"encoding/binary"
"errors"
"fmt"
+ "math"
"strings"
)
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
+
+ // Optimisation (?): 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()
+}
+
+// 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
+}
+
+// low64 returns the least significant 64 bits of natural number z.
+func low64(z nat) uint64 {
+ if len(z) == 0 {
+ return 0
+ }
+ if _W == 32 && len(z) > 1 {
+ return uint64(z[1])<<32 | uint64(z[0])
+ }
+ return uint64(z[0])
+}
+
+// quotToFloat returns the non-negative IEEE 754 double-precision
+// 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 quotToFloat(a, b nat) (f float64, exact bool) {
+ // 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<<53, 1<<55).
+ // (54 bits if A<B when they are left-aligned, 55 bits if A>=B.)
+ // This is 2 or 3 more than the float64 mantissa field width of 52:
+ // - the optional extra bit is shifted away in step 3 below.
+ // - the high-order 1 is omitted in float64 "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 := 54 - 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 54 bits, re-do division by b2<<1
+ // (in effect---we accomplish this incrementally).
+ if mantissa>>54 == 1 {
+ if mantissa&1 == 1 {
+ haveRem = true
+ }
+ mantissa >>= 1
+ exp++
+ }
+ if mantissa>>53 != 1 {
+ panic("expected exactly 54 bits of result")
+ }
+
+ // 4. Rounding.
+ if -1022-52 <= exp && exp <= -1022 {
+ // Denormal case; lose 'shift' bits of precision.
+ shift := uint64(-1022 - (exp - 1)) // [1..53)
+ lostbits := mantissa & (1<<shift - 1)
+ haveRem = haveRem || lostbits != 0
+ mantissa >>= shift
+ exp = -1023 + 2
+ }
+ // Round q using round-half-to-even.
+ exact = !haveRem
+ if mantissa&1 != 0 {
+ exact = false
+ if haveRem || mantissa&2 != 0 {
+ if mantissa++; mantissa >= 1<<54 {
+ // Complete rollover 11...1 => 100...0, so shift is safe
+ mantissa >>= 1
+ exp++
+ }
+ }
+ }
+ mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 2^53.
+
+ f = math.Ldexp(float64(mantissa), exp-53)
+ if math.IsInf(f, 0) {
+ exact = false
+ }
+ return
+}
+
+// Float64 returns the nearest float64 value to z.
+// If z is exactly representable as a float64, Float64 returns exact=true.
+// If z is negative, so too is f, even if f==0.
+func (z *Rat) Float64() (f float64, exact bool) {
+ b := z.b.abs
+ if len(b) == 0 {
+ b = b.set(natOne) // materialize denominator
+ }
+ f, exact = quotToFloat(z.a.abs, b)
+ if z.a.neg {
+ f = -f
+ }
+ return
+}
+
// SetFrac sets z to a/b and returns z.
func (z *Rat) SetFrac(a, b *Int) *Rat {
z.a.neg = a.neg != b.neg
"bytes"
"encoding/gob"
"fmt"
+ "math"
+ "strconv"
+ "strings"
"testing"
)
t.Errorf("3) got %s want %s", x, q53)
}
}
+
+// Test inputs to Rat.SetString. The optional prefix "slow:" skips
+// checks found to be slow for certain large rationals.
+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",
+ "1e400000",
+ "-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",
+ "slow:1e-400000",
+ // way too small, negative
+ "-1e-350",
+ "slow:-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",
+ "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.
+ "slow: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",
+}
+
+func TestFloat64SpecialCases(t *testing.T) {
+ for _, input := range float64inputs {
+ slow := strings.HasPrefix(input, "slow:")
+ if slow {
+ input = input[len("slow:"):]
+ }
+
+ 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) || slow {
+ continue
+ }
+
+ // 2. Check f is best approximation to r.
+ if !checkIsBestApprox(t, f, r) {
+ // Append context information.
+ t.Errorf("(input was %q)", input)
+ }
+
+ // 3. Check f->R->f roundtrip is non-lossy.
+ checkNonLossyRoundtrip(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 = %b, want %b", input, exact, wasExact)
+ }
+ }
+}
+
+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,
+ }
+ const winc, einc = 5, 100 // quick test (<1s)
+ //const winc, einc = 1, 1 // soak test (~75s)
+ for _, sign := range "+-" {
+ for _, a := range add {
+ for wid := uint64(0); wid < 60; wid += winc {
+ b := int64(1<<wid + a)
+ if sign == '-' {
+ b = -b
+ }
+ for exp := -1100; exp < 1100; exp += einc {
+ num, den := NewInt(b), NewInt(1)
+ if exp > 0 {
+ num.Lsh(num, uint(exp))
+ } else {
+ den.Lsh(den, uint(-exp))
+ }
+ r := new(Rat).SetFrac(num, den)
+ f, _ := r.Float64()
+
+ if !checkIsBestApprox(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)
+ }
+
+ checkNonLossyRoundtrip(t, f)
+ }
+ }
+ }
+ }
+}
+
+// TestFloat64NonFinite 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)
+ }
+ }
+}
+
+// checkNonLossyRoundtrip checks that a float->Rat->float roundtrip is
+// non-lossy for finite f.
+func checkNonLossyRoundtrip(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)
+}
+
+// checkIsBestApprox checks that f is the best possible float64
+// approximation of r.
+// Returns true on success.
+func checkIsBestApprox(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 && !isEven(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 && !isEven(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 isEven(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)
+ }
+ }
+}