",
+
+ // way too large
+ "1e310",
+ "-1e310",
+ "1e400",
+ "-1e400",
+ "long:1e400000",
+ "long:-1e400000",
+
+ // denormalized
+ "1e-305",
+ "1e-306",
+ "1e-307",
+ "1e-308",
+ "1e-309",
+ "1e-310",
+ "1e-322",
+ // smallest denormal
+ "5e-324",
+ "4e-324",
+ "3e-324",
+ // too small
+ "2e-324",
+ // way too small
+ "1e-350",
+ "long:1e-400000",
+ // way too small, negative
+ "-1e-350",
+ "long:-1e-400000",
+
+ // try to overflow exponent
+ // [Disabled: too slow and memory-hungry with rationals.]
+ // "1e-4294967296",
+ // "1e+4294967296",
+ // "1e-18446744073709551616",
+ // "1e+18446744073709551616",
+
+ // http://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
+ "2.2250738585072012e-308",
+ // http://www.exploringbinary.com/php-hangs-on-numeric-value-2-2250738585072011e-308/
+ "2.2250738585072011e-308",
+
+ // A very large number (initially wrongly parsed by the fast algorithm).
+ "4.630813248087435e+307",
+
+ // A different kind of very large number.
+ "22.222222222222222",
+ "long:2." + strings.Repeat("2", 4000) + "e+1",
+
+ // Exactly halfway between 1 and math.Nextafter(1, 2).
+ // Round to even (down).
+ "1.00000000000000011102230246251565404236316680908203125",
+ // Slightly lower; still round down.
+ "1.00000000000000011102230246251565404236316680908203124",
+ // Slightly higher; round up.
+ "1.00000000000000011102230246251565404236316680908203126",
+ // Slightly higher, but you have to read all the way to the end.
+ "long:1.00000000000000011102230246251565404236316680908203125" + strings.Repeat("0", 10000) + "1",
+
+ // Smallest denormal, 2^(-1022-52)
+ "4.940656458412465441765687928682213723651e-324",
+ // Half of smallest denormal, 2^(-1022-53)
+ "2.470328229206232720882843964341106861825e-324",
+ // A little more than the exact half of smallest denormal
+ // 2^-1075 + 2^-1100. (Rounds to 1p-1074.)
+ "2.470328302827751011111470718709768633275e-324",
+ // The exact halfway between smallest normal and largest denormal:
+ // 2^-1022 - 2^-1075. (Rounds to 2^-1022.)
+ "2.225073858507201136057409796709131975935e-308",
+
+ "1152921504606846975", // 1<<60 - 1
+ "-1152921504606846975", // -(1<<60 - 1)
+ "1152921504606846977", // 1<<60 + 1
+ "-1152921504606846977", // -(1<<60 + 1)
+
+ "1/3",
+}
+
+// isFinite reports whether f represents a finite rational value.
+// It is equivalent to !math.IsNan(f) && !math.IsInf(f, 0).
+func isFinite(f float64) bool {
+ return math.Abs(f) <= math.MaxFloat64
+}
+
+func TestFloat32SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if testing.Short() {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float32()
+
+ // 1. Check string -> Rat -> float32 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e64, _ := strconv.ParseFloat(input, 32)
+ e := float32(e64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float32bits(e) == math.Float32bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(float64(f)) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox32(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip32(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(float64(f)).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float32().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
+
+func TestFloat64SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ if strings.HasPrefix(input, "long:") {
+ if testing.Short() {
+ continue
+ }
+ input = input[len("long:"):]
+ }
+
+ r, ok := new(Rat).SetString(input)
+ if !ok {
+ t.Errorf("Rat.SetString(%q) failed", input)
+ continue
+ }
+ f, exact := r.Float64()
+
+ // 1. Check string -> Rat -> float64 conversions are
+ // consistent with strconv.ParseFloat.
+ // Skip this check if the input uses "a/b" rational syntax.
+ if !strings.Contains(input, "/") {
+ e, _ := strconv.ParseFloat(input, 64)
+
+ // Careful: negative Rats too small for
+ // float64 become -0, but Rat obviously cannot
+ // preserve the sign from SetString("-0").
+ switch {
+ case math.Float64bits(e) == math.Float64bits(f):
+ // Ok: bitwise equal.
+ case f == 0 && r.Num().BitLen() == 0:
+ // Ok: Rat(0) is equivalent to both +/- float64(0).
+ default:
+ t.Errorf("strconv.ParseFloat(%q) = %g (%b), want %g (%b); delta = %g", input, e, e, f, f, f-e)
+ }
+ }
+
+ if !isFinite(f) {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox64(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip64(t, f)
+
+ // 4. Check exactness using slow algorithm.
+ if wasExact := new(Rat).SetFloat64(f).Cmp(r) == 0; wasExact != exact {
+ t.Errorf("Rat.SetString(%q).Float64().exact = %t, want %t", input, exact, wasExact)
+ }
+ }
+}
--- /dev/null
+// generated by stringer -type=RoundingMode; DO NOT EDIT
+
+package big
+
+import "fmt"
+
+const _RoundingMode_name = "ToNearestEvenToNearestAwayToZeroAwayFromZeroToNegativeInfToPositiveInf"
+
+var _RoundingMode_index = [...]uint8{0, 13, 26, 32, 44, 57, 70}
+
+func (i RoundingMode) String() string {
+ if i+1 >= RoundingMode(len(_RoundingMode_index)) {
+ return fmt.Sprintf("RoundingMode(%d)", i)
+ }
+ return _RoundingMode_name[_RoundingMode_index[i]:_RoundingMode_index[i+1]]
+}
--- /dev/null
+#!/usr/bin/env bash
+
+# Copyright 2015 The Go Authors. All rights reserved.
+# Use of this source code is governed by a BSD-style
+# license that can be found in the LICENSE file.
+
+# Run this script to obtain an up-to-date vendored version of math/big.
+
+BIGDIR=../../../../math/big
+
+# Start from scratch.
+rm *.go
+
+# We don't want any assembly files.
+cp $BIGDIR/*.go .
+
+# Use pure Go arith ops w/o build tag.
+sed 's/^\/\/ \+build math_big_pure_go$//' arith_decl_pure.go > arith_decl.go
+rm arith_decl_pure.go
+
+# Test that it works
+go test -short