// string converts x to a string using digits from a charset; a digit with
// value d is represented by charset[d]. The conversion base is determined
-// by len(charset), which must be >= 2.
+// by len(charset), which must be >= 2 and <= 256.
func (x nat) string(charset string) string {
b := Word(len(charset))
// special cases
switch {
- case b < 2 || MaxBase < b:
+ case b < 2 || MaxBase > 256:
panic("illegal base")
case len(x) == 0:
return string(charset[0])
w >>= shift
nbits -= shift
}
+
} else {
- // determine "big base" as in 10^19 for 19 decimal digits in a 64 bit Word
- bb := Word(1) // big base is b**ndigits
- ndigits := 0 // number of base b digits
+ // determine "big base"; i.e., the largest possible value bb
+ // that is a power of base b and still fits into a Word
+ // (as in 10^19 for 19 decimal digits in a 64bit Word)
+ bb := b // big base is b**ndigits
+ ndigits := 1 // number of base b digits
for max := Word(_M / b); bb <= max; bb *= b {
ndigits++ // maximize ndigits where bb = b**ndigits, bb <= _M
}
// 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 in divisions
+ // preserve x, create local copy for use by convertWords
q := nat(nil).set(x)
- // convert q to string s in base b with index of MSD indicated by return value
- i = q.convertWords(0, i, s, charset, b, ndigits, bb, table)
+ // convert q to string s in base b
+ q.convertWords(s, charset, 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 zero := charset[0]; s[i] == zero; {
+ i++
+ }
}
return string(s[i:])
}
-// Convert words of q to base b digits in s directly using iterated nat/Word divison to extract
-// low-order Words and indirectly by recursive subdivision and nat/nat division by tabulated
-// divisors.
+// 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 divison.
//
-// The direct method processes n Words by n divW() calls, each of which visits every Word in the
+// 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.
-// Indirect conversion divides q by its approximate square root, yielding two parts, each half
-// the size of q. Using the direct 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
+// 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
-// direct approach. This threshold is represented by leafSize. Benchmarking of leafSize in 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
// specfic hardware.
//
-// lo and hi index character array s. conversion starts with the LSD at hi and moves down toward
-// the MSD, which will be at s[0] or s[1]. lo == 0 signals span includes the most significant word.
-//
-func (q nat) convertWords(lo, hi int, s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) int {
- // indirect conversion: split larger blocks to reduce quadratic expense of iterated nat/W division
- if leafSize > 0 && len(q) > leafSize && table != nil {
+func (q nat) convertWords(s []byte, charset string, 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 {
// 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[lo:partition] and s[partition:hi]
- partition := hi - table[index].ndigits
- r.convertWords(partition, hi, s, charset, b, ndigits, bb, table[0:index])
- hi = partition // i.e., q.convertWords(lo, partition, s, charset, b, ndigits, bb, table[0:index+1])
+ // convert subblocks and collect results in s[:h] and s[h:]
+ h := len(s) - table[index].ndigits
+ r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
+ s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
}
- } // having split any large blocks now process the remaining small block
+ }
- // direct conversion: process smaller blocks monolithically to avoid overhead of nat/nat division
+ // 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 mod as mul vs div)
+ 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)
- if lo == 0 && len(q) == 0 {
- // skip leading zeros in most-significant group of digits
- for j := 0; j < ndigits && r != 0; j++ {
- hi--
- t := r / 10
- s[hi] = charset[r-(t<<3+t<<1)] // 8*t + 2*t = 10*t; r - 10*int(r/10) = r mod 10
- r = t
- }
- } else {
- for j := 0; j < ndigits && hi > lo; j++ {
- hi--
- t := r / 10
- s[hi] = charset[r-(t<<3+t<<1)] // 8*t + 2*t = 10*t; r - 10*int(r/10) = r mod 10
- r = t
- }
+ 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] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
+ r = t
}
}
} else {
for len(q) > 0 {
- // extract least significant group of digits
+ // extract least significant, base bb "digit"
q, r = q.divW(q, bb)
- if lo == 0 && len(q) == 0 {
- // skip leading zeros in most-significant group of digits
- for j := 0; j < ndigits && r != 0; j++ {
- hi--
- s[hi] = charset[r%b]
- r = r / b
- }
- } else {
- for j := 0; j < ndigits && hi > lo; j++ {
- hi--
- s[hi] = charset[r%b]
- r = r / b
- }
+ for j := 0; j < ndigits && i > 0; j++ {
+ i--
+ s[i] = charset[r%b]
+ r /= b
}
}
}
- // prepend high-order zeroes when q has been normalized to a short number of Words.
- // however, do not prepend zeroes when converting the most dignificant digits.
- if lo != 0 { // if not MSD
- zero := charset[0]
- for hi > lo { // while need more leading zeroes
- hi--
- s[hi] = zero
- }
+ // prepend high-order zeroes
+ zero := charset[0]
+ for i > 0 { // while need more leading zeroes
+ i--
+ s[i] = zero
}
-
- // return index of most significant output digit in s[] (stored in lowest index)
- return hi
}
-// Split blocks greater than leafSize Words (or set to 0 to disable indirect conversion)
+// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
// Benchmark and configure leafSize using: gotest -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
ndigits int // digit length of divisor in terms of output base digits
}
-const maxCache = 64 // maximum number of divisors in a single table
-var cacheBase10 [maxCache]divisor // cached divisors for base 10
-var cacheLock sync.Mutex // defense against concurrent table extensions
+var cacheBase10 [64]divisor // cached divisors for base 10
+var cacheLock sync.Mutex // protects cacheBase10
+
+// 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 build table when indirect conversion is enabled and x is large
+ // 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 < maxCache; words <<= 1 {
+ for words := leafSize; words < m>>1 && k < len(cacheBase10); words <<= 1 {
k++
}
// create new table of divisors or extend and reuse existing table as appropriate
- var cached bool
var table []divisor
+ var cached bool
switch b {
case 10:
table = cacheBase10[0:k] // reuse old table for this conversion
cacheLock.Lock() // begin critical section
}
- var i int
+ // add new entries as needed
var larger nat
- for i < k && table[i].ndigits != 0 { // skip existing entries
- i++
- }
- for ; i < k; i++ { // add new entries
- if i == 0 {
- table[i].bbb = nat(nil).expWW(bb, Word(leafSize))
- table[i].ndigits = ndigits * leafSize
- } else {
- table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
- table[i].ndigits = 2 * table[i-1].ndigits
- }
+ for i := 0; i < k; i++ {
+ if table[i].ndigits == 0 {
+ if i == 0 {
+ table[i].bbb = nat(nil).expWW(bb, Word(leafSize))
+ table[i].ndigits = ndigits * leafSize
+ } else {
+ table[i].bbb = nat(nil).mul(table[i-1].bbb, 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++
- }
+ // 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()
+ table[i].nbits = table[i].bbb.bitLen()
+ }
}
if cached {
return z.norm()
}
-// calculate x**y for Word arguments y and y
-func (z nat) expWW(x, y Word) nat {
- return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
-}
-
// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
// If it returns true, n is prime with probability 1 - 1/4^reps.
// If it returns false, n is not prime.
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