/src/libgmp/mpn/toom32_mul.c

Source (jump to first uncovered line)
/* 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: 2024-11-21 06:47

Line	Count	Source (jump to first uncovered line)
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	2.82k	do { \
61	2.82k	mpn_mul_n (p, a, b, n); \
62	2.82k	} 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	942	{
70	942	mp_size_t n, s, t;
71	942	int vm1_neg;
72	942	mp_limb_t cy;
73	942	mp_limb_signed_t hi;
74	942	mp_limb_t ap1_hi, bp1_hi;
75
76	942	#define a0 ap
77	2.42k	#define a1 (ap + n)
78	1.88k	#define a2 (ap + 2 * n)
79	2.08k	#define b0 bp
80	2.51k	#define b1 (bp + n)
81
82		/* Required, to ensure that s + t >= n. */
83	942	ASSERT (bn + 2 <= an && an + 6 <= 3*bn);
84
85	942	n = 2 * an >= 3 * bn ? (an + 2) / (size_t) 3 : (bn + 1) >> 1;
86
87	942	s = an - 2 * n;
88	942	t = bn - n;
89
90	942	ASSERT (0 < s && s <= n);
91	942	ASSERT (0 < t && t <= n);
92	942	ASSERT (s + t >= n);
93
94		/* Product area of size an + bn = 3n + s + t >= 4n + 2. */
95	4.34k	#define ap1 (pp) /* n, most significant limb in ap1_hi */
96	1.40k	#define bp1 (pp + n) /* n, most significant bit in bp1_hi */
97	942	#define am1 (pp + 2n) / n, most significant bit in hi */
98	942	#define bm1 (pp + 3n) / n */
99	7.59k	#define v1 (scratch) /* 2n + 1 */
100	4.71k	#define vm1 (pp) /* 2n + 1 */
101	942	#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	942	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	942	if (ap1_hi == 0 && mpn_cmp (ap1, a1, n) < 0)
125	380	{
126	380	ASSERT_NOCARRY (mpn_sub_n (am1, a1, ap1, n));
127	380	hi = 0;
128	380	vm1_neg = 1;
129	380	}
130	562	else
131	562	{
132	562	hi = ap1_hi - mpn_sub_n (am1, ap1, a1, n);
133	562	vm1_neg = 0;
134	562	}
135	942	ap1_hi += mpn_add_n (ap1, ap1, a1, n);
136	942	#endif
137
138		/* Compute bp1 = b0 + b1 and bm1 = b0 - b1. */
139	942	if (t == n)
140	433	{
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	433	bp1_hi = mpn_add_n (bp1, b0, b1, n);
154
155	433	if (mpn_cmp (b0, b1, n) < 0)
156	29	{
157	29	ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, n));
158	29	vm1_neg ^= 1;
159	29	}
160	404	else
161	404	{
162	404	ASSERT_NOCARRY (mpn_sub_n (bm1, b0, b1, n));
163	404	}
164	433	#endif
165	433	}
166	509	else
167	509	{
168		/* FIXME: Should still use mpn_add_n_sub_n for the main part. */
169	509	bp1_hi = mpn_add (bp1, b0, n, b1, t);
170
171	509	if (mpn_zero_p (b0 + t, n - t) && mpn_cmp (b0, b1, t) < 0)
172	84	{
173	84	ASSERT_NOCARRY (mpn_sub_n (bm1, b1, b0, t));
174	84	MPN_ZERO (bm1 + t, n - t);
175	84	vm1_neg ^= 1;
176	84	}
177	425	else
178	425	{
179	425	ASSERT_NOCARRY (mpn_sub (bm1, b0, n, b1, t));
180	425	}
181	509	}
182
183	942	TOOM32_MUL_N_REC (v1, ap1, bp1, n, scratch_out);
184	942	if (ap1_hi == 1)
185	462	{
186	462	cy = mpn_add_n (v1 + n, v1 + n, bp1, n);
187	462	}
188	480	else if (ap1_hi > 1) /* ap1_hi == 2 */
189	16	{
190	16	#if HAVE_NATIVE_mpn_addlsh1_n_ip1
191	16	cy = mpn_addlsh1_n_ip1 (v1 + n, bp1, n);
192		#else
193		cy = mpn_addmul_1 (v1 + n, bp1, n, CNST_LIMB(2));
194		#endif
195	16	}
196	464	else
197	464	cy = 0;
198	942	if (bp1_hi != 0)
199	36	cy += ap1_hi + mpn_add_n (v1 + n, v1 + n, ap1, n);
200	942	v1[2 * n] = cy;
201
202	942	TOOM32_MUL_N_REC (vm1, am1, bm1, n, scratch_out);
203	942	if (hi)
204	3	hi = mpn_add_n (vm1+n, vm1+n, bm1, n);
205
206	942	vm1[2*n] = hi;
207
208		/* v1 <-- (v1 + vm1) / 2 = x0 + x2 */
209	942	if (vm1_neg)
210	407	{
211	407	#if HAVE_NATIVE_mpn_rsh1sub_n
212	407	mpn_rsh1sub_n (v1, v1, vm1, 2*n+1);
213		#else
214		mpn_sub_n (v1, v1, vm1, 2*n+1);
215		ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
216		#endif
217	407	}
218	535	else
219	535	{
220	535	#if HAVE_NATIVE_mpn_rsh1add_n
221	535	mpn_rsh1add_n (v1, v1, vm1, 2*n+1);
222		#else
223		mpn_add_n (v1, v1, vm1, 2*n+1);
224		ASSERT_NOCARRY (mpn_rshift (v1, v1, 2*n+1, 1));
225		#endif
226	535	}
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	942	hi = vm1[2*n];
255	942	cy = mpn_add_n (pp + 2*n, v1, v1 + n, n);
256	942	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	942	if (vm1_neg)
261	407	{
262	407	cy = mpn_add_n (v1, v1, vm1, n);
263	407	hi += mpn_add_nc (pp + 2n, pp + 2n, vm1 + n, n, cy);
264	407	MPN_INCR_U (v1 + n, n+1, hi);
265	407	}
266	535	else
267	535	{
268	535	cy = mpn_sub_n (v1, v1, vm1, n);
269	535	hi += mpn_sub_nc (pp + 2n, pp + 2n, vm1 + n, n, cy);
270	535	MPN_DECR_U (v1 + n, n+1, hi);
271	535	}
272
273	942	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	942	if (s > t) mpn_mul (pp+3*n, a2, s, b1, t);
276	730	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	942	cy = mpn_sub_n (pp + n, pp + n, pp+3*n, n);
303	942	hi = scratch[2*n] + cy;
304
305	942	cy = mpn_sub_nc (pp + 2n, pp + 2n, pp, n, cy);
306	942	hi -= mpn_sub_nc (pp + 3*n, scratch + n, pp + n, n, cy);
307
308	942	hi += mpn_add (pp + n, pp + n, 3*n, scratch, n);
309
310		/* FIXME: Is support for s + t == n needed? */
311	942	if (LIKELY (s + t > n))
312	942	{
313	942	hi -= mpn_sub (pp + 2n, pp + 2n, 2n, pp + 4n, s+t-n);
314
315	942	ASSERT (hi >= 0); /* contribution of the middle terms >= 0 */
316	942	MPN_INCR_U (pp + 4*n, s+t-n, hi);
317	942	}
318	0	else
319	0	ASSERT (hi == 0);
320	942	}