/src/gmp/mpn/toom32_mul.c

Source
/* mpn_toom32_mul -- Multiply {ap,an} and {bp,bn} where an is nominally 1.5
   times as large as bn.  Or more accurately, bn < an < 3bn.

   Contributed to the GNU project by Torbjorn Granlund.
   Improvements by Marco Bodrato and Niels Möller.

   The idea of applying Toom to unbalanced multiplication is due to Marco
   Bodrato and Alberto Zanoni.

   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 2006-2010, 2020, 2021 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"

/* Evaluate in: -1, 0, +1, +inf

  <-s-><--n--><--n-->
   ___ ______ ______
  |a2_|___a1_|___a0_|
  |_b1_|___b0_|
  <-t--><--n-->

  v0  =  a0         * b0      #   A(0)*B(0)
  v1  = (a0+ a1+ a2)*(b0+ b1) #   A(1)*B(1)      ah  <= 2  bh <= 1
  vm1 = (a0- a1+ a2)*(b0- b1) #  A(-1)*B(-1)    |ah| <= 1  bh = 0
  vinf=          a2 *     b1  # A(inf)*B(inf)
*/

#define TOOM32_MUL_N_REC(p, a, b, n, ws)        \
  do {                 \
    mpn_mul_n (p, a, b, n);            \
  } while (0)

void
mpn_toom32_mul (mp_ptr pp,
    mp_srcptr ap, mp_size_t an,
    mp_srcptr bp, mp_size_t bn,
    mp_ptr scratch)
{
  mp_size_t n, s, t;
  int vm1_neg;
  mp_limb_t cy;
  mp_limb_signed_t hi;
  mp_limb_t ap1_hi, bp1_hi;

#define a0  ap
#define a1  (ap + n)
#define a2  (ap + 2 * n)
#define b0  bp
#define b1  (bp + n)

  /* Required, to ensure that s + t >= n. */
  ASSERT (bn + 2 <= an && an + 6 <= 3*bn);

  n = 2 * an >= 3 * bn ? (an + 2) / (size_t) 3 : (bn + 1) >> 1;

  s = an - 2 * n;
  t = bn - n;

  ASSERT (0 < s && s <= n);
  ASSERT (0 < t && t <= n);
  ASSERT (s + t >= n);

  /* Product area of size an + bn = 3*n + s + t >= 4*n + 2. */
#define ap1 (pp)    /* n, most significant limb in ap1_hi */
#define bp1 (pp + n)    /* n, most significant bit in bp1_hi */
#define am1 (pp + 2*n)    /* n, most significant bit in hi */
#define bm1 (pp + 3*n)    /* n */
#define v1 (scratch)    /* 2n + 1 */
#define vm1 (pp)    /* 2n + 1 */
#define scratch_out (scratch + 2*n + 1) /* Currently unused. */

  /* Scratch need: 2*n + 1 + scratch for the recursive multiplications. */

  /* FIXME: Keep v1[2*n] and vm1[2*n] in scalar variables? */

  /* Compute ap1 = a0 + a1 + a2, am1 = a0 - a1 + a2 */
  ap1_hi = mpn_add (ap1, a0, n, a2, s);
#if HAVE_NATIVE_mpn_add_n_sub_n
  if (ap1_hi == 0 && mpn_cmp (ap1, a1, n) < 0)
    {
      ap1_hi = mpn_add_n_sub_n (ap1, am1, a1, ap1, n) >> 1;
      hi = 0;
      vm1_neg = 1;
    }
  else
    {
      cy = mpn_add_n_sub_n (ap1, am1, ap1, a1, n);
      hi = ap1_hi - (cy & 1);
      ap1_hi += (cy >> 1);
      vm1_neg = 0;
    }
#else
  if (ap1_hi == 0 && mpn_cmp (ap1, a1, n) < 0)
    {
      ASSERT_NOCARRY (mpn_sub_n (am1, a1, ap1, n));
      hi = 0;
      vm1_neg = 1;
    }
  else
    {
      hi = ap1_hi - mpn_sub_n (am1, ap1, a1, n);
      vm1_neg = 0;
    }
  ap1_hi += mpn_add_n (ap1, ap1, a1, n);
#endif

  /* Compute bp1 = b0 + b1 and bm1 = b0 - b1. */
  if (t == n)
    {
#if HAVE_NATIVE_mpn_add_n_sub_n
      if (mpn_cmp (b0, b1, n) < 0)
  {
    cy = mpn_add_n_sub_n (bp1, bm1, b1, b0, n);
    vm1_neg ^= 1;
  }
      else
  {
    cy = mpn_add_n_sub_n (bp1, bm1, b0, b1, n);
  }
      bp1_hi = cy >> 1;
#else
      bp1_hi = mpn_add_n (bp1, b0, b1, n);

      if (mpn_cmp (b0, b1, n) < 0)
  {
    ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, n));
    vm1_neg ^= 1;
  }
      else
  {
    ASSERT_NOCARRY (mpn_sub_n (bm1, b0, b1, n));
  }
#endif
    }
  else
    {
      /* FIXME: Should still use mpn_add_n_sub_n for the main part. */
      bp1_hi = mpn_add (bp1, b0, n, b1, t);

      if (mpn_zero_p (b0 + t, n - t) && mpn_cmp (b0, b1, t) < 0)
  {
    ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, t));
    MPN_ZERO (bm1 + t, n - t);
    vm1_neg ^= 1;
  }
      else
  {
    ASSERT_NOCARRY (mpn_sub (bm1, b0, n, b1, t));
  }
    }

  TOOM32_MUL_N_REC (v1, ap1, bp1, n, scratch_out);
  if (ap1_hi == 1)
    {
      cy = mpn_add_n (v1 + n, v1 + n, bp1, n);
    }
  else if (ap1_hi > 1) /* ap1_hi == 2 */
    {
#if HAVE_NATIVE_mpn_addlsh1_n_ip1
      cy = mpn_addlsh1_n_ip1 (v1 + n, bp1, n);
#else
      cy = mpn_addmul_1 (v1 + n, bp1, n, CNST_LIMB(2));
#endif
    }
  else
    cy = 0;
  if (bp1_hi != 0)
    cy += ap1_hi + mpn_add_n (v1 + n, v1 + n, ap1, n);
  v1[2 * n] = cy;

  TOOM32_MUL_N_REC (vm1, am1, bm1, n, scratch_out);
  if (hi)
    hi = mpn_add_n (vm1+n, vm1+n, bm1, n);

  vm1[2*n] = hi;

  /* v1 <-- (v1 + vm1) / 2 = x0 + x2 */
  if (vm1_neg)
    {
#if HAVE_NATIVE_mpn_rsh1sub_n
      mpn_rsh1sub_n (v1, v1, vm1, 2*n+1);
#else
      mpn_sub_n (v1, v1, vm1, 2*n+1);
      ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
#endif
    }
  else
    {
#if HAVE_NATIVE_mpn_rsh1add_n
      mpn_rsh1add_n (v1, v1, vm1, 2*n+1);
#else
      mpn_add_n (v1, v1, vm1, 2*n+1);
      ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
#endif
    }

  /* We get x1 + x3 = (x0 + x2) - (x0 - x1 + x2 - x3), and hence

     y = x1 + x3 + (x0 + x2) * B
       = (x0 + x2) * B + (x0 + x2) - vm1.

     y is 3*n + 1 limbs, y = y0 + y1 B + y2 B^2. We store them as
     follows: y0 at scratch, y1 at pp + 2*n, and y2 at scratch + n
     (already in place, except for carry propagation).

     We thus add

   B^3  B^2   B    1
    |    |    |    |
   +-----+----+
 + |  x0 + x2 |
   +----+-----+----+
 +      |  x0 + x2 |
  +----------+
 -      |  vm1     |
 --+----++----+----+-
   | y2  | y1 | y0 |
   +-----+----+----+

  Since we store y0 at the same location as the low half of x0 + x2, we
  need to do the middle sum first. */

  hi = vm1[2*n];
  cy = mpn_add_n (pp + 2*n, v1, v1 + n, n);
  MPN_INCR_U (v1 + n, n + 1, cy + v1[2*n]);

  /* FIXME: Can we get rid of this second vm1_neg conditional by
     swapping the location of +1 and -1 values? */
  if (vm1_neg)
    {
      cy = mpn_add_n (v1, v1, vm1, n);
      hi += mpn_add_nc (pp + 2*n, pp + 2*n, vm1 + n, n, cy);
      MPN_INCR_U (v1 + n, n+1, hi);
    }
  else
    {
      cy = mpn_sub_n (v1, v1, vm1, n);
      hi += mpn_sub_nc (pp + 2*n, pp + 2*n, vm1 + n, n, cy);
      MPN_DECR_U (v1 + n, n+1, hi);
    }

  TOOM32_MUL_N_REC (pp, a0, b0, n, scratch_out);
  /* vinf, s+t limbs.  Use mpn_mul for now, to handle unbalanced operands */
  if (s > t)  mpn_mul (pp+3*n, a2, s, b1, t);
  else        mpn_mul (pp+3*n, b1, t, a2, s);

  /* Remaining interpolation.

     y * B + x0 + x3 B^3 - x0 B^2 - x3 B
     = (x1 + x3) B + (x0 + x2) B^2 + x0 + x3 B^3 - x0 B^2 - x3 B
     = y0 B + y1 B^2 + y3 B^3 + Lx0 + H x0 B
       + L x3 B^3 + H x3 B^4 - Lx0 B^2 - H x0 B^3 - L x3 B - H x3 B^2
     = L x0 + (y0 + H x0 - L x3) B + (y1 - L x0 - H x3) B^2
       + (y2 - (H x0 - L x3)) B^3 + H x3 B^4

    B^4       B^3       B^2        B         1
 |         |         |         |         |         |
   +-------+                   +---------+---------+
   |  Hx3  |                   | Hx0-Lx3 |    Lx0  |
   +------+----------+---------+---------+---------+
    |    y2    |  y1     |   y0    |
    ++---------+---------+---------+
    -| Hx0-Lx3 | - Lx0   |
     +---------+---------+
          | - Hx3  |
          +--------+

    We must take into account the carry from Hx0 - Lx3.
  */

  cy = mpn_sub_n (pp + n, pp + n, pp+3*n, n);
  hi = scratch[2*n] + cy;

  cy = mpn_sub_nc (pp + 2*n, pp + 2*n, pp, n, cy);
  hi -= mpn_sub_nc (pp + 3*n, scratch + n, pp + n, n, cy);

  hi += mpn_add (pp + n, pp + n, 3*n, scratch, n);

  /* FIXME: Is support for s + t == n needed? */
  if (LIKELY (s + t > n))
    {
      hi -= mpn_sub (pp + 2*n, pp + 2*n, 2*n, pp + 4*n, s+t-n);

      ASSERT (hi >= 0); /* contribution of the middle terms >= 0 */
      MPN_INCR_U (pp + 4*n, s+t-n, hi);
    }
  else
    ASSERT (hi == 0);
}

Coverage Report

Created: 2026-11-12 07:00

Line	Count	Source
1		/* mpn_toom32_mul -- Multiply {ap,an} and {bp,bn} where an is nominally 1.5
2		times as large as bn. Or more accurately, bn < an < 3bn.
3
4		Contributed to the GNU project by Torbjorn Granlund.
5		Improvements by Marco Bodrato and Niels Möller.
6
7		The idea of applying Toom to unbalanced multiplication is due to Marco
8		Bodrato and Alberto Zanoni.
9
10		THE FUNCTION IN THIS FILE IS INTERNAL WITH A MUTABLE INTERFACE. IT IS ONLY
11		SAFE TO REACH IT THROUGH DOCUMENTED INTERFACES. IN FACT, IT IS ALMOST
12		GUARANTEED THAT IT WILL CHANGE OR DISAPPEAR IN A FUTURE GNU MP RELEASE.
13
14		Copyright 2006-2010, 2020, 2021 Free Software Foundation, Inc.
15
16		This file is part of the GNU MP Library.
17
18		The GNU MP Library is free software; you can redistribute it and/or modify
19		it under the terms of either:
20
21		* the GNU Lesser General Public License as published by the Free
22		Software Foundation; either version 3 of the License, or (at your
23		option) any later version.
24
25		or
26
27		* the GNU General Public License as published by the Free Software
28		Foundation; either version 2 of the License, or (at your option) any
29		later version.
30
31		or both in parallel, as here.
32
33		The GNU MP Library is distributed in the hope that it will be useful, but
34		WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
35		or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
36		for more details.
37
38		You should have received copies of the GNU General Public License and the
39		GNU Lesser General Public License along with the GNU MP Library. If not,
40		see https://www.gnu.org/licenses/. */
41
42
43		#include "gmp-impl.h"
44
45		/* Evaluate in: -1, 0, +1, +inf
46
47		<-s-><--n--><--n-->
48		___ ______ ______
49		\|a2_\|___a1_\|___a0_\|
50		\|_b1_\|___b0_\|
51		<-t--><--n-->
52
53		v0 = a0 * b0 # A(0)*B(0)
54		v1 = (a0+ a1+ a2)(b0+ b1) # A(1)B(1) ah <= 2 bh <= 1
55		vm1 = (a0- a1+ a2)(b0- b1) # A(-1)B(-1) \|ah\| <= 1 bh = 0
56		vinf= a2 * b1 # A(inf)*B(inf)
57		*/
58
59		#define TOOM32_MUL_N_REC(p, a, b, n, ws) \
60	15.5k	do { \
61	15.5k	mpn_mul_n (p, a, b, n); \
62	15.5k	} while (0)
63
64		void
65		mpn_toom32_mul (mp_ptr pp,
66		mp_srcptr ap, mp_size_t an,
67		mp_srcptr bp, mp_size_t bn,
68		mp_ptr scratch)
69	5.18k	{
70	5.18k	mp_size_t n, s, t;
71	5.18k	int vm1_neg;
72	5.18k	mp_limb_t cy;
73	5.18k	mp_limb_signed_t hi;
74	5.18k	mp_limb_t ap1_hi, bp1_hi;
75
76	5.18k	#define a0 ap
77	13.0k	#define a1 (ap + n)
78	10.3k	#define a2 (ap + 2 * n)
79	11.0k	#define b0 bp
80	13.1k	#define b1 (bp + n)
81
82		/* Required, to ensure that s + t >= n. */
83	5.18k	ASSERT (bn + 2 <= an && an + 6 <= 3*bn);
84
85	5.18k	n = 2 * an >= 3 * bn ? (an + 2) / (size_t) 3 : (bn + 1) >> 1;
86
87	5.18k	s = an - 2 * n;
88	5.18k	t = bn - n;
89
90	5.18k	ASSERT (0 < s && s <= n);
91	5.18k	ASSERT (0 < t && t <= n);
92	5.18k	ASSERT (s + t >= n);
93
94		/* Product area of size an + bn = 3n + s + t >= 4n + 2. */
95	23.7k	#define ap1 (pp) /* n, most significant limb in ap1_hi */
96	7.35k	#define bp1 (pp + n) /* n, most significant bit in bp1_hi */
97	5.18k	#define am1 (pp + 2n) / n, most significant bit in hi */
98	5.18k	#define bm1 (pp + 3n) / n */
99	41.0k	#define v1 (scratch) /* 2n + 1 */
100	26.1k	#define vm1 (pp) /* 2n + 1 */
101	5.18k	#define scratch_out (scratch + 2n + 1) / Currently unused. */
102
103		/* Scratch need: 2n + 1 + scratch for the recursive multiplications. /
104
105		/* FIXME: Keep v1[2n] and vm1[2n] in scalar variables? */
106
107		/* Compute ap1 = a0 + a1 + a2, am1 = a0 - a1 + a2 */
108	5.18k	ap1_hi = mpn_add (ap1, a0, n, a2, s);
109		#if HAVE_NATIVE_mpn_add_n_sub_n
110		if (ap1_hi == 0 && mpn_cmp (ap1, a1, n) < 0)
111		{
112		ap1_hi = mpn_add_n_sub_n (ap1, am1, a1, ap1, n) >> 1;
113		hi = 0;
114		vm1_neg = 1;
115		}
116		else
117		{
118		cy = mpn_add_n_sub_n (ap1, am1, ap1, a1, n);
119		hi = ap1_hi - (cy & 1);
120		ap1_hi += (cy >> 1);
121		vm1_neg = 0;
122		}
123		#else
124	5.18k	if (ap1_hi == 0 && mpn_cmp (ap1, a1, n) < 0)
125	2.14k	{
126	2.14k	ASSERT_NOCARRY (mpn_sub_n (am1, a1, ap1, n));
127	2.14k	hi = 0;
128	2.14k	vm1_neg = 1;
129	2.14k	}
130	3.04k	else
131	3.04k	{
132	3.04k	hi = ap1_hi - mpn_sub_n (am1, ap1, a1, n);
133	3.04k	vm1_neg = 0;
134	3.04k	}
135	5.18k	ap1_hi += mpn_add_n (ap1, ap1, a1, n);
136	5.18k	#endif
137
138		/* Compute bp1 = b0 + b1 and bm1 = b0 - b1. */
139	5.18k	if (t == n)
140	2.09k	{
141		#if HAVE_NATIVE_mpn_add_n_sub_n
142		if (mpn_cmp (b0, b1, n) < 0)
143		{
144		cy = mpn_add_n_sub_n (bp1, bm1, b1, b0, n);
145		vm1_neg ^= 1;
146		}
147		else
148		{
149		cy = mpn_add_n_sub_n (bp1, bm1, b0, b1, n);
150		}
151		bp1_hi = cy >> 1;
152		#else
153	2.09k	bp1_hi = mpn_add_n (bp1, b0, b1, n);
154
155	2.09k	if (mpn_cmp (b0, b1, n) < 0)
156	259	{
157	259	ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, n));
158	259	vm1_neg ^= 1;
159	259	}
160	1.83k	else
161	1.83k	{
162	1.83k	ASSERT_NOCARRY (mpn_sub_n (bm1, b0, b1, n));
163	1.83k	}
164	2.09k	#endif
165	2.09k	}
166	3.08k	else
167	3.08k	{
168		/* FIXME: Should still use mpn_add_n_sub_n for the main part. */
169	3.08k	bp1_hi = mpn_add (bp1, b0, n, b1, t);
170
171	3.08k	if (mpn_zero_p (b0 + t, n - t) && mpn_cmp (b0, b1, t) < 0)
172	385	{
173	385	ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, t));
174	385	MPN_ZERO (bm1 + t, n - t);
175	385	vm1_neg ^= 1;
176	385	}
177	2.70k	else
178	2.70k	{
179	2.70k	ASSERT_NOCARRY (mpn_sub (bm1, b0, n, b1, t));
180	2.70k	}
181	3.08k	}
182
183	5.18k	TOOM32_MUL_N_REC (v1, ap1, bp1, n, scratch_out);
184	5.18k	if (ap1_hi == 1)
185	1.95k	{
186	1.95k	cy = mpn_add_n (v1 + n, v1 + n, bp1, n);
187	1.95k	}
188	3.23k	else if (ap1_hi > 1) /* ap1_hi == 2 */
189	218	{
190		#if HAVE_NATIVE_mpn_addlsh1_n_ip1
191		cy = mpn_addlsh1_n_ip1 (v1 + n, bp1, n);
192		#else
193	218	cy = mpn_addmul_1 (v1 + n, bp1, n, CNST_LIMB(2));
194	218	#endif
195	218	}
196	3.01k	else
197	3.01k	cy = 0;
198	5.18k	if (bp1_hi != 0)
199	311	cy += ap1_hi + mpn_add_n (v1 + n, v1 + n, ap1, n);
200	5.18k	v1[2 * n] = cy;
201
202	5.18k	TOOM32_MUL_N_REC (vm1, am1, bm1, n, scratch_out);
203	5.18k	if (hi)
204	119	hi = mpn_add_n (vm1+n, vm1+n, bm1, n);
205
206	5.18k	vm1[2*n] = hi;
207
208		/* v1 <-- (v1 + vm1) / 2 = x0 + x2 */
209	5.18k	if (vm1_neg)
210	2.31k	{
211		#if HAVE_NATIVE_mpn_rsh1sub_n
212		mpn_rsh1sub_n (v1, v1, vm1, 2*n+1);
213		#else
214	2.31k	mpn_sub_n (v1, v1, vm1, 2*n+1);
215	2.31k	ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
216	2.31k	#endif
217	2.31k	}
218	2.86k	else
219	2.86k	{
220		#if HAVE_NATIVE_mpn_rsh1add_n
221		mpn_rsh1add_n (v1, v1, vm1, 2*n+1);
222		#else
223	2.86k	mpn_add_n (v1, v1, vm1, 2*n+1);
224	2.86k	ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
225	2.86k	#endif
226	2.86k	}
227
228		/* We get x1 + x3 = (x0 + x2) - (x0 - x1 + x2 - x3), and hence
229
230		y = x1 + x3 + (x0 + x2) * B
231		= (x0 + x2) * B + (x0 + x2) - vm1.
232
233		y is 3*n + 1 limbs, y = y0 + y1 B + y2 B^2. We store them as
234		follows: y0 at scratch, y1 at pp + 2*n, and y2 at scratch + n
235		(already in place, except for carry propagation).
236
237		We thus add
238
239		B^3 B^2 B 1
240		\| \| \| \|
241		+-----+----+
242		+ \| x0 + x2 \|
243		+----+-----+----+
244		+ \| x0 + x2 \|
245		+----------+
246		- \| vm1 \|
247		--+----++----+----+-
248		\| y2 \| y1 \| y0 \|
249		+-----+----+----+
250
251		Since we store y0 at the same location as the low half of x0 + x2, we
252		need to do the middle sum first. */
253
254	5.18k	hi = vm1[2*n];
255	5.18k	cy = mpn_add_n (pp + 2*n, v1, v1 + n, n);
256	5.18k	MPN_INCR_U (v1 + n, n + 1, cy + v1[2*n]);
257
258		/* FIXME: Can we get rid of this second vm1_neg conditional by
259		swapping the location of +1 and -1 values? */
260	5.18k	if (vm1_neg)
261	2.31k	{
262	2.31k	cy = mpn_add_n (v1, v1, vm1, n);
263	2.31k	hi += mpn_add_nc (pp + 2n, pp + 2n, vm1 + n, n, cy);
264	2.31k	MPN_INCR_U (v1 + n, n+1, hi);
265	2.31k	}
266	2.86k	else
267	2.86k	{
268	2.86k	cy = mpn_sub_n (v1, v1, vm1, n);
269	2.86k	hi += mpn_sub_nc (pp + 2n, pp + 2n, vm1 + n, n, cy);
270	2.86k	MPN_DECR_U (v1 + n, n+1, hi);
271	2.86k	}
272
273	5.18k	TOOM32_MUL_N_REC (pp, a0, b0, n, scratch_out);
274		/* vinf, s+t limbs. Use mpn_mul for now, to handle unbalanced operands */
275	5.18k	if (s > t) mpn_mul (pp+3*n, a2, s, b1, t);
276	3.98k	else mpn_mul (pp+3*n, b1, t, a2, s);
277
278		/* Remaining interpolation.
279
280		y * B + x0 + x3 B^3 - x0 B^2 - x3 B
281		= (x1 + x3) B + (x0 + x2) B^2 + x0 + x3 B^3 - x0 B^2 - x3 B
282		= y0 B + y1 B^2 + y3 B^3 + Lx0 + H x0 B
283		+ L x3 B^3 + H x3 B^4 - Lx0 B^2 - H x0 B^3 - L x3 B - H x3 B^2
284		= L x0 + (y0 + H x0 - L x3) B + (y1 - L x0 - H x3) B^2
285		+ (y2 - (H x0 - L x3)) B^3 + H x3 B^4
286
287		B^4 B^3 B^2 B 1
288		\| \| \| \| \| \|
289		+-------+ +---------+---------+
290		\| Hx3 \| \| Hx0-Lx3 \| Lx0 \|
291		+------+----------+---------+---------+---------+
292		\| y2 \| y1 \| y0 \|
293		++---------+---------+---------+
294		-\| Hx0-Lx3 \| - Lx0 \|
295		+---------+---------+
296		\| - Hx3 \|
297		+--------+
298
299		We must take into account the carry from Hx0 - Lx3.
300		*/
301
302	5.18k	cy = mpn_sub_n (pp + n, pp + n, pp+3*n, n);
303	5.18k	hi = scratch[2*n] + cy;
304
305	5.18k	cy = mpn_sub_nc (pp + 2n, pp + 2n, pp, n, cy);
306	5.18k	hi -= mpn_sub_nc (pp + 3*n, scratch + n, pp + n, n, cy);
307
308	5.18k	hi += mpn_add (pp + n, pp + n, 3*n, scratch, n);
309
310		/* FIXME: Is support for s + t == n needed? */
311	5.18k	if (LIKELY (s + t > n))
312	5.18k	{
313	5.18k	hi -= mpn_sub (pp + 2n, pp + 2n, 2n, pp + 4n, s+t-n);
314
315	5.18k	ASSERT (hi >= 0); /* contribution of the middle terms >= 0 */
316	5.18k	MPN_INCR_U (pp + 4*n, s+t-n, hi);
317	5.18k	}
318	0	else
319	0	ASSERT (hi == 0);
320	5.18k	}