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-rw-r--r--src/math/big/natconv.go512
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diff --git a/src/math/big/natconv.go b/src/math/big/natconv.go
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+// 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 nat-to-string conversion functions.
+
+package big
+
+import (
+ "errors"
+ "fmt"
+ "io"
+ "math"
+ "math/bits"
+ "sync"
+)
+
+const digits = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"
+
+// Note: MaxBase = len(digits), but it must remain an untyped rune constant
+// for API compatibility.
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 10 + ('z' - 'a' + 1) + ('Z' - 'A' + 1)
+const maxBaseSmall = 10 + ('z' - 'a' + 1)
+
+// maxPow returns (b**n, n) such that b**n is the largest power b**n <= _M.
+// For instance maxPow(10) == (1e19, 19) for 19 decimal digits in a 64bit Word.
+// In other words, at most n digits in base b fit into a Word.
+// TODO(gri) replace this with a table, generated at build time.
+func maxPow(b Word) (p Word, n int) {
+ p, n = b, 1 // assuming b <= _M
+ for max := _M / b; p <= max; {
+ // p == b**n && p <= max
+ p *= b
+ n++
+ }
+ // p == b**n && p <= _M
+ return
+}
+
+// pow returns x**n for n > 0, and 1 otherwise.
+func pow(x Word, n int) (p Word) {
+ // n == sum of bi * 2**i, for 0 <= i < imax, and bi is 0 or 1
+ // thus x**n == product of x**(2**i) for all i where bi == 1
+ // (Russian Peasant Method for exponentiation)
+ p = 1
+ for n > 0 {
+ if n&1 != 0 {
+ p *= x
+ }
+ x *= x
+ n >>= 1
+ }
+ return
+}
+
+// scan errors
+var (
+ errNoDigits = errors.New("number has no digits")
+ errInvalSep = errors.New("'_' must separate successive digits")
+)
+
+// scan scans the number corresponding to the longest possible prefix
+// from r representing an unsigned number in a given conversion base.
+// scan returns the corresponding natural number res, the actual base b,
+// a digit count, and a read or syntax error err, if any.
+//
+// For base 0, an underscore character ``_'' may appear between a base
+// prefix and an adjacent digit, and between successive digits; such
+// underscores do not change the value of the number, or the returned
+// digit count. Incorrect placement of underscores is reported as an
+// error if there are no other errors. If base != 0, underscores are
+// not recognized and thus terminate scanning like any other character
+// that is not a valid radix point or digit.
+//
+// number = mantissa | prefix pmantissa .
+// prefix = "0" [ "b" | "B" | "o" | "O" | "x" | "X" ] .
+// mantissa = digits "." [ digits ] | digits | "." digits .
+// pmantissa = [ "_" ] digits "." [ digits ] | [ "_" ] digits | "." digits .
+// digits = digit { [ "_" ] digit } .
+// digit = "0" ... "9" | "a" ... "z" | "A" ... "Z" .
+//
+// Unless fracOk is set, the base argument must be 0 or a value between
+// 2 and MaxBase. If fracOk is set, the base argument must be one of
+// 0, 2, 8, 10, or 16. Providing an invalid base argument leads to a run-
+// time panic.
+//
+// For base 0, the number prefix determines the actual base: A prefix of
+// ``0b'' or ``0B'' selects base 2, ``0o'' or ``0O'' selects base 8, and
+// ``0x'' or ``0X'' selects base 16. If fracOk is false, a ``0'' prefix
+// (immediately followed by digits) selects base 8 as well. Otherwise,
+// the selected base is 10 and no prefix is accepted.
+//
+// If fracOk is set, a period followed by a fractional part is permitted.
+// The result value is computed as if there were no period present; and
+// the count value is used to determine the fractional part.
+//
+// For bases <= 36, lower and upper case letters are considered the same:
+// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
+// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
+// values 36 to 61.
+//
+// A result digit count > 0 corresponds to the number of (non-prefix) digits
+// parsed. A digit count <= 0 indicates the presence of a period (if fracOk
+// is set, only), and -count is the number of fractional digits found.
+// In this case, the actual value of the scanned number is res * b**count.
+//
+func (z nat) scan(r io.ByteScanner, base int, fracOk bool) (res nat, b, count int, err error) {
+ // reject invalid bases
+ baseOk := base == 0 ||
+ !fracOk && 2 <= base && base <= MaxBase ||
+ fracOk && (base == 2 || base == 8 || base == 10 || base == 16)
+ if !baseOk {
+ panic(fmt.Sprintf("invalid number base %d", base))
+ }
+
+ // prev encodes the previously seen char: it is one
+ // of '_', '0' (a digit), or '.' (anything else). A
+ // valid separator '_' may only occur after a digit
+ // and if base == 0.
+ prev := '.'
+ invalSep := false
+
+ // one char look-ahead
+ ch, err := r.ReadByte()
+
+ // determine actual base
+ b, prefix := base, 0
+ if base == 0 {
+ // actual base is 10 unless there's a base prefix
+ b = 10
+ if err == nil && ch == '0' {
+ prev = '0'
+ count = 1
+ ch, err = r.ReadByte()
+ if err == nil {
+ // possibly one of 0b, 0B, 0o, 0O, 0x, 0X
+ switch ch {
+ case 'b', 'B':
+ b, prefix = 2, 'b'
+ case 'o', 'O':
+ b, prefix = 8, 'o'
+ case 'x', 'X':
+ b, prefix = 16, 'x'
+ default:
+ if !fracOk {
+ b, prefix = 8, '0'
+ }
+ }
+ if prefix != 0 {
+ count = 0 // prefix is not counted
+ if prefix != '0' {
+ ch, err = r.ReadByte()
+ }
+ }
+ }
+ }
+ }
+
+ // convert string
+ // Algorithm: Collect digits in groups of at most n digits in di
+ // and then use mulAddWW for every such group to add them to the
+ // result.
+ z = z[:0]
+ b1 := Word(b)
+ bn, n := maxPow(b1) // at most n digits in base b1 fit into Word
+ di := Word(0) // 0 <= di < b1**i < bn
+ i := 0 // 0 <= i < n
+ dp := -1 // position of decimal point
+ for err == nil {
+ if ch == '.' && fracOk {
+ fracOk = false
+ if prev == '_' {
+ invalSep = true
+ }
+ prev = '.'
+ dp = count
+ } else if ch == '_' && base == 0 {
+ if prev != '0' {
+ invalSep = true
+ }
+ prev = '_'
+ } else {
+ // convert rune into digit value d1
+ var d1 Word
+ switch {
+ case '0' <= ch && ch <= '9':
+ d1 = Word(ch - '0')
+ case 'a' <= ch && ch <= 'z':
+ d1 = Word(ch - 'a' + 10)
+ case 'A' <= ch && ch <= 'Z':
+ if b <= maxBaseSmall {
+ d1 = Word(ch - 'A' + 10)
+ } else {
+ d1 = Word(ch - 'A' + maxBaseSmall)
+ }
+ default:
+ d1 = MaxBase + 1
+ }
+ if d1 >= b1 {
+ r.UnreadByte() // ch does not belong to number anymore
+ break
+ }
+ prev = '0'
+ count++
+
+ // collect d1 in di
+ di = di*b1 + d1
+ i++
+
+ // if di is "full", add it to the result
+ if i == n {
+ z = z.mulAddWW(z, bn, di)
+ di = 0
+ i = 0
+ }
+ }
+
+ ch, err = r.ReadByte()
+ }
+
+ if err == io.EOF {
+ err = nil
+ }
+
+ // other errors take precedence over invalid separators
+ if err == nil && (invalSep || prev == '_') {
+ err = errInvalSep
+ }
+
+ if count == 0 {
+ // no digits found
+ if prefix == '0' {
+ // there was only the octal prefix 0 (possibly followed by separators and digits > 7);
+ // interpret as decimal 0
+ return z[:0], 10, 1, err
+ }
+ err = errNoDigits // fall through; result will be 0
+ }
+
+ // add remaining digits to result
+ if i > 0 {
+ z = z.mulAddWW(z, pow(b1, i), di)
+ }
+ res = z.norm()
+
+ // adjust count for fraction, if any
+ if dp >= 0 {
+ // 0 <= dp <= count
+ count = dp - count
+ }
+
+ return
+}
+
+// utoa converts x to an ASCII representation in the given base;
+// base must be between 2 and MaxBase, inclusive.
+func (x nat) utoa(base int) []byte {
+ return x.itoa(false, base)
+}
+
+// itoa is like utoa but it prepends a '-' if neg && x != 0.
+func (x nat) itoa(neg bool, base int) []byte {
+ if base < 2 || base > MaxBase {
+ panic("invalid base")
+ }
+
+ // x == 0
+ if len(x) == 0 {
+ return []byte("0")
+ }
+ // len(x) > 0
+
+ // allocate buffer for conversion
+ i := int(float64(x.bitLen())/math.Log2(float64(base))) + 1 // off by 1 at most
+ if neg {
+ i++
+ }
+ s := make([]byte, i)
+
+ // convert power of two and non power of two bases separately
+ if b := Word(base); b == b&-b {
+ // shift is base b digit size in bits
+ shift := uint(bits.TrailingZeros(uint(b))) // shift > 0 because b >= 2
+ mask := Word(1<<shift - 1)
+ w := x[0] // current word
+ nbits := uint(_W) // number of unprocessed bits in w
+
+ // convert less-significant words (include leading zeros)
+ for k := 1; k < len(x); k++ {
+ // convert full digits
+ for nbits >= shift {
+ i--
+ s[i] = digits[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ // convert any partial leading digit and advance to next word
+ if nbits == 0 {
+ // no partial digit remaining, just advance
+ w = x[k]
+ nbits = _W
+ } else {
+ // partial digit in current word w (== x[k-1]) and next word x[k]
+ w |= x[k] << nbits
+ i--
+ s[i] = digits[w&mask]
+
+ // advance
+ w = x[k] >> (shift - nbits)
+ nbits = _W - (shift - nbits)
+ }
+ }
+
+ // convert digits of most-significant word w (omit leading zeros)
+ for w != 0 {
+ i--
+ s[i] = digits[w&mask]
+ w >>= shift
+ }
+
+ } else {
+ bb, ndigits := maxPow(b)
+
+ // construct table of successive squares of bb*leafSize to use in subdivisions
+ // result (table != nil) <=> (len(x) > leafSize > 0)
+ table := divisors(len(x), b, ndigits, bb)
+
+ // preserve x, create local copy for use by convertWords
+ q := nat(nil).set(x)
+
+ // convert q to string s in base b
+ q.convertWords(s, b, ndigits, bb, table)
+
+ // strip leading zeros
+ // (x != 0; thus s must contain at least one non-zero digit
+ // and the loop will terminate)
+ i = 0
+ for s[i] == '0' {
+ i++
+ }
+ }
+
+ if neg {
+ i--
+ s[i] = '-'
+ }
+
+ return s[i:]
+}
+
+// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
+// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
+// repeated nat/Word division.
+//
+// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
+// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
+// Recursive conversion divides q by its approximate square root, yielding two parts, each half
+// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
+// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
+// is made better by splitting the subblocks recursively. Best is to split blocks until one more
+// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
+// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
+// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
+// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
+// specific hardware.
+//
+func (q nat) convertWords(s []byte, b Word, ndigits int, bb Word, table []divisor) {
+ // split larger blocks recursively
+ if table != nil {
+ // len(q) > leafSize > 0
+ var r nat
+ index := len(table) - 1
+ for len(q) > leafSize {
+ // find divisor close to sqrt(q) if possible, but in any case < q
+ maxLength := q.bitLen() // ~= log2 q, or at of least largest possible q of this bit length
+ minLength := maxLength >> 1 // ~= log2 sqrt(q)
+ for index > 0 && table[index-1].nbits > minLength {
+ index-- // desired
+ }
+ if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
+ index--
+ if index < 0 {
+ panic("internal inconsistency")
+ }
+ }
+
+ // split q into the two digit number (q'*bbb + r) to form independent subblocks
+ q, r = q.div(r, q, table[index].bbb)
+
+ // convert subblocks and collect results in s[:h] and s[h:]
+ h := len(s) - table[index].ndigits
+ r.convertWords(s[h:], b, ndigits, bb, table[0:index])
+ s = s[:h] // == q.convertWords(s, b, ndigits, bb, table[0:index+1])
+ }
+ }
+
+ // having split any large blocks now process the remaining (small) block iteratively
+ i := len(s)
+ var r Word
+ if b == 10 {
+ // hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ // avoid % computation since r%10 == r - int(r/10)*10;
+ // this appears to be faster for BenchmarkString10000Base10
+ // and smaller strings (but a bit slower for larger ones)
+ t := r / 10
+ s[i] = '0' + byte(r-t*10)
+ r = t
+ }
+ }
+ } else {
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb)
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ s[i] = digits[r%b]
+ r /= b
+ }
+ }
+ }
+
+ // prepend high-order zeros
+ for i > 0 { // while need more leading zeros
+ i--
+ s[i] = '0'
+ }
+}
+
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
+// Benchmark and configure leafSize using: go test -bench="Leaf"
+// 8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
+// 8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
+var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
+
+type divisor struct {
+ bbb nat // divisor
+ nbits int // bit length of divisor (discounting leading zeros) ~= log2(bbb)
+ ndigits int // digit length of divisor in terms of output base digits
+}
+
+var cacheBase10 struct {
+ sync.Mutex
+ table [64]divisor // cached divisors for base 10
+}
+
+// expWW computes x**y
+func (z nat) expWW(x, y Word) nat {
+ return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
+}
+
+// construct table of powers of bb*leafSize to use in subdivisions
+func divisors(m int, b Word, ndigits int, bb Word) []divisor {
+ // only compute table when recursive conversion is enabled and x is large
+ if leafSize == 0 || m <= leafSize {
+ return nil
+ }
+
+ // determine k where (bb**leafSize)**(2**k) >= sqrt(x)
+ k := 1
+ for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
+ k++
+ }
+
+ // reuse and extend existing table of divisors or create new table as appropriate
+ var table []divisor // for b == 10, table overlaps with cacheBase10.table
+ if b == 10 {
+ cacheBase10.Lock()
+ table = cacheBase10.table[0:k] // reuse old table for this conversion
+ } else {
+ table = make([]divisor, k) // create new table for this conversion
+ }
+
+ // extend table
+ if table[k-1].ndigits == 0 {
+ // add new entries as needed
+ var larger nat
+ for i := 0; i < k; i++ {
+ if table[i].ndigits == 0 {
+ if i == 0 {
+ table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
+ table[0].ndigits = ndigits * leafSize
+ } else {
+ table[i].bbb = nat(nil).sqr(table[i-1].bbb)
+ table[i].ndigits = 2 * table[i-1].ndigits
+ }
+
+ // optimization: exploit aggregated extra bits in macro blocks
+ larger = nat(nil).set(table[i].bbb)
+ for mulAddVWW(larger, larger, b, 0) == 0 {
+ table[i].bbb = table[i].bbb.set(larger)
+ table[i].ndigits++
+ }
+
+ table[i].nbits = table[i].bbb.bitLen()
+ }
+ }
+ }
+
+ if b == 10 {
+ cacheBase10.Unlock()
+ }
+
+ return table
+}
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