Source: extfloat.go in package strconv


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

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.

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) {

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.

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

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

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.

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
	}

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
	}

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
	}

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
	}

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.

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')

evaluate whether we should stop.

		if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance {
			d.nd = i + 1
			d.dp = integerDigits + exp10

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

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 {

Approximation is too wide.

		return false
	}
	for currentDiff+ulpDecimal/2+ulpBinary < targetDiff {
		d.d[d.nd-1]--
		currentDiff += ulpDecimal
	}

we have two choices, and don't know what to do.

		return false
	}

we went too far

		return false
	}

the number has actually reached zero.

		d.nd = 0
		d.dp = 0
	}
	return true