449 lines
11 KiB
Go
449 lines
11 KiB
Go
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// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// TODO: use build tags once a low-level public API has been established in
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// package strconv.
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// Binary to decimal floating point conversion.
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// Algorithm:
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// 1) store mantissa in multiprecision decimal
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// 2) shift decimal by exponent
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// 3) read digits out & format
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package number
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import "math"
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var optimize = true
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// TODO: move elsewhere?
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type floatInfo struct {
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mantbits uint
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expbits uint
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bias int
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}
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var float32info = floatInfo{23, 8, -127}
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var float64info = floatInfo{52, 11, -1023}
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// genericFtoa converts the floating-point number f to a string,
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// according to the format fmt and precision prec. It rounds the
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// result assuming that the original was obtained from a floating-point
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// value of bitSize bits (32 for float32, 64 for float64).
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//
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// The format fmt is one of
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// 'b' (-ddddp±ddd, a binary exponent),
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// 'e' (-d.dddde±dd, a decimal exponent),
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// 'E' (-d.ddddE±dd, a decimal exponent),
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// 'f' (-ddd.dddd, no exponent),
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// 'g' ('e' for large exponents, 'f' otherwise), or
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// 'G' ('E' for large exponents, 'f' otherwise).
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//
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// The precision prec controls the number of digits
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// (excluding the exponent) printed by the 'e', 'E', 'f', 'g', and 'G' formats.
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// For 'e', 'E', and 'f' it is the number of digits after the decimal point.
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// For 'g' and 'G' it is the total number of digits.
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// The special precision -1 uses the smallest number of digits
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// necessary such that ParseFloat will return f exactly.
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func genericFtoa(dst []byte, val float64, fmt byte, prec, bitSize int) []byte {
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var bits uint64
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var flt *floatInfo
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switch bitSize {
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case 32:
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bits = uint64(math.Float32bits(float32(val)))
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flt = &float32info
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case 64:
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bits = math.Float64bits(val)
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flt = &float64info
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default:
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panic("strconv: illegal AppendFloat/FormatFloat bitSize")
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}
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neg := bits>>(flt.expbits+flt.mantbits) != 0
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exp := int(bits>>flt.mantbits) & (1<<flt.expbits - 1)
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mant := bits & (uint64(1)<<flt.mantbits - 1)
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switch exp {
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case 1<<flt.expbits - 1:
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// Inf, NaN
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var s string
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switch {
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case mant != 0:
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s = "NaN"
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case neg:
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s = "-Inf"
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default:
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s = "+Inf"
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}
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return append(dst, s...)
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case 0:
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// denormalized
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exp++
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default:
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// add implicit top bit
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mant |= uint64(1) << flt.mantbits
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}
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exp += flt.bias
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// Pick off easy binary format.
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if fmt == 'b' {
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return fmtB(dst, neg, mant, exp, flt)
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}
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if !optimize {
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return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
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}
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var digs decimalSlice
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ok := false
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// Negative precision means "only as much as needed to be exact."
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shortest := prec < 0
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if shortest {
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// Try Grisu3 algorithm.
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f := new(extFloat)
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lower, upper := f.AssignComputeBounds(mant, exp, neg, flt)
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var buf [32]byte
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digs.d = buf[:]
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ok = f.ShortestDecimal(&digs, &lower, &upper)
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if !ok {
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return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
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}
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// Precision for shortest representation mode.
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switch fmt {
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case 'e', 'E':
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prec = max(digs.nd-1, 0)
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case 'f':
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prec = max(digs.nd-digs.dp, 0)
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case 'g', 'G':
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prec = digs.nd
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}
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} else if fmt != 'f' {
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// Fixed number of digits.
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digits := prec
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switch fmt {
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case 'e', 'E':
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digits++
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case 'g', 'G':
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if prec == 0 {
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prec = 1
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}
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digits = prec
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}
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if digits <= 15 {
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// try fast algorithm when the number of digits is reasonable.
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var buf [24]byte
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digs.d = buf[:]
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f := extFloat{mant, exp - int(flt.mantbits), neg}
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ok = f.FixedDecimal(&digs, digits)
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}
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}
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if !ok {
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return bigFtoa(dst, prec, fmt, neg, mant, exp, flt)
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}
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return formatDigits(dst, shortest, neg, digs, prec, fmt)
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}
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// bigFtoa uses multiprecision computations to format a float.
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func bigFtoa(dst []byte, prec int, fmt byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
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d := new(decimal)
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d.Assign(mant)
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d.Shift(exp - int(flt.mantbits))
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var digs decimalSlice
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shortest := prec < 0
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if shortest {
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roundShortest(d, mant, exp, flt)
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digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
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// Precision for shortest representation mode.
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switch fmt {
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case 'e', 'E':
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prec = digs.nd - 1
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case 'f':
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prec = max(digs.nd-digs.dp, 0)
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case 'g', 'G':
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prec = digs.nd
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}
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} else {
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// Round appropriately.
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switch fmt {
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case 'e', 'E':
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d.Round(prec + 1)
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case 'f':
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d.Round(d.dp + prec)
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case 'g', 'G':
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if prec == 0 {
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prec = 1
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}
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d.Round(prec)
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}
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digs = decimalSlice{d: d.d[:], nd: d.nd, dp: d.dp}
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}
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return formatDigits(dst, shortest, neg, digs, prec, fmt)
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}
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func formatDigits(dst []byte, shortest bool, neg bool, digs decimalSlice, prec int, fmt byte) []byte {
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switch fmt {
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case 'e', 'E':
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return fmtE(dst, neg, digs, prec, fmt)
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case 'f':
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return fmtF(dst, neg, digs, prec)
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case 'g', 'G':
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// trailing fractional zeros in 'e' form will be trimmed.
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eprec := prec
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if eprec > digs.nd && digs.nd >= digs.dp {
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eprec = digs.nd
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}
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// %e is used if the exponent from the conversion
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// is less than -4 or greater than or equal to the precision.
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// if precision was the shortest possible, use precision 6 for this decision.
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if shortest {
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eprec = 6
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}
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exp := digs.dp - 1
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if exp < -4 || exp >= eprec {
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if prec > digs.nd {
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prec = digs.nd
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}
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return fmtE(dst, neg, digs, prec-1, fmt+'e'-'g')
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}
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if prec > digs.dp {
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prec = digs.nd
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}
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return fmtF(dst, neg, digs, max(prec-digs.dp, 0))
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}
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// unknown format
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return append(dst, '%', fmt)
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}
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// roundShortest rounds d (= mant * 2^exp) to the shortest number of digits
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// that will let the original floating point value be precisely reconstructed.
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func roundShortest(d *decimal, mant uint64, exp int, flt *floatInfo) {
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// If mantissa is zero, the number is zero; stop now.
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if mant == 0 {
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d.nd = 0
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return
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}
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// Compute upper and lower such that any decimal number
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// between upper and lower (possibly inclusive)
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// will round to the original floating point number.
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// We may see at once that the number is already shortest.
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//
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// Suppose d is not denormal, so that 2^exp <= d < 10^dp.
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// The closest shorter number is at least 10^(dp-nd) away.
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// The lower/upper bounds computed below are at distance
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// at most 2^(exp-mantbits).
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//
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// So the number is already shortest if 10^(dp-nd) > 2^(exp-mantbits),
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// or equivalently log2(10)*(dp-nd) > exp-mantbits.
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// It is true if 332/100*(dp-nd) >= exp-mantbits (log2(10) > 3.32).
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minexp := flt.bias + 1 // minimum possible exponent
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if exp > minexp && 332*(d.dp-d.nd) >= 100*(exp-int(flt.mantbits)) {
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// The number is already shortest.
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return
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}
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// d = mant << (exp - mantbits)
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// Next highest floating point number is mant+1 << exp-mantbits.
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// Our upper bound is halfway between, mant*2+1 << exp-mantbits-1.
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upper := new(decimal)
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upper.Assign(mant*2 + 1)
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upper.Shift(exp - int(flt.mantbits) - 1)
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// d = mant << (exp - mantbits)
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// Next lowest floating point number is mant-1 << exp-mantbits,
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// unless mant-1 drops the significant bit and exp is not the minimum exp,
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// in which case the next lowest is mant*2-1 << exp-mantbits-1.
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// Either way, call it mantlo << explo-mantbits.
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// Our lower bound is halfway between, mantlo*2+1 << explo-mantbits-1.
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var mantlo uint64
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var explo int
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if mant > 1<<flt.mantbits || exp == minexp {
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mantlo = mant - 1
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explo = exp
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} else {
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mantlo = mant*2 - 1
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explo = exp - 1
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}
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lower := new(decimal)
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lower.Assign(mantlo*2 + 1)
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lower.Shift(explo - int(flt.mantbits) - 1)
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// The upper and lower bounds are possible outputs only if
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// the original mantissa is even, so that IEEE round-to-even
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// would round to the original mantissa and not the neighbors.
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inclusive := mant%2 == 0
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// Now we can figure out the minimum number of digits required.
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// Walk along until d has distinguished itself from upper and lower.
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for i := 0; i < d.nd; i++ {
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l := byte('0') // lower digit
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if i < lower.nd {
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l = lower.d[i]
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}
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m := d.d[i] // middle digit
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u := byte('0') // upper digit
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if i < upper.nd {
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u = upper.d[i]
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}
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// Okay to round down (truncate) if lower has a different digit
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// or if lower is inclusive and is exactly the result of rounding
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// down (i.e., and we have reached the final digit of lower).
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okdown := l != m || inclusive && i+1 == lower.nd
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// Okay to round up if upper has a different digit and either upper
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// is inclusive or upper is bigger than the result of rounding up.
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okup := m != u && (inclusive || m+1 < u || i+1 < upper.nd)
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// If it's okay to do either, then round to the nearest one.
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// If it's okay to do only one, do it.
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switch {
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case okdown && okup:
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d.Round(i + 1)
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return
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case okdown:
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d.RoundDown(i + 1)
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return
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case okup:
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d.RoundUp(i + 1)
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return
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}
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}
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}
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type decimalSlice struct {
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d []byte
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nd, dp int
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neg bool
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}
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// %e: -d.ddddde±dd
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func fmtE(dst []byte, neg bool, d decimalSlice, prec int, fmt byte) []byte {
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// sign
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if neg {
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dst = append(dst, '-')
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}
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// first digit
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ch := byte('0')
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if d.nd != 0 {
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ch = d.d[0]
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}
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dst = append(dst, ch)
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// .moredigits
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if prec > 0 {
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dst = append(dst, '.')
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i := 1
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m := min(d.nd, prec+1)
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if i < m {
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dst = append(dst, d.d[i:m]...)
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i = m
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}
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for ; i <= prec; i++ {
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dst = append(dst, '0')
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}
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}
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// e±
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dst = append(dst, fmt)
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exp := d.dp - 1
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if d.nd == 0 { // special case: 0 has exponent 0
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exp = 0
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}
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if exp < 0 {
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ch = '-'
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exp = -exp
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} else {
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ch = '+'
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}
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dst = append(dst, ch)
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// dd or ddd
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switch {
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case exp < 10:
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dst = append(dst, '0', byte(exp)+'0')
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case exp < 100:
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dst = append(dst, byte(exp/10)+'0', byte(exp%10)+'0')
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default:
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dst = append(dst, byte(exp/100)+'0', byte(exp/10)%10+'0', byte(exp%10)+'0')
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}
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return dst
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}
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|
|
// %f: -ddddddd.ddddd
|
||
|
|
func fmtF(dst []byte, neg bool, d decimalSlice, prec int) []byte {
|
||
|
|
// sign
|
||
|
|
if neg {
|
||
|
|
dst = append(dst, '-')
|
||
|
|
}
|
||
|
|
|
||
|
|
// integer, padded with zeros as needed.
|
||
|
|
if d.dp > 0 {
|
||
|
|
m := min(d.nd, d.dp)
|
||
|
|
dst = append(dst, d.d[:m]...)
|
||
|
|
for ; m < d.dp; m++ {
|
||
|
|
dst = append(dst, '0')
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
dst = append(dst, '0')
|
||
|
|
}
|
||
|
|
|
||
|
|
// fraction
|
||
|
|
if prec > 0 {
|
||
|
|
dst = append(dst, '.')
|
||
|
|
for i := 0; i < prec; i++ {
|
||
|
|
ch := byte('0')
|
||
|
|
if j := d.dp + i; 0 <= j && j < d.nd {
|
||
|
|
ch = d.d[j]
|
||
|
|
}
|
||
|
|
dst = append(dst, ch)
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return dst
|
||
|
|
}
|
||
|
|
|
||
|
|
// %b: -ddddddddp±ddd
|
||
|
|
func fmtB(dst []byte, neg bool, mant uint64, exp int, flt *floatInfo) []byte {
|
||
|
|
// sign
|
||
|
|
if neg {
|
||
|
|
dst = append(dst, '-')
|
||
|
|
}
|
||
|
|
|
||
|
|
// mantissa
|
||
|
|
dst, _ = formatBits(dst, mant, 10, false, true)
|
||
|
|
|
||
|
|
// p
|
||
|
|
dst = append(dst, 'p')
|
||
|
|
|
||
|
|
// ±exponent
|
||
|
|
exp -= int(flt.mantbits)
|
||
|
|
if exp >= 0 {
|
||
|
|
dst = append(dst, '+')
|
||
|
|
}
|
||
|
|
dst, _ = formatBits(dst, uint64(exp), 10, exp < 0, true)
|
||
|
|
|
||
|
|
return dst
|
||
|
|
}
|
||
|
|
|
||
|
|
func min(a, b int) int {
|
||
|
|
if a < b {
|
||
|
|
return a
|
||
|
|
}
|
||
|
|
return b
|
||
|
|
}
|
||
|
|
|
||
|
|
func max(a, b int) int {
|
||
|
|
if a > b {
|
||
|
|
return a
|
||
|
|
}
|
||
|
|
return b
|
||
|
|
}
|