{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"},
{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"},
}
}
+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() {
{"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},
{"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},
{"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) {
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.
//
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.
// 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
// 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
}
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)
+}