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/**
* Copyright 2014 Paul Querna
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*/
/* Portions of this file are on Go stdlib's strconv/atof.go */
// Copyright 2009 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.
package internal
// decimal to binary floating point conversion.
// Algorithm:
// 1) Store input in multiprecision decimal.
// 2) Multiply/divide decimal by powers of two until in range [0.5, 1)
// 3) Multiply by 2^precision and round to get mantissa.
import "math"
var optimize = true // can change for testing
func equalIgnoreCase(s1 []byte, s2 []byte) bool {
if len(s1) != len(s2) {
return false
}
for i := 0; i < len(s1); i++ {
c1 := s1[i]
if 'A' <= c1 && c1 <= 'Z' {
c1 += 'a' - 'A'
}
c2 := s2[i]
if 'A' <= c2 && c2 <= 'Z' {
c2 += 'a' - 'A'
}
if c1 != c2 {
return false
}
}
return true
}
func special(s []byte) (f float64, ok bool) {
if len(s) == 0 {
return
}
switch s[0] {
default:
return
case '+':
if equalIgnoreCase(s, []byte("+inf")) || equalIgnoreCase(s, []byte("+infinity")) {
return math.Inf(1), true
}
case '-':
if equalIgnoreCase(s, []byte("-inf")) || equalIgnoreCase(s, []byte("-infinity")) {
return math.Inf(-1), true
}
case 'n', 'N':
if equalIgnoreCase(s, []byte("nan")) {
return math.NaN(), true
}
case 'i', 'I':
if equalIgnoreCase(s, []byte("inf")) || equalIgnoreCase(s, []byte("infinity")) {
return math.Inf(1), true
}
}
return
}
func (b *decimal) set(s []byte) (ok bool) {
i := 0
b.neg = false
b.trunc = false
// optional sign
if i >= len(s) {
return
}
switch {
case s[i] == '+':
i++
case s[i] == '-':
b.neg = true
i++
}
// digits
sawdot := false
sawdigits := false
for ; i < len(s); i++ {
switch {
case s[i] == '.':
if sawdot {
return
}
sawdot = true
b.dp = b.nd
continue
case '0' <= s[i] && s[i] <= '9':
sawdigits = true
if s[i] == '0' && b.nd == 0 { // ignore leading zeros
b.dp--
continue
}
if b.nd < len(b.d) {
b.d[b.nd] = s[i]
b.nd++
} else if s[i] != '0' {
b.trunc = true
}
continue
}
break
}
if !sawdigits {
return
}
if !sawdot {
b.dp = b.nd
}
// optional exponent moves decimal point.
// if we read a very large, very long number,
// just be sure to move the decimal point by
// a lot (say, 100000). it doesn't matter if it's
// not the exact number.
if i < len(s) && (s[i] == 'e' || s[i] == 'E') {
i++
if i >= len(s) {
return
}
esign := 1
if s[i] == '+' {
i++
} else if s[i] == '-' {
i++
esign = -1
}
if i >= len(s) || s[i] < '0' || s[i] > '9' {
return
}
e := 0
for ; i < len(s) && '0' <= s[i] && s[i] <= '9'; i++ {
if e < 10000 {
e = e*10 + int(s[i]) - '0'
}
}
b.dp += e * esign
}
if i != len(s) {
return
}
ok = true
return
}
// readFloat reads a decimal mantissa and exponent from a float
// string representation. It sets ok to false if the number could
// not fit return types or is invalid.
func readFloat(s []byte) (mantissa uint64, exp int, neg, trunc, ok bool) {
const uint64digits = 19
i := 0
// optional sign
if i >= len(s) {
return
}
switch {
case s[i] == '+':
i++
case s[i] == '-':
neg = true
i++
}
// digits
sawdot := false
sawdigits := false
nd := 0
ndMant := 0
dp := 0
for ; i < len(s); i++ {
switch c := s[i]; true {
case c == '.':
if sawdot {
return
}
sawdot = true
dp = nd
continue
case '0' <= c && c <= '9':
sawdigits = true
if c == '0' && nd == 0 { // ignore leading zeros
dp--
continue
}
nd++
if ndMant < uint64digits {
mantissa *= 10
mantissa += uint64(c - '0')
ndMant++
} else if s[i] != '0' {
trunc = true
}
continue
}
break
}
if !sawdigits {
return
}
if !sawdot {
dp = nd
}
// optional exponent moves decimal point.
// if we read a very large, very long number,
// just be sure to move the decimal point by
// a lot (say, 100000). it doesn't matter if it's
// not the exact number.
if i < len(s) && (s[i] == 'e' || s[i] == 'E') {
i++
if i >= len(s) {
return
}
esign := 1
if s[i] == '+' {
i++
} else if s[i] == '-' {
i++
esign = -1
}
if i >= len(s) || s[i] < '0' || s[i] > '9' {
return
}
e := 0
for ; i < len(s) && '0' <= s[i] && s[i] <= '9'; i++ {
if e < 10000 {
e = e*10 + int(s[i]) - '0'
}
}
dp += e * esign
}
if i != len(s) {
return
}
exp = dp - ndMant
ok = true
return
}
// decimal power of ten to binary power of two.
var powtab = []int{1, 3, 6, 9, 13, 16, 19, 23, 26}
func (d *decimal) floatBits(flt *floatInfo) (b uint64, overflow bool) {
var exp int
var mant uint64
// Zero is always a special case.
if d.nd == 0 {
mant = 0
exp = flt.bias
goto out
}
// Obvious overflow/underflow.
// These bounds are for 64-bit floats.
// Will have to change if we want to support 80-bit floats in the future.
if d.dp > 310 {
goto overflow
}
if d.dp < -330 {
// zero
mant = 0
exp = flt.bias
goto out
}
// Scale by powers of two until in range [0.5, 1.0)
exp = 0
for d.dp > 0 {
var n int
if d.dp >= len(powtab) {
n = 27
} else {
n = powtab[d.dp]
}
d.Shift(-n)
exp += n
}
for d.dp < 0 || d.dp == 0 && d.d[0] < '5' {
var n int
if -d.dp >= len(powtab) {
n = 27
} else {
n = powtab[-d.dp]
}
d.Shift(n)
exp -= n
}
// Our range is [0.5,1) but floating point range is [1,2).
exp--
// Minimum representable exponent is flt.bias+1.
// If the exponent is smaller, move it up and
// adjust d accordingly.
if exp < flt.bias+1 {
n := flt.bias + 1 - exp
d.Shift(-n)
exp += n
}
if exp-flt.bias >= 1<<flt.expbits-1 {
goto overflow
}
// Extract 1+flt.mantbits bits.
d.Shift(int(1 + flt.mantbits))
mant = d.RoundedInteger()
// Rounding might have added a bit; shift down.
if mant == 2<<flt.mantbits {
mant >>= 1
exp++
if exp-flt.bias >= 1<<flt.expbits-1 {
goto overflow
}
}
// Denormalized?
if mant&(1<<flt.mantbits) == 0 {
exp = flt.bias
}
goto out
overflow:
// ±Inf
mant = 0
exp = 1<<flt.expbits - 1 + flt.bias
overflow = true
out:
// Assemble bits.
bits := mant & (uint64(1)<<flt.mantbits - 1)
bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
if d.neg {
bits |= 1 << flt.mantbits << flt.expbits
}
return bits, overflow
}
// Exact powers of 10.
var float64pow10 = []float64{
1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
1e20, 1e21, 1e22,
}
var float32pow10 = []float32{1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10}
// If possible to convert decimal representation to 64-bit float f exactly,
// entirely in floating-point math, do so, avoiding the expense of decimalToFloatBits.
// Three common cases:
// value is exact integer
// value is exact integer * exact power of ten
// value is exact integer / exact power of ten
// These all produce potentially inexact but correctly rounded answers.
func atof64exact(mantissa uint64, exp int, neg bool) (f float64, ok bool) {
if mantissa>>float64info.mantbits != 0 {
return
}
f = float64(mantissa)
if neg {
f = -f
}
switch {
case exp == 0:
// an integer.
return f, true
// Exact integers are <= 10^15.
// Exact powers of ten are <= 10^22.
case exp > 0 && exp <= 15+22: // int * 10^k
// If exponent is big but number of digits is not,
// can move a few zeros into the integer part.
if exp > 22 {
f *= float64pow10[exp-22]
exp = 22
}
if f > 1e15 || f < -1e15 {
// the exponent was really too large.
return
}
return f * float64pow10[exp], true
case exp < 0 && exp >= -22: // int / 10^k
return f / float64pow10[-exp], true
}
return
}
// If possible to compute mantissa*10^exp to 32-bit float f exactly,
// entirely in floating-point math, do so, avoiding the machinery above.
func atof32exact(mantissa uint64, exp int, neg bool) (f float32, ok bool) {
if mantissa>>float32info.mantbits != 0 {
return
}
f = float32(mantissa)
if neg {
f = -f
}
switch {
case exp == 0:
return f, true
// Exact integers are <= 10^7.
// Exact powers of ten are <= 10^10.
case exp > 0 && exp <= 7+10: // int * 10^k
// If exponent is big but number of digits is not,
// can move a few zeros into the integer part.
if exp > 10 {
f *= float32pow10[exp-10]
exp = 10
}
if f > 1e7 || f < -1e7 {
// the exponent was really too large.
return
}
return f * float32pow10[exp], true
case exp < 0 && exp >= -10: // int / 10^k
return f / float32pow10[-exp], true
}
return
}
const fnParseFloat = "ParseFloat"
func atof32(s []byte) (f float32, err error) {
if val, ok := special(s); ok {
return float32(val), nil
}
if optimize {
// Parse mantissa and exponent.
mantissa, exp, neg, trunc, ok := readFloat(s)
if ok {
// Try pure floating-point arithmetic conversion.
if !trunc {
if f, ok := atof32exact(mantissa, exp, neg); ok {
return f, nil
}
}
// Try another fast path.
ext := new(extFloat)
if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float32info); ok {
b, ovf := ext.floatBits(&float32info)
f = math.Float32frombits(uint32(b))
if ovf {
err = rangeError(fnParseFloat, string(s))
}
return f, err
}
}
}
var d decimal
if !d.set(s) {
return 0, syntaxError(fnParseFloat, string(s))
}
b, ovf := d.floatBits(&float32info)
f = math.Float32frombits(uint32(b))
if ovf {
err = rangeError(fnParseFloat, string(s))
}
return f, err
}
func atof64(s []byte) (f float64, err error) {
if val, ok := special(s); ok {
return val, nil
}
if optimize {
// Parse mantissa and exponent.
mantissa, exp, neg, trunc, ok := readFloat(s)
if ok {
// Try pure floating-point arithmetic conversion.
if !trunc {
if f, ok := atof64exact(mantissa, exp, neg); ok {
return f, nil
}
}
// Try another fast path.
ext := new(extFloat)
if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float64info); ok {
b, ovf := ext.floatBits(&float64info)
f = math.Float64frombits(b)
if ovf {
err = rangeError(fnParseFloat, string(s))
}
return f, err
}
}
}
var d decimal
if !d.set(s) {
return 0, syntaxError(fnParseFloat, string(s))
}
b, ovf := d.floatBits(&float64info)
f = math.Float64frombits(b)
if ovf {
err = rangeError(fnParseFloat, string(s))
}
return f, err
}
// ParseFloat converts the string s to a floating-point number
// with the precision specified by bitSize: 32 for float32, or 64 for float64.
// When bitSize=32, the result still has type float64, but it will be
// convertible to float32 without changing its value.
//
// If s is well-formed and near a valid floating point number,
// ParseFloat returns the nearest floating point number rounded
// using IEEE754 unbiased rounding.
//
// The errors that ParseFloat returns have concrete type *NumError
// and include err.Num = s.
//
// If s is not syntactically well-formed, ParseFloat returns err.Err = ErrSyntax.
//
// If s is syntactically well-formed but is more than 1/2 ULP
// away from the largest floating point number of the given size,
// ParseFloat returns f = ±Inf, err.Err = ErrRange.
func ParseFloat(s []byte, bitSize int) (f float64, err error) {
if bitSize == 32 {
f1, err1 := atof32(s)
return float64(f1), err1
}
f1, err1 := atof64(s)
return f1, err1
}
// oroginal: strconv/decimal.go, but not exported, and needed for PareFloat.
// Copyright 2009 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.
// Multiprecision decimal numbers.
// For floating-point formatting only; not general purpose.
// Only operations are assign and (binary) left/right shift.
// Can do binary floating point in multiprecision decimal precisely
// because 2 divides 10; cannot do decimal floating point
// in multiprecision binary precisely.
type decimal struct {
d [800]byte // digits
nd int // number of digits used
dp int // decimal point
neg bool
trunc bool // discarded nonzero digits beyond d[:nd]
}
func (a *decimal) String() string {
n := 10 + a.nd
if a.dp > 0 {
n += a.dp
}
if a.dp < 0 {
n += -a.dp
}
buf := make([]byte, n)
w := 0
switch {
case a.nd == 0:
return "0"
case a.dp <= 0:
// zeros fill space between decimal point and digits
buf[w] = '0'
w++
buf[w] = '.'
w++
w += digitZero(buf[w : w+-a.dp])
w += copy(buf[w:], a.d[0:a.nd])
case a.dp < a.nd:
// decimal point in middle of digits
w += copy(buf[w:], a.d[0:a.dp])
buf[w] = '.'
w++
w += copy(buf[w:], a.d[a.dp:a.nd])
default:
// zeros fill space between digits and decimal point
w += copy(buf[w:], a.d[0:a.nd])
w += digitZero(buf[w : w+a.dp-a.nd])
}
return string(buf[0:w])
}
func digitZero(dst []byte) int {
for i := range dst {
dst[i] = '0'
}
return len(dst)
}
// trim trailing zeros from number.
// (They are meaningless; the decimal point is tracked
// independent of the number of digits.)
func trim(a *decimal) {
for a.nd > 0 && a.d[a.nd-1] == '0' {
a.nd--
}
if a.nd == 0 {
a.dp = 0
}
}
// Assign v to a.
func (a *decimal) Assign(v uint64) {
var buf [24]byte
// Write reversed decimal in buf.
n := 0
for v > 0 {
v1 := v / 10
v -= 10 * v1
buf[n] = byte(v + '0')
n++
v = v1
}
// Reverse again to produce forward decimal in a.d.
a.nd = 0
for n--; n >= 0; n-- {
a.d[a.nd] = buf[n]
a.nd++
}
a.dp = a.nd
trim(a)
}
// Maximum shift that we can do in one pass without overflow.
// Signed int has 31 bits, and we have to be able to accommodate 9<<k.
const maxShift = 27
// Binary shift right (* 2) by k bits. k <= maxShift to avoid overflow.
func rightShift(a *decimal, k uint) {
r := 0 // read pointer
w := 0 // write pointer
// Pick up enough leading digits to cover first shift.
n := 0
for ; n>>k == 0; r++ {
if r >= a.nd {
if n == 0 {
// a == 0; shouldn't get here, but handle anyway.
a.nd = 0
return
}
for n>>k == 0 {
n = n * 10
r++
}
break
}
c := int(a.d[r])
n = n*10 + c - '0'
}
a.dp -= r - 1
// Pick up a digit, put down a digit.
for ; r < a.nd; r++ {
c := int(a.d[r])
dig := n >> k
n -= dig << k
a.d[w] = byte(dig + '0')
w++
n = n*10 + c - '0'
}
// Put down extra digits.
for n > 0 {
dig := n >> k
n -= dig << k
if w < len(a.d) {
a.d[w] = byte(dig + '0')
w++
} else if dig > 0 {
a.trunc = true
}
n = n * 10
}
a.nd = w
trim(a)
}
// Cheat sheet for left shift: table indexed by shift count giving
// number of new digits that will be introduced by that shift.
//
// For example, leftcheats[4] = {2, "625"}. That means that
// if we are shifting by 4 (multiplying by 16), it will add 2 digits
// when the string prefix is "625" through "999", and one fewer digit
// if the string prefix is "000" through "624".
//
// Credit for this trick goes to Ken.
type leftCheat struct {
delta int // number of new digits
cutoff string // minus one digit if original < a.
}
var leftcheats = []leftCheat{
// Leading digits of 1/2^i = 5^i.
// 5^23 is not an exact 64-bit floating point number,
// so have to use bc for the math.
/*
seq 27 | sed 's/^/5^/' | bc |
awk 'BEGIN{ print "\tleftCheat{ 0, \"\" }," }
{
log2 = log(2)/log(10)
printf("\tleftCheat{ %d, \"%s\" },\t// * %d\n",
int(log2*NR+1), $0, 2**NR)
}'
*/
{0, ""},
{1, "5"}, // * 2
{1, "25"}, // * 4
{1, "125"}, // * 8
{2, "625"}, // * 16
{2, "3125"}, // * 32
{2, "15625"}, // * 64
{3, "78125"}, // * 128
{3, "390625"}, // * 256
{3, "1953125"}, // * 512
{4, "9765625"}, // * 1024
{4, "48828125"}, // * 2048
{4, "244140625"}, // * 4096
{4, "1220703125"}, // * 8192
{5, "6103515625"}, // * 16384
{5, "30517578125"}, // * 32768
{5, "152587890625"}, // * 65536
{6, "762939453125"}, // * 131072
{6, "3814697265625"}, // * 262144
{6, "19073486328125"}, // * 524288
{7, "95367431640625"}, // * 1048576
{7, "476837158203125"}, // * 2097152
{7, "2384185791015625"}, // * 4194304
{7, "11920928955078125"}, // * 8388608
{8, "59604644775390625"}, // * 16777216
{8, "298023223876953125"}, // * 33554432
{8, "1490116119384765625"}, // * 67108864
{9, "7450580596923828125"}, // * 134217728
}
// Is the leading prefix of b lexicographically less than s?
func prefixIsLessThan(b []byte, s string) bool {
for i := 0; i < len(s); i++ {
if i >= len(b) {
return true
}
if b[i] != s[i] {
return b[i] < s[i]
}
}
return false
}
// Binary shift left (/ 2) by k bits. k <= maxShift to avoid overflow.
func leftShift(a *decimal, k uint) {
delta := leftcheats[k].delta
if prefixIsLessThan(a.d[0:a.nd], leftcheats[k].cutoff) {
delta--
}
r := a.nd // read index
w := a.nd + delta // write index
n := 0
// Pick up a digit, put down a digit.
for r--; r >= 0; r-- {
n += (int(a.d[r]) - '0') << k
quo := n / 10
rem := n - 10*quo
w--
if w < len(a.d) {
a.d[w] = byte(rem + '0')
} else if rem != 0 {
a.trunc = true
}
n = quo
}
// Put down extra digits.
for n > 0 {
quo := n / 10
rem := n - 10*quo
w--
if w < len(a.d) {
a.d[w] = byte(rem + '0')
} else if rem != 0 {
a.trunc = true
}
n = quo
}
a.nd += delta
if a.nd >= len(a.d) {
a.nd = len(a.d)
}
a.dp += delta
trim(a)
}
// Binary shift left (k > 0) or right (k < 0).
func (a *decimal) Shift(k int) {
switch {
case a.nd == 0:
// nothing to do: a == 0
case k > 0:
for k > maxShift {
leftShift(a, maxShift)
k -= maxShift
}
leftShift(a, uint(k))
case k < 0:
for k < -maxShift {
rightShift(a, maxShift)
k += maxShift
}
rightShift(a, uint(-k))
}
}
// If we chop a at nd digits, should we round up?
func shouldRoundUp(a *decimal, nd int) bool {
if nd < 0 || nd >= a.nd {
return false
}
if a.d[nd] == '5' && nd+1 == a.nd { // exactly halfway - round to even
// if we truncated, a little higher than what's recorded - always round up
if a.trunc {
return true
}
return nd > 0 && (a.d[nd-1]-'0')%2 != 0
}
// not halfway - digit tells all
return a.d[nd] >= '5'
}
// Round a to nd digits (or fewer).
// If nd is zero, it means we're rounding
// just to the left of the digits, as in
// 0.09 -> 0.1.
func (a *decimal) Round(nd int) {
if nd < 0 || nd >= a.nd {
return
}
if shouldRoundUp(a, nd) {
a.RoundUp(nd)
} else {
a.RoundDown(nd)
}
}
// Round a down to nd digits (or fewer).
func (a *decimal) RoundDown(nd int) {
if nd < 0 || nd >= a.nd {
return
}
a.nd = nd
trim(a)
}
// Round a up to nd digits (or fewer).
func (a *decimal) RoundUp(nd int) {
if nd < 0 || nd >= a.nd {
return
}
// round up
for i := nd - 1; i >= 0; i-- {
c := a.d[i]
if c < '9' { // can stop after this digit
a.d[i]++
a.nd = i + 1
return
}
}
// Number is all 9s.
// Change to single 1 with adjusted decimal point.
a.d[0] = '1'
a.nd = 1
a.dp++
}
// Extract integer part, rounded appropriately.
// No guarantees about overflow.
func (a *decimal) RoundedInteger() uint64 {
if a.dp > 20 {
return 0xFFFFFFFFFFFFFFFF
}
var i int
n := uint64(0)
for i = 0; i < a.dp && i < a.nd; i++ {
n = n*10 + uint64(a.d[i]-'0')
}
for ; i < a.dp; i++ {
n *= 10
}
if shouldRoundUp(a, a.dp) {
n++
}
return n
}
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