568 lines
16 KiB
Go
568 lines
16 KiB
Go
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// Copyright 2021 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package strconv
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import (
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"math/bits"
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)
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// binary to decimal conversion using the Ryū algorithm.
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//
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// See Ulf Adams, "Ryū: Fast Float-to-String Conversion" (doi:10.1145/3192366.3192369)
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//
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// Fixed precision formatting is a variant of the original paper's
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// algorithm, where a single multiplication by 10^k is required,
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// sharing the same rounding guarantees.
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// ryuFtoaFixed32 formats mant*(2^exp) with prec decimal digits.
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func ryuFtoaFixed32(d *decimalSlice, mant uint32, exp int, prec int) {
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if prec < 0 {
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panic("ryuFtoaFixed32 called with negative prec")
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}
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if prec > 9 {
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panic("ryuFtoaFixed32 called with prec > 9")
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}
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// Zero input.
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if mant == 0 {
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d.nd, d.dp = 0, 0
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return
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}
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// Renormalize to a 25-bit mantissa.
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e2 := exp
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if b := bits.Len32(mant); b < 25 {
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mant <<= uint(25 - b)
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e2 += int(b) - 25
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}
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// Choose an exponent such that rounded mant*(2^e2)*(10^q) has
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// at least prec decimal digits, i.e
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// mant*(2^e2)*(10^q) >= 10^(prec-1)
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// Because mant >= 2^24, it is enough to choose:
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// 2^(e2+24) >= 10^(-q+prec-1)
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// or q = -mulByLog2Log10(e2+24) + prec - 1
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q := -mulByLog2Log10(e2+24) + prec - 1
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// Now compute mant*(2^e2)*(10^q).
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// Is it an exact computation?
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// Only small positive powers of 10 are exact (5^28 has 66 bits).
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exact := q <= 27 && q >= 0
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di, dexp2, d0 := mult64bitPow10(mant, e2, q)
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if dexp2 >= 0 {
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panic("not enough significant bits after mult64bitPow10")
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}
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// As a special case, computation might still be exact, if exponent
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// was negative and if it amounts to computing an exact division.
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// In that case, we ignore all lower bits.
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// Note that division by 10^11 cannot be exact as 5^11 has 26 bits.
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if q < 0 && q >= -10 && divisibleByPower5(uint64(mant), -q) {
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exact = true
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d0 = true
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}
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// Remove extra lower bits and keep rounding info.
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extra := uint(-dexp2)
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extraMask := uint32(1<<extra - 1)
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di, dfrac := di>>extra, di&extraMask
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roundUp := false
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if exact {
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// If we computed an exact product, d + 1/2
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// should round to d+1 if 'd' is odd.
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roundUp = dfrac > 1<<(extra-1) ||
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(dfrac == 1<<(extra-1) && !d0) ||
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(dfrac == 1<<(extra-1) && d0 && di&1 == 1)
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} else {
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// otherwise, d+1/2 always rounds up because
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// we truncated below.
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roundUp = dfrac>>(extra-1) == 1
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}
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if dfrac != 0 {
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d0 = false
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}
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// Proceed to the requested number of digits
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formatDecimal(d, uint64(di), !d0, roundUp, prec)
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// Adjust exponent
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d.dp -= q
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}
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// ryuFtoaFixed64 formats mant*(2^exp) with prec decimal digits.
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func ryuFtoaFixed64(d *decimalSlice, mant uint64, exp int, prec int) {
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if prec > 18 {
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panic("ryuFtoaFixed64 called with prec > 18")
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}
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// Zero input.
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if mant == 0 {
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d.nd, d.dp = 0, 0
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return
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}
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// Renormalize to a 55-bit mantissa.
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e2 := exp
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if b := bits.Len64(mant); b < 55 {
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mant = mant << uint(55-b)
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e2 += int(b) - 55
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}
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// Choose an exponent such that rounded mant*(2^e2)*(10^q) has
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// at least prec decimal digits, i.e
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// mant*(2^e2)*(10^q) >= 10^(prec-1)
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// Because mant >= 2^54, it is enough to choose:
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// 2^(e2+54) >= 10^(-q+prec-1)
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// or q = -mulByLog2Log10(e2+54) + prec - 1
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//
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// The minimal required exponent is -mulByLog2Log10(1025)+18 = -291
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// The maximal required exponent is mulByLog2Log10(1074)+18 = 342
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q := -mulByLog2Log10(e2+54) + prec - 1
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// Now compute mant*(2^e2)*(10^q).
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// Is it an exact computation?
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// Only small positive powers of 10 are exact (5^55 has 128 bits).
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exact := q <= 55 && q >= 0
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di, dexp2, d0 := mult128bitPow10(mant, e2, q)
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if dexp2 >= 0 {
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panic("not enough significant bits after mult128bitPow10")
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}
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// As a special case, computation might still be exact, if exponent
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// was negative and if it amounts to computing an exact division.
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// In that case, we ignore all lower bits.
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// Note that division by 10^23 cannot be exact as 5^23 has 54 bits.
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if q < 0 && q >= -22 && divisibleByPower5(mant, -q) {
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exact = true
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d0 = true
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}
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// Remove extra lower bits and keep rounding info.
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extra := uint(-dexp2)
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extraMask := uint64(1<<extra - 1)
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di, dfrac := di>>extra, di&extraMask
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roundUp := false
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if exact {
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// If we computed an exact product, d + 1/2
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// should round to d+1 if 'd' is odd.
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roundUp = dfrac > 1<<(extra-1) ||
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(dfrac == 1<<(extra-1) && !d0) ||
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(dfrac == 1<<(extra-1) && d0 && di&1 == 1)
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} else {
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// otherwise, d+1/2 always rounds up because
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// we truncated below.
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roundUp = dfrac>>(extra-1) == 1
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}
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if dfrac != 0 {
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d0 = false
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}
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// Proceed to the requested number of digits
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formatDecimal(d, di, !d0, roundUp, prec)
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// Adjust exponent
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d.dp -= q
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}
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var uint64pow10 = [...]uint64{
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1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
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1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
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}
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// formatDecimal fills d with at most prec decimal digits
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// of mantissa m. The boolean trunc indicates whether m
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// is truncated compared to the original number being formatted.
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func formatDecimal(d *decimalSlice, m uint64, trunc bool, roundUp bool, prec int) {
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max := uint64pow10[prec]
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trimmed := 0
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for m >= max {
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a, b := m/10, m%10
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m = a
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trimmed++
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if b > 5 {
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roundUp = true
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} else if b < 5 {
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roundUp = false
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} else { // b == 5
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// round up if there are trailing digits,
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// or if the new value of m is odd (round-to-even convention)
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roundUp = trunc || m&1 == 1
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}
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if b != 0 {
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trunc = true
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}
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}
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if roundUp {
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m++
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}
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if m >= max {
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// Happens if di was originally 99999....xx
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m /= 10
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trimmed++
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}
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// render digits (similar to formatBits)
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n := uint(prec)
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d.nd = int(prec)
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v := m
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for v >= 100 {
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var v1, v2 uint64
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if v>>32 == 0 {
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v1, v2 = uint64(uint32(v)/100), uint64(uint32(v)%100)
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} else {
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v1, v2 = v/100, v%100
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}
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n -= 2
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d.d[n+1] = smallsString[2*v2+1]
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d.d[n+0] = smallsString[2*v2+0]
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v = v1
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}
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if v > 0 {
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n--
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d.d[n] = smallsString[2*v+1]
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}
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if v >= 10 {
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n--
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d.d[n] = smallsString[2*v]
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}
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for d.d[d.nd-1] == '0' {
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d.nd--
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trimmed++
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}
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d.dp = d.nd + trimmed
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}
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// ryuFtoaShortest formats mant*2^exp with prec decimal digits.
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func ryuFtoaShortest(d *decimalSlice, mant uint64, exp int, flt *floatInfo) {
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if mant == 0 {
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d.nd, d.dp = 0, 0
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return
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}
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// If input is an exact integer with fewer bits than the mantissa,
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// the previous and next integer are not admissible representations.
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if exp <= 0 && bits.TrailingZeros64(mant) >= -exp {
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mant >>= uint(-exp)
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ryuDigits(d, mant, mant, mant, true, false)
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return
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}
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ml, mc, mu, e2 := computeBounds(mant, exp, flt)
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if e2 == 0 {
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ryuDigits(d, ml, mc, mu, true, false)
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return
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}
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// Find 10^q *larger* than 2^-e2
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q := mulByLog2Log10(-e2) + 1
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// We are going to multiply by 10^q using 128-bit arithmetic.
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// The exponent is the same for all 3 numbers.
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var dl, dc, du uint64
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var dl0, dc0, du0 bool
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if flt == &float32info {
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var dl32, dc32, du32 uint32
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dl32, _, dl0 = mult64bitPow10(uint32(ml), e2, q)
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dc32, _, dc0 = mult64bitPow10(uint32(mc), e2, q)
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du32, e2, du0 = mult64bitPow10(uint32(mu), e2, q)
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dl, dc, du = uint64(dl32), uint64(dc32), uint64(du32)
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} else {
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dl, _, dl0 = mult128bitPow10(ml, e2, q)
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dc, _, dc0 = mult128bitPow10(mc, e2, q)
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du, e2, du0 = mult128bitPow10(mu, e2, q)
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}
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if e2 >= 0 {
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panic("not enough significant bits after mult128bitPow10")
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}
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// Is it an exact computation?
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if q > 55 {
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// Large positive powers of ten are not exact
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dl0, dc0, du0 = false, false, false
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}
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if q < 0 && q >= -24 {
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// Division by a power of ten may be exact.
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// (note that 5^25 is a 59-bit number so division by 5^25 is never exact).
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if divisibleByPower5(ml, -q) {
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dl0 = true
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}
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if divisibleByPower5(mc, -q) {
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dc0 = true
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}
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if divisibleByPower5(mu, -q) {
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du0 = true
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}
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}
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// Express the results (dl, dc, du)*2^e2 as integers.
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// Extra bits must be removed and rounding hints computed.
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extra := uint(-e2)
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extraMask := uint64(1<<extra - 1)
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// Now compute the floored, integral base 10 mantissas.
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dl, fracl := dl>>extra, dl&extraMask
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dc, fracc := dc>>extra, dc&extraMask
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du, fracu := du>>extra, du&extraMask
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// Is it allowed to use 'du' as a result?
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// It is always allowed when it is truncated, but also
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// if it is exact and the original binary mantissa is even
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// When disallowed, we can subtract 1.
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uok := !du0 || fracu > 0
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if du0 && fracu == 0 {
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uok = mant&1 == 0
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}
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if !uok {
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du--
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}
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// Is 'dc' the correctly rounded base 10 mantissa?
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// The correct rounding might be dc+1
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cup := false // don't round up.
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if dc0 {
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// If we computed an exact product, the half integer
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// should round to next (even) integer if 'dc' is odd.
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cup = fracc > 1<<(extra-1) ||
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(fracc == 1<<(extra-1) && dc&1 == 1)
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} else {
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// otherwise, the result is a lower truncation of the ideal
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// result.
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cup = fracc>>(extra-1) == 1
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}
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// Is 'dl' an allowed representation?
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// Only if it is an exact value, and if the original binary mantissa
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// was even.
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lok := dl0 && fracl == 0 && (mant&1 == 0)
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if !lok {
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dl++
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}
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// We need to remember whether the trimmed digits of 'dc' are zero.
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c0 := dc0 && fracc == 0
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// render digits
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ryuDigits(d, dl, dc, du, c0, cup)
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d.dp -= q
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}
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// mulByLog2Log10 returns math.Floor(x * log(2)/log(10)) for an integer x in
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// the range -1600 <= x && x <= +1600.
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//
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// The range restriction lets us work in faster integer arithmetic instead of
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// slower floating point arithmetic. Correctness is verified by unit tests.
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func mulByLog2Log10(x int) int {
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// log(2)/log(10) ≈ 0.30102999566 ≈ 78913 / 2^18
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return (x * 78913) >> 18
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}
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|
|
// mulByLog10Log2 returns math.Floor(x * log(10)/log(2)) for an integer x in
|
|||
|
|
// the range -500 <= x && x <= +500.
|
|||
|
|
//
|
|||
|
|
// The range restriction lets us work in faster integer arithmetic instead of
|
|||
|
|
// slower floating point arithmetic. Correctness is verified by unit tests.
|
|||
|
|
func mulByLog10Log2(x int) int {
|
|||
|
|
// log(10)/log(2) ≈ 3.32192809489 ≈ 108853 / 2^15
|
|||
|
|
return (x * 108853) >> 15
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// computeBounds returns a floating-point vector (l, c, u)×2^e2
|
|||
|
|
// where the mantissas are 55-bit (or 26-bit) integers, describing the interval
|
|||
|
|
// represented by the input float64 or float32.
|
|||
|
|
func computeBounds(mant uint64, exp int, flt *floatInfo) (lower, central, upper uint64, e2 int) {
|
|||
|
|
if mant != 1<<flt.mantbits || exp == flt.bias+1-int(flt.mantbits) {
|
|||
|
|
// regular case (or denormals)
|
|||
|
|
lower, central, upper = 2*mant-1, 2*mant, 2*mant+1
|
|||
|
|
e2 = exp - 1
|
|||
|
|
return
|
|||
|
|
} else {
|
|||
|
|
// border of an exponent
|
|||
|
|
lower, central, upper = 4*mant-1, 4*mant, 4*mant+2
|
|||
|
|
e2 = exp - 2
|
|||
|
|
return
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
func ryuDigits(d *decimalSlice, lower, central, upper uint64,
|
|||
|
|
c0, cup bool) {
|
|||
|
|
lhi, llo := divmod1e9(lower)
|
|||
|
|
chi, clo := divmod1e9(central)
|
|||
|
|
uhi, ulo := divmod1e9(upper)
|
|||
|
|
if uhi == 0 {
|
|||
|
|
// only low digits (for denormals)
|
|||
|
|
ryuDigits32(d, llo, clo, ulo, c0, cup, 8)
|
|||
|
|
} else if lhi < uhi {
|
|||
|
|
// truncate 9 digits at once.
|
|||
|
|
if llo != 0 {
|
|||
|
|
lhi++
|
|||
|
|
}
|
|||
|
|
c0 = c0 && clo == 0
|
|||
|
|
cup = (clo > 5e8) || (clo == 5e8 && cup)
|
|||
|
|
ryuDigits32(d, lhi, chi, uhi, c0, cup, 8)
|
|||
|
|
d.dp += 9
|
|||
|
|
} else {
|
|||
|
|
d.nd = 0
|
|||
|
|
// emit high part
|
|||
|
|
n := uint(9)
|
|||
|
|
for v := chi; v > 0; {
|
|||
|
|
v1, v2 := v/10, v%10
|
|||
|
|
v = v1
|
|||
|
|
n--
|
|||
|
|
d.d[n] = byte(v2 + '0')
|
|||
|
|
}
|
|||
|
|
d.d = d.d[n:]
|
|||
|
|
d.nd = int(9 - n)
|
|||
|
|
// emit low part
|
|||
|
|
ryuDigits32(d, llo, clo, ulo,
|
|||
|
|
c0, cup, d.nd+8)
|
|||
|
|
}
|
|||
|
|
// trim trailing zeros
|
|||
|
|
for d.nd > 0 && d.d[d.nd-1] == '0' {
|
|||
|
|
d.nd--
|
|||
|
|
}
|
|||
|
|
// trim initial zeros
|
|||
|
|
for d.nd > 0 && d.d[0] == '0' {
|
|||
|
|
d.nd--
|
|||
|
|
d.dp--
|
|||
|
|
d.d = d.d[1:]
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// ryuDigits32 emits decimal digits for a number less than 1e9.
|
|||
|
|
func ryuDigits32(d *decimalSlice, lower, central, upper uint32,
|
|||
|
|
c0, cup bool, endindex int) {
|
|||
|
|
if upper == 0 {
|
|||
|
|
d.dp = endindex + 1
|
|||
|
|
return
|
|||
|
|
}
|
|||
|
|
trimmed := 0
|
|||
|
|
// Remember last trimmed digit to check for round-up.
|
|||
|
|
// c0 will be used to remember zeroness of following digits.
|
|||
|
|
cNextDigit := 0
|
|||
|
|
for upper > 0 {
|
|||
|
|
// Repeatedly compute:
|
|||
|
|
// l = Ceil(lower / 10^k)
|
|||
|
|
// c = Round(central / 10^k)
|
|||
|
|
// u = Floor(upper / 10^k)
|
|||
|
|
// and stop when c goes out of the (l, u) interval.
|
|||
|
|
l := (lower + 9) / 10
|
|||
|
|
c, cdigit := central/10, central%10
|
|||
|
|
u := upper / 10
|
|||
|
|
if l > u {
|
|||
|
|
// don't trim the last digit as it is forbidden to go below l
|
|||
|
|
// other, trim and exit now.
|
|||
|
|
break
|
|||
|
|
}
|
|||
|
|
// Check that we didn't cross the lower boundary.
|
|||
|
|
// The case where l < u but c == l-1 is essentially impossible,
|
|||
|
|
// but may happen if:
|
|||
|
|
// lower = ..11
|
|||
|
|
// central = ..19
|
|||
|
|
// upper = ..31
|
|||
|
|
// and means that 'central' is very close but less than
|
|||
|
|
// an integer ending with many zeros, and usually
|
|||
|
|
// the "round-up" logic hides the problem.
|
|||
|
|
if l == c+1 && c < u {
|
|||
|
|
c++
|
|||
|
|
cdigit = 0
|
|||
|
|
cup = false
|
|||
|
|
}
|
|||
|
|
trimmed++
|
|||
|
|
// Remember trimmed digits of c
|
|||
|
|
c0 = c0 && cNextDigit == 0
|
|||
|
|
cNextDigit = int(cdigit)
|
|||
|
|
lower, central, upper = l, c, u
|
|||
|
|
}
|
|||
|
|
// should we round up?
|
|||
|
|
if trimmed > 0 {
|
|||
|
|
cup = cNextDigit > 5 ||
|
|||
|
|
(cNextDigit == 5 && !c0) ||
|
|||
|
|
(cNextDigit == 5 && c0 && central&1 == 1)
|
|||
|
|
}
|
|||
|
|
if central < upper && cup {
|
|||
|
|
central++
|
|||
|
|
}
|
|||
|
|
// We know where the number ends, fill directly
|
|||
|
|
endindex -= trimmed
|
|||
|
|
v := central
|
|||
|
|
n := endindex
|
|||
|
|
for n > d.nd {
|
|||
|
|
v1, v2 := v/100, v%100
|
|||
|
|
d.d[n] = smallsString[2*v2+1]
|
|||
|
|
d.d[n-1] = smallsString[2*v2+0]
|
|||
|
|
n -= 2
|
|||
|
|
v = v1
|
|||
|
|
}
|
|||
|
|
if n == d.nd {
|
|||
|
|
d.d[n] = byte(v + '0')
|
|||
|
|
}
|
|||
|
|
d.nd = endindex + 1
|
|||
|
|
d.dp = d.nd + trimmed
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// mult64bitPow10 takes a floating-point input with a 25-bit
|
|||
|
|
// mantissa and multiplies it with 10^q. The resulting mantissa
|
|||
|
|
// is m*P >> 57 where P is a 64-bit element of the detailedPowersOfTen tables.
|
|||
|
|
// It is typically 31 or 32-bit wide.
|
|||
|
|
// The returned boolean is true if all trimmed bits were zero.
|
|||
|
|
//
|
|||
|
|
// That is:
|
|||
|
|
// m*2^e2 * round(10^q) = resM * 2^resE + ε
|
|||
|
|
// exact = ε == 0
|
|||
|
|
func mult64bitPow10(m uint32, e2, q int) (resM uint32, resE int, exact bool) {
|
|||
|
|
if q == 0 {
|
|||
|
|
// P == 1<<63
|
|||
|
|
return m << 6, e2 - 6, true
|
|||
|
|
}
|
|||
|
|
if q < detailedPowersOfTenMinExp10 || detailedPowersOfTenMaxExp10 < q {
|
|||
|
|
// This never happens due to the range of float32/float64 exponent
|
|||
|
|
panic("mult64bitPow10: power of 10 is out of range")
|
|||
|
|
}
|
|||
|
|
pow := detailedPowersOfTen[q-detailedPowersOfTenMinExp10][1]
|
|||
|
|
if q < 0 {
|
|||
|
|
// Inverse powers of ten must be rounded up.
|
|||
|
|
pow += 1
|
|||
|
|
}
|
|||
|
|
hi, lo := bits.Mul64(uint64(m), pow)
|
|||
|
|
e2 += mulByLog10Log2(q) - 63 + 57
|
|||
|
|
return uint32(hi<<7 | lo>>57), e2, lo<<7 == 0
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// mult128bitPow10 takes a floating-point input with a 55-bit
|
|||
|
|
// mantissa and multiplies it with 10^q. The resulting mantissa
|
|||
|
|
// is m*P >> 119 where P is a 128-bit element of the detailedPowersOfTen tables.
|
|||
|
|
// It is typically 63 or 64-bit wide.
|
|||
|
|
// The returned boolean is true is all trimmed bits were zero.
|
|||
|
|
//
|
|||
|
|
// That is:
|
|||
|
|
// m*2^e2 * round(10^q) = resM * 2^resE + ε
|
|||
|
|
// exact = ε == 0
|
|||
|
|
func mult128bitPow10(m uint64, e2, q int) (resM uint64, resE int, exact bool) {
|
|||
|
|
if q == 0 {
|
|||
|
|
// P == 1<<127
|
|||
|
|
return m << 8, e2 - 8, true
|
|||
|
|
}
|
|||
|
|
if q < detailedPowersOfTenMinExp10 || detailedPowersOfTenMaxExp10 < q {
|
|||
|
|
// This never happens due to the range of float32/float64 exponent
|
|||
|
|
panic("mult128bitPow10: power of 10 is out of range")
|
|||
|
|
}
|
|||
|
|
pow := detailedPowersOfTen[q-detailedPowersOfTenMinExp10]
|
|||
|
|
if q < 0 {
|
|||
|
|
// Inverse powers of ten must be rounded up.
|
|||
|
|
pow[0] += 1
|
|||
|
|
}
|
|||
|
|
e2 += mulByLog10Log2(q) - 127 + 119
|
|||
|
|
|
|||
|
|
// long multiplication
|
|||
|
|
l1, l0 := bits.Mul64(m, pow[0])
|
|||
|
|
h1, h0 := bits.Mul64(m, pow[1])
|
|||
|
|
mid, carry := bits.Add64(l1, h0, 0)
|
|||
|
|
h1 += carry
|
|||
|
|
return h1<<9 | mid>>55, e2, mid<<9 == 0 && l0 == 0
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
func divisibleByPower5(m uint64, k int) bool {
|
|||
|
|
if m == 0 {
|
|||
|
|
return true
|
|||
|
|
}
|
|||
|
|
for i := 0; i < k; i++ {
|
|||
|
|
if m%5 != 0 {
|
|||
|
|
return false
|
|||
|
|
}
|
|||
|
|
m /= 5
|
|||
|
|
}
|
|||
|
|
return true
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// divmod1e9 computes quotient and remainder of division by 1e9,
|
|||
|
|
// avoiding runtime uint64 division on 32-bit platforms.
|
|||
|
|
func divmod1e9(x uint64) (uint32, uint32) {
|
|||
|
|
if !host32bit {
|
|||
|
|
return uint32(x / 1e9), uint32(x % 1e9)
|
|||
|
|
}
|
|||
|
|
// Use the same sequence of operations as the amd64 compiler.
|
|||
|
|
hi, _ := bits.Mul64(x>>1, 0x89705f4136b4a598) // binary digits of 1e-9
|
|||
|
|
q := hi >> 28
|
|||
|
|
return uint32(q), uint32(x - q*1e9)
|
|||
|
|
}
|