strconv: implement Ryū-like algorithm for fixed precision ftoa

This patch implements a simplified version of Ulf Adams,
"Ryū: Fast Float-to-String Conversion" (doi:10.1145/3192366.3192369)
for formatting floating-point numbers with a fixed number of decimal
digits.

It uses the same principles but does not need to handle
the complex task of finding a shortest representation.
This allows to handle a few more cases than Grisu3, notably
formatting with up to 18 significant digits.

name                         old time/op  new time/op  delta
AppendFloat/32Fixed8Hard-4   72.0ns ± 2%  56.0ns ± 2%  -22.28%  (p=0.000 n=10+10)
AppendFloat/32Fixed9Hard-4   74.8ns ± 0%  64.2ns ± 2%  -14.16%  (p=0.000 n=8+10)
AppendFloat/64Fixed1-4       60.4ns ± 1%  54.2ns ± 1%  -10.31%  (p=0.000 n=10+9)
AppendFloat/64Fixed2-4       66.3ns ± 1%  53.3ns ± 1%  -19.54%  (p=0.000 n=10+9)
AppendFloat/64Fixed3-4       61.0ns ± 1%  55.0ns ± 2%   -9.80%  (p=0.000 n=9+10)
AppendFloat/64Fixed4-4       66.9ns ± 0%  52.0ns ± 2%  -22.20%  (p=0.000 n=8+10)
AppendFloat/64Fixed12-4      95.5ns ± 1%  76.2ns ± 3%  -20.19%  (p=0.000 n=10+9)
AppendFloat/64Fixed16-4      1.62µs ± 0%  0.07µs ± 2%  -95.69%  (p=0.000 n=10+10)
AppendFloat/64Fixed12Hard-4  1.27µs ± 1%  0.07µs ± 1%  -94.83%  (p=0.000 n=9+9)
AppendFloat/64Fixed17Hard-4  3.68µs ± 1%  0.08µs ± 2%  -97.86%  (p=0.000 n=10+9)
AppendFloat/64Fixed18Hard-4  3.67µs ± 0%  3.72µs ± 1%   +1.44%  (p=0.000 n=9+10)

Updates #15672

Change-Id: I160963e141dd48287ad8cf57bcc3c686277788e8
Reviewed-on: https://go-review.googlesource.com/c/go/+/170079
Reviewed-by: Emmanuel Odeke <emmanuel@orijtech.com>
Trust: Emmanuel Odeke <emmanuel@orijtech.com>
Trust: Nigel Tao <nigeltao@golang.org>
Trust: Robert Griesemer <gri@golang.org>
Run-TryBot: Emmanuel Odeke <emmanuel@orijtech.com>
TryBot-Result: Go Bot <gobot@golang.org>

src/strconv/ftoa.go

@@ -143,12 +143,15 @@ func genericFtoa(dst []byte, val float64, fmt byte, prec, bitSize int) []byte {
 			}
 			digits = prec
 		}
-		if digits <= 15 {
+		var buf [24]byte
-			// try fast algorithm when the number of digits is reasonable.
+		if bitSize == 32 && digits <= 9 {
-			var buf [24]byte
 			digs.d = buf[:]
-			f := extFloat{mant, exp - int(flt.mantbits), neg}
+			ryuFtoaFixed32(&digs, uint32(mant), exp-int(flt.mantbits), digits)
-			ok = f.FixedDecimal(&digs, digits)
+			ok = true
+		} else if digits <= 18 {
+			digs.d = buf[:]
+			ryuFtoaFixed64(&digs, mant, exp-int(flt.mantbits), digits)
+			ok = true
 		}
 	}
 	if !ok {

src/strconv/ftoa_test.go

@@ -77,6 +77,14 @@ var ftoatests = []ftoaTest{
 	{1.2345e6, 'f', 5, "1234500.00000"},
 	{1.2345e6, 'g', 5, "1.2345e+06"},
 
+	// Round to even
+	{1.2345e6, 'e', 3, "1.234e+06"},
+	{1.2355e6, 'e', 3, "1.236e+06"},
+	{1.2345, 'f', 3, "1.234"},
+	{1.2355, 'f', 3, "1.236"},
+	{1234567890123456.5, 'e', 15, "1.234567890123456e+15"},
+	{1234567890123457.5, 'e', 15, "1.234567890123458e+15"},
+
 	{1e23, 'e', 17, "9.99999999999999916e+22"},
 	{1e23, 'f', 17, "99999999999999991611392.00000000000000000"},
 	{1e23, 'g', 17, "9.9999999999999992e+22"},
@@ -241,11 +249,19 @@ var ftoaBenches = []struct {
 	{"32Point", 339.7784, 'g', -1, 32},
 	{"32Exp", -5.09e25, 'g', -1, 32},
 	{"32NegExp", -5.11e-25, 'g', -1, 32},
+	{"32Fixed8Hard", math.Ldexp(15961084, -125), 'e', 8, 32},
+	{"32Fixed9Hard", math.Ldexp(14855922, -83), 'e', 9, 32},
 
 	{"64Fixed1", 123456, 'e', 3, 64},
 	{"64Fixed2", 123.456, 'e', 3, 64},
 	{"64Fixed3", 1.23456e+78, 'e', 3, 64},
 	{"64Fixed4", 1.23456e-78, 'e', 3, 64},
+	{"64Fixed12", 1.23456e-78, 'e', 12, 64},
+	{"64Fixed16", 1.23456e-78, 'e', 16, 64},
+	// From testdata/testfp.txt
+	{"64Fixed12Hard", math.Ldexp(6965949469487146, -249), 'e', 12, 64},
+	{"64Fixed17Hard", math.Ldexp(8887055249355788, 665), 'e', 17, 64},
+	{"64Fixed18Hard", math.Ldexp(6994187472632449, 690), 'e', 18, 64},
 
 	// Trigger slow path (see issue #15672).
 	{"Slowpath64", 622666234635.3213e-320, 'e', -1, 64},

src/strconv/ftoaryu.go

@@ -0,0 +1,311 @@
+// 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 += int(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 += int(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
+}
+
+// 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 = int(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
+}
+
+// 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
+}
+
+// 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 {
+		return m << 7, e2 - 7, 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 {
+		return m << 9, e2 - 9, 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
+}

src/strconv/ftoaryu_test.go

@@ -0,0 +1,31 @@
+// 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_test
+
+import (
+	"math"
+	. "strconv"
+	"testing"
+)
+
+func TestMulByLog2Log10(t *testing.T) {
+	for x := -1600; x <= +1600; x++ {
+		iMath := MulByLog2Log10(x)
+		fMath := int(math.Floor(float64(x) * math.Ln2 / math.Ln10))
+		if iMath != fMath {
+			t.Errorf("mulByLog2Log10(%d) failed: %d vs %d\n", x, iMath, fMath)
+		}
+	}
+}
+
+func TestMulByLog10Log2(t *testing.T) {
+	for x := -500; x <= +500; x++ {
+		iMath := MulByLog10Log2(x)
+		fMath := int(math.Floor(float64(x) * math.Ln10 / math.Ln2))
+		if iMath != fMath {
+			t.Errorf("mulByLog10Log2(%d) failed: %d vs %d\n", x, iMath, fMath)
+		}
+	}
+}

src/strconv/internal_test.go

@@ -21,3 +21,11 @@ func SetOptimize(b bool) bool {
 func ParseFloatPrefix(s string, bitSize int) (float64, int, error) {
 	return parseFloatPrefix(s, bitSize)
 }
+
+func MulByLog2Log10(x int) int {
+	return mulByLog2Log10(x)
+}
+
+func MulByLog10Log2(x int) int {
+	return mulByLog10Log2(x)
+}