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.