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Diffstat (limited to 'src/math/big/ftoa.go')
| -rw-r--r-- | src/math/big/ftoa.go | 536 |
1 files changed, 536 insertions, 0 deletions
diff --git a/src/math/big/ftoa.go b/src/math/big/ftoa.go new file mode 100644 index 0000000..5506e6e --- /dev/null +++ b/src/math/big/ftoa.go @@ -0,0 +1,536 @@ +// Copyright 2015 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +// This file implements Float-to-string conversion functions. +// It is closely following the corresponding implementation +// in strconv/ftoa.go, but modified and simplified for Float. + +package big + +import ( + "bytes" + "fmt" + "strconv" +) + +// Text converts the floating-point number x to a string according +// to the given format and precision prec. The format is one of: +// +// 'e' -d.dddde±dd, decimal exponent, at least two (possibly 0) exponent digits +// 'E' -d.ddddE±dd, decimal exponent, at least two (possibly 0) exponent digits +// 'f' -ddddd.dddd, no exponent +// 'g' like 'e' for large exponents, like 'f' otherwise +// 'G' like 'E' for large exponents, like 'f' otherwise +// 'x' -0xd.dddddp±dd, hexadecimal mantissa, decimal power of two exponent +// 'p' -0x.dddp±dd, hexadecimal mantissa, decimal power of two exponent (non-standard) +// 'b' -ddddddp±dd, decimal mantissa, decimal power of two exponent (non-standard) +// +// For the power-of-two exponent formats, the mantissa is printed in normalized form: +// +// 'x' hexadecimal mantissa in [1, 2), or 0 +// 'p' hexadecimal mantissa in [½, 1), or 0 +// 'b' decimal integer mantissa using x.Prec() bits, or 0 +// +// Note that the 'x' form is the one used by most other languages and libraries. +// +// If format is a different character, Text returns a "%" followed by the +// unrecognized format character. +// +// The precision prec controls the number of digits (excluding the exponent) +// printed by the 'e', 'E', 'f', 'g', 'G', and 'x' formats. +// For 'e', 'E', 'f', and 'x', it is the number of digits after the decimal point. +// For 'g' and 'G' it is the total number of digits. A negative precision selects +// the smallest number of decimal digits necessary to identify the value x uniquely +// using x.Prec() mantissa bits. +// The prec value is ignored for the 'b' and 'p' formats. +func (x *Float) Text(format byte, prec int) string { + cap := 10 // TODO(gri) determine a good/better value here + if prec > 0 { + cap += prec + } + return string(x.Append(make([]byte, 0, cap), format, prec)) +} + +// String formats x like x.Text('g', 10). +// (String must be called explicitly, Float.Format does not support %s verb.) +func (x *Float) String() string { + return x.Text('g', 10) +} + +// Append appends to buf the string form of the floating-point number x, +// as generated by x.Text, and returns the extended buffer. +func (x *Float) Append(buf []byte, fmt byte, prec int) []byte { + // sign + if x.neg { + buf = append(buf, '-') + } + + // Inf + if x.form == inf { + if !x.neg { + buf = append(buf, '+') + } + return append(buf, "Inf"...) + } + + // pick off easy formats + switch fmt { + case 'b': + return x.fmtB(buf) + case 'p': + return x.fmtP(buf) + case 'x': + return x.fmtX(buf, prec) + } + + // Algorithm: + // 1) convert Float to multiprecision decimal + // 2) round to desired precision + // 3) read digits out and format + + // 1) convert Float to multiprecision decimal + var d decimal // == 0.0 + if x.form == finite { + // x != 0 + d.init(x.mant, int(x.exp)-x.mant.bitLen()) + } + + // 2) round to desired precision + shortest := false + if prec < 0 { + shortest = true + roundShortest(&d, x) + // Precision for shortest representation mode. + switch fmt { + case 'e', 'E': + prec = len(d.mant) - 1 + case 'f': + prec = max(len(d.mant)-d.exp, 0) + case 'g', 'G': + prec = len(d.mant) + } + } else { + // round appropriately + switch fmt { + case 'e', 'E': + // one digit before and number of digits after decimal point + d.round(1 + prec) + case 'f': + // number of digits before and after decimal point + d.round(d.exp + prec) + case 'g', 'G': + if prec == 0 { + prec = 1 + } + d.round(prec) + } + } + + // 3) read digits out and format + switch fmt { + case 'e', 'E': + return fmtE(buf, fmt, prec, d) + case 'f': + return fmtF(buf, prec, d) + case 'g', 'G': + // trim trailing fractional zeros in %e format + eprec := prec + if eprec > len(d.mant) && len(d.mant) >= d.exp { + eprec = len(d.mant) + } + // %e is used if the exponent from the conversion + // is less than -4 or greater than or equal to the precision. + // If precision was the shortest possible, use eprec = 6 for + // this decision. + if shortest { + eprec = 6 + } + exp := d.exp - 1 + if exp < -4 || exp >= eprec { + if prec > len(d.mant) { + prec = len(d.mant) + } + return fmtE(buf, fmt+'e'-'g', prec-1, d) + } + if prec > d.exp { + prec = len(d.mant) + } + return fmtF(buf, max(prec-d.exp, 0), d) + } + + // unknown format + if x.neg { + buf = buf[:len(buf)-1] // sign was added prematurely - remove it again + } + return append(buf, '%', fmt) +} + +func roundShortest(d *decimal, x *Float) { + // if the mantissa is zero, the number is zero - stop now + if len(d.mant) == 0 { + return + } + + // Approach: All numbers in the interval [x - 1/2ulp, x + 1/2ulp] + // (possibly exclusive) round to x for the given precision of x. + // Compute the lower and upper bound in decimal form and find the + // shortest decimal number d such that lower <= d <= upper. + + // TODO(gri) strconv/ftoa.do describes a shortcut in some cases. + // See if we can use it (in adjusted form) here as well. + + // 1) Compute normalized mantissa mant and exponent exp for x such + // that the lsb of mant corresponds to 1/2 ulp for the precision of + // x (i.e., for mant we want x.prec + 1 bits). + mant := nat(nil).set(x.mant) + exp := int(x.exp) - mant.bitLen() + s := mant.bitLen() - int(x.prec+1) + switch { + case s < 0: + mant = mant.shl(mant, uint(-s)) + case s > 0: + mant = mant.shr(mant, uint(+s)) + } + exp += s + // x = mant * 2**exp with lsb(mant) == 1/2 ulp of x.prec + + // 2) Compute lower bound by subtracting 1/2 ulp. + var lower decimal + var tmp nat + lower.init(tmp.sub(mant, natOne), exp) + + // 3) Compute upper bound by adding 1/2 ulp. + var upper decimal + upper.init(tmp.add(mant, natOne), exp) + + // The upper and lower bounds are possible outputs only if + // the original mantissa is even, so that ToNearestEven rounding + // would round to the original mantissa and not the neighbors. + inclusive := mant[0]&2 == 0 // test bit 1 since original mantissa was shifted by 1 + + // Now we can figure out the minimum number of digits required. + // Walk along until d has distinguished itself from upper and lower. + for i, m := range d.mant { + l := lower.at(i) + u := upper.at(i) + + // Okay to round down (truncate) if lower has a different digit + // or if lower is inclusive and is exactly the result of rounding + // down (i.e., and we have reached the final digit of lower). + okdown := l != m || inclusive && i+1 == len(lower.mant) + + // Okay to round up if upper has a different digit and either upper + // is inclusive or upper is bigger than the result of rounding up. + okup := m != u && (inclusive || m+1 < u || i+1 < len(upper.mant)) + + // If it's okay to do either, then round to the nearest one. + // If it's okay to do only one, do it. + switch { + case okdown && okup: + d.round(i + 1) + return + case okdown: + d.roundDown(i + 1) + return + case okup: + d.roundUp(i + 1) + return + } + } +} + +// %e: d.ddddde±dd +func fmtE(buf []byte, fmt byte, prec int, d decimal) []byte { + // first digit + ch := byte('0') + if len(d.mant) > 0 { + ch = d.mant[0] + } + buf = append(buf, ch) + + // .moredigits + if prec > 0 { + buf = append(buf, '.') + i := 1 + m := min(len(d.mant), prec+1) + if i < m { + buf = append(buf, d.mant[i:m]...) + i = m + } + for ; i <= prec; i++ { + buf = append(buf, '0') + } + } + + // e± + buf = append(buf, fmt) + var exp int64 + if len(d.mant) > 0 { + exp = int64(d.exp) - 1 // -1 because first digit was printed before '.' + } + if exp < 0 { + ch = '-' + exp = -exp + } else { + ch = '+' + } + buf = append(buf, ch) + + // dd...d + if exp < 10 { + buf = append(buf, '0') // at least 2 exponent digits + } + return strconv.AppendInt(buf, exp, 10) +} + +// %f: ddddddd.ddddd +func fmtF(buf []byte, prec int, d decimal) []byte { + // integer, padded with zeros as needed + if d.exp > 0 { + m := min(len(d.mant), d.exp) + buf = append(buf, d.mant[:m]...) + for ; m < d.exp; m++ { + buf = append(buf, '0') + } + } else { + buf = append(buf, '0') + } + + // fraction + if prec > 0 { + buf = append(buf, '.') + for i := 0; i < prec; i++ { + buf = append(buf, d.at(d.exp+i)) + } + } + + return buf +} + +// fmtB appends the string of x in the format mantissa "p" exponent +// with a decimal mantissa and a binary exponent, or 0" if x is zero, +// and returns the extended buffer. +// The mantissa is normalized such that is uses x.Prec() bits in binary +// representation. +// The sign of x is ignored, and x must not be an Inf. +// (The caller handles Inf before invoking fmtB.) +func (x *Float) fmtB(buf []byte) []byte { + if x.form == zero { + return append(buf, '0') + } + + if debugFloat && x.form != finite { + panic("non-finite float") + } + // x != 0 + + // adjust mantissa to use exactly x.prec bits + m := x.mant + switch w := uint32(len(x.mant)) * _W; { + case w < x.prec: + m = nat(nil).shl(m, uint(x.prec-w)) + case w > x.prec: + m = nat(nil).shr(m, uint(w-x.prec)) + } + + buf = append(buf, m.utoa(10)...) + buf = append(buf, 'p') + e := int64(x.exp) - int64(x.prec) + if e >= 0 { + buf = append(buf, '+') + } + return strconv.AppendInt(buf, e, 10) +} + +// fmtX appends the string of x in the format "0x1." mantissa "p" exponent +// with a hexadecimal mantissa and a binary exponent, or "0x0p0" if x is zero, +// and returns the extended buffer. +// A non-zero mantissa is normalized such that 1.0 <= mantissa < 2.0. +// The sign of x is ignored, and x must not be an Inf. +// (The caller handles Inf before invoking fmtX.) +func (x *Float) fmtX(buf []byte, prec int) []byte { + if x.form == zero { + buf = append(buf, "0x0"...) + if prec > 0 { + buf = append(buf, '.') + for i := 0; i < prec; i++ { + buf = append(buf, '0') + } + } + buf = append(buf, "p+00"...) + return buf + } + + if debugFloat && x.form != finite { + panic("non-finite float") + } + + // round mantissa to n bits + var n uint + if prec < 0 { + n = 1 + (x.MinPrec()-1+3)/4*4 // round MinPrec up to 1 mod 4 + } else { + n = 1 + 4*uint(prec) + } + // n%4 == 1 + x = new(Float).SetPrec(n).SetMode(x.mode).Set(x) + + // adjust mantissa to use exactly n bits + m := x.mant + switch w := uint(len(x.mant)) * _W; { + case w < n: + m = nat(nil).shl(m, n-w) + case w > n: + m = nat(nil).shr(m, w-n) + } + exp64 := int64(x.exp) - 1 // avoid wrap-around + + hm := m.utoa(16) + if debugFloat && hm[0] != '1' { + panic("incorrect mantissa: " + string(hm)) + } + buf = append(buf, "0x1"...) + if len(hm) > 1 { + buf = append(buf, '.') + buf = append(buf, hm[1:]...) + } + + buf = append(buf, 'p') + if exp64 >= 0 { + buf = append(buf, '+') + } else { + exp64 = -exp64 + buf = append(buf, '-') + } + // Force at least two exponent digits, to match fmt. + if exp64 < 10 { + buf = append(buf, '0') + } + return strconv.AppendInt(buf, exp64, 10) +} + +// fmtP appends the string of x in the format "0x." mantissa "p" exponent +// with a hexadecimal mantissa and a binary exponent, or "0" if x is zero, +// and returns the extended buffer. +// The mantissa is normalized such that 0.5 <= 0.mantissa < 1.0. +// The sign of x is ignored, and x must not be an Inf. +// (The caller handles Inf before invoking fmtP.) +func (x *Float) fmtP(buf []byte) []byte { + if x.form == zero { + return append(buf, '0') + } + + if debugFloat && x.form != finite { + panic("non-finite float") + } + // x != 0 + + // remove trailing 0 words early + // (no need to convert to hex 0's and trim later) + m := x.mant + i := 0 + for i < len(m) && m[i] == 0 { + i++ + } + m = m[i:] + + buf = append(buf, "0x."...) + buf = append(buf, bytes.TrimRight(m.utoa(16), "0")...) + buf = append(buf, 'p') + if x.exp >= 0 { + buf = append(buf, '+') + } + return strconv.AppendInt(buf, int64(x.exp), 10) +} + +func min(x, y int) int { + if x < y { + return x + } + return y +} + +var _ fmt.Formatter = &floatZero // *Float must implement fmt.Formatter + +// Format implements fmt.Formatter. It accepts all the regular +// formats for floating-point numbers ('b', 'e', 'E', 'f', 'F', +// 'g', 'G', 'x') as well as 'p' and 'v'. See (*Float).Text for the +// interpretation of 'p'. The 'v' format is handled like 'g'. +// Format also supports specification of the minimum precision +// in digits, the output field width, as well as the format flags +// '+' and ' ' for sign control, '0' for space or zero padding, +// and '-' for left or right justification. See the fmt package +// for details. +func (x *Float) Format(s fmt.State, format rune) { + prec, hasPrec := s.Precision() + if !hasPrec { + prec = 6 // default precision for 'e', 'f' + } + + switch format { + case 'e', 'E', 'f', 'b', 'p', 'x': + // nothing to do + case 'F': + // (*Float).Text doesn't support 'F'; handle like 'f' + format = 'f' + case 'v': + // handle like 'g' + format = 'g' + fallthrough + case 'g', 'G': + if !hasPrec { + prec = -1 // default precision for 'g', 'G' + } + default: + fmt.Fprintf(s, "%%!%c(*big.Float=%s)", format, x.String()) + return + } + var buf []byte + buf = x.Append(buf, byte(format), prec) + if len(buf) == 0 { + buf = []byte("?") // should never happen, but don't crash + } + // len(buf) > 0 + + var sign string + switch { + case buf[0] == '-': + sign = "-" + buf = buf[1:] + case buf[0] == '+': + // +Inf + sign = "+" + if s.Flag(' ') { + sign = " " + } + buf = buf[1:] + case s.Flag('+'): + sign = "+" + case s.Flag(' '): + sign = " " + } + + var padding int + if width, hasWidth := s.Width(); hasWidth && width > len(sign)+len(buf) { + padding = width - len(sign) - len(buf) + } + + switch { + case s.Flag('0') && !x.IsInf(): + // 0-padding on left + writeMultiple(s, sign, 1) + writeMultiple(s, "0", padding) + s.Write(buf) + case s.Flag('-'): + // padding on right + writeMultiple(s, sign, 1) + s.Write(buf) + writeMultiple(s, " ", padding) + default: + // padding on left + writeMultiple(s, " ", padding) + writeMultiple(s, sign, 1) + s.Write(buf) + } +} |