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// Copyright 2025 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 strconv
// Binary to decimal conversion using the Dragonbox algorithm by Junekey Jeon.
//
// Fixed precision format is not supported by the Dragonbox algorithm
// so we continue to use Ryū-printf for this purpose.
// See https://github.com/jk-jeon/dragonbox/issues/38 for more details.
//
// For binary to decimal rounding, uses round to nearest, tie to even.
// For decimal to binary rounding, assumes round to nearest, tie to even.
//
// The original paper by Junekey Jeon can be found at:
// https://github.com/jk-jeon/dragonbox/blob/d5dc40ae6a3f1a4559cda816738df2d6255b4e24/other_files/Dragonbox.pdf
//
// The reference implementation in C++ by Junekey Jeon can be found at:
// https://github.com/jk-jeon/dragonbox/blob/6c7c925b571d54486b9ffae8d9d18a822801cbda/subproject/simple/include/simple_dragonbox.h
// dragonboxFtoa computes the decimal significand and exponent
// from the binary significand and exponent using the Dragonbox algorithm
// and formats the decimal floating point number in d.
func dboxFtoa(d *decimalSlice, mant uint64, exp int, denorm bool, bitSize int) {
if bitSize == 32 {
dboxFtoa32(d, uint32(mant), exp, denorm)
return
}
dboxFtoa64(d, mant, exp, denorm)
}
func dboxFtoa64(d *decimalSlice, mant uint64, exp int, denorm bool) {
if mant == 1<<float64MantBits && !denorm {
// Algorithm 5.6 (page 24).
k0 := -mulLog10_2MinusLog10_4Over3(exp)
pb := dboxPow64(k0, exp)
xi, zi := dboxRange64(pb.phi, pb.beta)
if exp != 2 && exp != 3 {
xi++
}
q := zi / 10
if xi <= q*10 {
q, zeros := trimZeros(q)
dboxDigits(d, q, -k0+1+zeros)
return
}
yru := dboxRoundUp64(pb.phi, pb.beta)
if exp == -77 && yru%2 != 0 {
yru--
} else if yru < xi {
yru++
}
dboxDigits(d, yru, -k0)
return
}
// κ = 2 for float64 (section 5.1.3)
const (
κ = 2
p10κ = 100 // 10**κ
p10κ1 = p10κ * 10 // 10**(κ+1)
)
// Algorithm 5.2 (page 15).
k0 := -mulLog10_2(exp)
pb := dboxPow64(κ+k0, exp)
zi, exact := dboxMulPow64(uint64(mant*2+1)<<pb.beta, pb.phi)
s, r := zi/p10κ1, uint32(zi%p10κ1)
δi := dboxDelta64(pb.phi, pb.beta)
if r < δi {
if r != 0 || !exact || mant%2 == 0 {
s, zeros := trimZeros(s)
dboxDigits(d, s, -k0+1+zeros)
return
}
s--
r = p10κ * 10
} else if r == δi {
parity, exact := dboxParity64(uint64(mant*2-1), pb.phi, pb.beta)
if parity || (exact && mant%2 == 0) {
s, zeros := trimZeros(s)
dboxDigits(d, s, -k0+1+zeros)
return
}
}
// Algorithm 5.4 (page 18).
D := r + p10κ/2 - δi/2
t, rho := D/p10κ, D%p10κ
yru := 10*s + uint64(t)
if rho == 0 {
parity, exact := dboxParity64(mant*2, pb.phi, pb.beta)
if parity != ((D-p10κ/2)%2 != 0) || (exact && yru%2 != 0) {
yru--
}
}
dboxDigits(d, yru, -k0)
}
// Almost identical to dragonboxFtoa64.
// This is kept as a separate copy to minimize runtime overhead.
func dboxFtoa32(d *decimalSlice, mant uint32, exp int, denorm bool) {
if mant == 1<<float32MantBits && !denorm {
// Algorithm 5.6 (page 24).
k0 := -mulLog10_2MinusLog10_4Over3(exp)
phi, beta := dboxPow32(k0, exp)
xi, zi := dboxRange32(phi, beta)
if exp != 2 && exp != 3 {
xi++
}
q := zi / 10
if xi <= q*10 {
q, zeros := trimZeros(uint64(q))
dboxDigits(d, q, -k0+1+zeros)
return
}
yru := dboxRoundUp32(phi, beta)
if exp == -77 && yru%2 != 0 {
yru--
} else if yru < xi {
yru++
}
dboxDigits(d, uint64(yru), -k0)
return
}
// κ = 1 for float32 (section 5.1.3)
const (
κ = 1
p10κ = 10
p10κ1 = p10κ * 10
)
// Algorithm 5.2 (page 15).
k0 := -mulLog10_2(exp)
phi, beta := dboxPow32(κ+k0, exp)
zi, exact := dboxMulPow32(uint32(mant*2+1)<<beta, phi)
s, r := zi/p10κ1, uint32(zi%p10κ1)
δi := dboxDelta32(phi, beta)
if r < δi {
if r != 0 || !exact || mant%2 == 0 {
s, zeros := trimZeros(uint64(s))
dboxDigits(d, s, -k0+1+zeros)
return
}
s--
r = p10κ * 10
} else if r == δi {
parity, exact := dboxParity32(uint32(mant*2-1), phi, beta)
if parity || (exact && mant%2 == 0) {
s, zeros := trimZeros(uint64(s))
dboxDigits(d, s, -k0+1+zeros)
return
}
}
// Algorithm 5.4 (page 18).
D := r + p10κ/2 - δi/2
t, rho := D/p10κ, D%p10κ
yru := 10*s + uint32(t)
if rho == 0 {
parity, exact := dboxParity32(mant*2, phi, beta)
if parity != ((D-p10κ/2)%2 != 0) || (exact && yru%2 != 0) {
yru--
}
}
dboxDigits(d, uint64(yru), -k0)
}
// dboxDigits emits decimal digits of mant in d for float64
// and adjusts the decimal point based on exp.
func dboxDigits(d *decimalSlice, mant uint64, exp int) {
i := formatBase10(d.d, mant)
d.d = d.d[i:]
d.nd = len(d.d)
d.dp = d.nd + exp
}
// uadd128 returns the full 128 bits of u + n.
func uadd128(u uint128, n uint64) uint128 {
sum := uint64(u.Lo + n)
// Check if lo is wrapped around.
if sum < u.Lo {
u.Hi++
}
u.Lo = sum
return u
}
// umul64 returns the full 64 bits of x * y.
func umul64(x, y uint32) uint64 {
return uint64(x) * uint64(y)
}
// umul96Upper64 returns the upper 64 bits (out of 96 bits) of x * y.
func umul96Upper64(x uint32, y uint64) uint64 {
yh := uint32(y >> 32)
yl := uint32(y)
xyh := umul64(x, yh)
xyl := umul64(x, yl)
return xyh + (xyl >> 32)
}
// umul96Lower64 returns the lower 64 bits (out of 96 bits) of x * y.
func umul96Lower64(x uint32, y uint64) uint64 {
return uint64(uint64(x) * y)
}
// umul128Upper64 returns the upper 64 bits (out of 128 bits) of x * y.
func umul128Upper64(x, y uint64) uint64 {
a := uint32(x >> 32)
b := uint32(x)
c := uint32(y >> 32)
d := uint32(y)
ac := umul64(a, c)
bc := umul64(b, c)
ad := umul64(a, d)
bd := umul64(b, d)
intermediate := (bd >> 32) + uint64(uint32(ad)) + uint64(uint32(bc))
return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32)
}
// umul192Upper128 returns the upper 128 bits (out of 192 bits) of x * y.
func umul192Upper128(x uint64, y uint128) uint128 {
r := umul128(x, y.Hi)
t := umul128Upper64(x, y.Lo)
return uadd128(r, t)
}
// umul192Lower128 returns the lower 128 bits (out of 192 bits) of x * y.
func umul192Lower128(x uint64, y uint128) uint128 {
high := x * y.Hi
highLow := umul128(x, y.Lo)
return uint128{uint64(high + highLow.Hi), highLow.Lo}
}
// dboxMulPow64 computes x^(i), y^(i), z^(i)
// from the precomputed value of φ̃k for float64
// and also checks if x^(f), y^(f), z^(f) == 0 (section 5.2.1).
func dboxMulPow64(u uint64, phi uint128) (uint64, bool) {
r := umul192Upper128(u, phi)
intPart := r.Hi
isInt := r.Lo == 0
return intPart, isInt
}
// dboxMulPow32 computes x^(i), y^(i), z^(i)
// from the precomputed value of φ̃k for float32
// and also checks if x^(f), y^(f), z^(f) == 0 (section 5.2.1).
func dboxMulPow32(u uint32, phi uint64) (uint32, bool) {
r := umul96Upper64(u, phi)
intPart := uint32(r >> 32)
isInt := uint32(r) == 0
return intPart, isInt
}
// dboxParity64 computes only the parity of x^(i), y^(i), z^(i)
// from the precomputed value of φ̃k for float64
// and also checks if x^(f), y^(f), z^(f) = 0 (section 5.2.1).
func dboxParity64(mant2 uint64, phi uint128, beta int) (bool, bool) {
r := umul192Lower128(mant2, phi)
parity := ((r.Hi >> (64 - beta)) & 1) != 0
isInt := ((uint64(r.Hi << beta)) | (r.Lo >> (64 - beta))) == 0
return parity, isInt
}
// dboxParity32 computes only the parity of x^(i), y^(i), z^(i)
// from the precomputed value of φ̃k for float32
// and also checks if x^(f), y^(f), z^(f) = 0 (section 5.2.1).
func dboxParity32(mant2 uint32, phi uint64, beta int) (bool, bool) {
r := umul96Lower64(mant2, phi)
parity := ((r >> (64 - beta)) & 1) != 0
isInt := uint32(r>>(32-beta)) == 0
return parity, isInt
}
// dboxDelta64 returns δ^(i) from the precomputed value of φ̃k for float64.
func dboxDelta64(phi uint128, beta int) uint32 {
return uint32(phi.Hi >> (64 - 1 - beta))
}
// dboxDelta32 returns δ^(i) from the precomputed value of φ̃k for float32.
func dboxDelta32(phi uint64, beta int) uint32 {
return uint32(phi >> (64 - 1 - beta))
}
// mulLog10_2MinusLog10_4Over3 computes
// ⌊e*log10(2)-log10(4/3)⌋ = ⌊log10(2^e)-log10(4/3)⌋ (section 6.3).
func mulLog10_2MinusLog10_4Over3(e int) int {
// e should be in the range [-2985, 2936].
return (e*631305 - 261663) >> 21
}
const (
floatMantBits64 = 52 // p = 52 for float64.
floatMantBits32 = 23 // p = 23 for float32.
)
// dboxRange64 returns the left and right float64 endpoints.
func dboxRange64(phi uint128, beta int) (uint64, uint64) {
left := (phi.Hi - (phi.Hi >> (floatMantBits64 + 2))) >> (64 - floatMantBits64 - 1 - beta)
right := (phi.Hi + (phi.Hi >> (floatMantBits64 + 1))) >> (64 - floatMantBits64 - 1 - beta)
return left, right
}
// dboxRange32 returns the left and right float32 endpoints.
func dboxRange32(phi uint64, beta int) (uint32, uint32) {
left := uint32((phi - (phi >> (floatMantBits32 + 2))) >> (64 - floatMantBits32 - 1 - beta))
right := uint32((phi + (phi >> (floatMantBits32 + 1))) >> (64 - floatMantBits32 - 1 - beta))
return left, right
}
// dboxRoundUp64 computes the round up of y (i.e., y^(ru)).
func dboxRoundUp64(phi uint128, beta int) uint64 {
return (phi.Hi>>(128/2-floatMantBits64-2-beta) + 1) / 2
}
// dboxRoundUp32 computes the round up of y (i.e., y^(ru)).
func dboxRoundUp32(phi uint64, beta int) uint32 {
return uint32(phi>>(64-floatMantBits32-2-beta)+1) / 2
}
// dboxPow64 gets the precomputed value of φ̃̃k for float64.
func dboxPow64(k, e int) phiBeta {
powr := intPow10(k)
phi, e1 := powr.mant, powr.exp
if k < 0 || k > 55 {
phi.Lo++
}
beta := e + e1 - 1
return phiBeta{phi, beta}
}
// dboxPow32 gets the precomputed value of φ̃̃k for float32.
func dboxPow32(k, e int) (uint64, int) {
powr := intPow10(k)
m, e1 := powr.mant, powr.exp
if k < 0 || k > 27 {
m.Hi++
}
exp := e + e1 - 1
return m.Hi, exp
}