+++ /dev/null
-// 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
-}
-
-// 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,
-}
-
-// 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
-}