Factorial/Go
An implementation of one of Peter Luschny's fast factorial algorithms. Fast as this algorithm is, I believe there is room to speed it up more with parallelization and attention to cache effects. The Go library has a nice Karatsuba multiplier but it is yet single threaded. <lang go>package main
import (
"math/big" "fmt" "time"
)
func main() {
const max = 1e6 var s sieve s.sieve(max)
// some trivial cases b := big.NewInt(1) tf(&s, 0, b) tf(&s, 1, b) tf(&s, 3, b.SetInt64(6)) tf(&s, 10, b.SetInt64(3628800))
// 20 is the first number that exercises the split factorial code tf(&s, 20, b.SetInt64(2432902008176640000))
// 65 is the first number that exercises the split odd swing code
b.SetString("8247650592082470666723170306785496252186258551345437492922123134388955774976000000000000000", 10)
tf(&s, 65, b)
// 402 is the first number that exercises the split primorial code
b.SetString("10322493151921465164081017511444523549144957788957729070658850054871632028467255601190963314928373192348001901396930189622367360453148777593779130493841936873495349332423413459470518031076600468677681086479354644916620480632630350145970538235260826120203515476630017152557002993632050731959317164706296917171625287200618560036028326143938282329483693985566225033103398611546364400484246579470387915281737632989645795534475998050620039413447425490893877731061666015468384131920640823824733578473025588407103553854530737735183050931478983505845362197959913863770041359352031682005647007823330600995250982455385703739491695583970372977196372367980241040180516191489137558020294105537577853569647066137370488100581103217089054291400441697731894590238418118698720784367447615471616000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", 10)
tf(&s, 402, b)
df(&s, 800) df(&s, 1e5) df(&s, max)
}
func tf(s *sieve, n uint, answer *big.Int) {
f := s.factorial(n)
if f.Cmp(answer) == 0 {
fmt.Printf("%d! pass.\n", n)
} else {
fmt.Printf("%d! fail.\nExpected %s\nFound %s\n",
n, answer.String(), f.String())
}
}
func df(s *sieve, n uint) {
start := time.Now()
a := s.factorial(n)
stop := time.Now()
fmt.Printf("n = %d -> factorial %v\n", n, stop.Sub(start))
dtrunc := int64(float64(a.BitLen())*.30103) - 10
var first, rest big.Int
rest.Exp(first.SetInt64(10), rest.SetInt64(dtrunc), nil)
first.Quo(a, &rest)
fstr := first.String()
fmt.Printf("%d! begins %s... and has %d digits.\n",
n, fstr, int64(len(fstr))+dtrunc)
}
func (s *sieve) factorial(n uint) *big.Int {
if n < 20 {
var r big.Int
return r.MulRange(1, int64(n))
}
if int64(n) > s.limit {
return nil
}
r := s.factorialS(n) return r.Lsh(r, n-bitCount32(uint32(n)))
}
func (s *sieve) factorialS(n uint) (swing *big.Int) {
if n < 2 {
return big.NewInt(1)
}
f2 := s.factorialS(n / 2) // recurse f2.Mul(f2, f2) // square
if n < uint(len(smallOddSwing)) {
swing = big.NewInt(smallOddSwing[n])
} else {
swing = s.oddSwing(n)
}
return swing.Mul(swing, f2)
}
func (s *sieve) oddSwing(k uint) *big.Int {
if k < uint(len(smallOddSwing)) {
return big.NewInt(smallOddSwing[k])
}
factors := make([]int64, k/2)
rootK := int64(floorSqrt(uint64(k)))
var i int
s.iterateFunc(3, rootK, func(p int64) bool {
q := int64(k) / p
for q > 0 {
if q&1 == 1 {
factors[i] = p
i++
}
q /= p
}
return false
})
s.iterateFunc(rootK+1, int64(k/3), func(p int64) bool {
if (int64(k) / p & 1) == 1 {
factors[i] = p
i++
}
return false
})
r := product(factors[0:i])
return r.Mul(r, s.primorial(int64(k/2+1), int64(k)))
}
func (s *sieve) primorial(m, n int64) *big.Int {
if m > n {
return big.NewInt(1)
}
if n-m < 200 {
var r, r2 big.Int
r.SetInt64(1)
s.iterateFunc(m, n, func(p int64) bool {
r.Mul(&r, r2.SetInt64(p))
return false
})
return &r
}
h := (m + n) / 2 r := s.primorial(m, h) return r.Mul(r, s.primorial(h+1, n))
}
type sieve struct {
limit int64 isComposite []uint64
}
func (s *sieve) iterateFunc(min, max int64, visitor func(prime int64) (terminate bool)) (ok bool) {
if max > s.limit {
return false // Max larger than sieve
}
if min < 2 {
min = 2
}
if min > max {
return true
}
if min == 2 && max >= 2 {
if visitor(2) {
return true
}
}
if min <= 3 && max >= 3 {
if visitor(3) {
return true
}
}
absPos := (min+(min+1)%2)/3 - 1 index := absPos / bitsPerInt bitPos := absPos % bitsPerInt prime := 5 + 3*(bitsPerInt*index+bitPos) - bitPos&1 inc := bitPos&1*2 + 2
for prime <= max {
bitField := s.isComposite[index] >> uint64(bitPos)
index++
for ; bitPos < bitsPerInt; bitPos++ {
if bitField&1 == 0 {
if visitor(prime) {
return true
}
}
prime += inc
if prime > max {
return true
}
inc = 6 - inc
bitField >>= 1
}
bitPos = 0
}
return true
}
// constants dependent on the word size of sieve.isComposite. const (
bitsPerInt = 64 mask = bitsPerInt - 1 log2Int = 6
)
func (s *sieve) sieve(n int64) {
if n <= 0 {
*s = sieve{}
}
s.limit = n s.isComposite = make([]uint64, n/(3*bitsPerInt)+1)
var (
d1, d2, p1, p2, s1, s2 uint64 = 8, 8, 3, 7, 7, 3
l, c, max uint64 = 0, 1, uint64(n) / 3
toggle bool
)
for s1 < max {
if (s.isComposite[l>>log2Int] & (1 << (l & mask))) == 0 {
inc := p1 + p2
for c = s1; c < max; c += inc {
s.isComposite[c>>log2Int] |= 1 << (c & mask)
}
for c = s1 + s2; c < max; c += inc {
s.isComposite[c>>log2Int] |= 1 << (c & mask)
}
}
l++
if toggle {
toggle = false
s1 += d1
d2 += 8
p1 += 2
p2 += 6
s2 = p1
} else {
toggle = true
s1 += d2
d1 += 16
p1 += 2
p2 += 2
s2 = p2
}
}
return
}
var smallOddSwing []int64 = []int64{1, 1, 1, 3, 3, 15, 5,
35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395, 12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075, 35102025, 5014575, 145422675, 9694845, 300540195, 300540195, 9917826435, 583401555, 20419054425, 2268783825, 83945001525, 4418157975, 172308161025, 34461632205, 1412926920405, 67282234305, 2893136075115, 263012370465, 11835556670925, 514589420475, 24185702762325, 8061900920775, 395033145117975, 15801325804719, 805867616040669, 61989816618513, 3285460280781189, 121683714103007, 6692604275665385, 956086325095055, 54496920530418135, 1879204156221315, 110873045217057585, 7391536347803839, 450883717216034179, 14544636039226909, 916312070471295267, 916312070471295267}
func bitCount32(w uint32) uint {
const (
ff = 1<<32 - 1
mask1 = ff / 3
mask3 = ff / 5
maskf = ff / 17
maskp = ff / 255
)
w -= w >> 1 & mask1
w = w&mask3 + w>>2&mask3
w = (w + w>>4) & maskf
return uint(w * maskp >> 24)
}
func floorSqrt(n uint64) (a uint64) {
for b := n; ; {
a := b
b = (n/a + a) / 2
if b >= a {
return a
}
}
return 0
}
func product(seq []int64) *big.Int {
if len(seq) <= 20 {
var b big.Int
sprod := big.NewInt(seq[0])
for _, s := range seq[1:] {
b.SetInt64(s)
sprod.Mul(sprod, &b)
}
return sprod
}
halfLen := len(seq) / 2 lprod := product(seq[0:halfLen]) rprod := product(seq[halfLen:]) return lprod.Mul(lprod, rprod)
} </lang> Output: I did 800 because a few others had computed it. Answers come pretty fast up to 10^5 but slow down after that. Times shown are for computing the factorial only. Producing the printable base 10 representation takes longer and isn't fun to wait for.
0! pass. 1! pass. 3! pass. 10! pass. 20! pass. 65! pass. 402! pass. n = 800 -> factorial 318us 800! begins 7710530113... and has 1977 digits. n = 100000 -> factorial 508.322ms 100000! begins 28242294079... and has 456574 digits. n = 1000000 -> factorial 3m33.118625s 1000000! begins 8263931688... and has 5565709 digits.