diff options
Diffstat (limited to 'src/strconv/ftoaryu.go')
| -rw-r--r-- | src/strconv/ftoaryu.go | 569 |
1 files changed, 569 insertions, 0 deletions
diff --git a/src/strconv/ftoaryu.go b/src/strconv/ftoaryu.go new file mode 100644 index 0000000..2e7bf71 --- /dev/null +++ b/src/strconv/ftoaryu.go @@ -0,0 +1,569 @@ +// Copyright 2021 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 + +import ( + "math/bits" +) + +// binary to decimal conversion using the Ryū algorithm. +// +// See Ulf Adams, "Ryū: Fast Float-to-String Conversion" (doi:10.1145/3192366.3192369) +// +// Fixed precision formatting is a variant of the original paper's +// algorithm, where a single multiplication by 10^k is required, +// sharing the same rounding guarantees. + +// ryuFtoaFixed32 formats mant*(2^exp) with prec decimal digits. +func ryuFtoaFixed32(d *decimalSlice, mant uint32, exp int, prec int) { + if prec < 0 { + panic("ryuFtoaFixed32 called with negative prec") + } + if prec > 9 { + panic("ryuFtoaFixed32 called with prec > 9") + } + // Zero input. + if mant == 0 { + d.nd, d.dp = 0, 0 + return + } + // Renormalize to a 25-bit mantissa. + e2 := exp + if b := bits.Len32(mant); b < 25 { + mant <<= uint(25 - b) + e2 += b - 25 + } + // Choose an exponent such that rounded mant*(2^e2)*(10^q) has + // at least prec decimal digits, i.e + // mant*(2^e2)*(10^q) >= 10^(prec-1) + // Because mant >= 2^24, it is enough to choose: + // 2^(e2+24) >= 10^(-q+prec-1) + // or q = -mulByLog2Log10(e2+24) + prec - 1 + q := -mulByLog2Log10(e2+24) + prec - 1 + + // Now compute mant*(2^e2)*(10^q). + // Is it an exact computation? + // Only small positive powers of 10 are exact (5^28 has 66 bits). + exact := q <= 27 && q >= 0 + + di, dexp2, d0 := mult64bitPow10(mant, e2, q) + if dexp2 >= 0 { + panic("not enough significant bits after mult64bitPow10") + } + // As a special case, computation might still be exact, if exponent + // was negative and if it amounts to computing an exact division. + // In that case, we ignore all lower bits. + // Note that division by 10^11 cannot be exact as 5^11 has 26 bits. + if q < 0 && q >= -10 && divisibleByPower5(uint64(mant), -q) { + exact = true + d0 = true + } + // Remove extra lower bits and keep rounding info. + extra := uint(-dexp2) + extraMask := uint32(1<<extra - 1) + + di, dfrac := di>>extra, di&extraMask + roundUp := false + if exact { + // If we computed an exact product, d + 1/2 + // should round to d+1 if 'd' is odd. + roundUp = dfrac > 1<<(extra-1) || + (dfrac == 1<<(extra-1) && !d0) || + (dfrac == 1<<(extra-1) && d0 && di&1 == 1) + } else { + // otherwise, d+1/2 always rounds up because + // we truncated below. + roundUp = dfrac>>(extra-1) == 1 + } + if dfrac != 0 { + d0 = false + } + // Proceed to the requested number of digits + formatDecimal(d, uint64(di), !d0, roundUp, prec) + // Adjust exponent + d.dp -= q +} + +// ryuFtoaFixed64 formats mant*(2^exp) with prec decimal digits. +func ryuFtoaFixed64(d *decimalSlice, mant uint64, exp int, prec int) { + if prec > 18 { + panic("ryuFtoaFixed64 called with prec > 18") + } + // Zero input. + if mant == 0 { + d.nd, d.dp = 0, 0 + return + } + // Renormalize to a 55-bit mantissa. + e2 := exp + if b := bits.Len64(mant); b < 55 { + mant = mant << uint(55-b) + e2 += b - 55 + } + // Choose an exponent such that rounded mant*(2^e2)*(10^q) has + // at least prec decimal digits, i.e + // mant*(2^e2)*(10^q) >= 10^(prec-1) + // Because mant >= 2^54, it is enough to choose: + // 2^(e2+54) >= 10^(-q+prec-1) + // or q = -mulByLog2Log10(e2+54) + prec - 1 + // + // The minimal required exponent is -mulByLog2Log10(1025)+18 = -291 + // The maximal required exponent is mulByLog2Log10(1074)+18 = 342 + q := -mulByLog2Log10(e2+54) + prec - 1 + + // Now compute mant*(2^e2)*(10^q). + // Is it an exact computation? + // Only small positive powers of 10 are exact (5^55 has 128 bits). + exact := q <= 55 && q >= 0 + + di, dexp2, d0 := mult128bitPow10(mant, e2, q) + if dexp2 >= 0 { + panic("not enough significant bits after mult128bitPow10") + } + // As a special case, computation might still be exact, if exponent + // was negative and if it amounts to computing an exact division. + // In that case, we ignore all lower bits. + // Note that division by 10^23 cannot be exact as 5^23 has 54 bits. + if q < 0 && q >= -22 && divisibleByPower5(mant, -q) { + exact = true + d0 = true + } + // Remove extra lower bits and keep rounding info. + extra := uint(-dexp2) + extraMask := uint64(1<<extra - 1) + + di, dfrac := di>>extra, di&extraMask + roundUp := false + if exact { + // If we computed an exact product, d + 1/2 + // should round to d+1 if 'd' is odd. + roundUp = dfrac > 1<<(extra-1) || + (dfrac == 1<<(extra-1) && !d0) || + (dfrac == 1<<(extra-1) && d0 && di&1 == 1) + } else { + // otherwise, d+1/2 always rounds up because + // we truncated below. + roundUp = dfrac>>(extra-1) == 1 + } + if dfrac != 0 { + d0 = false + } + // Proceed to the requested number of digits + formatDecimal(d, di, !d0, roundUp, prec) + // Adjust exponent + d.dp -= q +} + +var uint64pow10 = [...]uint64{ + 1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, + 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, +} + +// formatDecimal fills d with at most prec decimal digits +// of mantissa m. The boolean trunc indicates whether m +// is truncated compared to the original number being formatted. +func formatDecimal(d *decimalSlice, m uint64, trunc bool, roundUp bool, prec int) { + max := uint64pow10[prec] + trimmed := 0 + for m >= max { + a, b := m/10, m%10 + m = a + trimmed++ + if b > 5 { + roundUp = true + } else if b < 5 { + roundUp = false + } else { // b == 5 + // round up if there are trailing digits, + // or if the new value of m is odd (round-to-even convention) + roundUp = trunc || m&1 == 1 + } + if b != 0 { + trunc = true + } + } + if roundUp { + m++ + } + if m >= max { + // Happens if di was originally 99999....xx + m /= 10 + trimmed++ + } + // render digits (similar to formatBits) + n := uint(prec) + d.nd = prec + v := m + for v >= 100 { + var v1, v2 uint64 + if v>>32 == 0 { + v1, v2 = uint64(uint32(v)/100), uint64(uint32(v)%100) + } else { + v1, v2 = v/100, v%100 + } + n -= 2 + d.d[n+1] = smallsString[2*v2+1] + d.d[n+0] = smallsString[2*v2+0] + v = v1 + } + if v > 0 { + n-- + d.d[n] = smallsString[2*v+1] + } + if v >= 10 { + n-- + d.d[n] = smallsString[2*v] + } + for d.d[d.nd-1] == '0' { + d.nd-- + trimmed++ + } + d.dp = d.nd + trimmed +} + +// ryuFtoaShortest formats mant*2^exp with prec decimal digits. +func ryuFtoaShortest(d *decimalSlice, mant uint64, exp int, flt *floatInfo) { + if mant == 0 { + d.nd, d.dp = 0, 0 + return + } + // If input is an exact integer with fewer bits than the mantissa, + // the previous and next integer are not admissible representations. + if exp <= 0 && bits.TrailingZeros64(mant) >= -exp { + mant >>= uint(-exp) + ryuDigits(d, mant, mant, mant, true, false) + return + } + ml, mc, mu, e2 := computeBounds(mant, exp, flt) + if e2 == 0 { + ryuDigits(d, ml, mc, mu, true, false) + return + } + // Find 10^q *larger* than 2^-e2 + q := mulByLog2Log10(-e2) + 1 + + // We are going to multiply by 10^q using 128-bit arithmetic. + // The exponent is the same for all 3 numbers. + var dl, dc, du uint64 + var dl0, dc0, du0 bool + if flt == &float32info { + var dl32, dc32, du32 uint32 + dl32, _, dl0 = mult64bitPow10(uint32(ml), e2, q) + dc32, _, dc0 = mult64bitPow10(uint32(mc), e2, q) + du32, e2, du0 = mult64bitPow10(uint32(mu), e2, q) + dl, dc, du = uint64(dl32), uint64(dc32), uint64(du32) + } else { + dl, _, dl0 = mult128bitPow10(ml, e2, q) + dc, _, dc0 = mult128bitPow10(mc, e2, q) + du, e2, du0 = mult128bitPow10(mu, e2, q) + } + if e2 >= 0 { + panic("not enough significant bits after mult128bitPow10") + } + // Is it an exact computation? + if q > 55 { + // Large positive powers of ten are not exact + dl0, dc0, du0 = false, false, false + } + if q < 0 && q >= -24 { + // Division by a power of ten may be exact. + // (note that 5^25 is a 59-bit number so division by 5^25 is never exact). + if divisibleByPower5(ml, -q) { + dl0 = true + } + if divisibleByPower5(mc, -q) { + dc0 = true + } + if divisibleByPower5(mu, -q) { + du0 = true + } + } + // Express the results (dl, dc, du)*2^e2 as integers. + // Extra bits must be removed and rounding hints computed. + extra := uint(-e2) + extraMask := uint64(1<<extra - 1) + // Now compute the floored, integral base 10 mantissas. + dl, fracl := dl>>extra, dl&extraMask + dc, fracc := dc>>extra, dc&extraMask + du, fracu := du>>extra, du&extraMask + // Is it allowed to use 'du' as a result? + // It is always allowed when it is truncated, but also + // if it is exact and the original binary mantissa is even + // When disallowed, we can subtract 1. + uok := !du0 || fracu > 0 + if du0 && fracu == 0 { + uok = mant&1 == 0 + } + if !uok { + du-- + } + // Is 'dc' the correctly rounded base 10 mantissa? + // The correct rounding might be dc+1 + cup := false // don't round up. + if dc0 { + // If we computed an exact product, the half integer + // should round to next (even) integer if 'dc' is odd. + cup = fracc > 1<<(extra-1) || + (fracc == 1<<(extra-1) && dc&1 == 1) + } else { + // otherwise, the result is a lower truncation of the ideal + // result. + cup = fracc>>(extra-1) == 1 + } + // Is 'dl' an allowed representation? + // Only if it is an exact value, and if the original binary mantissa + // was even. + lok := dl0 && fracl == 0 && (mant&1 == 0) + if !lok { + dl++ + } + // We need to remember whether the trimmed digits of 'dc' are zero. + c0 := dc0 && fracc == 0 + // render digits + ryuDigits(d, dl, dc, du, c0, cup) + d.dp -= q +} + +// mulByLog2Log10 returns math.Floor(x * log(2)/log(10)) for an integer x in +// the range -1600 <= x && x <= +1600. +// +// The range restriction lets us work in faster integer arithmetic instead of +// slower floating point arithmetic. Correctness is verified by unit tests. +func mulByLog2Log10(x int) int { + // log(2)/log(10) ≈ 0.30102999566 ≈ 78913 / 2^18 + return (x * 78913) >> 18 +} + +// 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) +} |