How to calculate Double Factorial?

[Paul Zimmermann]

> Date: Wed, 14 Jan 2009 14:51:25 +0900
> From: "Bruce M. Axtens" <bruce.axtens at gmail.com>
> 
> Given Wikipedia's description of a Double Factorial 
> <http://en.wikipedia.org/wiki/Factorial#Double_factorial>, what would be 
> the fastest way of implementing it in GMP?
> 
> --Bruce.

the code below contains two implementations, a trivial but slow one
(double_fac_ui_ref) and a complex but fast one (mpz_fac_ui2). Some timings
on a 2.4Ghz Core 2:

clafoutis% ./double_fac_ui 1000000
double_fac_ui took 78648ms
mpz_fac_ui2 took 722ms

Challenge: do better than mpz_fac_ui2 for n=1000000.

Paul Zimmermann

#include <stdio.h>
#include <stdlib.h>
#include <sys/types.h>
#include <sys/resource.h>
#include "gmp.h"

/* Computes double factorial:
   http://en.wikipedia.org/wiki/Factorial#Double_factorial

   Paul Zimmerman, 14 January 2009
   
   Based on algorithm page 226 of the book
   "Fast Algorithms, A Multitape Turing Machine Implementation",
   by A. Scho"nhage, A. F. W. Grotefeld and E. Vetter,
   BI-Wissenschaftsverlag, 1994.
*/

int cputime (void);
void mpz_fac_ui4 (mpz_ptr, unsigned long int *, unsigned long int);
void mpz_fac_ui3 (mpz_ptr, unsigned long int *, unsigned long int *,
                  unsigned long int *, unsigned long int);
void mpz_fac_ui2 (mpz_ptr, unsigned long int);

int
cputime ()
{
  struct rusage rus;

  getrusage (0, &rus);
  return rus.ru_utime.tv_sec * 1000 + rus.ru_utime.tv_usec / 1000;
}

/* result <- q[0]*q[1]*...*q[n-1] by binary splitting */
void
mpz_fac_ui4 (mpz_ptr result, unsigned long int *q, unsigned long int n)
{
  unsigned long int i, n0, n1;
  mpz_t tmp;

  if (n == 0)
    {
      mpz_set_ui (result, 1);
      return;
    }
  
  if (n <= 3)
    {
      mpz_set_ui (result, q[0]);
      for (i = 1; i < n; i++)
        mpz_mul_ui (result, result, q[i]);
      return;
    }

  n0 = n / 2;
  n1 = n - n0;
  mpz_fac_ui4 (result, q, n1);
  mpz_init (tmp);
  mpz_fac_ui4 (tmp, q + n1, n0);
  mpz_mul (result, result, tmp);
  mpz_clear (tmp);
}

/* put p[0]^e[0]*...*p[n-1]^e[n-1] in result,
   assuming e[0] >= e[1] >= ... >= e[n-1] > 0.

   Uses auxiliary table q[0...n-1].
*/
void
mpz_fac_ui3 (mpz_ptr result, unsigned long int *p, unsigned long int *e,
             unsigned long int *q, unsigned long int n)
{
  unsigned long int i, j, mask;
  mpz_t tmp;

  /* find largest power of two <= the exponent of two */
  for (mask = 1; e[0] >= mask << 1; mask <<= 1);
  mpz_init (tmp);
  mpz_set_ui (result, 1);
  while (mask)
    {
      /* put part corresponding to bit mask set in q[] */
      for (i = 0, j = 0; i < n; i++)
        if (e[i] & mask)
          q[j++] = p[i];
      mpz_fac_ui4 (tmp, q, j);
      mpz_mul (result, result, result);
      mpz_mul (result, result, tmp);
      mask >>= 1;
    }
  mpz_clear (tmp);
}

void
mpz_fac_ui2 (mpz_ptr result, unsigned long int n)
{
  unsigned long int *p;
  unsigned long int *e;
  unsigned long int *q;
  unsigned long int i, j, k;

  if (n <= 1)
    {
      mpz_set_ui (result, 1);
      return;
    }

  p = (unsigned long int*) malloc ((n + 1) * sizeof(unsigned long int));
  e = (unsigned long int*) malloc ((n + 1) * sizeof(unsigned long int));
  for (i = 2; i <= n; i++)
    {
      p[i] = i;
      e[i] = (n + 1 - i) & 1;
    }
  for (i = 2; i < n / 2; i++)
    {
      if (p[i] != 1) /* i is prime */
        {
          /* remove factor i in 2i, 3i, ... */
          for (j = 2 * i; j <= n; j += i)
            {
              p[j] /= i;
              e[i] += (n + 1 - j) & 1;
            }
          /* remove factor i in i^2, 2i^2, ..., i^3, 2i^3, ... */
          for (k = i; k <= n;)
            {
              /* check that k * i does not overflow */
              if (k > (~(0UL)) / i)
                break;
              k *= i;
              for (j = k; j <= n; j += k)
                {
                  p[j] /= i;
                  e[i] += (n + 1 - j) & 1;
                }
            }
        }
    }
  /* shrink table */
  for (i = 2, j = 0; i <= n; i++)
    {
      if (p[i] != 1)
        {
          p[j] = p[i];
          e[j] = e[i];
          j++;
        }
    }
  q = (unsigned long int*) malloc (j * sizeof(unsigned long int));
  mpz_fac_ui3 (result, p + 1, e + 1, q, j - 1);
  /* treat apart exponent of two */
  mpz_mul_2exp (result, result, e[0]);
  free (q);
  free (p);
  free (e);
}

void
double_fac_ui_ref (mpz_t f, unsigned long n)
{
  mpz_set_ui (f, 1);
  while (n > 1)
    {
      mpz_mul_ui (f, f, n);
      n -= 2;
    }
}

int
main (int argc, char *argv[])
{
  int n = atoi (argv[1]);
  mpz_t f, f2;
  int st;

  mpz_init (f);
  mpz_init (f2);

  st = cputime ();
  double_fac_ui_ref (f, n);
  printf ("double_fac_ui took %dms\n", cputime () - st);

  st = cputime ();
  mpz_fac_ui2 (f2, n);
  printf ("mpz_fac_ui2 took %dms\n", cputime () - st);

  if (mpz_cmp (f, f2))
    {
      fprintf (stderr, "f and f2 differ\n");
      exit (1);
    }

  mpz_clear (f);
  mpz_clear (f2);

  return 0;
}