// 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 scans the number corresponding to the longest possible prefix
// from r representing an unsigned number in a given conversion base.
// It returns the corresponding natural number res, the actual base b,
// a digit count, and a read or syntax error err, if any.
//
// number = [ prefix ] mantissa .
// prefix = "0" [ "x" | "X" | "b" | "B" ] .
// mantissa = 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, 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
// ``0x'' or ``0X'' selects base 16; if fracOk is not set, the ``0'' prefix
// selects base 8, and a ``0b'' or ``0B'' prefix selects base 2. Otherwise
// the selected base is 10 and no prefix is accepted.
//
// If fracOk is set, an octal prefix is ignored (a leading ``0'' simply
// stands for a zero digit), and 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 illegal bases
baseOk := base == 0 ||
!fracOk && 2 <= base && base <= MaxBase ||
fracOk && (base == 2 || base == 10 || base == 16)
if !baseOk {
panic(fmt.Sprintf("illegal number base %d", base))
}
// one char look-ahead
ch, err := r.ReadByte()
if err != nil {
return
}
// determine actual base
b = base
if base == 0 {
// actual base is 10 unless there's a base prefix
b = 10
if ch == '0' {
count = 1
switch ch, err = r.ReadByte(); err {
case nil:
// possibly one of 0x, 0X, 0b, 0B
if !fracOk {
b = 8
}
switch ch {
case 'x', 'X':
b = 16
case 'b', 'B':
b = 2
}
switch b {
case 16, 2:
count = 0 // prefix is not counted
if ch, err = r.ReadByte(); err != nil {
// io.EOF is also an error in this case
return
}
case 8:
count = 0 // prefix is not counted
}
case io.EOF:
// input is "0"
res = z[:0]
err = nil
return
default:
// read error
return
}
}
}
// 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 {
if fracOk && ch == '.' {
fracOk = false
dp = count
// advance
if ch, err = r.ReadByte(); err != nil {
if err == io.EOF {
err = nil
break
}
return
}
}
// 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
}
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
}
// advance
if ch, err = r.ReadByte(); err != nil {
if err == io.EOF {
err = nil
break
}
return
}
}
if count == 0 {
// no digits found
switch {
case base == 0 && b == 8:
// there was only the octal prefix 0 (possibly followed by digits > 7);
// count as one digit and return base 10, not 8
count = 1
b = 10
case base != 0 || b != 8:
// there was neither a mantissa digit nor the octal prefix 0
err = errors.New("syntax error scanning number")
}
return
}
// count > 0
// add remaining digits to result
if i > 0 {
z = z.mulAddWW(z, pow(b1, i), di)
}
res = z.norm()
// adjust for fraction, if any
if dp >= 0 {
// 0 <= dp <= count > 0
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
}