// Copyright 2011 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"
)
// An extFloat represents an extended floating-point number, with more
// precision than a float64. It does not try to save bits: the
// number represented by the structure is mant*(2^exp), with a negative
// sign if neg is true.
type extFloat struct {
mant uint64
exp int
neg bool
}
// Powers of ten taken from double-conversion library.
// https://code.google.com/p/double-conversion/
const (
firstPowerOfTen = -348
stepPowerOfTen = 8
)
var smallPowersOfTen = [...]extFloat{
{1 << 63, -63, false}, // 1
{0xa << 60, -60, false}, // 1e1
{0x64 << 57, -57, false}, // 1e2
{0x3e8 << 54, -54, false}, // 1e3
{0x2710 << 50, -50, false}, // 1e4
{0x186a0 << 47, -47, false}, // 1e5
{0xf4240 << 44, -44, false}, // 1e6
{0x989680 << 40, -40, false}, // 1e7
}
var powersOfTen = [...]extFloat{
{0xfa8fd5a0081c0288, -1220, false}, // 10^-348
{0xbaaee17fa23ebf76, -1193, false}, // 10^-340
{0x8b16fb203055ac76, -1166, false}, // 10^-332
{0xcf42894a5dce35ea, -1140, false}, // 10^-324
{0x9a6bb0aa55653b2d, -1113, false}, // 10^-316
{0xe61acf033d1a45df, -1087, false}, // 10^-308
{0xab70fe17c79ac6ca, -1060, false}, // 10^-300
{0xff77b1fcbebcdc4f, -1034, false}, // 10^-292
{0xbe5691ef416bd60c, -1007, false}, // 10^-284
{0x8dd01fad907ffc3c, -980, false}, // 10^-276
{0xd3515c2831559a83, -954, false}, // 10^-268
{0x9d71ac8fada6c9b5, -927, false}, // 10^-260
{0xea9c227723ee8bcb, -901, false}, // 10^-252
{0xaecc49914078536d, -874, false}, // 10^-244
{0x823c12795db6ce57, -847, false}, // 10^-236
{0xc21094364dfb5637, -821, false}, // 10^-228
{0x9096ea6f3848984f, -794, false}, // 10^-220
{0xd77485cb25823ac7, -768, false}, // 10^-212
{0xa086cfcd97bf97f4, -741, false}, // 10^-204
{0xef340a98172aace5, -715, false}, // 10^-196
{0xb23867fb2a35b28e, -688, false}, // 10^-188
{0x84c8d4dfd2c63f3b, -661, false}, // 10^-180
{0xc5dd44271ad3cdba, -635, false}, // 10^-172
{0x936b9fcebb25c996, -608, false}, // 10^-164
{0xdbac6c247d62a584, -582, false}, // 10^-156
{0xa3ab66580d5fdaf6, -555, false}, // 10^-148
{0xf3e2f893dec3f126, -529, false}, // 10^-140
{0xb5b5ada8aaff80b8, -502, false}, // 10^-132
{0x87625f056c7c4a8b, -475, false}, // 10^-124
{0xc9bcff6034c13053, -449, false}, // 10^-116
{0x964e858c91ba2655, -422, false}, // 10^-108
{0xdff9772470297ebd, -396, false}, // 10^-100
{0xa6dfbd9fb8e5b88f, -369, false}, // 10^-92
{0xf8a95fcf88747d94, -343, false}, // 10^-84
{0xb94470938fa89bcf, -316, false}, // 10^-76
{0x8a08f0f8bf0f156b, -289, false}, // 10^-68
{0xcdb02555653131b6, -263, false}, // 10^-60
{0x993fe2c6d07b7fac, -236, false}, // 10^-52
{0xe45c10c42a2b3b06, -210, false}, // 10^-44
{0xaa242499697392d3, -183, false}, // 10^-36
{0xfd87b5f28300ca0e, -157, false}, // 10^-28
{0xbce5086492111aeb, -130, false}, // 10^-20
{0x8cbccc096f5088cc, -103, false}, // 10^-12
{0xd1b71758e219652c, -77, false}, // 10^-4
{0x9c40000000000000, -50, false}, // 10^4
{0xe8d4a51000000000, -24, false}, // 10^12
{0xad78ebc5ac620000, 3, false}, // 10^20
{0x813f3978f8940984, 30, false}, // 10^28
{0xc097ce7bc90715b3, 56, false}, // 10^36
{0x8f7e32ce7bea5c70, 83, false}, // 10^44
{0xd5d238a4abe98068, 109, false}, // 10^52
{0x9f4f2726179a2245, 136, false}, // 10^60
{0xed63a231d4c4fb27, 162, false}, // 10^68
{0xb0de65388cc8ada8, 189, false}, // 10^76
{0x83c7088e1aab65db, 216, false}, // 10^84
{0xc45d1df942711d9a, 242, false}, // 10^92
{0x924d692ca61be758, 269, false}, // 10^100
{0xda01ee641a708dea, 295, false}, // 10^108
{0xa26da3999aef774a, 322, false}, // 10^116
{0xf209787bb47d6b85, 348, false}, // 10^124
{0xb454e4a179dd1877, 375, false}, // 10^132
{0x865b86925b9bc5c2, 402, false}, // 10^140
{0xc83553c5c8965d3d, 428, false}, // 10^148
{0x952ab45cfa97a0b3, 455, false}, // 10^156
{0xde469fbd99a05fe3, 481, false}, // 10^164
{0xa59bc234db398c25, 508, false}, // 10^172
{0xf6c69a72a3989f5c, 534, false}, // 10^180
{0xb7dcbf5354e9bece, 561, false}, // 10^188
{0x88fcf317f22241e2, 588, false}, // 10^196
{0xcc20ce9bd35c78a5, 614, false}, // 10^204
{0x98165af37b2153df, 641, false}, // 10^212
{0xe2a0b5dc971f303a, 667, false}, // 10^220
{0xa8d9d1535ce3b396, 694, false}, // 10^228
{0xfb9b7cd9a4a7443c, 720, false}, // 10^236
{0xbb764c4ca7a44410, 747, false}, // 10^244
{0x8bab8eefb6409c1a, 774, false}, // 10^252
{0xd01fef10a657842c, 800, false}, // 10^260
{0x9b10a4e5e9913129, 827, false}, // 10^268
{0xe7109bfba19c0c9d, 853, false}, // 10^276
{0xac2820d9623bf429, 880, false}, // 10^284
{0x80444b5e7aa7cf85, 907, false}, // 10^292
{0xbf21e44003acdd2d, 933, false}, // 10^300
{0x8e679c2f5e44ff8f, 960, false}, // 10^308
{0xd433179d9c8cb841, 986, false}, // 10^316
{0x9e19db92b4e31ba9, 1013, false}, // 10^324
{0xeb96bf6ebadf77d9, 1039, false}, // 10^332
{0xaf87023b9bf0ee6b, 1066, false}, // 10^340
}
// floatBits returns the bits of the float64 that best approximates
// the extFloat passed as receiver. Overflow is set to true if
// the resulting float64 is ±Inf.
func (f *extFloat) floatBits(flt *floatInfo) (bits uint64, overflow bool) {
f.Normalize()
exp := f.exp + 63
// Exponent too small.
if exp < flt.bias+1 {
n := flt.bias + 1 - exp
f.mant >>= uint(n)
exp += n
}
// Extract 1+flt.mantbits bits from the 64-bit mantissa.
mant := f.mant >> (63 - flt.mantbits)
if f.mant&(1<<(62-flt.mantbits)) != 0 {
// Round up.
mant += 1
}
// Rounding might have added a bit; shift down.
if mant == 2<<flt.mantbits {
mant >>= 1
exp++
}
// Infinities.
if exp-flt.bias >= 1<<flt.expbits-1 {
// ±Inf
mant = 0
exp = 1<<flt.expbits - 1 + flt.bias
overflow = true
} else if mant&(1<<flt.mantbits) == 0 {
// Denormalized?
exp = flt.bias
}
// Assemble bits.
bits = mant & (uint64(1)<<flt.mantbits - 1)
bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
if f.neg {
bits |= 1 << (flt.mantbits + flt.expbits)
}
return
}
// AssignComputeBounds sets f to the floating point value
// defined by mant, exp and precision given by flt. It returns
// lower, upper such that any number in the closed interval
// [lower, upper] is converted back to the same floating point number.
func (f *extFloat) AssignComputeBounds(mant uint64, exp int, neg bool, flt *floatInfo) (lower, upper extFloat) {
f.mant = mant
f.exp = exp - int(flt.mantbits)
f.neg = neg
if f.exp <= 0 && mant == (mant>>uint(-f.exp))<<uint(-f.exp) {
// An exact integer
f.mant >>= uint(-f.exp)
f.exp = 0
return *f, *f
}
expBiased := exp - flt.bias
upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg}
if mant != 1<<flt.mantbits || expBiased == 1 {
lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg}
} else {
lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg}
}
return
}
// Normalize normalizes f so that the highest bit of the mantissa is
// set, and returns the number by which the mantissa was left-shifted.
func (f *extFloat) Normalize() uint {
// bits.LeadingZeros64 would return 64
if f.mant == 0 {
return 0
}
shift := bits.LeadingZeros64(f.mant)
f.mant <<= uint(shift)
f.exp -= shift
return uint(shift)
}
// Multiply sets f to the product f*g: the result is correctly rounded,
// but not normalized.
func (f *extFloat) Multiply(g extFloat) {
hi, lo := bits.Mul64(f.mant, g.mant)
// Round up.
f.mant = hi + (lo >> 63)
f.exp = f.exp + g.exp + 64
}
var uint64pow10 = [...]uint64{
1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
}
// AssignDecimal sets f to an approximate value mantissa*10^exp. It
// reports whether the value represented by f is guaranteed to be the
// best approximation of d after being rounded to a float64 or
// float32 depending on flt.
func (f *extFloat) AssignDecimal(mantissa uint64, exp10 int, neg bool, trunc bool, flt *floatInfo) (ok bool) {
const uint64digits = 19
const errorscale = 8
errors := 0 // An upper bound for error, computed in errorscale*ulp.
if trunc {
// the decimal number was truncated.
errors += errorscale / 2
}
f.mant = mantissa
f.exp = 0
f.neg = neg
// Multiply by powers of ten.
i := (exp10 - firstPowerOfTen) / stepPowerOfTen
if exp10 < firstPowerOfTen || i >= len(powersOfTen) {
return false
}
adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen
// We multiply by exp%step
if adjExp < uint64digits && mantissa < uint64pow10[uint64digits-adjExp] {
// We can multiply the mantissa exactly.
f.mant *= uint64pow10[adjExp]
f.Normalize()
} else {
f.Normalize()
f.Multiply(smallPowersOfTen[adjExp])
errors += errorscale / 2
}
// We multiply by 10 to the exp - exp%step.
f.Multiply(powersOfTen[i])
if errors > 0 {
errors += 1
}
errors += errorscale / 2
// Normalize
shift := f.Normalize()
errors <<= shift
// Now f is a good approximation of the decimal.
// Check whether the error is too large: that is, if the mantissa
// is perturbated by the error, the resulting float64 will change.
// The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits.
//
// In many cases the approximation will be good enough.
denormalExp := flt.bias - 63
var extrabits uint
if f.exp <= denormalExp {
// f.mant * 2^f.exp is smaller than 2^(flt.bias+1).
extrabits = 63 - flt.mantbits + 1 + uint(denormalExp-f.exp)
} else {
extrabits = 63 - flt.mantbits
}
halfway := uint64(1) << (extrabits - 1)
mant_extra := f.mant & (1<<extrabits - 1)
// Do a signed comparison here! If the error estimate could make
// the mantissa round differently for the conversion to double,
// then we can't give a definite answer.
if int64(halfway)-int64(errors) < int64(mant_extra) &&
int64(mant_extra) < int64(halfway)+int64(errors) {
return false
}
return true
}
// Frexp10 is an analogue of math.Frexp for decimal powers. It scales
// f by an approximate power of ten 10^-exp, and returns exp10, so
// that f*10^exp10 has the same value as the old f, up to an ulp,
// as well as the index of 10^-exp in the powersOfTen table.
func (f *extFloat) frexp10() (exp10, index int) {
// The constants expMin and expMax constrain the final value of the
// binary exponent of f. We want a small integral part in the result
// because finding digits of an integer requires divisions, whereas
// digits of the fractional part can be found by repeatedly multiplying
// by 10.
const expMin = -60
const expMax = -32
// Find power of ten such that x * 10^n has a binary exponent
// between expMin and expMax.
approxExp10 := ((expMin+expMax)/2 - f.exp) * 28 / 93 // log(10)/log(2) is close to 93/28.
i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen
Loop:
for {
exp := f.exp + powersOfTen[i].exp + 64
switch {
case exp < expMin:
i++
case exp > expMax:
i--
default:
break Loop
}
}
// Apply the desired decimal shift on f. It will have exponent
// in the desired range. This is multiplication by 10^-exp10.
f.Multiply(powersOfTen[i])
return -(firstPowerOfTen + i*stepPowerOfTen), i
}
// frexp10Many applies a common shift by a power of ten to a, b, c.
func frexp10Many(a, b, c *extFloat) (exp10 int) {
exp10, i := c.frexp10()
a.Multiply(powersOfTen[i])
b.Multiply(powersOfTen[i])
return
}
// FixedDecimal stores in d the first n significant digits
// of the decimal representation of f. It returns false
// if it cannot be sure of the answer.
func (f *extFloat) FixedDecimal(d *decimalSlice, n int) bool {
if f.mant == 0 {
d.nd = 0
d.dp = 0
d.neg = f.neg
return true
}
if n == 0 {
panic("strconv: internal error: extFloat.FixedDecimal called with n == 0")
}
// Multiply by an appropriate power of ten to have a reasonable
// number to process.
f.Normalize()
exp10, _ := f.frexp10()
shift := uint(-f.exp)
integer := uint32(f.mant >> shift)
fraction := f.mant - (uint64(integer) << shift)
ε := uint64(1) // ε is the uncertainty we have on the mantissa of f.
// Write exactly n digits to d.
needed := n // how many digits are left to write.
integerDigits := 0 // the number of decimal digits of integer.
pow10 := uint64(1) // the power of ten by which f was scaled.
for i, pow := 0, uint64(1); i < 20; i++ {
if pow > uint64(integer) {
integerDigits = i
break
}
pow *= 10
}
rest := integer
if integerDigits > needed {
// the integral part is already large, trim the last digits.
pow10 = uint64pow10[integerDigits-needed]
integer /= uint32(pow10)
rest -= integer * uint32(pow10)
} else {
rest = 0
}
// Write the digits of integer: the digits of rest are omitted.
var buf [32]byte
pos := len(buf)
for v := integer; v > 0; {
v1 := v / 10
v -= 10 * v1
pos--
buf[pos] = byte(v + '0')
v = v1
}
for i := pos; i < len(buf); i++ {
d.d[i-pos] = buf[i]
}
nd := len(buf) - pos
d.nd = nd
d.dp = integerDigits + exp10
needed -= nd
if needed > 0 {
if rest != 0 || pow10 != 1 {
panic("strconv: internal error, rest != 0 but needed > 0")
}
// Emit digits for the fractional part. Each time, 10*fraction
// fits in a uint64 without overflow.
for needed > 0 {
fraction *= 10
ε *= 10 // the uncertainty scales as we multiply by ten.
if 2*ε > 1<<shift {
// the error is so large it could modify which digit to write, abort.
return false
}
digit := fraction >> shift
d.d[nd] = byte(digit + '0')
fraction -= digit << shift
nd++
needed--
}
d.nd = nd
}
// We have written a truncation of f (a numerator / 10^d.dp). The remaining part
// can be interpreted as a small number (< 1) to be added to the last digit of the
// numerator.
//
// If rest > 0, the amount is:
// (rest<<shift | fraction) / (pow10 << shift)
// fraction being known with a ±ε uncertainty.
// The fact that n > 0 guarantees that pow10 << shift does not overflow a uint64.
//
// If rest = 0, pow10 == 1 and the amount is
// fraction / (1 << shift)
// fraction being known with a ±ε uncertainty.
//
// We pass this information to the rounding routine for adjustment.
ok := adjustLastDigitFixed(d, uint64(rest)<<shift|fraction, pow10, shift, ε)
if !ok {
return false
}
// Trim trailing zeros.
for i := d.nd - 1; i >= 0; i-- {
if d.d[i] != '0' {
d.nd = i + 1
break
}
}
return true
}
// adjustLastDigitFixed assumes d contains the representation of the integral part
// of some number, whose fractional part is num / (den << shift). The numerator
// num is only known up to an uncertainty of size ε, assumed to be less than
// (den << shift)/2.
//
// It will increase the last digit by one to account for correct rounding, typically
// when the fractional part is greater than 1/2, and will return false if ε is such
// that no correct answer can be given.
func adjustLastDigitFixed(d *decimalSlice, num, den uint64, shift uint, ε uint64) bool {
if num > den<<shift {
panic("strconv: num > den<<shift in adjustLastDigitFixed")
}
if 2*ε > den<<shift {
panic("strconv: ε > (den<<shift)/2")
}
if 2*(num+ε) < den<<shift {
return true
}
if 2*(num-ε) > den<<shift {
// increment d by 1.
i := d.nd - 1
for ; i >= 0; i-- {
if d.d[i] == '9' {
d.nd--
} else {
break
}
}
if i < 0 {
d.d[0] = '1'
d.nd = 1
d.dp++
} else {
d.d[i]++
}
return true
}
return false
}
// ShortestDecimal stores in d the shortest decimal representation of f
// which belongs to the open interval (lower, upper), where f is supposed
// to lie. It returns false whenever the result is unsure. The implementation
// uses the Grisu3 algorithm.
func (f *extFloat) ShortestDecimal(d *decimalSlice, lower, upper *extFloat) bool {
if f.mant == 0 {
d.nd = 0
d.dp = 0
d.neg = f.neg
return true
}
if f.exp == 0 && *lower == *f && *lower == *upper {
// an exact integer.
var buf [24]byte
n := len(buf) - 1
for v := f.mant; v > 0; {
v1 := v / 10
v -= 10 * v1
buf[n] = byte(v + '0')
n--
v = v1
}
nd := len(buf) - n - 1
for i := 0; i < nd; i++ {
d.d[i] = buf[n+1+i]
}
d.nd, d.dp = nd, nd
for d.nd > 0 && d.d[d.nd-1] == '0' {
d.nd--
}
if d.nd == 0 {
d.dp = 0
}
d.neg = f.neg
return true
}
upper.Normalize()
// Uniformize exponents.
if f.exp > upper.exp {
f.mant <<= uint(f.exp - upper.exp)
f.exp = upper.exp
}
if lower.exp > upper.exp {
lower.mant <<= uint(lower.exp - upper.exp)
lower.exp = upper.exp
}
exp10 := frexp10Many(lower, f, upper)
// Take a safety margin due to rounding in frexp10Many, but we lose precision.
upper.mant++
lower.mant--
// The shortest representation of f is either rounded up or down, but
// in any case, it is a truncation of upper.
shift := uint(-upper.exp)
integer := uint32(upper.mant >> shift)
fraction := upper.mant - (uint64(integer) << shift)
// How far we can go down from upper until the result is wrong.
allowance := upper.mant - lower.mant
// How far we should go to get a very precise result.
targetDiff := upper.mant - f.mant
// Count integral digits: there are at most 10.
var integerDigits int
for i, pow := 0, uint64(1); i < 20; i++ {
if pow > uint64(integer) {
integerDigits = i
break
}
pow *= 10
}
for i := 0; i < integerDigits; i++ {
pow := uint64pow10[integerDigits-i-1]
digit := integer / uint32(pow)
d.d[i] = byte(digit + '0')
integer -= digit * uint32(pow)
// evaluate whether we should stop.
if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance {
d.nd = i + 1
d.dp = integerDigits + exp10
d.neg = f.neg
// Sometimes allowance is so large the last digit might need to be
// decremented to get closer to f.
return adjustLastDigit(d, currentDiff, targetDiff, allowance, pow<<shift, 2)
}
}
d.nd = integerDigits
d.dp = d.nd + exp10
d.neg = f.neg
// Compute digits of the fractional part. At each step fraction does not
// overflow. The choice of minExp implies that fraction is less than 2^60.
var digit int
multiplier := uint64(1)
for {
fraction *= 10
multiplier *= 10
digit = int(fraction >> shift)
d.d[d.nd] = byte(digit + '0')
d.nd++
fraction -= uint64(digit) << shift
if fraction < allowance*multiplier {
// We are in the admissible range. Note that if allowance is about to
// overflow, that is, allowance > 2^64/10, the condition is automatically
// true due to the limited range of fraction.
return adjustLastDigit(d,
fraction, targetDiff*multiplier, allowance*multiplier,
1<<shift, multiplier*2)
}
}
}
// adjustLastDigit modifies d = x-currentDiff*ε, to get closest to
// d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε.
// It assumes that a decimal digit is worth ulpDecimal*ε, and that
// all data is known with an error estimate of ulpBinary*ε.
func adjustLastDigit(d *decimalSlice, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool {
if ulpDecimal < 2*ulpBinary {
// Approximation is too wide.
return false
}
for currentDiff+ulpDecimal/2+ulpBinary < targetDiff {
d.d[d.nd-1]--
currentDiff += ulpDecimal
}
if currentDiff+ulpDecimal <= targetDiff+ulpDecimal/2+ulpBinary {
// we have two choices, and don't know what to do.
return false
}
if currentDiff < ulpBinary || currentDiff > maxDiff-ulpBinary {
// we went too far
return false
}
if d.nd == 1 && d.d[0] == '0' {
// the number has actually reached zero.
d.nd = 0
d.dp = 0
}
return true
}
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