extfloat.go

Documentation: strconv

  // 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
  
  // 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.
  // http://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() (shift uint) {
  	mant, exp := f.mant, f.exp
  	if mant == 0 {
  		return 0
  	}
  	if mant>>(64-32) == 0 {
  		mant <<= 32
  		exp -= 32
  	}
  	if mant>>(64-16) == 0 {
  		mant <<= 16
  		exp -= 16
  	}
  	if mant>>(64-8) == 0 {
  		mant <<= 8
  		exp -= 8
  	}
  	if mant>>(64-4) == 0 {
  		mant <<= 4
  		exp -= 4
  	}
  	if mant>>(64-2) == 0 {
  		mant <<= 2
  		exp -= 2
  	}
  	if mant>>(64-1) == 0 {
  		mant <<= 1
  		exp -= 1
  	}
  	shift = uint(f.exp - exp)
  	f.mant, f.exp = mant, exp
  	return
  }
  
  // Multiply sets f to the product f*g: the result is correctly rounded,
  // but not normalized.
  func (f *extFloat) Multiply(g extFloat) {
  	fhi, flo := f.mant>>32, uint64(uint32(f.mant))
  	ghi, glo := g.mant>>32, uint64(uint32(g.mant))
  
  	// Cross products.
  	cross1 := fhi * glo
  	cross2 := flo * ghi
  
  	// f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo
  	f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32)
  	rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32)
  	// Round up.
  	rem += (1 << 31)
  
  	f.mant += (rem >> 32)
  	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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