mpir/mpn/generic/mul_n.c

/* mpn_mul_n and helper function -- Multiply/square natural numbers.

   THE HELPER FUNCTIONS IN THIS FILE (meaning everything except mpn_mul_n)
   ARE INTERNAL FUNCTIONS WITH MUTABLE INTERFACES.  IT IS ONLY SAFE TO REACH
   THEM THROUGH DOCUMENTED INTERFACES.  IN FACT, IT IS ALMOST GUARANTEED
   THAT THEY'LL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.


Copyright 1991, 1993, 1994, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003,
2005, Free Software Foundation, Inc.

Copyright 2009 Jason Moxham

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 the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.

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 Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library; see the file COPYING.LIB.  If not, write to
the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
MA 02110-1301, USA. */

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


/* Multiplies using 3 half-sized mults and so on recursively.
 * p[0..2*n-1] := product of a[0..n-1] and b[0..n-1].
 * No overlap of p[...] with a[...] or b[...].
 * ws is workspace.
 */

void
mpn_kara_mul_n (mp_ptr p, mp_srcptr a, mp_srcptr b, mp_size_t n, mp_ptr ws)
{
  mp_limb_t w, w0, w1,c;
  mp_size_t n2;
  mp_srcptr x, y;
  mp_size_t i;
  int sign, ci;

  n2 = n >> 1;
  ASSERT (n2 > 0);

  if ((n & 1) != 0)
    {
      /* Odd length. */
      mp_size_t n1, n3, nm1;

      n3 = n - n2;

      sign = 0;
      w = a[n2];
      if (w != 0)
	w -= mpn_sub_n (p, a, a + n3, n2);
      else
	{
	  i = n2;
	  do
	    {
	      --i;
	      w0 = a[i];
	      w1 = a[n3 + i];
	    }
	  while (w0 == w1 && i != 0);
	  if (w0 < w1)
	    {
	      x = a + n3;
	      y = a;
	      sign = ~0;
	    }
	  else
	    {
	      x = a;
	      y = a + n3;
	    }
	  mpn_sub_n (p, x, y, n2);
	}
      p[n2] = w;

      w = b[n2];
      if (w != 0)
	w -= mpn_sub_n (p + n3, b, b + n3, n2);
      else
	{
	  i = n2;
	  do
	    {
	      --i;
	      w0 = b[i];
	      w1 = b[n3 + i];
	    }
	  while (w0 == w1 && i != 0);
	  if (w0 < w1)
	    {
	      x = b + n3;
	      y = b;
	      sign = ~sign;
	    }
	  else
	    {
	      x = b;
	      y = b + n3;
	    }
	  mpn_sub_n (p + n3, x, y, n2);
	}
      p[n] = w;

      n1 = n + 1;
      if (n2 < MUL_KARATSUBA_THRESHOLD)
	{
	  if (n3 < MUL_KARATSUBA_THRESHOLD)
	    {
	      mpn_mul_basecase (ws, p, n3, p + n3, n3);
	      mpn_mul_basecase (p, a, n3, b, n3);
	    }
	  else
	    {
	      mpn_kara_mul_n (ws, p, p + n3, n3, ws + n1);
	      mpn_kara_mul_n (p, a, b, n3, ws + n1);
	    }
	  mpn_mul_basecase (p + n1, a + n3, n2, b + n3, n2);
	}
      else
	{
	  mpn_kara_mul_n (ws, p, p + n3, n3, ws + n1);
	  mpn_kara_mul_n (p, a, b, n3, ws + n1);
	  mpn_kara_mul_n (p + n1, a + n3, b + n3, n2, ws + n1);
	}

      nm1 = n - 1;
      if (sign)
	{
	 #if HAVE_NATIVE_mpn_addadd_n
	   c=mpn_addadd_n(ws,p+n1,p,ws,nm1);
  	   #if GMP_NAIL_BITS==0
  	     add_ssaaaa(ws[n],ws[nm1],ws[n],ws[nm1],p[n],p[nm1]);
  	     add_ssaaaa(ws[n],ws[nm1],ws[n],ws[nm1],0,c);
  	   #else
  	     mpn_add_n(ws+nm1, p+nm1, ws+nm1 ,2);
  	     mpn_add_1(ws+nm1,ws+nm1,2,c);
  	   #endif
	 #else
	   mpn_add_n (ws, p, ws, n1);
	   c=mpn_add_n (ws, p + n1, ws, nm1);
	   if(c){mp_limb_t x=(ws[nm1]+1) & GMP_NUMB_MASK;ws[nm1]=x;if(x==0)ws[n]=(ws[n]+1) & GMP_NUMB_MASK;}
	 #endif
	}
      else
	{
	 #if HAVE_NATIVE_mpn_addsub_n
           ci=mpn_addsub_n(ws,p+n1,p,ws,nm1);
           #if GMP_NAIL_BITS==0
             sub_ddmmss(ws[n],ws[nm1],p[n],p[nm1],ws[n],ws[nm1]);//mpn_sub_n(ws+nm1,p+nm1,ws+nm1,2);
             if(ci==-1)sub_ddmmss(ws[n],ws[nm1],ws[n],ws[nm1],0,1);//mpn_sub_1(ws+nm1,ws+nm1,2,1);
             if(ci== 1)add_ssaaaa(ws[n],ws[nm1],ws[n],ws[nm1],0,1);//mpn_add_1(ws+nm1,ws+nm1,2,1);
           #else
             mpn_sub_n(ws+nm1,p+nm1,ws+nm1,2);
             if(ci==-1)mpn_sub_1(ws+nm1,ws+nm1,2,1);
             if(ci== 1)mpn_add_1(ws+nm1,ws+nm1,2,1);
           #endif
         #else
	   mpn_sub_n (ws, p, ws, n1);
	   c=mpn_add_n (ws, p + n1, ws, nm1);
	   if(c){mp_limb_t x=(ws[nm1]+1) & GMP_NUMB_MASK;ws[nm1]=x;if(x==0)ws[n]=(ws[n]+1) & GMP_NUMB_MASK;}
         #endif
        }      
      if (mpn_add_n (p + n3, p + n3, ws, n1))
	{
	  mpn_incr_u (p + n1 + n3, 1);
	}
    }
  else
    {
      /* Even length. */
      i = n2;
      do
	{
	  --i;
	  w0 = a[i];
	  w1 = a[n2 + i];
	}
      while (w0 == w1 && i != 0);
      sign = 0;
      if (w0 < w1)
	{
	  x = a + n2;
	  y = a;
	  sign = ~0;
	}
      else
	{
	  x = a;
	  y = a + n2;
	}
      mpn_sub_n (p, x, y, n2);

      i = n2;
      do
	{
	  --i;
	  w0 = b[i];
	  w1 = b[n2 + i];
	}
      while (w0 == w1 && i != 0);
      if (w0 < w1)
	{
	  x = b + n2;
	  y = b;
	  sign = ~sign;
	}
      else
	{
	  x = b;
	  y = b + n2;
	}
      mpn_sub_n (p + n2, x, y, n2);

      /* Pointwise products. */
      if (n2 < MUL_KARATSUBA_THRESHOLD)
	{
	  mpn_mul_basecase (ws, p, n2, p + n2, n2);
	  mpn_mul_basecase (p, a, n2, b, n2);
	  mpn_mul_basecase (p + n, a + n2, n2, b + n2, n2);
	}
      else
	{
	  mpn_kara_mul_n (ws, p, p + n2, n2, ws + n);
	  mpn_kara_mul_n (p, a, b, n2, ws + n);
	  mpn_kara_mul_n (p + n, a + n2, b + n2, n2, ws + n);
	}

      /* Interpolate. */
          
      if (sign)
	{
	 #if HAVE_NATIVE_mpn_addadd_n
	 w=mpn_addadd_n(ws,p+n,p,ws,n);
	 #else 
	 w = mpn_add_n (ws, p, ws, n);
         w += mpn_add_n (ws, p + n, ws, n);
         #endif
        }
      else
	{
	 #if HAVE_NATIVE_mpn_addsub_n
	 w=mpn_addsub_n(ws,p+n,p,ws,n);
	 #else
	 w = -mpn_sub_n (ws, p, ws, n);
         w += mpn_add_n (ws, p + n, ws, n);
         #endif         
        }
     
      w += mpn_add_n (p + n2, p + n2, ws, n);
      MPN_INCR_U (p + n2 + n, 2 * n - (n2 + n), w);
    }
}

void
mpn_kara_sqr_n (mp_ptr p, mp_srcptr a, mp_size_t n, mp_ptr ws)
{
  mp_limb_t w, w0, w1, c;
  int ci;
  mp_size_t n2;
  mp_srcptr x, y;
  mp_size_t i;

  n2 = n >> 1;
  ASSERT (n2 > 0);

  if ((n & 1) != 0)
    {
      /* Odd length. */
      mp_size_t n1, n3, nm1;

      n3 = n - n2;

      w = a[n2];
      if (w != 0)
	w -= mpn_sub_n (p, a, a + n3, n2);
      else
	{
	  i = n2;
	  do
	    {
	      --i;
	      w0 = a[i];
	      w1 = a[n3 + i];
	    }
	  while (w0 == w1 && i != 0);
	  if (w0 < w1)
	    {
	      x = a + n3;
	      y = a;
	    }
	  else
	    {
	      x = a;
	      y = a + n3;
	    }
	  mpn_sub_n (p, x, y, n2);
	}
      p[n2] = w;

      n1 = n + 1;

      /* n2 is always either n3 or n3-1 so maybe the two sets of tests here
	 could be combined.  But that's not important, since the tests will
	 take a miniscule amount of time compared to the function calls.  */
      if (BELOW_THRESHOLD (n3, SQR_BASECASE_THRESHOLD))
	{
	  mpn_mul_basecase (ws, p, n3, p, n3);
	  mpn_mul_basecase (p,  a, n3, a, n3);
	}
      else if (BELOW_THRESHOLD (n3, SQR_KARATSUBA_THRESHOLD))
	{
	  mpn_sqr_basecase (ws, p, n3);
	  mpn_sqr_basecase (p,  a, n3);
	}
      else
	{
	  mpn_kara_sqr_n   (ws, p, n3, ws + n1);	 /* (x-y)^2 */
	  mpn_kara_sqr_n   (p,  a, n3, ws + n1);	 /* x^2	    */
	}
      if (BELOW_THRESHOLD (n2, SQR_BASECASE_THRESHOLD))
	mpn_mul_basecase (p + n1, a + n3, n2, a + n3, n2);
      else if (BELOW_THRESHOLD (n2, SQR_KARATSUBA_THRESHOLD))
	mpn_sqr_basecase (p + n1, a + n3, n2);
      else
	mpn_kara_sqr_n   (p + n1, a + n3, n2, ws + n1);	 /* y^2	    */

      /* Since x^2+y^2-(x-y)^2 = 2xy >= 0 there's no need to track the
	 borrow from mpn_sub_n.	 If it occurs then it'll be cancelled by a
	 carry from ws[n].  Further, since 2xy fits in n1 limbs there won't
	 be any carry out of ws[n] other than cancelling that borrow. */

      nm1=n-1;
      #if HAVE_NATIVE_mpn_addsub_n
        ci=mpn_addsub_n(ws,p+n1,p,ws,nm1);
        #if GMP_NAIL_BITS==0
          sub_ddmmss(ws[n],ws[nm1],p[n],p[nm1],ws[n],ws[nm1]);//mpn_sub_n(ws+nm1,p+nm1,ws+nm1,2);
          if(ci==-1)sub_ddmmss(ws[n],ws[nm1],ws[n],ws[nm1],0,1);//mpn_sub_1(ws+nm1,ws+nm1,2,1);
          if(ci==1)add_ssaaaa(ws[n],ws[nm1],ws[n],ws[nm1],0,1);//mpn_add_1(ws+nm1,ws+nm1,2,1);
        #else
          mpn_sub_n(ws+nm1,p+nm1,ws+nm1,2);
          if(ci==-1)mpn_sub_1(ws+nm1,ws+nm1,2,1);
          if(ci==1)mpn_add_1(ws+nm1,ws+nm1,2,1);        
        #endif
      #else
        mpn_sub_n (ws, p, ws, n1);/*   x^2-(x-y)^2   */
        c=mpn_add_n (ws, p + n1, ws, nm1);/*  x^2+y^2-(x-y)^2 = 2xy    */
        if(c){mp_limb_t x = (ws[nm1] + 1) & GMP_NUMB_MASK;ws[nm1] = x; if (x == 0) ws[n] = (ws[n] + 1) & GMP_NUMB_MASK;}
      #endif

      if (mpn_add_n (p + n3, p + n3, ws, n1))
	{
	  mpn_incr_u (p + n1 + n3, 1);
	}
    }
  else
    {
      /* Even length. */
      i = n2;
      do
	{
	  --i;
	  w0 = a[i];
	  w1 = a[n2 + i];
	}
      while (w0 == w1 && i != 0);
      if (w0 < w1)
	{
	  x = a + n2;
	  y = a;
	}
      else
	{
	  x = a;
	  y = a + n2;
	}
      mpn_sub_n (p, x, y, n2);

      /* Pointwise products. */
      if (BELOW_THRESHOLD (n2, SQR_BASECASE_THRESHOLD))
	{
	  mpn_mul_basecase (ws,    p,      n2, p,      n2);
	  mpn_mul_basecase (p,     a,      n2, a,      n2);
	  mpn_mul_basecase (p + n, a + n2, n2, a + n2, n2);
	}
      else if (BELOW_THRESHOLD (n2, SQR_KARATSUBA_THRESHOLD))
	{
	  mpn_sqr_basecase (ws,    p,      n2);
	  mpn_sqr_basecase (p,     a,      n2);
	  mpn_sqr_basecase (p + n, a + n2, n2);
	}
      else
	{
	  mpn_kara_sqr_n (ws,    p,      n2, ws + n);
	  mpn_kara_sqr_n (p,     a,      n2, ws + n);
	  mpn_kara_sqr_n (p + n, a + n2, n2, ws + n);
	}

      /* Interpolate. */
      #if HAVE_NATIVE_mpn_addsub_n
      w=mpn_addsub_n(ws,p+n,p,ws,n);      
      #else      
      w = -mpn_sub_n (ws, p, ws, n);
      w += mpn_add_n (ws, p + n, ws, n);
      #endif
      w += mpn_add_n (p + n2, p + n2, ws, n);
      MPN_INCR_U (p + n2 + n, 2 * n - (n2 + n), w);
    }
}

/******************************************************************************
 *                                                                            *
 *              Toom 3-way multiplication and squaring                        *
 *                                                                            *
 *****************************************************************************/

/* put in {c, 2n} where n = 2k+r the value of {v0,2k} (already in place)
   + B^k * [{v1, 2k+1} - {t1, 2k+1}]
   + B^(2k) * [{t2, 2k+1} - {v0+vinf, 2k}]
   + B^(3k) * [{t1, 2k+1} - {t2, 2k+1}]
   + B^(4k) * {vinf,2r} (high 2r-1 limbs already in place)
   where {t1, 2k+1} = (3*{v0,2k}+2*sa*{vm1,2k+1}+{v2,2k+1})/6-2*{vinf,2r}
	 {t2, 2k+1} = ({v1, 2k+1} + sa * {vm1, 2k+1})/2
   (sa is the sign of {vm1, 2k+1}).

   {vinf, 2r} stores the content of {v0, 2r} + {vinf, 2r}, with carry in cinf0.
   vinf0 is the low limb of vinf.

   ws is temporary space, and should have at least 2r limbs.

   Think about:

   The evaluated point a-b+c stands a good chance of having a zero carry
   limb, a+b+c would have a 1/4 chance, and 4*a+2*b+c a 1/8 chance, roughly.
   Perhaps this could be tested and stripped.  Doing so before recursing
   would be better than stripping at the start of mpn_toom3_mul_n/sqr_n,
   since then the recursion could be based on the new size.  Although in
   truth the kara vs toom3 crossover is never so exact that one limb either
   way makes a difference.

   A small value like 1 or 2 for the carry could perhaps also be handled
   with an add_n or addlsh1_n.  Would that be faster than an extra limb on a
   (recursed) multiply/square?
*/
static void
toom3_interpolate (mp_ptr c, mp_srcptr v1, mp_ptr v2, mp_ptr vm1,
		   mp_ptr vinf, mp_size_t k, mp_size_t r, int sa,
		   mp_limb_t vinf0, mp_limb_t cinf0, mp_ptr ws)
{
  mp_limb_t cy, saved;
  unsigned long twok = k + k;
  unsigned long kk1 = twok + 1;
  unsigned long twor = r + r;
  mp_ptr c1, c2, c3, c4, c5;
  mp_limb_t cout; /* final carry, should be zero at the end */

  c1 = c + k;
  c2 = c1 + k;
  c3 = c2 + k;
  c4 = c3 + k;
  c5 = c4 + k;

#define v0 (c)
  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
       v0      |vm1|       hi(vinf)       v1       v2+2vm1      vinf
							      +lo(v0) */

  ASSERT_NOCARRY (mpn_divexact_by3 (v2, v2, kk1));    /* v2 <- v2 / 3 */
#ifdef HAVE_NATIVE_mpn_rsh1add_n
  mpn_rsh1add_n (v2, v2, v0, twok); /* v2 <- (lo(v2)+v0) / 2, exact */
  cy = v2[twok] & 1; /* add high limb of v2 divided by 2 */
  v2[twok] >>= 1;
  MPN_INCR_U (v2 + twok - 1, 2, cy << (GMP_NUMB_BITS - 1));
#else
  v2[twok] += mpn_add_n (v2, v2, v0, twok);
  mpn_rshift1 (v2, v2, kk1);
#endif

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
       v0      |vm1|      hi(vinf)       v1    (3v0+2vm1+v2)    vinf
						    /6         +lo(v0) */

  /* vm1 <- t2 := (v1 + sa*vm1) / 2
     t2 = a0*b0+a0*b2+a1*b1+a2*b0+a2*b2 >= 0
     No carry comes out from {v1, kk1} +/- {vm1, kk1},
     and the division by two is exact */
  if (sa >= 0)
    {
#ifdef HAVE_NATIVE_mpn_rsh1add_n
      mpn_rsh1add_n (vm1, v1, vm1, kk1);
#else
      mpn_add_n (vm1, vm1, v1, kk1);
      mpn_rshift1 (vm1, vm1, kk1);
#endif
    }
  else
    {
#ifdef HAVE_NATIVE_mpn_rsh1sub_n
      mpn_rsh1sub_n (vm1, v1, vm1, kk1);
#else
      mpn_sub_n (vm1, v1, vm1, kk1);
      mpn_rshift1 (vm1, vm1, kk1);
#endif
    }

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
       v0       t2        hi(vinf)       v1         t1       vinf+lo(v0) */

  /* subtract 2*vinf to v2,
     result is t1 := a0*b0+a0*b2+a1*b1+a1*b2+a2*b0+a2*b1+a2*b2 >= 0 */
  saved = c4[0];
  c4[0] = vinf0;
#ifdef HAVE_NATIVE_mpn_sublsh1_n
  cy = mpn_sublsh1_n (v2, v2, c4, twor);
#else
  cy = mpn_lshift1 (ws, c4, twor);
  cy += mpn_sub_n (v2, v2, ws, twor);
#endif
  MPN_DECR_U (v2 + twor, kk1 - twor, cy);
  c4[0] = saved;

  /* subtract {t2, 2k+1} in {c+3k, 2k+1} i.e. in {t2+k, 2k+1}:
     by chunks of k limbs from right to left to avoid overlap */
#define t2 (vm1)
  /* a borrow may occur in one of the 2 following __GMPN_SUB_1 calls, but since
     the final result is nonnegative, it will be compensated later on */
  __GMPN_SUB_1 (cout, c5, c5, twor - k, t2[twok]);
  cy = mpn_sub_n (c4, c4, t2 + k, k);
  __GMPN_SUB_1 (cout, c5, c5, twor - k, cy);
  cy = mpn_sub_n (c3, c3, t2, k);
  __GMPN_SUB_1 (cout, c4, c4, twor, cy);

  /* don't forget to add vinf0 in {c+4k, ...} */
  __GMPN_ADD_1 (cout, c4, c4, twor, vinf0);

  /* c  c+k c+2k c+3k c+4k+1   t  t+2k+1 t+4k+2
     v0     t2        hi(vinf) v1 t1     vinf
		 -t2                    +lo(v0)
  */

  /* c  c+k c+2k c+3k c+4k     t  t+2k+1 t+4k+2
     v0     t2        vinf     v1 t1     vinf
		 -t2                    +lo(v0)
  */

  /* subtract v0+vinf in {c+2k, ...} */
  cy = cinf0 + mpn_sub_n (c2, c2, vinf, twor);
  if (twor < twok)
    {
      __GMPN_SUB_1 (cy, c2 + twor, c2 + twor, twok - twor, cy);
      cy += mpn_sub_n (c2 + twor, c2 + twor, v0 + twor, twok - twor);
    }
  __GMPN_SUB_1 (cout, c4, c4, twor, cy); /* 2n-4k = 2r */

  /* c   c+k  c+2k  c+3k  c+4k      t   t+2k+1  t+4k+2
     v0       t2          vinf      v1  t1      vinf
	      -v0   -t2                        +lo(v0)
	      -vinf                                    */

  /* subtract t1 in {c+k, ...} */
  cy = mpn_sub_n (c1, c1, v2, kk1);
  __GMPN_SUB_1 (cout, c3 + 1, c3 + 1, twor + k - 1, cy); /* 2n-(3k+1)=k+2r-1 */

  /* c   c+k  c+2k  c+3k  c+4k      t   t+2k+1  t+4k+2
     v0       t2          vinf      v1  t1      vinf
	 -t1  -v0   -t2
	      -vinf                                    */

  /* add t1 in {c+3k, ...} */
  cy = mpn_add_n (c3, c3, v2, kk1);
  __GMPN_ADD_1 (cout, c5 + 1, c5 + 1, twor - k - 1, cy); /* 2n-(5k+1) = 2r-k-1 */

  /* c   c+k  c+2k  c+3k  c+4k      t   t+2k+1  t+4k+2
     v0       t2    t1    vinf      v1  t1      vinf
	 -t1  -v0   -t2
	      -vinf                                    */

  /* add v1 in {c+k, ...} */
  cy = mpn_add_n (c1, c1, v1, kk1);
  __GMPN_ADD_1 (cout, c3 + 1, c3 + 1, twor + k - 1, cy); /* 2n-(3k+1) = 2r+k-1 */

  /* c   c+k  c+2k  c+3k  c+4k      t   t+2k+1  t+4k+2
     v0  v1   t2    t1    vinf      v1  t1      vinf
	 -t1  -v0   -t2
	      -vinf                                    */
#undef v0
#undef t2
}

#define TOOM3_MUL_REC(p, a, b, n, ws) \
  do {								\
    if (MUL_TOOM3_THRESHOLD / 3 < MUL_KARATSUBA_THRESHOLD	\
	&& BELOW_THRESHOLD (n, MUL_KARATSUBA_THRESHOLD))	\
      mpn_mul_basecase (p, a, n, b, n);				\
    else if (BELOW_THRESHOLD (n, MUL_TOOM3_THRESHOLD))		\
      mpn_kara_mul_n (p, a, b, n, ws);				\
    else							\
      mpn_toom3_mul_n (p, a, b, n, ws);				\
  } while (0)

#define TOOM3_SQR_REC(p, a, n, ws)				\
  do {								\
    if (SQR_TOOM3_THRESHOLD / 3 < SQR_BASECASE_THRESHOLD	\
	&& BELOW_THRESHOLD (n, SQR_BASECASE_THRESHOLD))		\
      mpn_mul_basecase (p, a, n, a, n);				\
    else if (SQR_TOOM3_THRESHOLD / 3 < SQR_KARATSUBA_THRESHOLD	\
	&& BELOW_THRESHOLD (n, SQR_KARATSUBA_THRESHOLD))	\
      mpn_sqr_basecase (p, a, n);				\
    else if (BELOW_THRESHOLD (n, SQR_TOOM3_THRESHOLD))		\
      mpn_kara_sqr_n (p, a, n, ws);				\
    else							\
      mpn_toom3_sqr_n (p, a, n, ws);				\
  } while (0)

/* The necessary temporary space T(n) satisfies T(n)=0 for n < THRESHOLD,
   and T(n) <= max(2n+2, 6k+3, 4k+3+T(k+1)) otherwise, where k = ceil(n/3).

   Assuming T(n) >= 2n, 6k+3 <= 4k+3+T(k+1).
   Similarly, 2n+2 <= 6k+2 <= 4k+3+T(k+1).

   With T(n) = 2n+S(n), this simplifies to S(n) <= 9 + S(k+1).
   Since THRESHOLD >= 17, we have n/(k+1) >= 19/8
   thus S(n) <= S(n/(19/8)) + 9 thus S(n) <= 9*log(n)/log(19/8) <= 8*log2(n).

   We need in addition 2*r for mpn_sublsh1_n, so the total is at most
   8/3*n+8*log2(n).
*/

void
mpn_toom3_mul_n (mp_ptr c, mp_srcptr a, mp_srcptr b, mp_size_t n, mp_ptr t)
{
  mp_size_t k, k1, kk1, r, twok, twor;
  mp_limb_t cy, cc, saved, vinf0, cinf0;
  mp_ptr trec;
  int sa, sb;
  mp_ptr c1, c2, c3, c4, c5;

  ASSERT(GMP_NUMB_BITS >= 6);
  ASSERT(n >= 17); /* so that r <> 0 and 5k+3 <= 2n */

  /*
  The algorithm is the following:

  0. k = ceil(n/3), r = n - 2k, B = 2^(GMP_NUMB_BITS), t = B^k
  1. split a and b in three parts each a0, a1, a2 and b0, b1, b2
     with a0, a1, b0, b1 of k limbs, and a2, b2 of r limbs
  2. v0   <- a0*b0
     v1   <- (a0+a1+a2)*(b0+b1+b2)
     v2   <- (a0+2*a1+4*a2)*(b0+2*b1+4*b2)
     vm1  <- (a0-a1+a2)*(b0-b1+b2)
     vinf <- a2*b2
     t1   <- (3*v0+2*vm1+v2)/6-2*vinf
     t2   <- (v1+vm1)/2
  3. result is c0+c1*t+c2*t^2+c3*t^3+c4*t^4 where
     c0   <- v0
     c1   <- v1 - t1
     c2   <- t2 - v0 - vinf
     c3   <- t1 - t2
     c4   <- vinf
  */

  k = (n + 2) / 3; /* ceil(n/3) */
  twok = 2 * k;
  k1 = k + 1;
  kk1 = k + k1;
  r = n - twok;   /* last chunk */
  twor = 2 * r;

  c1 = c + k;
  c2 = c1 + k;
  c3 = c2 + k;
  c4 = c3 + k;
  c5 = c4 + k;

  trec = t + 4 * k + 3; /* trec = v2 + (2k+2) */

  /* put a0+a2 in {c, k+1}, and b0+b2 in {c+k+1, k+1};
     put a0+a1+a2 in {c+2k+2, k+1} and b0+b1+b2 in {c+3k+3,k+1}
     [requires 4k+4 <= 2n, ie. n >= 8] */
  cy = mpn_add_n (c, a, a + twok, r);
  cc = mpn_add_n (c1 + 1, b, b + twok, r);
  if (r < k)
    {
      __GMPN_ADD_1 (cy, c + r, a + r, k - r, cy);
      __GMPN_ADD_1 (cc, c1 + 1 + r, b + r, k - r, cc);
    }
  c3[2] = (c1[0] = cy) + mpn_add_n (c2 + 2, c, a + k, k);
  c4[3] = (c2[1] = cc) + mpn_add_n (c3 + 3, c1 + 1, b + k, k);

#define v2 (t+2*k+1)
#define vinf (t+4*k+2)

  /* compute v1 := (a0+a1+a2)*(b0+b1+b2) in {t, 2k+1};
     since v1 < 9*B^(2k), v1 uses only 2k+1 words if GMP_NUMB_BITS >= 4 */
  TOOM3_MUL_REC (t, c2 + 2, c3 + 3, k1, trec);

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
					v1
  */

  /* put |a0-a1+a2| in {c, k+1} and |b0-b1+b2| in {c+4k+2,k+1} */
  /* sa = sign(a0-a1+a2) */
  sa = (c[k] != 0) ? 1 : mpn_cmp (c, a + k, k);
  c[k] = (sa >= 0) ? c[k] - mpn_sub_n (c, c, a + k, k)
		   : mpn_sub_n (c, a + k, c, k);
  /* b0+b2 is in {c+k+1, k+1} now */
  sb = (c2[1] != 0) ? 1 : mpn_cmp (c1 + 1, b + k, k);
  c5[2] = (sb >= 0) ? c2[1] - mpn_sub_n (c4 + 2, c1 + 1, b + k, k)
		    : mpn_sub_n (c4 + 2, b + k, c1 + 1, k);
  sa *= sb; /* sign of vm1 */

  /* compute vm1 := (a0-a1+a2)*(b0-b1+b2) in {c+2k, 2k+1};
     since |vm1| < 4*B^(2k), vm1 uses only 2k+1 limbs */
  TOOM3_MUL_REC (c2, c, c4 + 2, k1, trec);

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
		vm1                      v1
  */

  /* compute a0+2a1+4a2 in {c, k+1} and b0+2b1+4b2 in {c+4k+2, k+1}
     [requires 5k+3 <= 2n, i.e. n >= 17] */
#ifdef HAVE_NATIVE_mpn_addlsh1_n
  c1[0] = mpn_addlsh1_n (c, a + k, a + twok, r);
  c5[2] = mpn_addlsh1_n (c4 + 2, b + k, b + twok, r);
  if (r < k)
    {
      __GMPN_ADD_1 (c1[0], c + r, a + k + r, k - r, c1[0]);
      __GMPN_ADD_1 (c5[2], c4 + 2 + r, b + k + r, k - r, c5[2]);
    }
  c1[0] = 2 * c1[0] + mpn_addlsh1_n (c, a, c, k);
  c5[2] = 2 * c5[2] + mpn_addlsh1_n (c4 + 2, b, c4 + 2, k);
#else
  c[r] = mpn_lshift1 (c, a + twok, r);
  c4[r + 2] = mpn_lshift1 (c4 + 2, b + twok, r);
  if (r < k)
    {
      MPN_ZERO(c + r + 1, k - r);
      MPN_ZERO(c4 + r + 3, k - r);
    }
  c1[0] += mpn_add_n (c, c, a + k, k);
  c5[2] += mpn_add_n (c4 + 2, c4 + 2, b + k, k);
  mpn_lshift1 (c, c, k1);
  mpn_lshift1 (c4 + 2, c4 + 2, k1);
  c1[0] += mpn_add_n (c, c, a, k);
  c5[2] += mpn_add_n (c4 + 2, c4 + 2, b, k);
#endif

  /* compute v2 := (a0+2a1+4a2)*(b0+2b1+4b2) in {t+2k+1, 2k+1}
     v2 < 49*B^k so v2 uses at most 2k+1 limbs if GMP_NUMB_BITS >= 6 */
  TOOM3_MUL_REC (v2, c, c4 + 2, k1, trec);

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
		vm1                      v1         v2
  */

  /* compute v0 := a0*b0 in {c, 2k} */
  TOOM3_MUL_REC (c, a, b, k, trec);

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
       v0       vm1                      v1         v2
  */

  /* now compute (3v0+2vm1+v2)/6 = [v0 + (2vm1+v2)/3]/2
     v2 <- v2+2vm1 = 3*(a0*b0+2*a0*b2+2*a1*b1+2*a1*b2+2*a2*b0+2*a2*b1+6*a2*b2),
     thus 0 <= v2 < 51*B^(2k) < 2^6*B^(2k)
     Uses temporary space {t+4k+2,2k+1}, requires T(n) >= 6k+3.
  */
  if (sa >= 0)
    {
#ifdef HAVE_NATIVE_mpn_addlsh1_n
      mpn_addlsh1_n (v2, v2, c2, kk1);
#else
      /* we can use vinf=t+4k+2 as workspace since it is not full yet */
      mpn_lshift1 (vinf, c2, kk1);
      mpn_add_n (v2, v2, vinf, kk1);
#endif
    }
  else
    {
#ifdef HAVE_NATIVE_mpn_sublsh1_n
      mpn_sublsh1_n (v2, v2, c2, kk1);
#else
      /* we can use vinf=t+4k+2 as workspace since it is not full yet */
      mpn_lshift1 (vinf, c2, kk1);
      mpn_sub_n (v2, v2, vinf, kk1);
#endif
    }

  /* {c,2k} {c+2k,2k+1} {c+4k+1,2r-1} {t,2k+1} {t+2k+1,2k+1} {t+4k+2,2r}
       v0       vm1                      v1       v2+2vm1             */

  /* compute vinf := a2*b2 in {t+4k+2, 2r}: first put it in {c4, 2r},
     then copy it in {t+4k+2,2r} */
  saved = c4[0];
  TOOM3_MUL_REC (c4, a + twok, b + twok, r, trec);
  cinf0 = mpn_add_n (vinf, c4, c, twor); /* {v0,2r} + {vinf,2r} */
  vinf0 = c4[0];
  c4[0] = saved;

  toom3_interpolate (c, t, v2, c2, vinf, k, r, sa, vinf0, cinf0, vinf + twor);

#undef v2
#undef vinf
}

void
mpn_toom3_sqr_n (mp_ptr c, mp_srcptr a, mp_size_t n, mp_ptr t)
{
  mp_size_t k, k1, kk1, r, twok, twor;
  mp_limb_t cy, saved, vinf0, cinf0;
  mp_ptr trec;
  int sa;
  mp_ptr c1, c2, c3, c4;

  ASSERT(GMP_NUMB_BITS >= 6);
  ASSERT(n >= 17); /* so that r <> 0 and 5k+3 <= 2n */

  /* the algorithm is the same as mpn_mul_n_tc3, with b=a */

  k = (n + 2) / 3; /* ceil(n/3) */
  twok = 2 * k;
  k1 = k + 1;
  kk1 = k + k1;
  r = n - twok;   /* last chunk */
  twor = 2 * r;

  c1 = c + k;
  c2 = c1 + k;
  c3 = c2 + k;
  c4 = c3 + k;

  trec = t + 4 * k + 3; /* trec = v2 + (2k+2) */

  cy = mpn_add_n (c, a, a + twok, r);
  if (r < k)
    __GMPN_ADD_1 (cy, c + r, a + r, k - r, cy);
  c3[2] = (c1[0] = cy) + mpn_add_n (c2 + 2, c, a + k, k);

#define v2 (t+2*k+1)
#define vinf (t+4*k+2)

  TOOM3_SQR_REC (t, c2 + 2, k1, trec);

  sa = (c[k] != 0) ? 1 : mpn_cmp (c, a + k, k);
  c[k] = (sa >= 0) ? c[k] - mpn_sub_n (c, c, a + k, k)
    : mpn_sub_n (c, a + k, c, k);

  TOOM3_SQR_REC (c2, c, k1, trec);

#ifdef HAVE_NATIVE_mpn_addlsh1_n
  c1[0] = mpn_addlsh1_n (c, a + k, a + twok, r);
  if (r < k)
    __GMPN_ADD_1 (c1[0], c + r, a + k + r, k - r, c1[0]);
  c1[0] = 2 * c1[0] + mpn_addlsh1_n (c, a, c, k);
#else
  c[r] = mpn_lshift1 (c, a + twok, r);
  if (r < k)
    MPN_ZERO(c + r + 1, k - r);
  c1[0] += mpn_add_n (c, c, a + k, k);
  mpn_lshift1 (c, c, k1);
  c1[0] += mpn_add_n (c, c, a, k);
#endif

  TOOM3_SQR_REC (v2, c, k1, trec);

  TOOM3_SQR_REC (c, a, k, trec);

#ifdef HAVE_NATIVE_mpn_addlsh1_n
  mpn_addlsh1_n (v2, v2, c2, kk1);
#else
  mpn_lshift1 (t + 4 * k + 2, c2, kk1);
  mpn_add_n (v2, v2, t + 4 * k + 2, kk1);
#endif

  saved = c4[0];
  TOOM3_SQR_REC (c4, a + twok, r, trec);
  cinf0 = mpn_add_n (vinf, c4, c, twor);
  vinf0 = c4[0];
  c4[0] = saved;

  toom3_interpolate (c, t, v2, c2, vinf, k, r, 1, vinf0, cinf0, vinf + twor);

#undef v2
#undef vinf
}

void
mpn_mul_n (mp_ptr p, mp_srcptr a, mp_srcptr b, mp_size_t n)
{
  ASSERT (n >= 1);
  ASSERT (! MPN_OVERLAP_P (p, 2 * n, a, n));
  ASSERT (! MPN_OVERLAP_P (p, 2 * n, b, n));

  if (BELOW_THRESHOLD (n, MUL_KARATSUBA_THRESHOLD))
    {
      mpn_mul_basecase (p, a, n, b, n);
    }
  else if (BELOW_THRESHOLD (n, MUL_TOOM3_THRESHOLD))
    {
      /* Allocate workspace of fixed size on stack: fast! */
      mp_limb_t ws[MPN_KARA_MUL_N_TSIZE (MUL_TOOM3_THRESHOLD_LIMIT-1)];
      ASSERT (MUL_TOOM3_THRESHOLD <= MUL_TOOM3_THRESHOLD_LIMIT);
      mpn_kara_mul_n (p, a, b, n, ws);
    }
  else if (BELOW_THRESHOLD (n, MUL_TOOM4_THRESHOLD))
    {
      mp_ptr ws;
      TMP_SDECL;
      TMP_SMARK;
      ws = TMP_SALLOC_LIMBS (MPN_TOOM3_MUL_N_TSIZE (n));
      mpn_toom3_mul_n (p, a, b, n, ws);
      TMP_SFREE;
    }
  else if (BELOW_THRESHOLD (n, MUL_TOOM7_THRESHOLD))
    {
       mpn_toom4_mul_n (p, a, b, n);
    }
#if WANT_FFT || TUNE_PROGRAM_BUILD
  else if (BELOW_THRESHOLD (n, MUL_FFT_THRESHOLD))
    {
      mpn_toom7_mul_n (p, a, b, n);
    }
#endif
  else
#if WANT_FFT || TUNE_PROGRAM_BUILD
    {
      /* The current FFT code allocates its own space.  That should probably
	 change.  */
      mpn_mul_fft_full (p, a, n, b, n);
    }
#else
    {
      /* Toom7 for large operands. */
      mpn_toom7_mul_n (p, a, b, n);
    }
#endif
}

void
mpn_sqr_n (mp_ptr p, mp_srcptr a, mp_size_t n)
{
  ASSERT (n >= 1);
  ASSERT (! MPN_OVERLAP_P (p, 2 * n, a, n));

#if 0
  /* FIXME: Can this be removed? */
  if (n == 0)
    return;
#endif

  if (BELOW_THRESHOLD (n, SQR_BASECASE_THRESHOLD))
    { /* mul_basecase is faster than sqr_basecase on small sizes sometimes */
      mpn_mul_basecase (p, a, n, a, n);
    }
  else if (BELOW_THRESHOLD (n, SQR_KARATSUBA_THRESHOLD))
    {
      mpn_sqr_basecase (p, a, n);
    }
  else if (BELOW_THRESHOLD (n, SQR_TOOM3_THRESHOLD))
    {
      /* Allocate workspace of fixed size on stack: fast! */
      mp_limb_t ws[MPN_KARA_SQR_N_TSIZE (SQR_TOOM3_THRESHOLD_LIMIT-1)];
      ASSERT (SQR_TOOM3_THRESHOLD <= SQR_TOOM3_THRESHOLD_LIMIT);
      mpn_kara_sqr_n (p, a, n, ws);
    }
#if WANT_FFT || TUNE_PROGRAM_BUILD
  else if (BELOW_THRESHOLD (n, SQR_FFT_THRESHOLD))
#else
  else if (BELOW_THRESHOLD (n, MPN_TOOM3_MAX_N))
#endif
    {
      mp_ptr ws;
      TMP_SDECL;
      TMP_SMARK;
      ws = TMP_SALLOC_LIMBS (MPN_TOOM3_SQR_N_TSIZE (n));
      mpn_toom3_sqr_n (p, a, n, ws);
      TMP_SFREE;
    }
  else
#if WANT_FFT || TUNE_PROGRAM_BUILD
    {
      /* The current FFT code allocates its own space.  That should probably
	 change.  */
      mpn_mul_fft_full (p, a, n, a, n);
    }
#else
    {
      /* Toom3 for large operands.  Use workspace from the heap, as stack space
      may be limited.  Since n is at least MUL_TOOM3_THRESHOLD, multiplication
      will take much longer than malloc()/free().  */
      mp_ptr ws;  mp_size_t ws_size;
      ws_size = MPN_TOOM3_SQR_N_TSIZE (n);
      ws = __GMP_ALLOCATE_FUNC_LIMBS (ws_size);
      mpn_toom3_sqr_n (p, a, n, ws);
      __GMP_FREE_FUNC_LIMBS (ws, ws_size);
    }
#endif
}