ftoaryu.go

     1  // Copyright 2021 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  import (
     8  	"math/bits"
     9  )
    10  
    11  // binary to decimal conversion using the Ryū algorithm.
    12  //
    13  // See Ulf Adams, "Ryū: Fast Float-to-String Conversion" (doi:10.1145/3192366.3192369)
    14  //
    15  // Fixed precision formatting is a variant of the original paper's
    16  // algorithm, where a single multiplication by 10^k is required,
    17  // sharing the same rounding guarantees.
    18  
    19  // ryuFtoaFixed32 formats mant*(2^exp) with prec decimal digits.
    20  func ryuFtoaFixed32(d *decimalSlice, mant uint32, exp int, prec int) {
    21  	if prec < 0 {
    22  		panic("ryuFtoaFixed32 called with negative prec")
    23  	}
    24  	if prec > 9 {
    25  		panic("ryuFtoaFixed32 called with prec > 9")
    26  	}
    27  	// Zero input.
    28  	if mant == 0 {
    29  		d.nd, d.dp = 0, 0
    30  		return
    31  	}
    32  	// Renormalize to a 25-bit mantissa.
    33  	e2 := exp
    34  	if b := bits.Len32(mant); b < 25 {
    35  		mant <<= uint(25 - b)
    36  		e2 += b - 25
    37  	}
    38  	// Choose an exponent such that rounded mant*(2^e2)*(10^q) has
    39  	// at least prec decimal digits, i.e
    40  	//     mant*(2^e2)*(10^q) >= 10^(prec-1)
    41  	// Because mant >= 2^24, it is enough to choose:
    42  	//     2^(e2+24) >= 10^(-q+prec-1)
    43  	// or q = -mulLog10_2(e2+24) + prec - 1
    44  	q := -mulLog10_2(e2+24) + prec - 1
    45  
    46  	// Now compute mant*(2^e2)*(10^q).
    47  	// Is it an exact computation?
    48  	// Only small positive powers of 10 are exact (5^28 has 66 bits).
    49  	exact := q <= 27 && q >= 0
    50  
    51  	di, dexp2, d0 := mult64bitPow10(mant, e2, q)
    52  	if dexp2 >= 0 {
    53  		panic("not enough significant bits after mult64bitPow10")
    54  	}
    55  	// As a special case, computation might still be exact, if exponent
    56  	// was negative and if it amounts to computing an exact division.
    57  	// In that case, we ignore all lower bits.
    58  	// Note that division by 10^11 cannot be exact as 5^11 has 26 bits.
    59  	if q < 0 && q >= -10 && divisibleByPower5(uint64(mant), -q) {
    60  		exact = true
    61  		d0 = true
    62  	}
    63  	// Remove extra lower bits and keep rounding info.
    64  	extra := uint(-dexp2)
    65  	extraMask := uint32(1<<extra - 1)
    66  
    67  	di, dfrac := di>>extra, di&extraMask
    68  	roundUp := false
    69  	if exact {
    70  		// If we computed an exact product, d + 1/2
    71  		// should round to d+1 if 'd' is odd.
    72  		roundUp = dfrac > 1<<(extra-1) ||
    73  			(dfrac == 1<<(extra-1) && !d0) ||
    74  			(dfrac == 1<<(extra-1) && d0 && di&1 == 1)
    75  	} else {
    76  		// otherwise, d+1/2 always rounds up because
    77  		// we truncated below.
    78  		roundUp = dfrac>>(extra-1) == 1
    79  	}
    80  	if dfrac != 0 {
    81  		d0 = false
    82  	}
    83  	// Proceed to the requested number of digits
    84  	formatDecimal(d, uint64(di), !d0, roundUp, prec)
    85  	// Adjust exponent
    86  	d.dp -= q
    87  }
    88  
    89  // ryuFtoaFixed64 formats mant*(2^exp) with prec decimal digits.
    90  func ryuFtoaFixed64(d *decimalSlice, mant uint64, exp int, prec int) {
    91  	if prec > 18 {
    92  		panic("ryuFtoaFixed64 called with prec > 18")
    93  	}
    94  	// Zero input.
    95  	if mant == 0 {
    96  		d.nd, d.dp = 0, 0
    97  		return
    98  	}
    99  	// Renormalize to a 55-bit mantissa.
   100  	e2 := exp
   101  	if b := bits.Len64(mant); b < 55 {
   102  		mant = mant << uint(55-b)
   103  		e2 += b - 55
   104  	}
   105  	// Choose an exponent such that rounded mant*(2^e2)*(10^q) has
   106  	// at least prec decimal digits, i.e
   107  	//     mant*(2^e2)*(10^q) >= 10^(prec-1)
   108  	// Because mant >= 2^54, it is enough to choose:
   109  	//     2^(e2+54) >= 10^(-q+prec-1)
   110  	// or q = -mulLog10_2(e2+54) + prec - 1
   111  	//
   112  	// The minimal required exponent is -mulLog10_2(1025)+18 = -291
   113  	// The maximal required exponent is mulLog10_2(1074)+18 = 342
   114  	q := -mulLog10_2(e2+54) + prec - 1
   115  
   116  	// Now compute mant*(2^e2)*(10^q).
   117  	// Is it an exact computation?
   118  	// Only small positive powers of 10 are exact (5^55 has 128 bits).
   119  	exact := q <= 55 && q >= 0
   120  
   121  	di, dexp2, d0 := mult128bitPow10(mant, e2, q)
   122  	if dexp2 >= 0 {
   123  		panic("not enough significant bits after mult128bitPow10")
   124  	}
   125  	// As a special case, computation might still be exact, if exponent
   126  	// was negative and if it amounts to computing an exact division.
   127  	// In that case, we ignore all lower bits.
   128  	// Note that division by 10^23 cannot be exact as 5^23 has 54 bits.
   129  	if q < 0 && q >= -22 && divisibleByPower5(mant, -q) {
   130  		exact = true
   131  		d0 = true
   132  	}
   133  	// Remove extra lower bits and keep rounding info.
   134  	extra := uint(-dexp2)
   135  	extraMask := uint64(1<<extra - 1)
   136  
   137  	di, dfrac := di>>extra, di&extraMask
   138  	roundUp := false
   139  	if exact {
   140  		// If we computed an exact product, d + 1/2
   141  		// should round to d+1 if 'd' is odd.
   142  		roundUp = dfrac > 1<<(extra-1) ||
   143  			(dfrac == 1<<(extra-1) && !d0) ||
   144  			(dfrac == 1<<(extra-1) && d0 && di&1 == 1)
   145  	} else {
   146  		// otherwise, d+1/2 always rounds up because
   147  		// we truncated below.
   148  		roundUp = dfrac>>(extra-1) == 1
   149  	}
   150  	if dfrac != 0 {
   151  		d0 = false
   152  	}
   153  	// Proceed to the requested number of digits
   154  	formatDecimal(d, di, !d0, roundUp, prec)
   155  	// Adjust exponent
   156  	d.dp -= q
   157  }
   158  
   159  var uint64pow10 = [...]uint64{
   160  	1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
   161  	1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
   162  }
   163  
   164  // formatDecimal fills d with at most prec decimal digits
   165  // of mantissa m. The boolean trunc indicates whether m
   166  // is truncated compared to the original number being formatted.
   167  func formatDecimal(d *decimalSlice, m uint64, trunc bool, roundUp bool, prec int) {
   168  	max := uint64pow10[prec]
   169  	trimmed := 0
   170  	for m >= max {
   171  		a, b := m/10, m%10
   172  		m = a
   173  		trimmed++
   174  		if b > 5 {
   175  			roundUp = true
   176  		} else if b < 5 {
   177  			roundUp = false
   178  		} else { // b == 5
   179  			// round up if there are trailing digits,
   180  			// or if the new value of m is odd (round-to-even convention)
   181  			roundUp = trunc || m&1 == 1
   182  		}
   183  		if b != 0 {
   184  			trunc = true
   185  		}
   186  	}
   187  	if roundUp {
   188  		m++
   189  	}
   190  	if m >= max {
   191  		// Happens if di was originally 99999....xx
   192  		m /= 10
   193  		trimmed++
   194  	}
   195  	// render digits
   196  	formatBase10(d.d[:prec], m)
   197  	d.nd = prec
   198  	for d.d[d.nd-1] == '0' {
   199  		d.nd--
   200  		trimmed++
   201  	}
   202  	d.dp = d.nd + trimmed
   203  }
   204  
   205  // ryuFtoaShortest formats mant*2^exp with prec decimal digits.
   206  func ryuFtoaShortest(d *decimalSlice, mant uint64, exp int, flt *floatInfo) {
   207  	if mant == 0 {
   208  		d.nd, d.dp = 0, 0
   209  		return
   210  	}
   211  	// If input is an exact integer with fewer bits than the mantissa,
   212  	// the previous and next integer are not admissible representations.
   213  	if exp <= 0 && bits.TrailingZeros64(mant) >= -exp {
   214  		mant >>= uint(-exp)
   215  		ryuDigits(d, mant, mant, mant, true, false)
   216  		return
   217  	}
   218  	ml, mc, mu, e2 := computeBounds(mant, exp, flt)
   219  	if e2 == 0 {
   220  		ryuDigits(d, ml, mc, mu, true, false)
   221  		return
   222  	}
   223  	// Find 10^q *larger* than 2^-e2
   224  	q := mulLog10_2(-e2) + 1
   225  
   226  	// We are going to multiply by 10^q using 128-bit arithmetic.
   227  	// The exponent is the same for all 3 numbers.
   228  	var dl, dc, du uint64
   229  	var dl0, dc0, du0 bool
   230  	if flt == &float32info {
   231  		var dl32, dc32, du32 uint32
   232  		dl32, _, dl0 = mult64bitPow10(uint32(ml), e2, q)
   233  		dc32, _, dc0 = mult64bitPow10(uint32(mc), e2, q)
   234  		du32, e2, du0 = mult64bitPow10(uint32(mu), e2, q)
   235  		dl, dc, du = uint64(dl32), uint64(dc32), uint64(du32)
   236  	} else {
   237  		dl, _, dl0 = mult128bitPow10(ml, e2, q)
   238  		dc, _, dc0 = mult128bitPow10(mc, e2, q)
   239  		du, e2, du0 = mult128bitPow10(mu, e2, q)
   240  	}
   241  	if e2 >= 0 {
   242  		panic("not enough significant bits after mult128bitPow10")
   243  	}
   244  	// Is it an exact computation?
   245  	if q > 55 {
   246  		// Large positive powers of ten are not exact
   247  		dl0, dc0, du0 = false, false, false
   248  	}
   249  	if q < 0 && q >= -24 {
   250  		// Division by a power of ten may be exact.
   251  		// (note that 5^25 is a 59-bit number so division by 5^25 is never exact).
   252  		if divisibleByPower5(ml, -q) {
   253  			dl0 = true
   254  		}
   255  		if divisibleByPower5(mc, -q) {
   256  			dc0 = true
   257  		}
   258  		if divisibleByPower5(mu, -q) {
   259  			du0 = true
   260  		}
   261  	}
   262  	// Express the results (dl, dc, du)*2^e2 as integers.
   263  	// Extra bits must be removed and rounding hints computed.
   264  	extra := uint(-e2)
   265  	extraMask := uint64(1<<extra - 1)
   266  	// Now compute the floored, integral base 10 mantissas.
   267  	dl, fracl := dl>>extra, dl&extraMask
   268  	dc, fracc := dc>>extra, dc&extraMask
   269  	du, fracu := du>>extra, du&extraMask
   270  	// Is it allowed to use 'du' as a result?
   271  	// It is always allowed when it is truncated, but also
   272  	// if it is exact and the original binary mantissa is even
   273  	// When disallowed, we can subtract 1.
   274  	uok := !du0 || fracu > 0
   275  	if du0 && fracu == 0 {
   276  		uok = mant&1 == 0
   277  	}
   278  	if !uok {
   279  		du--
   280  	}
   281  	// Is 'dc' the correctly rounded base 10 mantissa?
   282  	// The correct rounding might be dc+1
   283  	cup := false // don't round up.
   284  	if dc0 {
   285  		// If we computed an exact product, the half integer
   286  		// should round to next (even) integer if 'dc' is odd.
   287  		cup = fracc > 1<<(extra-1) ||
   288  			(fracc == 1<<(extra-1) && dc&1 == 1)
   289  	} else {
   290  		// otherwise, the result is a lower truncation of the ideal
   291  		// result.
   292  		cup = fracc>>(extra-1) == 1
   293  	}
   294  	// Is 'dl' an allowed representation?
   295  	// Only if it is an exact value, and if the original binary mantissa
   296  	// was even.
   297  	lok := dl0 && fracl == 0 && (mant&1 == 0)
   298  	if !lok {
   299  		dl++
   300  	}
   301  	// We need to remember whether the trimmed digits of 'dc' are zero.
   302  	c0 := dc0 && fracc == 0
   303  	// render digits
   304  	ryuDigits(d, dl, dc, du, c0, cup)
   305  	d.dp -= q
   306  }
   307  
   308  // computeBounds returns a floating-point vector (l, c, u)×2^e2
   309  // where the mantissas are 55-bit (or 26-bit) integers, describing the interval
   310  // represented by the input float64 or float32.
   311  func computeBounds(mant uint64, exp int, flt *floatInfo) (lower, central, upper uint64, e2 int) {
   312  	if mant != 1<<flt.mantbits || exp == flt.bias+1-int(flt.mantbits) {
   313  		// regular case (or denormals)
   314  		lower, central, upper = 2*mant-1, 2*mant, 2*mant+1
   315  		e2 = exp - 1
   316  		return
   317  	} else {
   318  		// border of an exponent
   319  		lower, central, upper = 4*mant-1, 4*mant, 4*mant+2
   320  		e2 = exp - 2
   321  		return
   322  	}
   323  }
   324  
   325  func ryuDigits(d *decimalSlice, lower, central, upper uint64,
   326  	c0, cup bool) {
   327  	lhi, llo := uint32(lower/1e9), uint32(lower%1e9)
   328  	chi, clo := uint32(central/1e9), uint32(central%1e9)
   329  	uhi, ulo := uint32(upper/1e9), uint32(upper%1e9)
   330  	if uhi == 0 {
   331  		// only low digits (for denormals)
   332  		ryuDigits32(d, llo, clo, ulo, c0, cup, 8)
   333  	} else if lhi < uhi {
   334  		// truncate 9 digits at once.
   335  		if llo != 0 {
   336  			lhi++
   337  		}
   338  		c0 = c0 && clo == 0
   339  		cup = (clo > 5e8) || (clo == 5e8 && cup)
   340  		ryuDigits32(d, lhi, chi, uhi, c0, cup, 8)
   341  		d.dp += 9
   342  	} else {
   343  		d.nd = 0
   344  		// emit high part
   345  		n := uint(9)
   346  		for v := chi; v > 0; {
   347  			v1, v2 := v/10, v%10
   348  			v = v1
   349  			n--
   350  			d.d[n] = byte(v2 + '0')
   351  		}
   352  		d.d = d.d[n:]
   353  		d.nd = int(9 - n)
   354  		// emit low part
   355  		ryuDigits32(d, llo, clo, ulo,
   356  			c0, cup, d.nd+8)
   357  	}
   358  	// trim trailing zeros
   359  	for d.nd > 0 && d.d[d.nd-1] == '0' {
   360  		d.nd--
   361  	}
   362  	// trim initial zeros
   363  	for d.nd > 0 && d.d[0] == '0' {
   364  		d.nd--
   365  		d.dp--
   366  		d.d = d.d[1:]
   367  	}
   368  }
   369  
   370  // ryuDigits32 emits decimal digits for a number less than 1e9.
   371  func ryuDigits32(d *decimalSlice, lower, central, upper uint32,
   372  	c0, cup bool, endindex int) {
   373  	if upper == 0 {
   374  		d.dp = endindex + 1
   375  		return
   376  	}
   377  	trimmed := 0
   378  	// Remember last trimmed digit to check for round-up.
   379  	// c0 will be used to remember zeroness of following digits.
   380  	cNextDigit := 0
   381  	for upper > 0 {
   382  		// Repeatedly compute:
   383  		// l = Ceil(lower / 10^k)
   384  		// c = Round(central / 10^k)
   385  		// u = Floor(upper / 10^k)
   386  		// and stop when c goes out of the (l, u) interval.
   387  		l := (lower + 9) / 10
   388  		c, cdigit := central/10, central%10
   389  		u := upper / 10
   390  		if l > u {
   391  			// don't trim the last digit as it is forbidden to go below l
   392  			// other, trim and exit now.
   393  			break
   394  		}
   395  		// Check that we didn't cross the lower boundary.
   396  		// The case where l < u but c == l-1 is essentially impossible,
   397  		// but may happen if:
   398  		//    lower   = ..11
   399  		//    central = ..19
   400  		//    upper   = ..31
   401  		// and means that 'central' is very close but less than
   402  		// an integer ending with many zeros, and usually
   403  		// the "round-up" logic hides the problem.
   404  		if l == c+1 && c < u {
   405  			c++
   406  			cdigit = 0
   407  			cup = false
   408  		}
   409  		trimmed++
   410  		// Remember trimmed digits of c
   411  		c0 = c0 && cNextDigit == 0
   412  		cNextDigit = int(cdigit)
   413  		lower, central, upper = l, c, u
   414  	}
   415  	// should we round up?
   416  	if trimmed > 0 {
   417  		cup = cNextDigit > 5 ||
   418  			(cNextDigit == 5 && !c0) ||
   419  			(cNextDigit == 5 && c0 && central&1 == 1)
   420  	}
   421  	if central < upper && cup {
   422  		central++
   423  	}
   424  	// We know where the number ends, fill directly
   425  	endindex -= trimmed
   426  	v := central
   427  	n := endindex
   428  	for n > d.nd {
   429  		v1, v2 := v/100, v%100
   430  		d.d[n] = smalls[2*v2+1]
   431  		d.d[n-1] = smalls[2*v2+0]
   432  		n -= 2
   433  		v = v1
   434  	}
   435  	if n == d.nd {
   436  		d.d[n] = byte(v + '0')
   437  	}
   438  	d.nd = endindex + 1
   439  	d.dp = d.nd + trimmed
   440  }
   441  
   442  // mult64bitPow10 takes a floating-point input with a 25-bit
   443  // mantissa and multiplies it with 10^q. The resulting mantissa
   444  // is m*P >> 57 where P is a 64-bit truncated power of 10.
   445  // It is typically 31 or 32-bit wide.
   446  // The returned boolean is true if all trimmed bits were zero.
   447  //
   448  // That is:
   449  //
   450  //	m*2^e2 * round(10^q) = resM * 2^resE + ε
   451  //	exact = ε == 0
   452  func mult64bitPow10(m uint32, e2, q int) (resM uint32, resE int, exact bool) {
   453  	if q == 0 {
   454  		// P == 1<<63
   455  		return m << 6, e2 - 6, true
   456  	}
   457  	pow, exp2, ok := pow10(q)
   458  	if !ok {
   459  		// This never happens due to the range of float32/float64 exponent
   460  		panic("mult64bitPow10: power of 10 is out of range")
   461  	}
   462  	if q < 0 {
   463  		// Inverse powers of ten must be rounded up.
   464  		pow.Hi++
   465  	}
   466  	hi, lo := bits.Mul64(uint64(m), pow.Hi)
   467  	e2 += exp2 - 63 + 57
   468  	return uint32(hi<<7 | lo>>57), e2, lo<<7 == 0
   469  }
   470  
   471  // mult128bitPow10 takes a floating-point input with a 55-bit
   472  // mantissa and multiplies it with 10^q. The resulting mantissa
   473  // is m*P >> 119 where P is a 128-bit truncated power of 10.
   474  // It is typically 63 or 64-bit wide.
   475  // The returned boolean is true is all trimmed bits were zero.
   476  //
   477  // That is:
   478  //
   479  //	m*2^e2 * round(10^q) = resM * 2^resE + ε
   480  //	exact = ε == 0
   481  func mult128bitPow10(m uint64, e2, q int) (resM uint64, resE int, exact bool) {
   482  	if q == 0 {
   483  		// P == 1<<127
   484  		return m << 8, e2 - 8, true
   485  	}
   486  	pow, exp2, ok := pow10(q)
   487  	if !ok {
   488  		// This never happens due to the range of float32/float64 exponent
   489  		panic("mult128bitPow10: power of 10 is out of range")
   490  	}
   491  	if q < 0 {
   492  		// Inverse powers of ten must be rounded up.
   493  		pow.Lo++
   494  	}
   495  	e2 += exp2 - 127 + 119
   496  
   497  	hi, mid, lo := umul192(m, pow)
   498  	return hi<<9 | mid>>55, e2, mid<<9 == 0 && lo == 0
   499  }
   500  
   501  func divisibleByPower5(m uint64, k int) bool {
   502  	if m == 0 {
   503  		return true
   504  	}
   505  	for i := 0; i < k; i++ {
   506  		if m%5 != 0 {
   507  			return false
   508  		}
   509  		m /= 5
   510  	}
   511  	return true
   512  }
   513
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