strconv: Implement Ryū algorithm for ftoa shortest mode

This patch implements the algorithm from 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 is not a direct translation of the reference C implementation
but still follows the original paper. In particular, it uses full
128-bit powers of 10, which allows for more precision in the other
modes (fixed ftoa, atof).

name                              old time/op  new time/op   delta
AppendFloat/Decimal-4             49.6ns ± 3%   59.3ns ± 0%  +19.59%  (p=0.008 n=5+5)
AppendFloat/Float-4                122ns ± 1%     91ns ± 1%  -25.92%  (p=0.008 n=5+5)
AppendFloat/Exp-4                 89.3ns ± 1%  100.0ns ± 1%  +11.98%  (p=0.008 n=5+5)
AppendFloat/NegExp-4              88.3ns ± 2%   97.1ns ± 1%   +9.87%  (p=0.008 n=5+5)
AppendFloat/LongExp-4              143ns ± 2%    103ns ± 0%  -28.17%  (p=0.016 n=5+4)
AppendFloat/Big-4                  144ns ± 1%    110ns ± 1%  -23.26%  (p=0.008 n=5+5)
AppendFloat/BinaryExp-4           46.2ns ± 2%   46.0ns ± 1%     ~     (p=0.603 n=5+5)
AppendFloat/32Integer-4           49.1ns ± 1%   58.7ns ± 1%  +19.57%  (p=0.008 n=5+5)
AppendFloat/32ExactFraction-4     95.6ns ± 1%   88.6ns ± 1%   -7.30%  (p=0.008 n=5+5)
AppendFloat/32Point-4              122ns ± 1%     87ns ± 1%  -28.63%  (p=0.008 n=5+5)
AppendFloat/32Exp-4               88.6ns ± 2%   95.0ns ± 1%   +7.29%  (p=0.008 n=5+5)
AppendFloat/32NegExp-4            87.2ns ± 1%   91.3ns ± 1%   +4.63%  (p=0.008 n=5+5)
AppendFloat/32Shortest-4           107ns ± 1%     82ns ± 0%  -24.08%  (p=0.008 n=5+5)
AppendFloat/Slowpath64-4          1.00µs ± 1%   0.10µs ± 0%  -89.92%  (p=0.016 n=5+4)
AppendFloat/SlowpathDenormal64-4  34.1µs ± 3%    0.1µs ± 1%  -99.72%  (p=0.008 n=5+5)

Fixes #15672

Change-Id: Ib90dfa245f62490a6666671896013cf3f9a1fb22
Reviewed-on: https://go-review.googlesource.com/c/go/+/170080
Trust: Emmanuel Odeke <[email protected]>
Trust: Nigel Tao <[email protected]>
Run-TryBot: Emmanuel Odeke <[email protected]>
TryBot-Result: Go Bot <[email protected]>
Reviewed-by: Nigel Tao <[email protected]>

Author: remyoudompheng

src/strconv/ftoa.go

@@ -113,15 +113,11 @@ func genericFtoa(dst []byte, val float64, fmt byte, prec, bitSize int) []byte {
 	// Negative precision means "only as much as needed to be exact."
 	shortest := prec < 0
 	if shortest {
-		// Try Grisu3 algorithm.
-		f := new(extFloat)
-		lower, upper := f.AssignComputeBounds(mant, exp, neg, flt)
+		// Use Ryu algorithm.
 		var buf [32]byte
 		digs.d = buf[:]
-		ok = f.ShortestDecimal(&digs, &lower, &upper)
-		if !ok {
-			return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
-		}
+		ryuFtoaShortest(&digs, mant, exp-int(flt.mantbits), flt)
+		ok = true
 		// Precision for shortest representation mode.
 		switch fmt {
 		case 'e', 'E':

src/strconv/ftoa_test.go

@@ -40,6 +40,7 @@ var ftoatests = []ftoaTest{
 	{200000, 'x', -1, "0x1.86ap+17"},
 	{200000, 'X', -1, "0X1.86AP+17"},
 	{2000000, 'g', -1, "2e+06"},
+	{1e10, 'g', -1, "1e+10"},
 
 	// g conversion and zero suppression
 	{400, 'g', 2, "4e+02"},
@@ -84,6 +85,7 @@ var ftoatests = []ftoaTest{
 	{1.2355, 'f', 3, "1.236"},
 	{1234567890123456.5, 'e', 15, "1.234567890123456e+15"},
 	{1234567890123457.5, 'e', 15, "1.234567890123458e+15"},
+	{108678236358137.625, 'g', -1, "1.0867823635813762e+14"},
 
 	{1e23, 'e', 17, "9.99999999999999916e+22"},
 	{1e23, 'f', 17, "99999999999999991611392.00000000000000000"},
@@ -191,6 +193,25 @@ func TestFtoa(t *testing.T) {
 	}
 }
 
+func TestFtoaPowersOfTwo(t *testing.T) {
+	for exp := -2048; exp <= 2048; exp++ {
+		f := math.Ldexp(1, exp)
+		if !math.IsInf(f, 0) {
+			s := FormatFloat(f, 'e', -1, 64)
+			if x, _ := ParseFloat(s, 64); x != f {
+				t.Errorf("failed roundtrip %v => %s => %v", f, s, x)
+			}
+		}
+		f32 := float32(f)
+		if !math.IsInf(float64(f32), 0) {
+			s := FormatFloat(float64(f32), 'e', -1, 32)
+			if x, _ := ParseFloat(s, 32); float32(x) != f32 {
+				t.Errorf("failed roundtrip %v => %s => %v", f32, s, float32(x))
+			}
+		}
+	}
+}
+
 func TestFtoaRandom(t *testing.T) {
 	N := int(1e4)
 	if testing.Short() {
@@ -240,6 +261,7 @@ var ftoaBenches = []struct {
 	{"Float", 339.7784, 'g', -1, 64},
 	{"Exp", -5.09e75, 'g', -1, 64},
 	{"NegExp", -5.11e-95, 'g', -1, 64},
+	{"LongExp", 1.234567890123456e-78, 'g', -1, 64},
 
 	{"Big", 123456789123456789123456789, 'g', -1, 64},
 	{"BinaryExp", -1, 'b', -1, 64},
@@ -249,6 +271,7 @@ var ftoaBenches = []struct {
 	{"32Point", 339.7784, 'g', -1, 32},
 	{"32Exp", -5.09e25, 'g', -1, 32},
 	{"32NegExp", -5.11e-25, 'g', -1, 32},
+	{"32Shortest", 1.234567e-8, 'g', -1, 32},
 	{"32Fixed8Hard", math.Ldexp(15961084, -125), 'e', 8, 32},
 	{"32Fixed9Hard", math.Ldexp(14855922, -83), 'e', 9, 32},
 
@@ -264,7 +287,14 @@ var ftoaBenches = []struct {
 	{"64Fixed18Hard", math.Ldexp(6994187472632449, 690), 'e', 18, 64},
 
 	// Trigger slow path (see issue #15672).
-	{"Slowpath64", 622666234635.3213e-320, 'e', -1, 64},
+	// The shortest is: 8.034137530808823e+43
+	{"Slowpath64", 8.03413753080882349e+43, 'e', -1, 64},
+	// This denormal is pathological because the lower/upper
+	// halfways to neighboring floats are:
+	// 622666234635.321003e-320 ~= 622666234635.321e-320
+	// 622666234635.321497e-320 ~= 622666234635.3215e-320
+	// making it hard to find the 3rd digit
+	{"SlowpathDenormal64", 622666234635.3213e-320, 'e', -1, 64},
 }
 
 func BenchmarkFormatFloat(b *testing.B) {

src/strconv/ftoaryu.go

@@ -218,6 +218,109 @@ func formatDecimal(d *decimalSlice, m uint64, trunc bool, roundUp bool, prec int
 	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 substract 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.
 //
@@ -238,6 +341,140 @@ func mulByLog10Log2(x int) int {
 	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.
@@ -249,7 +486,8 @@ func mulByLog10Log2(x int) int {
 //     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
+		// 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
@@ -276,7 +514,8 @@ func mult64bitPow10(m uint32, e2, q int) (resM uint32, resE int, exact bool) {
 //     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
+		// 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
@@ -309,3 +548,15 @@ func divisibleByPower5(m uint64, k int) bool {
 	}
 	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)
+}