From e5a0d6399e00cec2a35b77a79d81a9435b0ee909 Mon Sep 17 00:00:00 2001 From: Robert Griesemer Date: Fri, 20 Mar 2015 13:35:02 -0700 Subject: [PATCH] cmd/internal/gc/big: vendored math/big for use by gc 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 Run-TryBot: Robert Griesemer TryBot-Result: Gobot Gobot Reviewed-by: Russ Cox --- src/cmd/internal/gc/big/accuracy_string.go | 16 + src/cmd/internal/gc/big/arith.go | 291 +++ src/cmd/internal/gc/big/arith_decl.go | 55 + src/cmd/internal/gc/big/arith_test.go | 456 +++++ src/cmd/internal/gc/big/bits_test.go | 224 +++ src/cmd/internal/gc/big/calibrate_test.go | 88 + src/cmd/internal/gc/big/decimal.go | 258 +++ src/cmd/internal/gc/big/decimal_test.go | 106 + src/cmd/internal/gc/big/example_test.go | 51 + src/cmd/internal/gc/big/float.go | 1702 +++++++++++++++++ src/cmd/internal/gc/big/float_test.go | 1682 ++++++++++++++++ src/cmd/internal/gc/big/floatconv.go | 350 ++++ src/cmd/internal/gc/big/floatconv_test.go | 391 ++++ src/cmd/internal/gc/big/floatexample_test.go | 137 ++ src/cmd/internal/gc/big/ftoa.go | 190 ++ src/cmd/internal/gc/big/gcd_test.go | 47 + src/cmd/internal/gc/big/hilbert_test.go | 160 ++ src/cmd/internal/gc/big/int.go | 845 ++++++++ src/cmd/internal/gc/big/int_test.go | 1348 +++++++++++++ src/cmd/internal/gc/big/intconv.go | 228 +++ src/cmd/internal/gc/big/intconv_test.go | 342 ++++ src/cmd/internal/gc/big/nat.go | 1155 +++++++++++ src/cmd/internal/gc/big/nat_test.go | 518 +++++ src/cmd/internal/gc/big/natconv.go | 495 +++++ src/cmd/internal/gc/big/natconv_test.go | 425 ++++ src/cmd/internal/gc/big/rat.go | 570 ++++++ src/cmd/internal/gc/big/rat_test.go | 736 +++++++ src/cmd/internal/gc/big/ratconv.go | 251 +++ src/cmd/internal/gc/big/ratconv_test.go | 451 +++++ .../internal/gc/big/roundingmode_string.go | 16 + src/cmd/internal/gc/big/vendor.bash | 22 + 31 files changed, 13606 insertions(+) create mode 100644 src/cmd/internal/gc/big/accuracy_string.go create mode 100644 src/cmd/internal/gc/big/arith.go create mode 100644 src/cmd/internal/gc/big/arith_decl.go create mode 100644 src/cmd/internal/gc/big/arith_test.go create mode 100644 src/cmd/internal/gc/big/bits_test.go create mode 100644 src/cmd/internal/gc/big/calibrate_test.go create mode 100644 src/cmd/internal/gc/big/decimal.go create mode 100644 src/cmd/internal/gc/big/decimal_test.go create mode 100644 src/cmd/internal/gc/big/example_test.go create mode 100644 src/cmd/internal/gc/big/float.go create mode 100644 src/cmd/internal/gc/big/float_test.go create mode 100644 src/cmd/internal/gc/big/floatconv.go create mode 100644 src/cmd/internal/gc/big/floatconv_test.go create mode 100644 src/cmd/internal/gc/big/floatexample_test.go create mode 100644 src/cmd/internal/gc/big/ftoa.go create mode 100644 src/cmd/internal/gc/big/gcd_test.go create mode 100644 src/cmd/internal/gc/big/hilbert_test.go create mode 100644 src/cmd/internal/gc/big/int.go create mode 100644 src/cmd/internal/gc/big/int_test.go create mode 100644 src/cmd/internal/gc/big/intconv.go create mode 100644 src/cmd/internal/gc/big/intconv_test.go create mode 100644 src/cmd/internal/gc/big/nat.go create mode 100644 src/cmd/internal/gc/big/nat_test.go create mode 100644 src/cmd/internal/gc/big/natconv.go create mode 100644 src/cmd/internal/gc/big/natconv_test.go create mode 100644 src/cmd/internal/gc/big/rat.go create mode 100644 src/cmd/internal/gc/big/rat_test.go create mode 100644 src/cmd/internal/gc/big/ratconv.go create mode 100644 src/cmd/internal/gc/big/ratconv_test.go create mode 100644 src/cmd/internal/gc/big/roundingmode_string.go create mode 100755 src/cmd/internal/gc/big/vendor.bash diff --git a/src/cmd/internal/gc/big/accuracy_string.go b/src/cmd/internal/gc/big/accuracy_string.go new file mode 100644 index 0000000000..647a1fb6d3 --- /dev/null +++ b/src/cmd/internal/gc/big/accuracy_string.go @@ -0,0 +1,16 @@ +// 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]] +} diff --git a/src/cmd/internal/gc/big/arith.go b/src/cmd/internal/gc/big/arith.go new file mode 100644 index 0000000000..328c85c4f7 --- /dev/null +++ b/src/cmd/internal/gc/big/arith.go @@ -0,0 +1,291 @@ +// 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<>(_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<>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 +} diff --git a/src/cmd/internal/gc/big/arith_decl.go b/src/cmd/internal/gc/big/arith_decl.go new file mode 100644 index 0000000000..0fc8d13ba8 --- /dev/null +++ b/src/cmd/internal/gc/big/arith_decl.go @@ -0,0 +1,55 @@ +// 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) +} diff --git a/src/cmd/internal/gc/big/arith_test.go b/src/cmd/internal/gc/big/arith_test.go new file mode 100644 index 0000000000..cd92dd7173 --- /dev/null +++ b/src/cmd/internal/gc/big/arith_test.go @@ -0,0 +1,456 @@ +// 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) } diff --git a/src/cmd/internal/gc/big/bits_test.go b/src/cmd/internal/gc/big/bits_test.go new file mode 100644 index 0000000000..3ce24222d7 --- /dev/null +++ b/src/cmd/internal/gc/big/bits_test.go @@ -0,0 +1,224 @@ +// 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) + } + } +} diff --git a/src/cmd/internal/gc/big/calibrate_test.go b/src/cmd/internal/gc/big/calibrate_test.go new file mode 100644 index 0000000000..f69ffbf5cf --- /dev/null +++ b/src/cmd/internal/gc/big/calibrate_test.go @@ -0,0 +1,88 @@ +// 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() + } +} diff --git a/src/cmd/internal/gc/big/decimal.go b/src/cmd/internal/gc/big/decimal.go new file mode 100644 index 0000000000..3d024dce68 --- /dev/null +++ b/src/cmd/internal/gc/big/decimal.go @@ -0,0 +1,258 @@ +// 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<= 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 == 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] +} diff --git a/src/cmd/internal/gc/big/decimal_test.go b/src/cmd/internal/gc/big/decimal_test.go new file mode 100644 index 0000000000..81e022a47d --- /dev/null +++ b/src/cmd/internal/gc/big/decimal_test.go @@ -0,0 +1,106 @@ +// 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) + } + } +} diff --git a/src/cmd/internal/gc/big/example_test.go b/src/cmd/internal/gc/big/example_test.go new file mode 100644 index 0000000000..078be47f95 --- /dev/null +++ b/src/cmd/internal/gc/big/example_test.go @@ -0,0 +1,51 @@ +// 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 +} diff --git a/src/cmd/internal/gc/big/float.go b/src/cmd/internal/gc/big/float.go new file mode 100644 index 0000000000..629510a18e --- /dev/null +++ b/src/cmd/internal/gc/big/float.go @@ -0,0 +1,1702 @@ +// 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< 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< 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 +} diff --git a/src/cmd/internal/gc/big/float_test.go b/src/cmd/internal/gc/big/float_test.go new file mode 100644 index 0000000000..9d101531de --- /dev/null +++ b/src/cmd/internal/gc/big/float_test.go @@ -0,0 +1,1682 @@ +// 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< y: + want = Above + } + if got != want { + t.Errorf("(%g).Cmp(%g) = %s; want %s", x, y, got, want) + } + } + } + } +} diff --git a/src/cmd/internal/gc/big/floatconv.go b/src/cmd/internal/gc/big/floatconv.go new file mode 100644 index 0000000000..8905718d29 --- /dev/null +++ b/src/cmd/internal/gc/big/floatconv.go @@ -0,0 +1,350 @@ +// 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) +} diff --git a/src/cmd/internal/gc/big/floatconv_test.go b/src/cmd/internal/gc/big/floatconv_test.go new file mode 100644 index 0000000000..17c8b14786 --- /dev/null +++ b/src/cmd/internal/gc/big/floatconv_test.go @@ -0,0 +1,391 @@ +// 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) + } + } + } +} diff --git a/src/cmd/internal/gc/big/floatexample_test.go b/src/cmd/internal/gc/big/floatexample_test.go new file mode 100644 index 0000000000..5e4a9cbe89 --- /dev/null +++ b/src/cmd/internal/gc/big/floatexample_test.go @@ -0,0 +1,137 @@ +// 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 '○' +} diff --git a/src/cmd/internal/gc/big/ftoa.go b/src/cmd/internal/gc/big/ftoa.go new file mode 100644 index 0000000000..5502eda69b --- /dev/null +++ b/src/cmd/internal/gc/big/ftoa.go @@ -0,0 +1,190 @@ +// 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 +} diff --git a/src/cmd/internal/gc/big/gcd_test.go b/src/cmd/internal/gc/big/gcd_test.go new file mode 100644 index 0000000000..c0b9f58300 --- /dev/null +++ b/src/cmd/internal/gc/big/gcd_test.go @@ -0,0 +1,47 @@ +// 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< 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) } diff --git a/src/cmd/internal/gc/big/hilbert_test.go b/src/cmd/internal/gc/big/hilbert_test.go new file mode 100644 index 0000000000..1a84341b3c --- /dev/null +++ b/src/cmd/internal/gc/big/hilbert_test.go @@ -0,0 +1,160 @@ +// 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) + } +} diff --git a/src/cmd/internal/gc/big/int.go b/src/cmd/internal/gc/big/int.go new file mode 100644 index 0000000000..0695d78973 --- /dev/null +++ b/src/cmd/internal/gc/big/int.go @@ -0,0 +1,845 @@ +// 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 +} diff --git a/src/cmd/internal/gc/big/int_test.go b/src/cmd/internal/gc/big/int_test.go new file mode 100644 index 0000000000..dff8057cf9 --- /dev/null +++ b/src/cmd/internal/gc/big/int_test.go @@ -0,0 +1,1348 @@ +// 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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xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD", + "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}, +} + +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) +} diff --git a/src/cmd/internal/gc/big/intconv.go b/src/cmd/internal/gc/big/intconv.go new file mode 100644 index 0000000000..9c68a22bed --- /dev/null +++ b/src/cmd/internal/gc/big/intconv.go @@ -0,0 +1,228 @@ +// 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 "" + 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, "") + 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 +} diff --git a/src/cmd/internal/gc/big/intconv_test.go b/src/cmd/internal/gc/big/intconv_test.go new file mode 100644 index 0000000000..2deb84b48f --- /dev/null +++ b/src/cmd/internal/gc/big/intconv_test.go @@ -0,0 +1,342 @@ +// 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 +}{ + {"", "%x", ""}, + {"", "%#x", ""}, + {"", "%#y", "%!y(big.Int=)"}, + + {"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 != "" { + 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) + } + } +} diff --git a/src/cmd/internal/gc/big/nat.go b/src/cmd/internal/gc/big/nat.go new file mode 100644 index 0000000000..2a279d186c --- /dev/null +++ b/src/cmd/internal/gc/big/nat.go @@ -0,0 +1,1155 @@ +// 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<= 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< 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<= 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() +} diff --git a/src/cmd/internal/gc/big/nat_test.go b/src/cmd/internal/gc/big/nat_test.go new file mode 100644 index 0000000000..b25a89f731 --- /dev/null +++ b/src/cmd/internal/gc/big/nat_test.go @@ -0,0 +1,518 @@ +// 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) + } + } + } + } +} diff --git a/src/cmd/internal/gc/big/natconv.go b/src/cmd/internal/gc/big/natconv.go new file mode 100644 index 0000000000..022dcfe38c --- /dev/null +++ b/src/cmd/internal/gc/big/natconv.go @@ -0,0 +1,495 @@ +// 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 { + 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 +} diff --git a/src/cmd/internal/gc/big/natconv_test.go b/src/cmd/internal/gc/big/natconv_test.go new file mode 100644 index 0000000000..f321fbc2df --- /dev/null +++ b/src/cmd/internal/gc/big/natconv_test.go @@ -0,0 +1,425 @@ +// 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 + } + } +} diff --git a/src/cmd/internal/gc/big/rat.go b/src/cmd/internal/gc/big/rat.go new file mode 100644 index 0000000000..748796c8ca --- /dev/null +++ b/src/cmd/internal/gc/big/rat.go @@ -0,0 +1,570 @@ +// 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<= 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 + 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< 100...0, so shift is safe + mantissa >>= 1 + exp++ + } + } + } + mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<= 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 + 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< 100...0, so shift is safe + mantissa >>= 1 + exp++ + } + } + } + mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1< 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 +} diff --git a/src/cmd/internal/gc/big/rat_test.go b/src/cmd/internal/gc/big/rat_test.go new file mode 100644 index 0000000000..012d0c47ec --- /dev/null +++ b/src/cmd/internal/gc/big/rat_test.go @@ -0,0 +1,736 @@ +// 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< 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< 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) + } + } +} diff --git a/src/cmd/internal/gc/big/ratconv.go b/src/cmd/internal/gc/big/ratconv.go new file mode 100644 index 0000000000..778077b96e --- /dev/null +++ b/src/cmd/internal/gc/big/ratconv.go @@ -0,0 +1,251 @@ +// 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 +} diff --git a/src/cmd/internal/gc/big/ratconv_test.go b/src/cmd/internal/gc/big/ratconv_test.go new file mode 100644 index 0000000000..16b3a19418 --- /dev/null +++ b/src/cmd/internal/gc/big/ratconv_test.go @@ -0,0 +1,451 @@ +// 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) + } + } +} diff --git a/src/cmd/internal/gc/big/roundingmode_string.go b/src/cmd/internal/gc/big/roundingmode_string.go new file mode 100644 index 0000000000..05024b8065 --- /dev/null +++ b/src/cmd/internal/gc/big/roundingmode_string.go @@ -0,0 +1,16 @@ +// 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]] +} diff --git a/src/cmd/internal/gc/big/vendor.bash b/src/cmd/internal/gc/big/vendor.bash new file mode 100755 index 0000000000..f7eca5e994 --- /dev/null +++ b/src/cmd/internal/gc/big/vendor.bash @@ -0,0 +1,22 @@ +#!/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 -- 2.48.1