dist/mpz/oddfac_1.c

/* mpz_oddfac_1(RESULT, N) -- Set RESULT to the odd factor of N!.

Contributed to the GNU project by Marco Bodrato.

THE FUNCTION IN THIS FILE IS INTERNAL WITH A MUTABLE INTERFACE.
IT IS ONLY SAFE TO REACH IT THROUGH DOCUMENTED INTERFACES.
IN FACT, IT IS ALMOST GUARANTEED THAT IT WILL CHANGE OR
DISAPPEAR IN A FUTURE GNU MP RELEASE.

Copyright 2010-2012, 2015-2017 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of either:

  * the GNU Lesser General Public License as published by the Free
    Software Foundation; either version 3 of the License, or (at your
    option) any later version.

or

  * the GNU General Public License as published by the Free Software
    Foundation; either version 2 of the License, or (at your option) any
    later version.

or both in parallel, as here.

The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received copies of the GNU General Public License and the
GNU Lesser General Public License along with the GNU MP Library.  If not,
see https://www.gnu.org/licenses/.  */

#include "gmp-impl.h"
#include "longlong.h"

/* TODO:
   - split this file in smaller parts with functions that can be recycled for different computations.
 */

/**************************************************************/
/* Section macros: common macros, for mswing/fac/bin (&sieve) */
/**************************************************************/

#define FACTOR_LIST_APPEND(PR, MAX_PR, VEC, I)			\
  if ((PR) > (MAX_PR)) {					\
    (VEC)[(I)++] = (PR);					\
    (PR) = 1;							\
  }

#define FACTOR_LIST_STORE(P, PR, MAX_PR, VEC, I)		\
  do {								\
    if ((PR) > (MAX_PR)) {					\
      (VEC)[(I)++] = (PR);					\
      (PR) = (P);						\
    } else							\
      (PR) *= (P);						\
  } while (0)

#define LOOP_ON_SIEVE_CONTINUE(prime,end,sieve)			\
    __max_i = (end);						\
								\
    do {							\
      ++__i;							\
      if (((sieve)[__index] & __mask) == 0)			\
	{							\
	  mp_limb_t prime;					\
	  prime = id_to_n(__i)

#define LOOP_ON_SIEVE_BEGIN(prime,start,end,off,sieve)		\
  do {								\
    mp_limb_t __mask, __index, __max_i, __i;			\
								\
    __i = (start)-(off);					\
    __index = __i / GMP_LIMB_BITS;				\
    __mask = CNST_LIMB(1) << (__i % GMP_LIMB_BITS);		\
    __i += (off);						\
								\
    LOOP_ON_SIEVE_CONTINUE(prime,end,sieve)

#define LOOP_ON_SIEVE_STOP					\
	}							\
      __mask = __mask << 1 | __mask >> (GMP_LIMB_BITS-1);	\
      __index += __mask & 1;					\
    }  while (__i <= __max_i)

#define LOOP_ON_SIEVE_END					\
    LOOP_ON_SIEVE_STOP;						\
  } while (0)

/*********************************************************/
/* Section sieve: sieving functions and tools for primes */
/*********************************************************/

#if WANT_ASSERT
static mp_limb_t
bit_to_n (mp_limb_t bit) { return (bit*3+4)|1; }
#endif

/* id_to_n (x) = bit_to_n (x-1) = (id*3+1)|1*/
static mp_limb_t
id_to_n  (mp_limb_t id)  { return id*3+1+(id&1); }

/* n_to_bit (n) = ((n-1)&(-CNST_LIMB(2)))/3U-1 */
static mp_limb_t
n_to_bit (mp_limb_t n) { return ((n-5)|1)/3U; }

#if WANT_ASSERT
static mp_size_t
primesieve_size (mp_limb_t n) { return n_to_bit(n) / GMP_LIMB_BITS + 1; }
#endif

/*********************************************************/
/* Section mswing: 2-multiswing factorial                */
/*********************************************************/

/* Returns an approximation of the sqare root of x.
 * It gives:
 *   limb_apprsqrt (x) ^ 2 <= x < (limb_apprsqrt (x)+1) ^ 2
 * or
 *   x <= limb_apprsqrt (x) ^ 2 <= x * 9/8
 */
static mp_limb_t
limb_apprsqrt (mp_limb_t x)
{
  int s;

  ASSERT (x > 2);
  count_leading_zeros (s, x);
  s = (GMP_LIMB_BITS - s) >> 1;
  return ((CNST_LIMB(1) << s) + (x >> s)) >> 1;
}

#if 0
/* A count-then-exponentiate variant for SWING_A_PRIME */
#define SWING_A_PRIME(P, N, PR, MAX_PR, VEC, I)		\
  do {							\
    mp_limb_t __q, __prime;				\
    int __exp;						\
    __prime = (P);					\
    __exp = 0;						\
    __q = (N);						\
    do {						\
      __q /= __prime;					\
      __exp += __q & 1;					\
    } while (__q >= __prime);				\
    if (__exp) { /* Store $prime^{exp}$ */		\
      for (__q = __prime; --__exp; __q *= __prime);	\
      FACTOR_LIST_STORE(__q, PR, MAX_PR, VEC, I);	\
    };							\
  } while (0)
#else
#define SWING_A_PRIME(P, N, PR, MAX_PR, VEC, I)	\
  do {						\
    mp_limb_t __q, __prime;			\
    __prime = (P);				\
    FACTOR_LIST_APPEND(PR, MAX_PR, VEC, I);	\
    __q = (N);					\
    do {					\
      __q /= __prime;				\
      if ((__q & 1) != 0) (PR) *= __prime;	\
    } while (__q >= __prime);			\
  } while (0)
#endif

#define SH_SWING_A_PRIME(P, N, PR, MAX_PR, VEC, I)	\
  do {							\
    mp_limb_t __prime;					\
    __prime = (P);					\
    if ((((N) / __prime) & 1) != 0)			\
      FACTOR_LIST_STORE(__prime, PR, MAX_PR, VEC, I);	\
  } while (0)

/* mpz_2multiswing_1 computes the odd part of the 2-multiswing
   factorial of the parameter n.  The result x is an odd positive
   integer so that multiswing(n,2) = x 2^a.

   Uses the algorithm described by Peter Luschny in "Divide, Swing and
   Conquer the Factorial!".

   The pointer sieve points to primesieve_size(n) limbs containing a
   bit-array where primes are marked as 0.
   Enough (FIXME: explain :-) limbs must be pointed by factors.
 */

static void
mpz_2multiswing_1 (mpz_ptr x, mp_limb_t n, mp_ptr sieve, mp_ptr factors)
{
  mp_limb_t prod, max_prod;
  mp_size_t j;

  ASSERT (n > 25);

  j = 0;
  prod  = -(n & 1);
  n &= ~ CNST_LIMB(1); /* n-1, if n is odd */

  prod = (prod & n) + 1; /* the original n, if it was odd, 1 otherwise */
  max_prod = GMP_NUMB_MAX / (n-1);

  /* Handle prime = 3 separately. */
  SWING_A_PRIME (3, n, prod, max_prod, factors, j);

  /* Swing primes from 5 to n/3 */
  {
    mp_limb_t s, l_max_prod;

    s = limb_apprsqrt(n);
    ASSERT (s >= 5);
    s = n_to_bit (s);
    ASSERT (bit_to_n (s+1) * bit_to_n (s+1) > n);
    ASSERT (s < n_to_bit (n / 3));
    LOOP_ON_SIEVE_BEGIN (prime, n_to_bit (5), s, 0,sieve);
    SWING_A_PRIME (prime, n, prod, max_prod, factors, j);
    LOOP_ON_SIEVE_STOP;

    ASSERT (max_prod <= GMP_NUMB_MAX / 3);

    l_max_prod = max_prod * 3;

    LOOP_ON_SIEVE_CONTINUE (prime, n_to_bit (n/3), sieve);
    SH_SWING_A_PRIME (prime, n, prod, l_max_prod, factors, j);
    LOOP_ON_SIEVE_END;
  }

  /* Store primes from (n+1)/2 to n */
  LOOP_ON_SIEVE_BEGIN (prime, n_to_bit (n >> 1) + 1, n_to_bit (n), 0,sieve);
  FACTOR_LIST_STORE (prime, prod, max_prod, factors, j);
  LOOP_ON_SIEVE_END;

  if (LIKELY (j != 0))
    {
      factors[j++] = prod;
      mpz_prodlimbs (x, factors, j);
    }
  else
    {
      ASSERT (ALLOC (x) > 0);
      PTR (x)[0] = prod;
      SIZ (x) = 1;
    }
}

#undef SWING_A_PRIME
#undef SH_SWING_A_PRIME
#undef LOOP_ON_SIEVE_END
#undef LOOP_ON_SIEVE_STOP
#undef LOOP_ON_SIEVE_BEGIN
#undef LOOP_ON_SIEVE_CONTINUE
#undef FACTOR_LIST_APPEND

/*********************************************************/
/* Section oddfac: odd factorial, needed also by binomial*/
/*********************************************************/

#if TUNE_PROGRAM_BUILD
#define FACTORS_PER_LIMB (GMP_NUMB_BITS / (LOG2C(FAC_DSC_THRESHOLD_LIMIT-1)+1))
#else
#define FACTORS_PER_LIMB (GMP_NUMB_BITS / (LOG2C(FAC_DSC_THRESHOLD-1)+1))
#endif

/* mpz_oddfac_1 computes the odd part of the factorial of the
   parameter n.  I.e. n! = x 2^a, where x is the returned value: an
   odd positive integer.

   If flag != 0 a square is skipped in the DSC part, e.g.
   if n is odd, n > FAC_DSC_THRESHOLD and flag = 1, x is set to n!!.

   If n is too small, flag is ignored, and an ASSERT can be triggered.

   TODO: FAC_DSC_THRESHOLD is used here with two different roles:
    - to decide when prime factorisation is needed,
    - to stop the recursion, once sieving is done.
   Maybe two thresholds can do a better job.
 */
void
mpz_oddfac_1 (mpz_ptr x, mp_limb_t n, unsigned flag)
{
  ASSERT (n <= GMP_NUMB_MAX);
  ASSERT (flag == 0 || (flag == 1 && n > ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 1 && ABOVE_THRESHOLD (n, FAC_DSC_THRESHOLD)));

  if (n <= ODD_FACTORIAL_TABLE_LIMIT)
    {
      MPZ_NEWALLOC (x, 1)[0] = __gmp_oddfac_table[n];
      SIZ (x) = 1;
    }
  else if (n <= ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 1)
    {
      mp_ptr   px;

      px = MPZ_NEWALLOC (x, 2);
      umul_ppmm (px[1], px[0], __gmp_odd2fac_table[(n - 1) >> 1], __gmp_oddfac_table[n >> 1]);
      SIZ (x) = 2;
    }
  else
    {
      unsigned s;
      mp_ptr   factors;

      s = 0;
      {
	mp_limb_t tn;
	mp_limb_t prod, max_prod, i;
	mp_size_t j;
	TMP_SDECL;

#if TUNE_PROGRAM_BUILD
	ASSERT (FAC_DSC_THRESHOLD_LIMIT >= FAC_DSC_THRESHOLD);
	ASSERT (FAC_DSC_THRESHOLD >= 2 * (ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 2));
#endif

	/* Compute the number of recursive steps for the DSC algorithm. */
	for (tn = n; ABOVE_THRESHOLD (tn, FAC_DSC_THRESHOLD); s++)
	  tn >>= 1;

	j = 0;

	TMP_SMARK;
	factors = TMP_SALLOC_LIMBS (1 + tn / FACTORS_PER_LIMB);
	ASSERT (tn >= FACTORS_PER_LIMB);

	prod = 1;
#if TUNE_PROGRAM_BUILD
	max_prod = GMP_NUMB_MAX / FAC_DSC_THRESHOLD_LIMIT;
#else
	max_prod = GMP_NUMB_MAX / FAC_DSC_THRESHOLD;
#endif

	ASSERT (tn > ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 1);
	do {
	  i = ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 2;
	  factors[j++] = ODD_DOUBLEFACTORIAL_TABLE_MAX;
	  do {
	    FACTOR_LIST_STORE (i, prod, max_prod, factors, j);
	    i += 2;
	  } while (i <= tn);
	  max_prod <<= 1;
	  tn >>= 1;
	} while (tn > ODD_DOUBLEFACTORIAL_TABLE_LIMIT + 1);

	factors[j++] = prod;
	factors[j++] = __gmp_odd2fac_table[(tn - 1) >> 1];
	factors[j++] = __gmp_oddfac_table[tn >> 1];
	mpz_prodlimbs (x, factors, j);

	TMP_SFREE;
      }

      if (s != 0)
	/* Use the algorithm described by Peter Luschny in "Divide,
	   Swing and Conquer the Factorial!".

	   Improvement: there are two temporary buffers, factors and
	   square, that are never used together; with a good estimate
	   of the maximal needed size, they could share a single
	   allocation.
	*/
	{
	  mpz_t mswing;
	  mp_ptr sieve;
	  mp_size_t size;
	  TMP_DECL;

	  TMP_MARK;

	  flag--;
	  size = n / GMP_NUMB_BITS + 4;
	  ASSERT (primesieve_size (n - 1) <= size - (size / 2 + 1));
	  /* 2-multiswing(n) < 2^(n-1)*sqrt(n/pi) < 2^(n+GMP_NUMB_BITS);
	     one more can be overwritten by mul, another for the sieve */
	  MPZ_TMP_INIT (mswing, size);
	  /* Initialize size, so that ASSERT can check it correctly. */
	  ASSERT_CODE (SIZ (mswing) = 0);

	  /* Put the sieve on the second half, it will be overwritten by the last mswing. */
	  sieve = PTR (mswing) + size / 2 + 1;

	  size = (gmp_primesieve (sieve, n - 1) + 1) / log_n_max (n) + 1;

	  factors = TMP_ALLOC_LIMBS (size);
	  do {
	    mp_ptr    square, px;
	    mp_size_t nx, ns;
	    mp_limb_t cy;
	    TMP_DECL;

	    s--;
	    ASSERT (ABSIZ (mswing) < ALLOC (mswing) / 2); /* Check: sieve has not been overwritten */
	    mpz_2multiswing_1 (mswing, n >> s, sieve, factors);

	    TMP_MARK;
	    nx = SIZ (x);
	    if (s == flag) {
	      size = nx;
	      square = TMP_ALLOC_LIMBS (size);
	      MPN_COPY (square, PTR (x), nx);
	    } else {
	      size = nx << 1;
	      square = TMP_ALLOC_LIMBS (size);
	      mpn_sqr (square, PTR (x), nx);
	      size -= (square[size - 1] == 0);
	    }
	    ns = SIZ (mswing);
	    nx = size + ns;
	    px = MPZ_NEWALLOC (x, nx);
	    ASSERT (ns <= size);
	    cy = mpn_mul (px, square, size, PTR(mswing), ns); /* n!= n$ * floor(n/2)!^2 */

	    SIZ(x) = nx - (cy == 0);
	    TMP_FREE;
	  } while (s != 0);
	  TMP_FREE;
	}
    }
}

#undef FACTORS_PER_LIMB
#undef FACTOR_LIST_STORE