mirror of git://gcc.gnu.org/git/gcc.git
493 lines
13 KiB
Go
493 lines
13 KiB
Go
// Copyright 2015 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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// This file implements nat-to-string conversion functions.
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package big
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import (
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"errors"
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"fmt"
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"io"
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"math"
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"sync"
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)
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const digits = "0123456789abcdefghijklmnopqrstuvwxyz"
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// Note: MaxBase = len(digits), but it must remain a rune constant
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// for API compatibility.
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// MaxBase is the largest number base accepted for string conversions.
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const MaxBase = 'z' - 'a' + 10 + 1
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// maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M.
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// For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word.
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// In other words, at most n digits in base b fit into a Word.
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// TODO(gri) replace this with a table, generated at build time.
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func maxPow(b Word) (p Word, n int) {
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p, n = b, 1 // assuming b <= _M
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for max := _M / b; p <= max; {
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// p == b**n && p <= max
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p *= b
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n++
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}
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// p == b**n && p <= _M
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return
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}
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// pow returns x**n for n > 0, and 1 otherwise.
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func pow(x Word, n int) (p Word) {
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// n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1
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// thus x**n == product of x**(2**i) for all i where bi == 1
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// (Russian Peasant Method for exponentiation)
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p = 1
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for n > 0 {
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if n&1 != 0 {
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p *= x
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}
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x *= x
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n >>= 1
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}
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return
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}
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// scan scans the number corresponding to the longest possible prefix
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// from r representing an unsigned number in a given conversion base.
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// It returns the corresponding natural number res, the actual base b,
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// a digit count, and a read or syntax error err, if any.
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//
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// number = [ prefix ] mantissa .
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// prefix = "0" [ "x" | "X" | "b" | "B" ] .
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// mantissa = digits | digits "." [ digits ] | "." digits .
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// digits = digit { digit } .
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// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
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//
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// Unless fracOk is set, the base argument must be 0 or a value between
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// 2 and MaxBase. If fracOk is set, the base argument must be one of
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// 0, 2, 10, or 16. Providing an invalid base argument leads to a run-
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// time panic.
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//
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// For base 0, the number prefix determines the actual base: A prefix of
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// ``0x'' or ``0X'' selects base 16; if fracOk is not set, the ``0'' prefix
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// selects base 8, and a ``0b'' or ``0B'' prefix selects base 2. Otherwise
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// the selected base is 10 and no prefix is accepted.
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//
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// If fracOk is set, an octal prefix is ignored (a leading ``0'' simply
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// stands for a zero digit), and a period followed by a fractional part
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// is permitted. The result value is computed as if there were no period
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// present; and the count value is used to determine the fractional part.
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//
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// A result digit count > 0 corresponds to the number of (non-prefix) digits
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// parsed. A digit count <= 0 indicates the presence of a period (if fracOk
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// is set, only), and -count is the number of fractional digits found.
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// In this case, the actual value of the scanned number is res * b**count.
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//
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func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) {
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// reject illegal bases
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baseOk := base == 0 ||
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!fracOk && 2 <= base && base <= MaxBase ||
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fracOk && (base == 2 || base == 10 || base == 16)
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if !baseOk {
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panic(fmt.Sprintf("illegal number base %d", base))
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}
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// one char look-ahead
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ch, err := r.ReadByte()
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if err != nil {
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return
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}
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// determine actual base
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b = base
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if base == 0 {
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// actual base is 10 unless there's a base prefix
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b = 10
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if ch == '0' {
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count = 1
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switch ch, err = r.ReadByte(); err {
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case nil:
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// possibly one of 0x, 0X, 0b, 0B
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if !fracOk {
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b = 8
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}
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switch ch {
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case 'x', 'X':
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b = 16
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case 'b', 'B':
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b = 2
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}
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switch b {
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case 16, 2:
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count = 0 // prefix is not counted
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if ch, err = r.ReadByte(); err != nil {
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// io.EOF is also an error in this case
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return
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}
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case 8:
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count = 0 // prefix is not counted
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}
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case io.EOF:
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// input is "0"
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res = z[:0]
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err = nil
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return
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default:
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// read error
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return
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}
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}
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}
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// convert string
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// Algorithm: Collect digits in groups of at most n digits in di
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// and then use mulAddWW for every such group to add them to the
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// result.
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z = z[:0]
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b1 := Word(b)
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bn, n := maxPow(b1) // at most n digits in base b1 fit into Word
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di := Word(0) // 0 <= di < b1**i < bn
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i := 0 // 0 <= i < n
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dp := -1 // position of decimal point
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for {
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if fracOk && ch == '.' {
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fracOk = false
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dp = count
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// advance
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if ch, err = r.ReadByte(); err != nil {
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if err == io.EOF {
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err = nil
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break
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}
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return
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}
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}
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// convert rune into digit value d1
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var d1 Word
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switch {
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case '0' <= ch && ch <= '9':
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d1 = Word(ch - '0')
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case 'a' <= ch && ch <= 'z':
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d1 = Word(ch - 'a' + 10)
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case 'A' <= ch && ch <= 'Z':
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d1 = Word(ch - 'A' + 10)
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default:
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d1 = MaxBase + 1
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}
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if d1 >= b1 {
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r.UnreadByte() // ch does not belong to number anymore
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break
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}
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count++
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// collect d1 in di
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di = di*b1 + d1
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i++
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// if di is "full", add it to the result
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if i == n {
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z = z.mulAddWW(z, bn, di)
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di = 0
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i = 0
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}
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// advance
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if ch, err = r.ReadByte(); err != nil {
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if err == io.EOF {
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err = nil
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break
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}
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return
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}
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}
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if count == 0 {
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// no digits found
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switch {
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case base == 0 && b == 8:
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// there was only the octal prefix 0 (possibly followed by digits > 7);
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// count as one digit and return base 10, not 8
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count = 1
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b = 10
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case base != 0 || b != 8:
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// there was neither a mantissa digit nor the octal prefix 0
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err = errors.New("syntax error scanning number")
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}
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return
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}
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// count > 0
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// add remaining digits to result
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if i > 0 {
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z = z.mulAddWW(z, pow(b1, i), di)
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}
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res = z.norm()
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// adjust for fraction, if any
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if dp >= 0 {
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// 0 <= dp <= count > 0
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count = dp - count
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}
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return
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}
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// utoa converts x to an ASCII representation in the given base;
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// base must be between 2 and MaxBase, inclusive.
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func (x nat) utoa(base int) []byte {
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return x.itoa(false, base)
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}
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// itoa is like utoa but it prepends a '-' if neg && x != 0.
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func (x nat) itoa(neg bool, base int) []byte {
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if base < 2 || base > MaxBase {
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panic("invalid base")
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}
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// x == 0
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if len(x) == 0 {
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return []byte("0")
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}
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// len(x) > 0
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// allocate buffer for conversion
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i := int(float64(x.bitLen())/math.Log2(float64(base))) + 1 // off by 1 at most
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if neg {
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i++
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}
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s := make([]byte, i)
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// convert power of two and non power of two bases separately
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if b := Word(base); b == b&-b {
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// shift is base b digit size in bits
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shift := trailingZeroBits(b) // shift > 0 because b >= 2
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mask := Word(1<<shift - 1)
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w := x[0] // current word
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nbits := uint(_W) // number of unprocessed bits in w
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// convert less-significant words (include leading zeros)
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for k := 1; k < len(x); k++ {
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// convert full digits
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for nbits >= shift {
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i--
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s[i] = digits[w&mask]
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w >>= shift
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nbits -= shift
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}
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// convert any partial leading digit and advance to next word
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if nbits == 0 {
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// no partial digit remaining, just advance
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w = x[k]
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nbits = _W
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} else {
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// partial digit in current word w (== x[k-1]) and next word x[k]
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w |= x[k] << nbits
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i--
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s[i] = digits[w&mask]
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// advance
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w = x[k] >> (shift - nbits)
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nbits = _W - (shift - nbits)
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}
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}
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// convert digits of most-significant word w (omit leading zeros)
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for w != 0 {
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i--
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s[i] = digits[w&mask]
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w >>= shift
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}
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} else {
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bb, ndigits := maxPow(b)
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// construct table of successive squares of bb*leafSize to use in subdivisions
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// result (table != nil) <=> (len(x) > leafSize > 0)
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table := divisors(len(x), b, ndigits, bb)
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// preserve x, create local copy for use by convertWords
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q := nat(nil).set(x)
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// convert q to string s in base b
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q.convertWords(s, b, ndigits, bb, table)
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// strip leading zeros
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// (x != 0; thus s must contain at least one non-zero digit
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// and the loop will terminate)
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i = 0
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for s[i] == '0' {
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i++
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}
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}
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if neg {
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i--
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s[i] = '-'
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}
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return s[i:]
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}
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// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
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// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
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// repeated nat/Word division.
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//
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// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
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// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
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// Recursive conversion divides q by its approximate square root, yielding two parts, each half
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// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
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// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
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// is made better by splitting the subblocks recursively. Best is to split blocks until one more
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// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
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// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
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// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
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// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
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// specific hardware.
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//
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func (q nat) convertWords(s []byte, b Word, ndigits int, bb Word, table []divisor) {
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// split larger blocks recursively
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if table != nil {
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// len(q) > leafSize > 0
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var r nat
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index := len(table) - 1
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for len(q) > leafSize {
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// find divisor close to sqrt(q) if possible, but in any case < q
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maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
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minLength := maxLength >> 1 // ~= log2 sqrt(q)
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for index > 0 && table[index-1].nbits > minLength {
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index-- // desired
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}
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if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
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index--
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if index < 0 {
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panic("internal inconsistency")
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}
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}
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// split q into the two digit number (q'*bbb + r) to form independent subblocks
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q, r = q.div(r, q, table[index].bbb)
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// convert subblocks and collect results in s[:h] and s[h:]
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h := len(s) - table[index].ndigits
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r.convertWords(s[h:], b, ndigits, bb, table[0:index])
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s = s[:h] // == q.convertWords(s, b, ndigits, bb, table[0:index+1])
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}
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}
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// having split any large blocks now process the remaining (small) block iteratively
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i := len(s)
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var r Word
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if b == 10 {
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// hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
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for len(q) > 0 {
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// extract least significant, base bb "digit"
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q, r = q.divW(q, bb)
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for j := 0; j < ndigits && i > 0; j++ {
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i--
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// avoid % computation since r%10 == r - int(r/10)*10;
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// this appears to be faster for BenchmarkString10000Base10
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// and smaller strings (but a bit slower for larger ones)
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t := r / 10
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s[i] = '0' + byte(r-t*10)
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r = t
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}
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}
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} else {
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for len(q) > 0 {
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// extract least significant, base bb "digit"
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q, r = q.divW(q, bb)
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for j := 0; j < ndigits && i > 0; j++ {
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i--
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s[i] = digits[r%b]
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r /= b
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}
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}
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}
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// prepend high-order zeros
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for i > 0 { // while need more leading zeros
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i--
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s[i] = '0'
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}
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}
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// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
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// Benchmark and configure leafSize using: go test -bench="Leaf"
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// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
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// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
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var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
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type divisor struct {
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bbb nat // divisor
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nbits int // bit length of divisor (discounting leading zeros) ~= log2(bbb)
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ndigits int // digit length of divisor in terms of output base digits
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}
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var cacheBase10 struct {
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sync.Mutex
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table [64]divisor // cached divisors for base 10
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}
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// expWW computes x**y
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func (z nat) expWW(x, y Word) nat {
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return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
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}
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// construct table of powers of bb*leafSize to use in subdivisions
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func divisors(m int, b Word, ndigits int, bb Word) []divisor {
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// only compute table when recursive conversion is enabled and x is large
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if leafSize == 0 || m <= leafSize {
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return nil
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}
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// determine k where (bb**leafSize)**(2**k) >= sqrt(x)
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k := 1
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for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
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k++
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}
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// reuse and extend existing table of divisors or create new table as appropriate
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var table []divisor // for b == 10, table overlaps with cacheBase10.table
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if b == 10 {
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cacheBase10.Lock()
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table = cacheBase10.table[0:k] // reuse old table for this conversion
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} else {
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table = make([]divisor, k) // create new table for this conversion
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}
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// extend table
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if table[k-1].ndigits == 0 {
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// add new entries as needed
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var larger nat
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for i := 0; i < k; i++ {
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if table[i].ndigits == 0 {
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if i == 0 {
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table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
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table[0].ndigits = ndigits * leafSize
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} else {
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table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
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table[i].ndigits = 2 * table[i-1].ndigits
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}
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// optimization: exploit aggregated extra bits in macro blocks
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larger = nat(nil).set(table[i].bbb)
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for mulAddVWW(larger, larger, b, 0) == 0 {
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table[i].bbb = table[i].bbb.set(larger)
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table[i].ndigits++
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}
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table[i].nbits = table[i].bbb.bitLen()
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}
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}
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}
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if b == 10 {
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cacheBase10.Unlock()
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}
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return table
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}
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