Source file src/strconv/ftoa.go

Documentation: strconv

     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  func FormatFloat(f float64, fmt byte, prec, bitSize int) string {
    48  	return string(genericFtoa(make([]byte, 0, max(prec+4, 24)), f, fmt, prec, bitSize))
    49  }
    50  
    51  // AppendFloat appends the string form of the floating-point number f,
    52  // as generated by FormatFloat, to dst and returns the extended buffer.
    53  func AppendFloat(dst []byte, f float64, fmt byte, prec, bitSize int) []byte {
    54  	return genericFtoa(dst, f, fmt, prec, bitSize)
    55  }
    56  
    57  func genericFtoa(dst []byte, val float64, fmt byte, prec, bitSize int) []byte {
    58  	var bits uint64
    59  	var flt *floatInfo
    60  	switch bitSize {
    61  	case 32:
    62  		bits = uint64(math.Float32bits(float32(val)))
    63  		flt = &float32info
    64  	case 64:
    65  		bits = math.Float64bits(val)
    66  		flt = &float64info
    67  	default:
    68  		panic("strconv: illegal AppendFloat/FormatFloat bitSize")
    69  	}
    70  
    71  	neg := bits>>(flt.expbits+flt.mantbits) != 0
    72  	exp := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
    73  	mant := bits & (uint64(1)<<flt.mantbits - 1)
    74  
    75  	switch exp {
    76  	case 1<<flt.expbits - 1:
    77  		// Inf, NaN
    78  		var s string
    79  		switch {
    80  		case mant != 0:
    81  			s = "NaN"
    82  		case neg:
    83  			s = "-Inf"
    84  		default:
    85  			s = "+Inf"
    86  		}
    87  		return append(dst, s...)
    88  
    89  	case 0:
    90  		// denormalized
    91  		exp++
    92  
    93  	default:
    94  		// add implicit top bit
    95  		mant |= uint64(1) << flt.mantbits
    96  	}
    97  	exp += flt.bias
    98  
    99  	// Pick off easy binary, hex formats.
   100  	if fmt == 'b' {
   101  		return fmtB(dst, neg, mant, exp, flt)
   102  	}
   103  	if fmt == 'x' || fmt == 'X' {
   104  		return fmtX(dst, prec, fmt, neg, mant, exp, flt)
   105  	}
   106  
   107  	if !optimize {
   108  		return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
   109  	}
   110  
   111  	var digs decimalSlice
   112  	ok := false
   113  	// Negative precision means "only as much as needed to be exact."
   114  	shortest := prec < 0
   115  	if shortest {
   116  		// Try Grisu3 algorithm.
   117  		f := new(extFloat)
   118  		lower, upper := f.AssignComputeBounds(mant, exp, neg, flt)
   119  		var buf [32]byte
   120  		digs.d = buf[:]
   121  		ok = f.ShortestDecimal(&digs, &lower, &upper)
   122  		if !ok {
   123  			return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
   124  		}
   125  		// Precision for shortest representation mode.
   126  		switch fmt {
   127  		case 'e', 'E':
   128  			prec = max(digs.nd-1, 0)
   129  		case 'f':
   130  			prec = max(digs.nd-digs.dp, 0)
   131  		case 'g', 'G':
   132  			prec = digs.nd
   133  		}
   134  	} else if fmt != 'f' {
   135  		// Fixed number of digits.
   136  		digits := prec
   137  		switch fmt {
   138  		case 'e', 'E':
   139  			digits++
   140  		case 'g', 'G':
   141  			if prec == 0 {
   142  				prec = 1
   143  			}
   144  			digits = prec
   145  		}
   146  		if digits <= 15 {
   147  			// try fast algorithm when the number of digits is reasonable.
   148  			var buf [24]byte
   149  			digs.d = buf[:]
   150  			f := extFloat{mant, exp - int(flt.mantbits), neg}
   151  			ok = f.FixedDecimal(&digs, digits)
   152  		}
   153  	}
   154  	if !ok {
   155  		return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
   156  	}
   157  	return formatDigits(dst, shortest, neg, digs, prec, fmt)
   158  }
   159  
   160  // bigFtoa uses multiprecision computations to format a float.
   161  func bigFtoa(dst []byte, prec int, fmt byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   162  	d := new(decimal)
   163  	d.Assign(mant)
   164  	d.Shift(exp - int(flt.mantbits))
   165  	var digs decimalSlice
   166  	shortest := prec < 0
   167  	if shortest {
   168  		roundShortest(d, mant, exp, flt)
   169  		digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
   170  		// Precision for shortest representation mode.
   171  		switch fmt {
   172  		case 'e', 'E':
   173  			prec = digs.nd - 1
   174  		case 'f':
   175  			prec = max(digs.nd-digs.dp, 0)
   176  		case 'g', 'G':
   177  			prec = digs.nd
   178  		}
   179  	} else {
   180  		// Round appropriately.
   181  		switch fmt {
   182  		case 'e', 'E':
   183  			d.Round(prec + 1)
   184  		case 'f':
   185  			d.Round(d.dp + prec)
   186  		case 'g', 'G':
   187  			if prec == 0 {
   188  				prec = 1
   189  			}
   190  			d.Round(prec)
   191  		}
   192  		digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
   193  	}
   194  	return formatDigits(dst, shortest, neg, digs, prec, fmt)
   195  }
   196  
   197  func formatDigits(dst []byte, shortest bool, neg bool, digs decimalSlice, prec int, fmt byte) []byte {
   198  	switch fmt {
   199  	case 'e', 'E':
   200  		return fmtE(dst, neg, digs, prec, fmt)
   201  	case 'f':
   202  		return fmtF(dst, neg, digs, prec)
   203  	case 'g', 'G':
   204  		// trailing fractional zeros in 'e' form will be trimmed.
   205  		eprec := prec
   206  		if eprec > digs.nd && digs.nd >= digs.dp {
   207  			eprec = digs.nd
   208  		}
   209  		// %e is used if the exponent from the conversion
   210  		// is less than -4 or greater than or equal to the precision.
   211  		// if precision was the shortest possible, use precision 6 for this decision.
   212  		if shortest {
   213  			eprec = 6
   214  		}
   215  		exp := digs.dp - 1
   216  		if exp < -4 || exp >= eprec {
   217  			if prec > digs.nd {
   218  				prec = digs.nd
   219  			}
   220  			return fmtE(dst, neg, digs, prec-1, fmt+'e'-'g')
   221  		}
   222  		if prec > digs.dp {
   223  			prec = digs.nd
   224  		}
   225  		return fmtF(dst, neg, digs, max(prec-digs.dp, 0))
   226  	}
   227  
   228  	// unknown format
   229  	return append(dst, '%', fmt)
   230  }
   231  
   232  // roundShortest rounds d (= mant * 2^exp) to the shortest number of digits
   233  // that will let the original floating point value be precisely reconstructed.
   234  func roundShortest(d *decimal, mant uint64, exp int, flt *floatInfo) {
   235  	// If mantissa is zero, the number is zero; stop now.
   236  	if mant == 0 {
   237  		d.nd = 0
   238  		return
   239  	}
   240  
   241  	// Compute upper and lower such that any decimal number
   242  	// between upper and lower (possibly inclusive)
   243  	// will round to the original floating point number.
   244  
   245  	// We may see at once that the number is already shortest.
   246  	//
   247  	// Suppose d is not denormal, so that 2^exp <= d < 10^dp.
   248  	// The closest shorter number is at least 10^(dp-nd) away.
   249  	// The lower/upper bounds computed below are at distance
   250  	// at most 2^(exp-mantbits).
   251  	//
   252  	// So the number is already shortest if 10^(dp-nd) > 2^(exp-mantbits),
   253  	// or equivalently log2(10)*(dp-nd) > exp-mantbits.
   254  	// It is true if 332/100*(dp-nd) >= exp-mantbits (log2(10) > 3.32).
   255  	minexp := flt.bias + 1 // minimum possible exponent
   256  	if exp > minexp && 332*(d.dp-d.nd) >= 100*(exp-int(flt.mantbits)) {
   257  		// The number is already shortest.
   258  		return
   259  	}
   260  
   261  	// d = mant << (exp - mantbits)
   262  	// Next highest floating point number is mant+1 << exp-mantbits.
   263  	// Our upper bound is halfway between, mant*2+1 << exp-mantbits-1.
   264  	upper := new(decimal)
   265  	upper.Assign(mant*2 + 1)
   266  	upper.Shift(exp - int(flt.mantbits) - 1)
   267  
   268  	// d = mant << (exp - mantbits)
   269  	// Next lowest floating point number is mant-1 << exp-mantbits,
   270  	// unless mant-1 drops the significant bit and exp is not the minimum exp,
   271  	// in which case the next lowest is mant*2-1 << exp-mantbits-1.
   272  	// Either way, call it mantlo << explo-mantbits.
   273  	// Our lower bound is halfway between, mantlo*2+1 << explo-mantbits-1.
   274  	var mantlo uint64
   275  	var explo int
   276  	if mant > 1<<flt.mantbits || exp == minexp {
   277  		mantlo = mant - 1
   278  		explo = exp
   279  	} else {
   280  		mantlo = mant*2 - 1
   281  		explo = exp - 1
   282  	}
   283  	lower := new(decimal)
   284  	lower.Assign(mantlo*2 + 1)
   285  	lower.Shift(explo - int(flt.mantbits) - 1)
   286  
   287  	// The upper and lower bounds are possible outputs only if
   288  	// the original mantissa is even, so that IEEE round-to-even
   289  	// would round to the original mantissa and not the neighbors.
   290  	inclusive := mant%2 == 0
   291  
   292  	// As we walk the digits we want to know whether rounding up would fall
   293  	// within the upper bound. This is tracked by upperdelta:
   294  	//
   295  	// If upperdelta == 0, the digits of d and upper are the same so far.
   296  	//
   297  	// If upperdelta == 1, we saw a difference of 1 between d and upper on a
   298  	// previous digit and subsequently only 9s for d and 0s for upper.
   299  	// (Thus rounding up may fall outside the bound, if it is exclusive.)
   300  	//
   301  	// If upperdelta == 2, then the difference is greater than 1
   302  	// and we know that rounding up falls within the bound.
   303  	var upperdelta uint8
   304  
   305  	// Now we can figure out the minimum number of digits required.
   306  	// Walk along until d has distinguished itself from upper and lower.
   307  	for ui := 0; ; ui++ {
   308  		// lower, d, and upper may have the decimal points at different
   309  		// places. In this case upper is the longest, so we iterate from
   310  		// ui==0 and start li and mi at (possibly) -1.
   311  		mi := ui - upper.dp + d.dp
   312  		if mi >= d.nd {
   313  			break
   314  		}
   315  		li := ui - upper.dp + lower.dp
   316  		l := byte('0') // lower digit
   317  		if li >= 0 && li < lower.nd {
   318  			l = lower.d[li]
   319  		}
   320  		m := byte('0') // middle digit
   321  		if mi >= 0 {
   322  			m = d.d[mi]
   323  		}
   324  		u := byte('0') // upper digit
   325  		if ui < upper.nd {
   326  			u = upper.d[ui]
   327  		}
   328  
   329  		// Okay to round down (truncate) if lower has a different digit
   330  		// or if lower is inclusive and is exactly the result of rounding
   331  		// down (i.e., and we have reached the final digit of lower).
   332  		okdown := l != m || inclusive && li+1 == lower.nd
   333  
   334  		switch {
   335  		case upperdelta == 0 && m+1 < u:
   336  			// Example:
   337  			// m = 12345xxx
   338  			// u = 12347xxx
   339  			upperdelta = 2
   340  		case upperdelta == 0 && m != u:
   341  			// Example:
   342  			// m = 12345xxx
   343  			// u = 12346xxx
   344  			upperdelta = 1
   345  		case upperdelta == 1 && (m != '9' || u != '0'):
   346  			// Example:
   347  			// m = 1234598x
   348  			// u = 1234600x
   349  			upperdelta = 2
   350  		}
   351  		// Okay to round up if upper has a different digit and either upper
   352  		// is inclusive or upper is bigger than the result of rounding up.
   353  		okup := upperdelta > 0 && (inclusive || upperdelta > 1 || ui+1 < upper.nd)
   354  
   355  		// If it's okay to do either, then round to the nearest one.
   356  		// If it's okay to do only one, do it.
   357  		switch {
   358  		case okdown && okup:
   359  			d.Round(mi + 1)
   360  			return
   361  		case okdown:
   362  			d.RoundDown(mi + 1)
   363  			return
   364  		case okup:
   365  			d.RoundUp(mi + 1)
   366  			return
   367  		}
   368  	}
   369  }
   370  
   371  type decimalSlice struct {
   372  	d      []byte
   373  	nd, dp int
   374  	neg    bool
   375  }
   376  
   377  // %e: -d.ddddde±dd
   378  func fmtE(dst []byte, neg bool, d decimalSlice, prec int, fmt byte) []byte {
   379  	// sign
   380  	if neg {
   381  		dst = append(dst, '-')
   382  	}
   383  
   384  	// first digit
   385  	ch := byte('0')
   386  	if d.nd != 0 {
   387  		ch = d.d[0]
   388  	}
   389  	dst = append(dst, ch)
   390  
   391  	// .moredigits
   392  	if prec > 0 {
   393  		dst = append(dst, '.')
   394  		i := 1
   395  		m := min(d.nd, prec+1)
   396  		if i < m {
   397  			dst = append(dst, d.d[i:m]...)
   398  			i = m
   399  		}
   400  		for ; i <= prec; i++ {
   401  			dst = append(dst, '0')
   402  		}
   403  	}
   404  
   405  	// e±
   406  	dst = append(dst, fmt)
   407  	exp := d.dp - 1
   408  	if d.nd == 0 { // special case: 0 has exponent 0
   409  		exp = 0
   410  	}
   411  	if exp < 0 {
   412  		ch = '-'
   413  		exp = -exp
   414  	} else {
   415  		ch = '+'
   416  	}
   417  	dst = append(dst, ch)
   418  
   419  	// dd or ddd
   420  	switch {
   421  	case exp < 10:
   422  		dst = append(dst, '0', byte(exp)+'0')
   423  	case exp < 100:
   424  		dst = append(dst, byte(exp/10)+'0', byte(exp%10)+'0')
   425  	default:
   426  		dst = append(dst, byte(exp/100)+'0', byte(exp/10)%10+'0', byte(exp%10)+'0')
   427  	}
   428  
   429  	return dst
   430  }
   431  
   432  // %f: -ddddddd.ddddd
   433  func fmtF(dst []byte, neg bool, d decimalSlice, prec int) []byte {
   434  	// sign
   435  	if neg {
   436  		dst = append(dst, '-')
   437  	}
   438  
   439  	// integer, padded with zeros as needed.
   440  	if d.dp > 0 {
   441  		m := min(d.nd, d.dp)
   442  		dst = append(dst, d.d[:m]...)
   443  		for ; m < d.dp; m++ {
   444  			dst = append(dst, '0')
   445  		}
   446  	} else {
   447  		dst = append(dst, '0')
   448  	}
   449  
   450  	// fraction
   451  	if prec > 0 {
   452  		dst = append(dst, '.')
   453  		for i := 0; i < prec; i++ {
   454  			ch := byte('0')
   455  			if j := d.dp + i; 0 <= j && j < d.nd {
   456  				ch = d.d[j]
   457  			}
   458  			dst = append(dst, ch)
   459  		}
   460  	}
   461  
   462  	return dst
   463  }
   464  
   465  // %b: -ddddddddp±ddd
   466  func fmtB(dst []byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   467  	// sign
   468  	if neg {
   469  		dst = append(dst, '-')
   470  	}
   471  
   472  	// mantissa
   473  	dst, _ = formatBits(dst, mant, 10, false, true)
   474  
   475  	// p
   476  	dst = append(dst, 'p')
   477  
   478  	// ±exponent
   479  	exp -= int(flt.mantbits)
   480  	if exp >= 0 {
   481  		dst = append(dst, '+')
   482  	}
   483  	dst, _ = formatBits(dst, uint64(exp), 10, exp < 0, true)
   484  
   485  	return dst
   486  }
   487  
   488  // %x: -0x1.yyyyyyyyp±ddd or -0x0p+0. (y is hex digit, d is decimal digit)
   489  func fmtX(dst []byte, prec int, fmt byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
   490  	if mant == 0 {
   491  		exp = 0
   492  	}
   493  
   494  	// Shift digits so leading 1 (if any) is at bit 1<<60.
   495  	mant <<= 60 - flt.mantbits
   496  	for mant != 0 && mant&(1<<60) == 0 {
   497  		mant <<= 1
   498  		exp--
   499  	}
   500  
   501  	// Round if requested.
   502  	if prec >= 0 && prec < 15 {
   503  		shift := uint(prec * 4)
   504  		extra := (mant << shift) & (1<<60 - 1)
   505  		mant >>= 60 - shift
   506  		if extra|(mant&1) > 1<<59 {
   507  			mant++
   508  		}
   509  		mant <<= 60 - shift
   510  		if mant&(1<<61) != 0 {
   511  			// Wrapped around.
   512  			mant >>= 1
   513  			exp++
   514  		}
   515  	}
   516  
   517  	hex := lowerhex
   518  	if fmt == 'X' {
   519  		hex = upperhex
   520  	}
   521  
   522  	// sign, 0x, leading digit
   523  	if neg {
   524  		dst = append(dst, '-')
   525  	}
   526  	dst = append(dst, '0', fmt, '0'+byte((mant>>60)&1))
   527  
   528  	// .fraction
   529  	mant <<= 4 // remove leading 0 or 1
   530  	if prec < 0 && mant != 0 {
   531  		dst = append(dst, '.')
   532  		for mant != 0 {
   533  			dst = append(dst, hex[(mant>>60)&15])
   534  			mant <<= 4
   535  		}
   536  	} else if prec > 0 {
   537  		dst = append(dst, '.')
   538  		for i := 0; i < prec; i++ {
   539  			dst = append(dst, hex[(mant>>60)&15])
   540  			mant <<= 4
   541  		}
   542  	}
   543  
   544  	// p±
   545  	ch := byte('P')
   546  	if fmt == lower(fmt) {
   547  		ch = 'p'
   548  	}
   549  	dst = append(dst, ch)
   550  	if exp < 0 {
   551  		ch = '-'
   552  		exp = -exp
   553  	} else {
   554  		ch = '+'
   555  	}
   556  	dst = append(dst, ch)
   557  
   558  	// dd or ddd or dddd
   559  	switch {
   560  	case exp < 100:
   561  		dst = append(dst, byte(exp/10)+'0', byte(exp%10)+'0')
   562  	case exp < 1000:
   563  		dst = append(dst, byte(exp/100)+'0', byte((exp/10)%10)+'0', byte(exp%10)+'0')
   564  	default:
   565  		dst = append(dst, byte(exp/1000)+'0', byte(exp/100)%10+'0', byte((exp/10)%10)+'0', byte(exp%10)+'0')
   566  	}
   567  
   568  	return dst
   569  }
   570  
   571  func min(a, b int) int {
   572  	if a < b {
   573  		return a
   574  	}
   575  	return b
   576  }
   577  
   578  func max(a, b int) int {
   579  	if a > b {
   580  		return a
   581  	}
   582  	return b
   583  }
   584  

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