extfloat.go

Documentation: strconv

     1  // Copyright 2011 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package strconv
     6  
     7  // An extFloat represents an extended floating-point number, with more
     8  // precision than a float64. It does not try to save bits: the
     9  // number represented by the structure is mant*(2^exp), with a negative
    10  // sign if neg is true.
    11  type extFloat struct {
    12  	mant uint64
    13  	exp  int
    14  	neg  bool
    15  }
    16  
    17  // Powers of ten taken from double-conversion library.
    18  // http://code.google.com/p/double-conversion/
    19  const (
    20  	firstPowerOfTen = -348
    21  	stepPowerOfTen  = 8
    22  )
    23  
    24  var smallPowersOfTen = [...]extFloat{
    25  	{1 << 63, -63, false},        // 1
    26  	{0xa << 60, -60, false},      // 1e1
    27  	{0x64 << 57, -57, false},     // 1e2
    28  	{0x3e8 << 54, -54, false},    // 1e3
    29  	{0x2710 << 50, -50, false},   // 1e4
    30  	{0x186a0 << 47, -47, false},  // 1e5
    31  	{0xf4240 << 44, -44, false},  // 1e6
    32  	{0x989680 << 40, -40, false}, // 1e7
    33  }
    34  
    35  var powersOfTen = [...]extFloat{
    36  	{0xfa8fd5a0081c0288, -1220, false}, // 10^-348
    37  	{0xbaaee17fa23ebf76, -1193, false}, // 10^-340
    38  	{0x8b16fb203055ac76, -1166, false}, // 10^-332
    39  	{0xcf42894a5dce35ea, -1140, false}, // 10^-324
    40  	{0x9a6bb0aa55653b2d, -1113, false}, // 10^-316
    41  	{0xe61acf033d1a45df, -1087, false}, // 10^-308
    42  	{0xab70fe17c79ac6ca, -1060, false}, // 10^-300
    43  	{0xff77b1fcbebcdc4f, -1034, false}, // 10^-292
    44  	{0xbe5691ef416bd60c, -1007, false}, // 10^-284
    45  	{0x8dd01fad907ffc3c, -980, false},  // 10^-276
    46  	{0xd3515c2831559a83, -954, false},  // 10^-268
    47  	{0x9d71ac8fada6c9b5, -927, false},  // 10^-260
    48  	{0xea9c227723ee8bcb, -901, false},  // 10^-252
    49  	{0xaecc49914078536d, -874, false},  // 10^-244
    50  	{0x823c12795db6ce57, -847, false},  // 10^-236
    51  	{0xc21094364dfb5637, -821, false},  // 10^-228
    52  	{0x9096ea6f3848984f, -794, false},  // 10^-220
    53  	{0xd77485cb25823ac7, -768, false},  // 10^-212
    54  	{0xa086cfcd97bf97f4, -741, false},  // 10^-204
    55  	{0xef340a98172aace5, -715, false},  // 10^-196
    56  	{0xb23867fb2a35b28e, -688, false},  // 10^-188
    57  	{0x84c8d4dfd2c63f3b, -661, false},  // 10^-180
    58  	{0xc5dd44271ad3cdba, -635, false},  // 10^-172
    59  	{0x936b9fcebb25c996, -608, false},  // 10^-164
    60  	{0xdbac6c247d62a584, -582, false},  // 10^-156
    61  	{0xa3ab66580d5fdaf6, -555, false},  // 10^-148
    62  	{0xf3e2f893dec3f126, -529, false},  // 10^-140
    63  	{0xb5b5ada8aaff80b8, -502, false},  // 10^-132
    64  	{0x87625f056c7c4a8b, -475, false},  // 10^-124
    65  	{0xc9bcff6034c13053, -449, false},  // 10^-116
    66  	{0x964e858c91ba2655, -422, false},  // 10^-108
    67  	{0xdff9772470297ebd, -396, false},  // 10^-100
    68  	{0xa6dfbd9fb8e5b88f, -369, false},  // 10^-92
    69  	{0xf8a95fcf88747d94, -343, false},  // 10^-84
    70  	{0xb94470938fa89bcf, -316, false},  // 10^-76
    71  	{0x8a08f0f8bf0f156b, -289, false},  // 10^-68
    72  	{0xcdb02555653131b6, -263, false},  // 10^-60
    73  	{0x993fe2c6d07b7fac, -236, false},  // 10^-52
    74  	{0xe45c10c42a2b3b06, -210, false},  // 10^-44
    75  	{0xaa242499697392d3, -183, false},  // 10^-36
    76  	{0xfd87b5f28300ca0e, -157, false},  // 10^-28
    77  	{0xbce5086492111aeb, -130, false},  // 10^-20
    78  	{0x8cbccc096f5088cc, -103, false},  // 10^-12
    79  	{0xd1b71758e219652c, -77, false},   // 10^-4
    80  	{0x9c40000000000000, -50, false},   // 10^4
    81  	{0xe8d4a51000000000, -24, false},   // 10^12
    82  	{0xad78ebc5ac620000, 3, false},     // 10^20
    83  	{0x813f3978f8940984, 30, false},    // 10^28
    84  	{0xc097ce7bc90715b3, 56, false},    // 10^36
    85  	{0x8f7e32ce7bea5c70, 83, false},    // 10^44
    86  	{0xd5d238a4abe98068, 109, false},   // 10^52
    87  	{0x9f4f2726179a2245, 136, false},   // 10^60
    88  	{0xed63a231d4c4fb27, 162, false},   // 10^68
    89  	{0xb0de65388cc8ada8, 189, false},   // 10^76
    90  	{0x83c7088e1aab65db, 216, false},   // 10^84
    91  	{0xc45d1df942711d9a, 242, false},   // 10^92
    92  	{0x924d692ca61be758, 269, false},   // 10^100
    93  	{0xda01ee641a708dea, 295, false},   // 10^108
    94  	{0xa26da3999aef774a, 322, false},   // 10^116
    95  	{0xf209787bb47d6b85, 348, false},   // 10^124
    96  	{0xb454e4a179dd1877, 375, false},   // 10^132
    97  	{0x865b86925b9bc5c2, 402, false},   // 10^140
    98  	{0xc83553c5c8965d3d, 428, false},   // 10^148
    99  	{0x952ab45cfa97a0b3, 455, false},   // 10^156
   100  	{0xde469fbd99a05fe3, 481, false},   // 10^164
   101  	{0xa59bc234db398c25, 508, false},   // 10^172
   102  	{0xf6c69a72a3989f5c, 534, false},   // 10^180
   103  	{0xb7dcbf5354e9bece, 561, false},   // 10^188
   104  	{0x88fcf317f22241e2, 588, false},   // 10^196
   105  	{0xcc20ce9bd35c78a5, 614, false},   // 10^204
   106  	{0x98165af37b2153df, 641, false},   // 10^212
   107  	{0xe2a0b5dc971f303a, 667, false},   // 10^220
   108  	{0xa8d9d1535ce3b396, 694, false},   // 10^228
   109  	{0xfb9b7cd9a4a7443c, 720, false},   // 10^236
   110  	{0xbb764c4ca7a44410, 747, false},   // 10^244
   111  	{0x8bab8eefb6409c1a, 774, false},   // 10^252
   112  	{0xd01fef10a657842c, 800, false},   // 10^260
   113  	{0x9b10a4e5e9913129, 827, false},   // 10^268
   114  	{0xe7109bfba19c0c9d, 853, false},   // 10^276
   115  	{0xac2820d9623bf429, 880, false},   // 10^284
   116  	{0x80444b5e7aa7cf85, 907, false},   // 10^292
   117  	{0xbf21e44003acdd2d, 933, false},   // 10^300
   118  	{0x8e679c2f5e44ff8f, 960, false},   // 10^308
   119  	{0xd433179d9c8cb841, 986, false},   // 10^316
   120  	{0x9e19db92b4e31ba9, 1013, false},  // 10^324
   121  	{0xeb96bf6ebadf77d9, 1039, false},  // 10^332
   122  	{0xaf87023b9bf0ee6b, 1066, false},  // 10^340
   123  }
   124  
   125  // floatBits returns the bits of the float64 that best approximates
   126  // the extFloat passed as receiver. Overflow is set to true if
   127  // the resulting float64 is ±Inf.
   128  func (f *extFloat) floatBits(flt *floatInfo) (bits uint64, overflow bool) {
   129  	f.Normalize()
   130  
   131  	exp := f.exp + 63
   132  
   133  	// Exponent too small.
   134  	if exp < flt.bias+1 {
   135  		n := flt.bias + 1 - exp
   136  		f.mant >>= uint(n)
   137  		exp += n
   138  	}
   139  
   140  	// Extract 1+flt.mantbits bits from the 64-bit mantissa.
   141  	mant := f.mant >> (63 - flt.mantbits)
   142  	if f.mant&(1<<(62-flt.mantbits)) != 0 {
   143  		// Round up.
   144  		mant += 1
   145  	}
   146  
   147  	// Rounding might have added a bit; shift down.
   148  	if mant == 2<<flt.mantbits {
   149  		mant >>= 1
   150  		exp++
   151  	}
   152  
   153  	// Infinities.
   154  	if exp-flt.bias >= 1<<flt.expbits-1 {
   155  		// ±Inf
   156  		mant = 0
   157  		exp = 1<<flt.expbits - 1 + flt.bias
   158  		overflow = true
   159  	} else if mant&(1<<flt.mantbits) == 0 {
   160  		// Denormalized?
   161  		exp = flt.bias
   162  	}
   163  	// Assemble bits.
   164  	bits = mant & (uint64(1)<<flt.mantbits - 1)
   165  	bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
   166  	if f.neg {
   167  		bits |= 1 << (flt.mantbits + flt.expbits)
   168  	}
   169  	return
   170  }
   171  
   172  // AssignComputeBounds sets f to the floating point value
   173  // defined by mant, exp and precision given by flt. It returns
   174  // lower, upper such that any number in the closed interval
   175  // [lower, upper] is converted back to the same floating point number.
   176  func (f *extFloat) AssignComputeBounds(mant uint64, exp int, neg bool, flt *floatInfo) (lower, upper extFloat) {
   177  	f.mant = mant
   178  	f.exp = exp - int(flt.mantbits)
   179  	f.neg = neg
   180  	if f.exp <= 0 && mant == (mant>>uint(-f.exp))<<uint(-f.exp) {
   181  		// An exact integer
   182  		f.mant >>= uint(-f.exp)
   183  		f.exp = 0
   184  		return *f, *f
   185  	}
   186  	expBiased := exp - flt.bias
   187  
   188  	upper = extFloat{mant: 2*f.mant + 1, exp: f.exp - 1, neg: f.neg}
   189  	if mant != 1<<flt.mantbits || expBiased == 1 {
   190  		lower = extFloat{mant: 2*f.mant - 1, exp: f.exp - 1, neg: f.neg}
   191  	} else {
   192  		lower = extFloat{mant: 4*f.mant - 1, exp: f.exp - 2, neg: f.neg}
   193  	}
   194  	return
   195  }
   196  
   197  // Normalize normalizes f so that the highest bit of the mantissa is
   198  // set, and returns the number by which the mantissa was left-shifted.
   199  func (f *extFloat) Normalize() (shift uint) {
   200  	mant, exp := f.mant, f.exp
   201  	if mant == 0 {
   202  		return 0
   203  	}
   204  	if mant>>(64-32) == 0 {
   205  		mant <<= 32
   206  		exp -= 32
   207  	}
   208  	if mant>>(64-16) == 0 {
   209  		mant <<= 16
   210  		exp -= 16
   211  	}
   212  	if mant>>(64-8) == 0 {
   213  		mant <<= 8
   214  		exp -= 8
   215  	}
   216  	if mant>>(64-4) == 0 {
   217  		mant <<= 4
   218  		exp -= 4
   219  	}
   220  	if mant>>(64-2) == 0 {
   221  		mant <<= 2
   222  		exp -= 2
   223  	}
   224  	if mant>>(64-1) == 0 {
   225  		mant <<= 1
   226  		exp -= 1
   227  	}
   228  	shift = uint(f.exp - exp)
   229  	f.mant, f.exp = mant, exp
   230  	return
   231  }
   232  
   233  // Multiply sets f to the product f*g: the result is correctly rounded,
   234  // but not normalized.
   235  func (f *extFloat) Multiply(g extFloat) {
   236  	fhi, flo := f.mant>>32, uint64(uint32(f.mant))
   237  	ghi, glo := g.mant>>32, uint64(uint32(g.mant))
   238  
   239  	// Cross products.
   240  	cross1 := fhi * glo
   241  	cross2 := flo * ghi
   242  
   243  	// f.mant*g.mant is fhi*ghi << 64 + (cross1+cross2) << 32 + flo*glo
   244  	f.mant = fhi*ghi + (cross1 >> 32) + (cross2 >> 32)
   245  	rem := uint64(uint32(cross1)) + uint64(uint32(cross2)) + ((flo * glo) >> 32)
   246  	// Round up.
   247  	rem += (1 << 31)
   248  
   249  	f.mant += (rem >> 32)
   250  	f.exp = f.exp + g.exp + 64
   251  }
   252  
   253  var uint64pow10 = [...]uint64{
   254  	1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
   255  	1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
   256  }
   257  
   258  // AssignDecimal sets f to an approximate value mantissa*10^exp. It
   259  // reports whether the value represented by f is guaranteed to be the
   260  // best approximation of d after being rounded to a float64 or
   261  // float32 depending on flt.
   262  func (f *extFloat) AssignDecimal(mantissa uint64, exp10 int, neg bool, trunc bool, flt *floatInfo) (ok bool) {
   263  	const uint64digits = 19
   264  	const errorscale = 8
   265  	errors := 0 // An upper bound for error, computed in errorscale*ulp.
   266  	if trunc {
   267  		// the decimal number was truncated.
   268  		errors += errorscale / 2
   269  	}
   270  
   271  	f.mant = mantissa
   272  	f.exp = 0
   273  	f.neg = neg
   274  
   275  	// Multiply by powers of ten.
   276  	i := (exp10 - firstPowerOfTen) / stepPowerOfTen
   277  	if exp10 < firstPowerOfTen || i >= len(powersOfTen) {
   278  		return false
   279  	}
   280  	adjExp := (exp10 - firstPowerOfTen) % stepPowerOfTen
   281  
   282  	// We multiply by exp%step
   283  	if adjExp < uint64digits && mantissa < uint64pow10[uint64digits-adjExp] {
   284  		// We can multiply the mantissa exactly.
   285  		f.mant *= uint64pow10[adjExp]
   286  		f.Normalize()
   287  	} else {
   288  		f.Normalize()
   289  		f.Multiply(smallPowersOfTen[adjExp])
   290  		errors += errorscale / 2
   291  	}
   292  
   293  	// We multiply by 10 to the exp - exp%step.
   294  	f.Multiply(powersOfTen[i])
   295  	if errors > 0 {
   296  		errors += 1
   297  	}
   298  	errors += errorscale / 2
   299  
   300  	// Normalize
   301  	shift := f.Normalize()
   302  	errors <<= shift
   303  
   304  	// Now f is a good approximation of the decimal.
   305  	// Check whether the error is too large: that is, if the mantissa
   306  	// is perturbated by the error, the resulting float64 will change.
   307  	// The 64 bits mantissa is 1 + 52 bits for float64 + 11 extra bits.
   308  	//
   309  	// In many cases the approximation will be good enough.
   310  	denormalExp := flt.bias - 63
   311  	var extrabits uint
   312  	if f.exp <= denormalExp {
   313  		// f.mant * 2^f.exp is smaller than 2^(flt.bias+1).
   314  		extrabits = 63 - flt.mantbits + 1 + uint(denormalExp-f.exp)
   315  	} else {
   316  		extrabits = 63 - flt.mantbits
   317  	}
   318  
   319  	halfway := uint64(1) << (extrabits - 1)
   320  	mant_extra := f.mant & (1<<extrabits - 1)
   321  
   322  	// Do a signed comparison here! If the error estimate could make
   323  	// the mantissa round differently for the conversion to double,
   324  	// then we can't give a definite answer.
   325  	if int64(halfway)-int64(errors) < int64(mant_extra) &&
   326  		int64(mant_extra) < int64(halfway)+int64(errors) {
   327  		return false
   328  	}
   329  	return true
   330  }
   331  
   332  // Frexp10 is an analogue of math.Frexp for decimal powers. It scales
   333  // f by an approximate power of ten 10^-exp, and returns exp10, so
   334  // that f*10^exp10 has the same value as the old f, up to an ulp,
   335  // as well as the index of 10^-exp in the powersOfTen table.
   336  func (f *extFloat) frexp10() (exp10, index int) {
   337  	// The constants expMin and expMax constrain the final value of the
   338  	// binary exponent of f. We want a small integral part in the result
   339  	// because finding digits of an integer requires divisions, whereas
   340  	// digits of the fractional part can be found by repeatedly multiplying
   341  	// by 10.
   342  	const expMin = -60
   343  	const expMax = -32
   344  	// Find power of ten such that x * 10^n has a binary exponent
   345  	// between expMin and expMax.
   346  	approxExp10 := ((expMin+expMax)/2 - f.exp) * 28 / 93 // log(10)/log(2) is close to 93/28.
   347  	i := (approxExp10 - firstPowerOfTen) / stepPowerOfTen
   348  Loop:
   349  	for {
   350  		exp := f.exp + powersOfTen[i].exp + 64
   351  		switch {
   352  		case exp < expMin:
   353  			i++
   354  		case exp > expMax:
   355  			i--
   356  		default:
   357  			break Loop
   358  		}
   359  	}
   360  	// Apply the desired decimal shift on f. It will have exponent
   361  	// in the desired range. This is multiplication by 10^-exp10.
   362  	f.Multiply(powersOfTen[i])
   363  
   364  	return -(firstPowerOfTen + i*stepPowerOfTen), i
   365  }
   366  
   367  // frexp10Many applies a common shift by a power of ten to a, b, c.
   368  func frexp10Many(a, b, c *extFloat) (exp10 int) {
   369  	exp10, i := c.frexp10()
   370  	a.Multiply(powersOfTen[i])
   371  	b.Multiply(powersOfTen[i])
   372  	return
   373  }
   374  
   375  // FixedDecimal stores in d the first n significant digits
   376  // of the decimal representation of f. It returns false
   377  // if it cannot be sure of the answer.
   378  func (f *extFloat) FixedDecimal(d *decimalSlice, n int) bool {
   379  	if f.mant == 0 {
   380  		d.nd = 0
   381  		d.dp = 0
   382  		d.neg = f.neg
   383  		return true
   384  	}
   385  	if n == 0 {
   386  		panic("strconv: internal error: extFloat.FixedDecimal called with n == 0")
   387  	}
   388  	// Multiply by an appropriate power of ten to have a reasonable
   389  	// number to process.
   390  	f.Normalize()
   391  	exp10, _ := f.frexp10()
   392  
   393  	shift := uint(-f.exp)
   394  	integer := uint32(f.mant >> shift)
   395  	fraction := f.mant - (uint64(integer) << shift)
   396  	ε := uint64(1) // ε is the uncertainty we have on the mantissa of f.
   397  
   398  	// Write exactly n digits to d.
   399  	needed := n        // how many digits are left to write.
   400  	integerDigits := 0 // the number of decimal digits of integer.
   401  	pow10 := uint64(1) // the power of ten by which f was scaled.
   402  	for i, pow := 0, uint64(1); i < 20; i++ {
   403  		if pow > uint64(integer) {
   404  			integerDigits = i
   405  			break
   406  		}
   407  		pow *= 10
   408  	}
   409  	rest := integer
   410  	if integerDigits > needed {
   411  		// the integral part is already large, trim the last digits.
   412  		pow10 = uint64pow10[integerDigits-needed]
   413  		integer /= uint32(pow10)
   414  		rest -= integer * uint32(pow10)
   415  	} else {
   416  		rest = 0
   417  	}
   418  
   419  	// Write the digits of integer: the digits of rest are omitted.
   420  	var buf [32]byte
   421  	pos := len(buf)
   422  	for v := integer; v > 0; {
   423  		v1 := v / 10
   424  		v -= 10 * v1
   425  		pos--
   426  		buf[pos] = byte(v + '0')
   427  		v = v1
   428  	}
   429  	for i := pos; i < len(buf); i++ {
   430  		d.d[i-pos] = buf[i]
   431  	}
   432  	nd := len(buf) - pos
   433  	d.nd = nd
   434  	d.dp = integerDigits + exp10
   435  	needed -= nd
   436  
   437  	if needed > 0 {
   438  		if rest != 0 || pow10 != 1 {
   439  			panic("strconv: internal error, rest != 0 but needed > 0")
   440  		}
   441  		// Emit digits for the fractional part. Each time, 10*fraction
   442  		// fits in a uint64 without overflow.
   443  		for needed > 0 {
   444  			fraction *= 10
   445  			ε *= 10 // the uncertainty scales as we multiply by ten.
   446  			if 2*ε > 1<<shift {
   447  				// the error is so large it could modify which digit to write, abort.
   448  				return false
   449  			}
   450  			digit := fraction >> shift
   451  			d.d[nd] = byte(digit + '0')
   452  			fraction -= digit << shift
   453  			nd++
   454  			needed--
   455  		}
   456  		d.nd = nd
   457  	}
   458  
   459  	// We have written a truncation of f (a numerator / 10^d.dp). The remaining part
   460  	// can be interpreted as a small number (< 1) to be added to the last digit of the
   461  	// numerator.
   462  	//
   463  	// If rest > 0, the amount is:
   464  	//    (rest<<shift | fraction) / (pow10 << shift)
   465  	//    fraction being known with a ±ε uncertainty.
   466  	//    The fact that n > 0 guarantees that pow10 << shift does not overflow a uint64.
   467  	//
   468  	// If rest = 0, pow10 == 1 and the amount is
   469  	//    fraction / (1 << shift)
   470  	//    fraction being known with a ±ε uncertainty.
   471  	//
   472  	// We pass this information to the rounding routine for adjustment.
   473  
   474  	ok := adjustLastDigitFixed(d, uint64(rest)<<shift|fraction, pow10, shift, ε)
   475  	if !ok {
   476  		return false
   477  	}
   478  	// Trim trailing zeros.
   479  	for i := d.nd - 1; i >= 0; i-- {
   480  		if d.d[i] != '0' {
   481  			d.nd = i + 1
   482  			break
   483  		}
   484  	}
   485  	return true
   486  }
   487  
   488  // adjustLastDigitFixed assumes d contains the representation of the integral part
   489  // of some number, whose fractional part is num / (den << shift). The numerator
   490  // num is only known up to an uncertainty of size ε, assumed to be less than
   491  // (den << shift)/2.
   492  //
   493  // It will increase the last digit by one to account for correct rounding, typically
   494  // when the fractional part is greater than 1/2, and will return false if ε is such
   495  // that no correct answer can be given.
   496  func adjustLastDigitFixed(d *decimalSlice, num, den uint64, shift uint, ε uint64) bool {
   497  	if num > den<<shift {
   498  		panic("strconv: num > den<<shift in adjustLastDigitFixed")
   499  	}
   500  	if 2*ε > den<<shift {
   501  		panic("strconv: ε > (den<<shift)/2")
   502  	}
   503  	if 2*(num+ε) < den<<shift {
   504  		return true
   505  	}
   506  	if 2*(num-ε) > den<<shift {
   507  		// increment d by 1.
   508  		i := d.nd - 1
   509  		for ; i >= 0; i-- {
   510  			if d.d[i] == '9' {
   511  				d.nd--
   512  			} else {
   513  				break
   514  			}
   515  		}
   516  		if i < 0 {
   517  			d.d[0] = '1'
   518  			d.nd = 1
   519  			d.dp++
   520  		} else {
   521  			d.d[i]++
   522  		}
   523  		return true
   524  	}
   525  	return false
   526  }
   527  
   528  // ShortestDecimal stores in d the shortest decimal representation of f
   529  // which belongs to the open interval (lower, upper), where f is supposed
   530  // to lie. It returns false whenever the result is unsure. The implementation
   531  // uses the Grisu3 algorithm.
   532  func (f *extFloat) ShortestDecimal(d *decimalSlice, lower, upper *extFloat) bool {
   533  	if f.mant == 0 {
   534  		d.nd = 0
   535  		d.dp = 0
   536  		d.neg = f.neg
   537  		return true
   538  	}
   539  	if f.exp == 0 && *lower == *f && *lower == *upper {
   540  		// an exact integer.
   541  		var buf [24]byte
   542  		n := len(buf) - 1
   543  		for v := f.mant; v > 0; {
   544  			v1 := v / 10
   545  			v -= 10 * v1
   546  			buf[n] = byte(v + '0')
   547  			n--
   548  			v = v1
   549  		}
   550  		nd := len(buf) - n - 1
   551  		for i := 0; i < nd; i++ {
   552  			d.d[i] = buf[n+1+i]
   553  		}
   554  		d.nd, d.dp = nd, nd
   555  		for d.nd > 0 && d.d[d.nd-1] == '0' {
   556  			d.nd--
   557  		}
   558  		if d.nd == 0 {
   559  			d.dp = 0
   560  		}
   561  		d.neg = f.neg
   562  		return true
   563  	}
   564  	upper.Normalize()
   565  	// Uniformize exponents.
   566  	if f.exp > upper.exp {
   567  		f.mant <<= uint(f.exp - upper.exp)
   568  		f.exp = upper.exp
   569  	}
   570  	if lower.exp > upper.exp {
   571  		lower.mant <<= uint(lower.exp - upper.exp)
   572  		lower.exp = upper.exp
   573  	}
   574  
   575  	exp10 := frexp10Many(lower, f, upper)
   576  	// Take a safety margin due to rounding in frexp10Many, but we lose precision.
   577  	upper.mant++
   578  	lower.mant--
   579  
   580  	// The shortest representation of f is either rounded up or down, but
   581  	// in any case, it is a truncation of upper.
   582  	shift := uint(-upper.exp)
   583  	integer := uint32(upper.mant >> shift)
   584  	fraction := upper.mant - (uint64(integer) << shift)
   585  
   586  	// How far we can go down from upper until the result is wrong.
   587  	allowance := upper.mant - lower.mant
   588  	// How far we should go to get a very precise result.
   589  	targetDiff := upper.mant - f.mant
   590  
   591  	// Count integral digits: there are at most 10.
   592  	var integerDigits int
   593  	for i, pow := 0, uint64(1); i < 20; i++ {
   594  		if pow > uint64(integer) {
   595  			integerDigits = i
   596  			break
   597  		}
   598  		pow *= 10
   599  	}
   600  	for i := 0; i < integerDigits; i++ {
   601  		pow := uint64pow10[integerDigits-i-1]
   602  		digit := integer / uint32(pow)
   603  		d.d[i] = byte(digit + '0')
   604  		integer -= digit * uint32(pow)
   605  		// evaluate whether we should stop.
   606  		if currentDiff := uint64(integer)<<shift + fraction; currentDiff < allowance {
   607  			d.nd = i + 1
   608  			d.dp = integerDigits + exp10
   609  			d.neg = f.neg
   610  			// Sometimes allowance is so large the last digit might need to be
   611  			// decremented to get closer to f.
   612  			return adjustLastDigit(d, currentDiff, targetDiff, allowance, pow<<shift, 2)
   613  		}
   614  	}
   615  	d.nd = integerDigits
   616  	d.dp = d.nd + exp10
   617  	d.neg = f.neg
   618  
   619  	// Compute digits of the fractional part. At each step fraction does not
   620  	// overflow. The choice of minExp implies that fraction is less than 2^60.
   621  	var digit int
   622  	multiplier := uint64(1)
   623  	for {
   624  		fraction *= 10
   625  		multiplier *= 10
   626  		digit = int(fraction >> shift)
   627  		d.d[d.nd] = byte(digit + '0')
   628  		d.nd++
   629  		fraction -= uint64(digit) << shift
   630  		if fraction < allowance*multiplier {
   631  			// We are in the admissible range. Note that if allowance is about to
   632  			// overflow, that is, allowance > 2^64/10, the condition is automatically
   633  			// true due to the limited range of fraction.
   634  			return adjustLastDigit(d,
   635  				fraction, targetDiff*multiplier, allowance*multiplier,
   636  				1<<shift, multiplier*2)
   637  		}
   638  	}
   639  }
   640  
   641  // adjustLastDigit modifies d = x-currentDiff*ε, to get closest to
   642  // d = x-targetDiff*ε, without becoming smaller than x-maxDiff*ε.
   643  // It assumes that a decimal digit is worth ulpDecimal*ε, and that
   644  // all data is known with an error estimate of ulpBinary*ε.
   645  func adjustLastDigit(d *decimalSlice, currentDiff, targetDiff, maxDiff, ulpDecimal, ulpBinary uint64) bool {
   646  	if ulpDecimal < 2*ulpBinary {
   647  		// Approximation is too wide.
   648  		return false
   649  	}
   650  	for currentDiff+ulpDecimal/2+ulpBinary < targetDiff {
   651  		d.d[d.nd-1]--
   652  		currentDiff += ulpDecimal
   653  	}
   654  	if currentDiff+ulpDecimal <= targetDiff+ulpDecimal/2+ulpBinary {
   655  		// we have two choices, and don't know what to do.
   656  		return false
   657  	}
   658  	if currentDiff < ulpBinary || currentDiff > maxDiff-ulpBinary {
   659  		// we went too far
   660  		return false
   661  	}
   662  	if d.nd == 1 && d.d[0] == '0' {
   663  		// the number has actually reached zero.
   664  		d.nd = 0
   665  		d.dp = 0
   666  	}
   667  	return true
   668  }
   669
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