ftoa.go

     1  // Copyright 2009 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  // Binary to decimal floating point conversion.
     6  // Algorithm:
     7  //   1) store mantissa in multiprecision decimal
     8  //   2) shift decimal by exponent
     9  //   3) read digits out & format
    10  
    11  package strconv
    12  
    13  import "math"
    14  
    15  // TODO: move elsewhere?
    16  type floatInfo struct {
    17  	mantbits uint
    18  	expbits  uint
    19  	bias     int
    20  }
    21  
    22  var float32info = floatInfo{23, 8, -127}
    23  var float64info = floatInfo{52, 11, -1023}
    24  
    25  // FormatFloat converts the floating-point number f to a string,
    26  // according to the format fmt and precision prec. It rounds the
    27  // result assuming that the original was obtained from a floating-point
    28  // value of bitSize bits (32 for float32, 64 for float64).
    29  //
    30  // The format fmt is one of
    31  //   - 'b' (-ddddp±ddd, a binary exponent),
    32  //   - 'e' (-d.dddde±dd, a decimal exponent),
    33  //   - 'E' (-d.ddddE±dd, a decimal exponent),
    34  //   - 'f' (-ddd.dddd, no exponent),
    35  //   - 'g' ('e' for large exponents, 'f' otherwise),
    36  //   - 'G' ('E' for large exponents, 'f' otherwise),
    37  //   - 'x' (-0xd.ddddp±ddd, a hexadecimal fraction and binary exponent), or
    38  //   - 'X' (-0Xd.ddddP±ddd, a hexadecimal fraction and binary exponent).
    39  //
    40  // The precision prec controls the number of digits (excluding the exponent)
    41  // printed by the 'e', 'E', 'f', 'g', 'G', 'x', and 'X' formats.
    42  // For 'e', 'E', 'f', 'x', and 'X', it is the number of digits after the decimal point.
    43  // For 'g' and 'G' it is the maximum number of significant digits (trailing
    44  // zeros are removed).
    45  // The special precision -1 uses the smallest number of digits
    46  // necessary such that ParseFloat will return f exactly.
    47  // The exponent is written as a decimal integer;
    48  // for all formats other than 'b', it will be at least two digits.
    49  func FormatFloat(f float64, fmt byte, prec, bitSize int) string {
    50  	return string(genericFtoa(make([]byte, 0, max(prec+4, 24)), f, fmt, prec, bitSize))
    51  }
    52  
    53  // AppendFloat appends the string form of the floating-point number f,
    54  // as generated by [FormatFloat], to dst and returns the extended buffer.
    55  func AppendFloat(dst []byte, f float64, fmt byte, prec, bitSize int) []byte {
    56  	return genericFtoa(dst, f, fmt, prec, bitSize)
    57  }
    58  
    59  func genericFtoa(dst []byte, val float64, fmt byte, prec, bitSize int) []byte {
    60  	var bits uint64
    61  	var flt *floatInfo
    62  	switch bitSize {
    63  	case 32:
    64  		bits = uint64(math.Float32bits(float32(val)))
    65  		flt = &float32info
    66  	case 64:
    67  		bits = math.Float64bits(val)
    68  		flt = &float64info
    69  	default:
    70  		panic("strconv: illegal AppendFloat/FormatFloat bitSize")
    71  	}
    72  
    73  	neg := bits>>(flt.expbits+flt.mantbits) != 0
    74  	exp := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
    75  	mant := bits & (uint64(1)<<flt.mantbits - 1)
    76  
    77  	switch exp {
    78  	case 1<<flt.expbits - 1:
    79  		// Inf, NaN
    80  		var s string
    81  		switch {
    82  		case mant != 0:
    83  			s = "NaN"
    84  		case neg:
    85  			s = "-Inf"
    86  		default:
    87  			s = "+Inf"
    88  		}
    89  		return append(dst, s...)
    90  
    91  	case 0:
    92  		// denormalized
    93  		exp++
    94  
    95  	default:
    96  		// add implicit top bit
    97  		mant |= uint64(1) << flt.mantbits
    98  	}
    99  	exp += flt.bias
   100  
   101  	// Pick off easy binary, hex formats.
   102  	if fmt == 'b' {
   103  		return fmtB(dst, neg, mant, exp, flt)
   104  	}
   105  	if fmt == 'x' || fmt == 'X' {
   106  		return fmtX(dst, prec, fmt, neg, mant, exp, flt)
   107  	}
   108  
   109  	if !optimize {
   110  		return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
   111  	}
   112  
   113  	var digs decimalSlice
   114  	ok := false
   115  	// Negative precision means "only as much as needed to be exact."
   116  	shortest := prec < 0
   117  	if shortest {
   118  		// Use Ryu algorithm.
   119  		var buf [32]byte
   120  		digs.d = buf[:]
   121  		ryuFtoaShortest(&digs, mant, exp-int(flt.mantbits), flt)
   122  		ok = true
   123  		// Precision for shortest representation mode.
   124  		switch fmt {
   125  		case 'e', 'E':
   126  			prec = max(digs.nd-1, 0)
   127  		case 'f':
   128  			prec = max(digs.nd-digs.dp, 0)
   129  		case 'g', 'G':
   130  			prec = digs.nd
   131  		}
   132  	} else if fmt != 'f' {
   133  		// Fixed number of digits.
   134  		digits := prec
   135  		switch fmt {
   136  		case 'e', 'E':
   137  			digits++
   138  		case 'g', 'G':
   139  			if prec == 0 {
   140  				prec = 1
   141  			}
   142  			digits = prec
   143  		default:
   144  			// Invalid mode.
   145  			digits = 1
   146  		}
   147  		var buf [24]byte
   148  		if bitSize == 32 && digits <= 9 {
   149  			digs.d = buf[:]
   150  			ryuFtoaFixed32(&digs, uint32(mant), exp-int(flt.mantbits), digits)
   151  			ok = true
   152  		} else if digits <= 18 {
   153  			digs.d = buf[:]
   154  			ryuFtoaFixed64(&digs, mant, exp-int(flt.mantbits), digits)
   155  			ok = true
   156  		}
   157  	}
   158  	if !ok {
   159  		return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
   160  	}
   161  	return formatDigits(dst, shortest, neg, digs, prec, fmt)
   162  }
   163  
   164  // bigFtoa uses multiprecision computations to format a float.
   165  func bigFtoa(dst []byte, prec int, fmt byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   166  	d := new(decimal)
   167  	d.Assign(mant)
   168  	d.Shift(exp - int(flt.mantbits))
   169  	var digs decimalSlice
   170  	shortest := prec < 0
   171  	if shortest {
   172  		roundShortest(d, mant, exp, flt)
   173  		digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
   174  		// Precision for shortest representation mode.
   175  		switch fmt {
   176  		case 'e', 'E':
   177  			prec = digs.nd - 1
   178  		case 'f':
   179  			prec = max(digs.nd-digs.dp, 0)
   180  		case 'g', 'G':
   181  			prec = digs.nd
   182  		}
   183  	} else {
   184  		// Round appropriately.
   185  		switch fmt {
   186  		case 'e', 'E':
   187  			d.Round(prec + 1)
   188  		case 'f':
   189  			d.Round(d.dp + prec)
   190  		case 'g', 'G':
   191  			if prec == 0 {
   192  				prec = 1
   193  			}
   194  			d.Round(prec)
   195  		}
   196  		digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
   197  	}
   198  	return formatDigits(dst, shortest, neg, digs, prec, fmt)
   199  }
   200  
   201  func formatDigits(dst []byte, shortest bool, neg bool, digs decimalSlice, prec int, fmt byte) []byte {
   202  	switch fmt {
   203  	case 'e', 'E':
   204  		return fmtE(dst, neg, digs, prec, fmt)
   205  	case 'f':
   206  		return fmtF(dst, neg, digs, prec)
   207  	case 'g', 'G':
   208  		// trailing fractional zeros in 'e' form will be trimmed.
   209  		eprec := prec
   210  		if eprec > digs.nd && digs.nd >= digs.dp {
   211  			eprec = digs.nd
   212  		}
   213  		// %e is used if the exponent from the conversion
   214  		// is less than -4 or greater than or equal to the precision.
   215  		// if precision was the shortest possible, use precision 6 for this decision.
   216  		if shortest {
   217  			eprec = 6
   218  		}
   219  		exp := digs.dp - 1
   220  		if exp < -4 || exp >= eprec {
   221  			if prec > digs.nd {
   222  				prec = digs.nd
   223  			}
   224  			return fmtE(dst, neg, digs, prec-1, fmt+'e'-'g')
   225  		}
   226  		if prec > digs.dp {
   227  			prec = digs.nd
   228  		}
   229  		return fmtF(dst, neg, digs, max(prec-digs.dp, 0))
   230  	}
   231  
   232  	// unknown format
   233  	return append(dst, '%', fmt)
   234  }
   235  
   236  // roundShortest rounds d (= mant * 2^exp) to the shortest number of digits
   237  // that will let the original floating point value be precisely reconstructed.
   238  func roundShortest(d *decimal, mant uint64, exp int, flt *floatInfo) {
   239  	// If mantissa is zero, the number is zero; stop now.
   240  	if mant == 0 {
   241  		d.nd = 0
   242  		return
   243  	}
   244  
   245  	// Compute upper and lower such that any decimal number
   246  	// between upper and lower (possibly inclusive)
   247  	// will round to the original floating point number.
   248  
   249  	// We may see at once that the number is already shortest.
   250  	//
   251  	// Suppose d is not denormal, so that 2^exp <= d < 10^dp.
   252  	// The closest shorter number is at least 10^(dp-nd) away.
   253  	// The lower/upper bounds computed below are at distance
   254  	// at most 2^(exp-mantbits).
   255  	//
   256  	// So the number is already shortest if 10^(dp-nd) > 2^(exp-mantbits),
   257  	// or equivalently log2(10)*(dp-nd) > exp-mantbits.
   258  	// It is true if 332/100*(dp-nd) >= exp-mantbits (log2(10) > 3.32).
   259  	minexp := flt.bias + 1 // minimum possible exponent
   260  	if exp > minexp && 332*(d.dp-d.nd) >= 100*(exp-int(flt.mantbits)) {
   261  		// The number is already shortest.
   262  		return
   263  	}
   264  
   265  	// d = mant << (exp - mantbits)
   266  	// Next highest floating point number is mant+1 << exp-mantbits.
   267  	// Our upper bound is halfway between, mant*2+1 << exp-mantbits-1.
   268  	upper := new(decimal)
   269  	upper.Assign(mant*2 + 1)
   270  	upper.Shift(exp - int(flt.mantbits) - 1)
   271  
   272  	// d = mant << (exp - mantbits)
   273  	// Next lowest floating point number is mant-1 << exp-mantbits,
   274  	// unless mant-1 drops the significant bit and exp is not the minimum exp,
   275  	// in which case the next lowest is mant*2-1 << exp-mantbits-1.
   276  	// Either way, call it mantlo << explo-mantbits.
   277  	// Our lower bound is halfway between, mantlo*2+1 << explo-mantbits-1.
   278  	var mantlo uint64
   279  	var explo int
   280  	if mant > 1<<flt.mantbits || exp == minexp {
   281  		mantlo = mant - 1
   282  		explo = exp
   283  	} else {
   284  		mantlo = mant*2 - 1
   285  		explo = exp - 1
   286  	}
   287  	lower := new(decimal)
   288  	lower.Assign(mantlo*2 + 1)
   289  	lower.Shift(explo - int(flt.mantbits) - 1)
   290  
   291  	// The upper and lower bounds are possible outputs only if
   292  	// the original mantissa is even, so that IEEE round-to-even
   293  	// would round to the original mantissa and not the neighbors.
   294  	inclusive := mant%2 == 0
   295  
   296  	// As we walk the digits we want to know whether rounding up would fall
   297  	// within the upper bound. This is tracked by upperdelta:
   298  	//
   299  	// If upperdelta == 0, the digits of d and upper are the same so far.
   300  	//
   301  	// If upperdelta == 1, we saw a difference of 1 between d and upper on a
   302  	// previous digit and subsequently only 9s for d and 0s for upper.
   303  	// (Thus rounding up may fall outside the bound, if it is exclusive.)
   304  	//
   305  	// If upperdelta == 2, then the difference is greater than 1
   306  	// and we know that rounding up falls within the bound.
   307  	var upperdelta uint8
   308  
   309  	// Now we can figure out the minimum number of digits required.
   310  	// Walk along until d has distinguished itself from upper and lower.
   311  	for ui := 0; ; ui++ {
   312  		// lower, d, and upper may have the decimal points at different
   313  		// places. In this case upper is the longest, so we iterate from
   314  		// ui==0 and start li and mi at (possibly) -1.
   315  		mi := ui - upper.dp + d.dp
   316  		if mi >= d.nd {
   317  			break
   318  		}
   319  		li := ui - upper.dp + lower.dp
   320  		l := byte('0') // lower digit
   321  		if li >= 0 && li < lower.nd {
   322  			l = lower.d[li]
   323  		}
   324  		m := byte('0') // middle digit
   325  		if mi >= 0 {
   326  			m = d.d[mi]
   327  		}
   328  		u := byte('0') // upper digit
   329  		if ui < upper.nd {
   330  			u = upper.d[ui]
   331  		}
   332  
   333  		// Okay to round down (truncate) if lower has a different digit
   334  		// or if lower is inclusive and is exactly the result of rounding
   335  		// down (i.e., and we have reached the final digit of lower).
   336  		okdown := l != m || inclusive && li+1 == lower.nd
   337  
   338  		switch {
   339  		case upperdelta == 0 && m+1 < u:
   340  			// Example:
   341  			// m = 12345xxx
   342  			// u = 12347xxx
   343  			upperdelta = 2
   344  		case upperdelta == 0 && m != u:
   345  			// Example:
   346  			// m = 12345xxx
   347  			// u = 12346xxx
   348  			upperdelta = 1
   349  		case upperdelta == 1 && (m != '9' || u != '0'):
   350  			// Example:
   351  			// m = 1234598x
   352  			// u = 1234600x
   353  			upperdelta = 2
   354  		}
   355  		// Okay to round up if upper has a different digit and either upper
   356  		// is inclusive or upper is bigger than the result of rounding up.
   357  		okup := upperdelta > 0 && (inclusive || upperdelta > 1 || ui+1 < upper.nd)
   358  
   359  		// If it's okay to do either, then round to the nearest one.
   360  		// If it's okay to do only one, do it.
   361  		switch {
   362  		case okdown && okup:
   363  			d.Round(mi + 1)
   364  			return
   365  		case okdown:
   366  			d.RoundDown(mi + 1)
   367  			return
   368  		case okup:
   369  			d.RoundUp(mi + 1)
   370  			return
   371  		}
   372  	}
   373  }
   374  
   375  type decimalSlice struct {
   376  	d      []byte
   377  	nd, dp int
   378  }
   379  
   380  // %e: -d.ddddde±dd
   381  func fmtE(dst []byte, neg bool, d decimalSlice, prec int, fmt byte) []byte {
   382  	// sign
   383  	if neg {
   384  		dst = append(dst, '-')
   385  	}
   386  
   387  	// first digit
   388  	ch := byte('0')
   389  	if d.nd != 0 {
   390  		ch = d.d[0]
   391  	}
   392  	dst = append(dst, ch)
   393  
   394  	// .moredigits
   395  	if prec > 0 {
   396  		dst = append(dst, '.')
   397  		i := 1
   398  		m := min(d.nd, prec+1)
   399  		if i < m {
   400  			dst = append(dst, d.d[i:m]...)
   401  			i = m
   402  		}
   403  		for ; i <= prec; i++ {
   404  			dst = append(dst, '0')
   405  		}
   406  	}
   407  
   408  	// e±
   409  	dst = append(dst, fmt)
   410  	exp := d.dp - 1
   411  	if d.nd == 0 { // special case: 0 has exponent 0
   412  		exp = 0
   413  	}
   414  	if exp < 0 {
   415  		ch = '-'
   416  		exp = -exp
   417  	} else {
   418  		ch = '+'
   419  	}
   420  	dst = append(dst, ch)
   421  
   422  	// dd or ddd
   423  	switch {
   424  	case exp < 10:
   425  		dst = append(dst, '0', byte(exp)+'0')
   426  	case exp < 100:
   427  		dst = append(dst, byte(exp/10)+'0', byte(exp%10)+'0')
   428  	default:
   429  		dst = append(dst, byte(exp/100)+'0', byte(exp/10)%10+'0', byte(exp%10)+'0')
   430  	}
   431  
   432  	return dst
   433  }
   434  
   435  // %f: -ddddddd.ddddd
   436  func fmtF(dst []byte, neg bool, d decimalSlice, prec int) []byte {
   437  	// sign
   438  	if neg {
   439  		dst = append(dst, '-')
   440  	}
   441  
   442  	// integer, padded with zeros as needed.
   443  	if d.dp > 0 {
   444  		m := min(d.nd, d.dp)
   445  		dst = append(dst, d.d[:m]...)
   446  		for ; m < d.dp; m++ {
   447  			dst = append(dst, '0')
   448  		}
   449  	} else {
   450  		dst = append(dst, '0')
   451  	}
   452  
   453  	// fraction
   454  	if prec > 0 {
   455  		dst = append(dst, '.')
   456  		for i := 0; i < prec; i++ {
   457  			ch := byte('0')
   458  			if j := d.dp + i; 0 <= j && j < d.nd {
   459  				ch = d.d[j]
   460  			}
   461  			dst = append(dst, ch)
   462  		}
   463  	}
   464  
   465  	return dst
   466  }
   467  
   468  // %b: -ddddddddp±ddd
   469  func fmtB(dst []byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   470  	// sign
   471  	if neg {
   472  		dst = append(dst, '-')
   473  	}
   474  
   475  	// mantissa
   476  	dst, _ = formatBits(dst, mant, 10, false, true)
   477  
   478  	// p
   479  	dst = append(dst, 'p')
   480  
   481  	// ±exponent
   482  	exp -= int(flt.mantbits)
   483  	if exp >= 0 {
   484  		dst = append(dst, '+')
   485  	}
   486  	dst, _ = formatBits(dst, uint64(exp), 10, exp < 0, true)
   487  
   488  	return dst
   489  }
   490  
   491  // %x: -0x1.yyyyyyyyp±ddd or -0x0p+0. (y is hex digit, d is decimal digit)
   492  func fmtX(dst []byte, prec int, fmt byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   493  	if mant == 0 {
   494  		exp = 0
   495  	}
   496  
   497  	// Shift digits so leading 1 (if any) is at bit 1<<60.
   498  	mant <<= 60 - flt.mantbits
   499  	for mant != 0 && mant&(1<<60) == 0 {
   500  		mant <<= 1
   501  		exp--
   502  	}
   503  
   504  	// Round if requested.
   505  	if prec >= 0 && prec < 15 {
   506  		shift := uint(prec * 4)
   507  		extra := (mant << shift) & (1<<60 - 1)
   508  		mant >>= 60 - shift
   509  		if extra|(mant&1) > 1<<59 {
   510  			mant++
   511  		}
   512  		mant <<= 60 - shift
   513  		if mant&(1<<61) != 0 {
   514  			// Wrapped around.
   515  			mant >>= 1
   516  			exp++
   517  		}
   518  	}
   519  
   520  	hex := lowerhex
   521  	if fmt == 'X' {
   522  		hex = upperhex
   523  	}
   524  
   525  	// sign, 0x, leading digit
   526  	if neg {
   527  		dst = append(dst, '-')
   528  	}
   529  	dst = append(dst, '0', fmt, '0'+byte((mant>>60)&1))
   530  
   531  	// .fraction
   532  	mant <<= 4 // remove leading 0 or 1
   533  	if prec < 0 && mant != 0 {
   534  		dst = append(dst, '.')
   535  		for mant != 0 {
   536  			dst = append(dst, hex[(mant>>60)&15])
   537  			mant <<= 4
   538  		}
   539  	} else if prec > 0 {
   540  		dst = append(dst, '.')
   541  		for i := 0; i < prec; i++ {
   542  			dst = append(dst, hex[(mant>>60)&15])
   543  			mant <<= 4
   544  		}
   545  	}
   546  
   547  	// p±
   548  	ch := byte('P')
   549  	if fmt == lower(fmt) {
   550  		ch = 'p'
   551  	}
   552  	dst = append(dst, ch)
   553  	if exp < 0 {
   554  		ch = '-'
   555  		exp = -exp
   556  	} else {
   557  		ch = '+'
   558  	}
   559  	dst = append(dst, ch)
   560  
   561  	// dd or ddd or dddd
   562  	switch {
   563  	case exp < 100:
   564  		dst = append(dst, byte(exp/10)+'0', byte(exp%10)+'0')
   565  	case exp < 1000:
   566  		dst = append(dst, byte(exp/100)+'0', byte((exp/10)%10)+'0', byte(exp%10)+'0')
   567  	default:
   568  		dst = append(dst, byte(exp/1000)+'0', byte(exp/100)%10+'0', byte((exp/10)%10)+'0', byte(exp%10)+'0')
   569  	}
   570  
   571  	return dst
   572  }
   573
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