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Commit be91bc29 authored by Robert Griesemer's avatar Robert Griesemer
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math/big: implement Rat.Float32 

Pending CL 101750048.
For submission after the 1.3 release.

Fixes #8065.

LGTM=adonovan
R=adonovan
CC=golang-codereviews
https://golang.org/cl/93550043
parent a9035ede
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Showing with 350 additions and 59 deletions
src/pkg/math/big/int.go
View file @ be91bc29
......@@ -510,10 +510,30 @@ func (z *Int) Scan(s fmt.ScanState, ch rune) error {
return err
}
// 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(x.Uint64())
v := int64(low64(x.abs))
if x.neg {
v = -v
}
......@@ -523,14 +543,7 @@ func (x *Int) Int64() int64 {
// Uint64 returns the uint64 representation of x.
// If x cannot be represented in a uint64, the result is undefined.
func (x *Int) Uint64() uint64 {
if len(x.abs) == 0 {
return 0
}
v := uint64(x.abs[0])
if _W == 32 && len(x.abs) > 1 {
v |= uint64(x.abs[1]) << 32
}
return v
return low64(x.abs)
}
// SetString sets z to the value of s, interpreted in the given base,
......
src/pkg/math/big/rat.go
View file @ be91bc29
......@@ -64,28 +64,125 @@ func (z *Rat) SetFloat64(f float64) *Rat {
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
}
// quotToFloat32 returns the non-negative float32 value
// nearest to the quotient a/b, using round-to-even in
// halfway cases. It does not mutate its arguments.
// Preconditions: b is non-zero; a and b have no common factors.
func quotToFloat32(a, b nat) (f float32, exact bool) {
const (
// float size in bits
Fsize = 32
// mantissa
Msize = 23
Msize1 = Msize + 1 // incl. implicit 1
Msize2 = Msize1 + 1
// exponent
Esize = Fsize - Msize1
Ebias = 1<<(Esize-1) - 1
Emin = 1 - Ebias
Emax = Ebias
)
// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
alen := a.bitLen()
if alen == 0 {
return 0, true
}
blen := b.bitLen()
if blen == 0 {
panic("division by zero")
}
// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
// This is 2 or 3 more than the float32 mantissa field width of Msize:
// - the optional extra bit is shifted away in step 3 below.
// - the high-order 1 is omitted in "normal" representation;
// - the low-order 1 will be used during rounding then discarded.
exp := alen - blen
var a2, b2 nat
a2 = a2.set(a)
b2 = b2.set(b)
if shift := Msize2 - exp; shift > 0 {
a2 = a2.shl(a2, uint(shift))
} else if shift < 0 {
b2 = b2.shl(b2, uint(-shift))
}
// 2. Compute quotient and remainder (q, r). NB: due to the
// extra shift, the low-order bit of q is logically the
// high-order bit of r.
var q nat
q, r := q.div(a2, a2, b2) // (recycle a2)
mantissa := low32(q)
haveRem := len(r) > 0 // mantissa&1 && !haveRem => remainder is exactly half
// low64 returns the least significant 64 bits of natural number z.
func low64(z nat) uint64 {
if len(z) == 0 {
return 0
// 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 _W == 32 && len(z) > 1 {
return uint64(z[1])<<32 | uint64(z[0])
if mantissa>>Msize1 != 1 {
panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
}
return uint64(z[0])
// 4. Rounding.
if Emin-Msize <= exp && exp <= Emin {
// Denormal case; lose 'shift' bits of precision.
shift := uint(Emin - (exp - 1)) // [1..Esize1)
lostbits := mantissa & (1<<shift - 1)
haveRem = haveRem || lostbits != 0
mantissa >>= shift
exp = 2 - Ebias // == exp + shift
}
// Round q using round-half-to-even.
exact = !haveRem
if mantissa&1 != 0 {
exact = false
if haveRem || mantissa&2 != 0 {
if mantissa++; mantissa >= 1<<Msize2 {
// Complete rollover 11...1 => 100...0, so shift is safe
mantissa >>= 1
exp++
}
}
}
mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
f = float32(math.Ldexp(float64(mantissa), exp-Msize1))
if math.IsInf(float64(f), 0) {
exact = false
}
return
}
// 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.
// quotToFloat64 returns the non-negative float64 value
// nearest to the quotient a/b, using round-to-even in
// halfway cases. It does not mutate its arguments.
// Preconditions: b is non-zero; a and b have no common factors.
func quotToFloat(a, b nat) (f float64, exact bool) {
func quotToFloat64(a, b nat) (f float64, exact bool) {
const (
// float size in bits
Fsize = 64
// mantissa
Msize = 52
Msize1 = Msize + 1 // incl. implicit 1
Msize2 = Msize1 + 1
// exponent
Esize = Fsize - Msize1
Ebias = 1<<(Esize-1) - 1
Emin = 1 - Ebias
Emax = Ebias
)
// TODO(adonovan): specialize common degenerate cases: 1.0, integers.
alen := a.bitLen()
if alen == 0 {
......@@ -96,17 +193,17 @@ func quotToFloat(a, b nat) (f float64, exact bool) {
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:
// 1. Left-shift A or B such that quotient A/B is in [1<<Msize1, 1<<(Msize2+1)
// (Msize2 bits if A < B when they are left-aligned, Msize2+1 bits if A >= B).
// This is 2 or 3 more than the float64 mantissa field width of Msize:
// - the optional extra bit is shifted away in step 3 below.
// - the high-order 1 is omitted in float64 "normal" representation;
// - 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 := 54 - exp; shift > 0 {
if shift := Msize2 - exp; shift > 0 {
a2 = a2.shl(a2, uint(shift))
} else if shift < 0 {
b2 = b2.shl(b2, uint(-shift))
......@@ -120,49 +217,65 @@ func quotToFloat(a, b nat) (f float64, exact bool) {
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
// 3. If quotient didn't fit in Msize2 bits, redo division by b2<<1
// (in effect---we accomplish this incrementally).
if mantissa>>54 == 1 {
if mantissa>>Msize2 == 1 {
if mantissa&1 == 1 {
haveRem = true
}
mantissa >>= 1
exp++
}
if mantissa>>53 != 1 {
panic("expected exactly 54 bits of result")
if mantissa>>Msize1 != 1 {
panic(fmt.Sprintf("expected exactly %d bits of result", Msize2))
}
// 4. Rounding.
if -1022-52 <= exp && exp <= -1022 {
if Emin-Msize <= exp && exp <= Emin {
// Denormal case; lose 'shift' bits of precision.
shift := uint64(-1022 - (exp - 1)) // [1..53)
shift := uint(Emin - (exp - 1)) // [1..Esize1)
lostbits := mantissa & (1<<shift - 1)
haveRem = haveRem || lostbits != 0
mantissa >>= shift
exp = -1023 + 2
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<<54 {
if mantissa++; mantissa >= 1<<Msize2 {
// Complete rollover 11...1 => 100...0, so shift is safe
mantissa >>= 1
exp++
}
}
}
mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 2^53.
mantissa >>= 1 // discard rounding bit. Mantissa now scaled by 1<<Msize1.
f = math.Ldexp(float64(mantissa), exp-53)
f = math.Ldexp(float64(mantissa), exp-Msize1)
if math.IsInf(f, 0) {
exact = false
}
return
}
// Float32 returns the nearest float32 value for x and a bool indicating
// whether f represents x exactly. If the magnitude of x is too large to
// be represented by a float32, f is an infinity and exact is false.
// The sign of f always matches the sign of x, even if f == 0.
func (x *Rat) Float32() (f float32, exact bool) {
b := x.b.abs
if len(b) == 0 {
b = b.set(natOne) // materialize denominator
}
f, exact = quotToFloat32(x.a.abs, b)
if x.a.neg {
f = -f
}
return
}
// Float64 returns the nearest float64 value for x and a bool indicating
// whether f represents x exactly. If the magnitude of x is too large to
// be represented by a float64, f is an infinity and exact is false.
......@@ -172,7 +285,7 @@ func (x *Rat) Float64() (f float64, exact bool) {
if len(b) == 0 {
b = b.set(natOne) // materialize denominator
}
f, exact = quotToFloat(x.a.abs, b)
f, exact = quotToFloat64(x.a.abs, b)
if x.a.neg {
f = -f
}
......
src/pkg/math/big/rat_test.go
View file @ be91bc29
......@@ -751,7 +751,6 @@ var float64inputs = []string{
// 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).
......@@ -761,7 +760,7 @@ var float64inputs = []string{
"22.222222222222222",
"long:2." + strings.Repeat("2", 4000) + "e+1",
// Exactly halfway between 1 and math.Nextafter(1, 2).
// Exactly halfway between 1 and math.Nextafter64(1, 2).
// Round to even (down).
"1.00000000000000011102230246251565404236316680908203125",
// Slightly lower; still round down.
......@@ -790,6 +789,68 @@ var float64inputs = []string{
"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:") {
......@@ -830,13 +891,13 @@ func TestFloat64SpecialCases(t *testing.T) {
}
// 2. Check f is best approximation to r.
if !checkIsBestApprox(t, f, r) {
if !checkIsBestApprox64(t, f, r) {
// Append context information.
t.Errorf("(input was %q)", input)
}
// 3. Check f->R->f roundtrip is non-lossy.
checkNonLossyRoundtrip(t, f)
checkNonLossyRoundtrip64(t, f)
// 4. Check exactness using slow algorithm.
if wasExact := new(Rat).SetFloat64(f).Cmp(r) == 0; wasExact != exact {
......@@ -845,6 +906,54 @@ func TestFloat64SpecialCases(t *testing.T) {
}
}
func TestFloat32Distribution(t *testing.T) {
// Generate a distribution of (sign, mantissa, exp) values
// broader than the float32 range, and check Rat.Float32()
// always picks the closest float32 approximation.
var add = []int64{
0,
1,
3,
5,
7,
9,
11,
}
var winc, einc = uint64(1), 1 // soak test (~1.5s on x86-64)
if testing.Short() {
winc, einc = 5, 15 // quick test (~60ms on x86-64)
}
for _, sign := range "+-" {
for _, a := range add {
for wid := uint64(0); wid < 30; wid += winc {
b := 1<<wid + a
if sign == '-' {
b = -b
}
for exp := -150; exp < 150; exp += einc {
num, den := NewInt(b), NewInt(1)
if exp > 0 {
num.Lsh(num, uint(exp))
} else {
den.Lsh(den, uint(-exp))
}
r := new(Rat).SetFrac(num, den)
f, _ := r.Float32()
if !checkIsBestApprox32(t, f, r) {
// Append context information.
t.Errorf("(input was mantissa %#x, exp %d; f = %g (%b); f ~ %g; r = %v)",
b, exp, f, f, math.Ldexp(float64(b), exp), r)
}
checkNonLossyRoundtrip32(t, f)
}
}
}
}
}
func TestFloat64Distribution(t *testing.T) {
// Generate a distribution of (sign, mantissa, exp) values
// broader than the float64 range, and check Rat.Float64()
......@@ -858,7 +967,7 @@ func TestFloat64Distribution(t *testing.T) {
9,
11,
}
var winc, einc = uint64(1), int(1) // soak test (~75s on x86-64)
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)
}
......@@ -866,7 +975,7 @@ func TestFloat64Distribution(t *testing.T) {
for _, sign := range "+-" {
for _, a := range add {
for wid := uint64(0); wid < 60; wid += winc {
b := int64(1<<wid + a)
b := 1<<wid + a
if sign == '-' {
b = -b
}
......@@ -880,20 +989,20 @@ func TestFloat64Distribution(t *testing.T) {
r := new(Rat).SetFrac(num, den)
f, _ := r.Float64()
if !checkIsBestApprox(t, f, r) {
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)
}
checkNonLossyRoundtrip(t, f)
checkNonLossyRoundtrip64(t, f)
}
}
}
}
}
// TestFloat64NonFinite checks that SetFloat64 of a non-finite value
// 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)} {
......@@ -904,9 +1013,27 @@ func TestSetFloat64NonFinite(t *testing.T) {
}
}
// checkNonLossyRoundtrip checks that a float->Rat->float roundtrip is
// checkNonLossyRoundtrip32 checks that a float->Rat->float roundtrip is
// non-lossy for finite f.
func checkNonLossyRoundtrip(t *testing.T, f float64) {
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
}
......@@ -928,10 +1055,47 @@ func delta(r *Rat, f float64) *Rat {
return d.Abs(d)
}
// checkIsBestApprox checks that f is the best possible float64
// 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 checkIsBestApprox(t *testing.T, f float64, r *Rat) bool {
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.
......@@ -939,8 +1103,8 @@ func checkIsBestApprox(t *testing.T, f float64, r *Rat) bool {
}
// 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))
f0 := math.Nextafter64(f, math.Inf(-1))
f1 := math.Nextafter64(f, math.Inf(+1))
// For f to be correct, r must be closer to f than to f0 or f1.
df := delta(r, f)
......@@ -954,18 +1118,19 @@ func checkIsBestApprox(t *testing.T, f float64, r *Rat) bool {
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) {
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 && !isEven(f) {
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 isEven(f float64) bool { return math.Float64bits(f)&1 == 0 }
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{
......
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