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strconv: Implement Ryū algorithm for ftoa shortest mode
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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]>
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@remyoudompheng @odeke-em
remyoudompheng authored and odeke-em committed Apr 15, 2021
1 parent 0184b44 commit 61a08fc
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Showing 3 changed files with 287 additions and 10 deletions.
10 changes: 3 additions & 7 deletions src/strconv/ftoa.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -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':
Expand Down
32 changes: 31 additions & 1 deletion src/strconv/ftoa_test.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -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"},
Expand Down Expand Up @@ -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"},
Expand Down Expand Up @@ -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() {
Expand Down Expand Up @@ -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},
Expand All @@ -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},

Expand All @@ -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) {
Expand Down
255 changes: 253 additions & 2 deletions src/strconv/ftoaryu.go
View file
Original file line number Diff line number Diff line change
Expand Up @@ -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.
//
Expand All @@ -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.
Expand All @@ -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
Expand All @@ -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
Expand Down Expand Up @@ -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)
}

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