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/* mpn_mul -- Multiply two natural numbers.
Contributed to the GNU project by Torbjorn Granlund.
Copyright 1991, 1993, 1994, 1996, 1997, 1999-2003, 2005-2007, 2009, 2010, 2012,
2014, 2019 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"
#ifndef MUL_BASECASE_MAX_UN
#define MUL_BASECASE_MAX_UN 500
#endif
/* Areas where the different toom algorithms can be called (extracted
from the t-toom*.c files, and ignoring small constant offsets):
1/6 1/5 1/4 4/13 1/3 3/8 2/5 5/11 1/2 3/5 2/3 3/4 4/5 1 vn/un
4/7 6/7
6/11
|--------------------| toom22 (small)
|| toom22 (large)
|xxxx| toom22 called
|-------------------------------------| toom32
|xxxxxxxxxxxxxxxx| | toom32 called
|------------| toom33
|x| toom33 called
|---------------------------------| | toom42
|xxxxxxxxxxxxxxxxxxxxxxxx| | toom42 called
|--------------------| toom43
|xxxxxxxxxx| toom43 called
|-----------------------------| toom52 (unused)
|--------| toom44
|xxxxxxxx| toom44 called
|--------------------| | toom53
|xxxxxx| toom53 called
|-------------------------| toom62 (unused)
|----------------| toom54 (unused)
|--------------------| toom63
|xxxxxxxxx| | toom63 called
|---------------------------------| toom6h
|xxxxxxxx| toom6h called
|-------------------------| toom8h (32 bit)
|------------------------------------------| toom8h (64 bit)
|xxxxxxxx| toom8h called
*/
#define TOOM33_OK(an,bn) (6 + 2 * an < 3 * bn)
#define TOOM44_OK(an,bn) (12 + 3 * an < 4 * bn)
/* Multiply the natural numbers u (pointed to by UP, with UN limbs) and v
(pointed to by VP, with VN limbs), and store the result at PRODP. The
result is UN + VN limbs. Return the most significant limb of the result.
NOTE: The space pointed to by PRODP is overwritten before finished with U
and V, so overlap is an error.
Argument constraints:
1. UN >= VN.
2. PRODP != UP and PRODP != VP, i.e. the destination must be distinct from
the multiplier and the multiplicand. */
/*
* The cutoff lines in the toomX2 and toomX3 code are now exactly between the
ideal lines of the surrounding algorithms. Is that optimal?
* The toomX3 code now uses a structure similar to the one of toomX2, except
that it loops longer in the unbalanced case. The result is that the
remaining area might have un < vn. Should we fix the toomX2 code in a
similar way?
* The toomX3 code is used for the largest non-FFT unbalanced operands. It
therefore calls mpn_mul recursively for certain cases.
* Allocate static temp space using THRESHOLD variables (except for toom44
when !WANT_FFT). That way, we can typically have no TMP_ALLOC at all.
* We sort ToomX2 algorithms together, assuming the toom22, toom32, toom42
have the same vn threshold. This is not true, we should actually use
mul_basecase for slightly larger operands for toom32 than for toom22, and
even larger for toom42.
* That problem is even more prevalent for toomX3. We therefore use special
THRESHOLD variables there.
*/
mp_limb_t
mpn_mul (mp_ptr prodp,
mp_srcptr up, mp_size_t un,
mp_srcptr vp, mp_size_t vn)
{
ASSERT (un >= vn);
ASSERT (vn >= 1);
ASSERT (! MPN_OVERLAP_P (prodp, un+vn, up, un));
ASSERT (! MPN_OVERLAP_P (prodp, un+vn, vp, vn));
if (BELOW_THRESHOLD (un, MUL_TOOM22_THRESHOLD))
{
/* When un (and thus vn) is below the toom22 range, do mul_basecase.
Test un and not vn here not to thwart the un >> vn code below.
This special case is not necessary, but cuts the overhead for the
smallest operands. */
mpn_mul_basecase (prodp, up, un, vp, vn);
}
else if (un == vn)
{
mpn_mul_n (prodp, up, vp, un);
}
else if (vn < MUL_TOOM22_THRESHOLD)
{ /* plain schoolbook multiplication */
/* Unless un is very large, or else if have an applicable mpn_mul_N,
perform basecase multiply directly. */
if (un <= MUL_BASECASE_MAX_UN
#if HAVE_NATIVE_mpn_mul_2
|| vn <= 2
#else
|| vn == 1
#endif
)
mpn_mul_basecase (prodp, up, un, vp, vn);
else
{
/* We have un >> MUL_BASECASE_MAX_UN > vn. For better memory
locality, split up[] into MUL_BASECASE_MAX_UN pieces and multiply
these pieces with the vp[] operand. After each such partial
multiplication (but the last) we copy the most significant vn
limbs into a temporary buffer since that part would otherwise be
overwritten by the next multiplication. After the next
multiplication, we add it back. This illustrates the situation:
-->vn<--
| |<------- un ------->|
_____________________|
X /|
/XX__________________/ |
_____________________ |
X / |
/XX__________________/ |
_____________________ |
/ / |
/____________________/ |
==================================================================
The parts marked with X are the parts whose sums are copied into
the temporary buffer. */
mp_limb_t tp[MUL_TOOM22_THRESHOLD_LIMIT];
mp_limb_t cy;
ASSERT (MUL_TOOM22_THRESHOLD <= MUL_TOOM22_THRESHOLD_LIMIT);
mpn_mul_basecase (prodp, up, MUL_BASECASE_MAX_UN, vp, vn);
prodp += MUL_BASECASE_MAX_UN;
MPN_COPY (tp, prodp, vn); /* preserve high triangle */
up += MUL_BASECASE_MAX_UN;
un -= MUL_BASECASE_MAX_UN;
while (un > MUL_BASECASE_MAX_UN)
{
mpn_mul_basecase (prodp, up, MUL_BASECASE_MAX_UN, vp, vn);
cy = mpn_add_n (prodp, prodp, tp, vn); /* add back preserved triangle */
mpn_incr_u (prodp + vn, cy);
prodp += MUL_BASECASE_MAX_UN;
MPN_COPY (tp, prodp, vn); /* preserve high triangle */
up += MUL_BASECASE_MAX_UN;
un -= MUL_BASECASE_MAX_UN;
}
if (un > vn)
{
mpn_mul_basecase (prodp, up, un, vp, vn);
}
else
{
ASSERT (un > 0);
mpn_mul_basecase (prodp, vp, vn, up, un);
}
cy = mpn_add_n (prodp, prodp, tp, vn); /* add back preserved triangle */
mpn_incr_u (prodp + vn, cy);
}
}
else if (BELOW_THRESHOLD (vn, MUL_TOOM33_THRESHOLD))
{
/* Use ToomX2 variants */
mp_ptr scratch;
TMP_SDECL; TMP_SMARK;
#define ITCH_TOOMX2 (9 * vn / 2 + GMP_NUMB_BITS * 2)
scratch = TMP_SALLOC_LIMBS (ITCH_TOOMX2);
ASSERT (mpn_toom22_mul_itch ((5*vn-1)/4, vn) <= ITCH_TOOMX2); /* 5vn/2+ */
ASSERT (mpn_toom32_mul_itch ((7*vn-1)/4, vn) <= ITCH_TOOMX2); /* 7vn/6+ */
ASSERT (mpn_toom42_mul_itch (3 * vn - 1, vn) <= ITCH_TOOMX2); /* 9vn/2+ */
#undef ITCH_TOOMX2
/* FIXME: This condition (repeated in the loop below) leaves from a vn*vn
square to a (3vn-1)*vn rectangle. Leaving such a rectangle is hardly
wise; we would get better balance by slightly moving the bound. We
will sometimes end up with un < vn, like in the X3 arm below. */
if (un >= 3 * vn)
{
mp_limb_t cy;
mp_ptr ws;
/* The maximum ws usage is for the mpn_mul result. */
ws = TMP_SALLOC_LIMBS (4 * vn);
mpn_toom42_mul (prodp, up, 2 * vn, vp, vn, scratch);
un -= 2 * vn;
up += 2 * vn;
prodp += 2 * vn;
while (un >= 3 * vn)
{
mpn_toom42_mul (ws, up, 2 * vn, vp, vn, scratch);
un -= 2 * vn;
up += 2 * vn;
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, 2 * vn);
mpn_incr_u (prodp + vn, cy);
prodp += 2 * vn;
}
/* vn <= un < 3vn */
if (4 * un < 5 * vn)
mpn_toom22_mul (ws, up, un, vp, vn, scratch);
else if (4 * un < 7 * vn)
mpn_toom32_mul (ws, up, un, vp, vn, scratch);
else
mpn_toom42_mul (ws, up, un, vp, vn, scratch);
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, un);
mpn_incr_u (prodp + vn, cy);
}
else
{
if (4 * un < 5 * vn)
mpn_toom22_mul (prodp, up, un, vp, vn, scratch);
else if (4 * un < 7 * vn)
mpn_toom32_mul (prodp, up, un, vp, vn, scratch);
else
mpn_toom42_mul (prodp, up, un, vp, vn, scratch);
}
TMP_SFREE;
}
else if (BELOW_THRESHOLD ((un + vn) >> 1, MUL_FFT_THRESHOLD) ||
BELOW_THRESHOLD (3 * vn, MUL_FFT_THRESHOLD))
{
/* Handle the largest operands that are not in the FFT range. The 2nd
condition makes very unbalanced operands avoid the FFT code (except
perhaps as coefficient products of the Toom code. */
if (BELOW_THRESHOLD (vn, MUL_TOOM44_THRESHOLD) || !TOOM44_OK (un, vn))
{
/* Use ToomX3 variants */
mp_ptr scratch;
TMP_DECL; TMP_MARK;
#define ITCH_TOOMX3 (4 * vn + GMP_NUMB_BITS)
scratch = TMP_ALLOC_LIMBS (ITCH_TOOMX3);
ASSERT (mpn_toom33_mul_itch ((7*vn-1)/6, vn) <= ITCH_TOOMX3); /* 7vn/2+ */
ASSERT (mpn_toom43_mul_itch ((3*vn-1)/2, vn) <= ITCH_TOOMX3); /* 9vn/4+ */
ASSERT (mpn_toom32_mul_itch ((7*vn-1)/4, vn) <= ITCH_TOOMX3); /* 7vn/6+ */
ASSERT (mpn_toom53_mul_itch ((11*vn-1)/6, vn) <= ITCH_TOOMX3); /* 11vn/3+ */
ASSERT (mpn_toom42_mul_itch ((5*vn-1)/2, vn) <= ITCH_TOOMX3); /* 15vn/4+ */
ASSERT (mpn_toom63_mul_itch ((5*vn-1)/2, vn) <= ITCH_TOOMX3); /* 15vn/4+ */
#undef ITCH_TOOMX3
if (2 * un >= 5 * vn)
{
mp_limb_t cy;
mp_ptr ws;
/* The maximum ws usage is for the mpn_mul result. */
ws = TMP_ALLOC_LIMBS (7 * vn >> 1);
if (BELOW_THRESHOLD (vn, MUL_TOOM42_TO_TOOM63_THRESHOLD))
mpn_toom42_mul (prodp, up, 2 * vn, vp, vn, scratch);
else
mpn_toom63_mul (prodp, up, 2 * vn, vp, vn, scratch);
un -= 2 * vn;
up += 2 * vn;
prodp += 2 * vn;
while (2 * un >= 5 * vn) /* un >= 2.5vn */
{
if (BELOW_THRESHOLD (vn, MUL_TOOM42_TO_TOOM63_THRESHOLD))
mpn_toom42_mul (ws, up, 2 * vn, vp, vn, scratch);
else
mpn_toom63_mul (ws, up, 2 * vn, vp, vn, scratch);
un -= 2 * vn;
up += 2 * vn;
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, 2 * vn);
mpn_incr_u (prodp + vn, cy);
prodp += 2 * vn;
}
/* vn / 2 <= un < 2.5vn */
if (un < vn)
mpn_mul (ws, vp, vn, up, un);
else
mpn_mul (ws, up, un, vp, vn);
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, un);
mpn_incr_u (prodp + vn, cy);
}
else
{
if (6 * un < 7 * vn)
mpn_toom33_mul (prodp, up, un, vp, vn, scratch);
else if (2 * un < 3 * vn)
{
if (BELOW_THRESHOLD (vn, MUL_TOOM32_TO_TOOM43_THRESHOLD))
mpn_toom32_mul (prodp, up, un, vp, vn, scratch);
else
mpn_toom43_mul (prodp, up, un, vp, vn, scratch);
}
else if (6 * un < 11 * vn)
{
if (4 * un < 7 * vn)
{
if (BELOW_THRESHOLD (vn, MUL_TOOM32_TO_TOOM53_THRESHOLD))
mpn_toom32_mul (prodp, up, un, vp, vn, scratch);
else
mpn_toom53_mul (prodp, up, un, vp, vn, scratch);
}
else
{
if (BELOW_THRESHOLD (vn, MUL_TOOM42_TO_TOOM53_THRESHOLD))
mpn_toom42_mul (prodp, up, un, vp, vn, scratch);
else
mpn_toom53_mul (prodp, up, un, vp, vn, scratch);
}
}
else
{
if (BELOW_THRESHOLD (vn, MUL_TOOM42_TO_TOOM63_THRESHOLD))
mpn_toom42_mul (prodp, up, un, vp, vn, scratch);
else
mpn_toom63_mul (prodp, up, un, vp, vn, scratch);
}
}
TMP_FREE;
}
else
{
mp_ptr scratch;
TMP_DECL; TMP_MARK;
if (BELOW_THRESHOLD (vn, MUL_TOOM6H_THRESHOLD))
{
scratch = TMP_SALLOC_LIMBS (mpn_toom44_mul_itch (un, vn));
mpn_toom44_mul (prodp, up, un, vp, vn, scratch);
}
else if (BELOW_THRESHOLD (vn, MUL_TOOM8H_THRESHOLD))
{
scratch = TMP_SALLOC_LIMBS (mpn_toom6h_mul_itch (un, vn));
mpn_toom6h_mul (prodp, up, un, vp, vn, scratch);
}
else
{
scratch = TMP_ALLOC_LIMBS (mpn_toom8h_mul_itch (un, vn));
mpn_toom8h_mul (prodp, up, un, vp, vn, scratch);
}
TMP_FREE;
}
}
else
{
if (un >= 8 * vn)
{
mp_limb_t cy;
mp_ptr ws;
TMP_DECL; TMP_MARK;
/* The maximum ws usage is for the mpn_mul result. */
ws = TMP_BALLOC_LIMBS (9 * vn >> 1);
mpn_fft_mul (prodp, up, 3 * vn, vp, vn);
un -= 3 * vn;
up += 3 * vn;
prodp += 3 * vn;
while (2 * un >= 7 * vn) /* un >= 3.5vn */
{
mpn_fft_mul (ws, up, 3 * vn, vp, vn);
un -= 3 * vn;
up += 3 * vn;
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, 3 * vn);
mpn_incr_u (prodp + vn, cy);
prodp += 3 * vn;
}
/* vn / 2 <= un < 3.5vn */
if (un < vn)
mpn_mul (ws, vp, vn, up, un);
else
mpn_mul (ws, up, un, vp, vn);
cy = mpn_add_n (prodp, prodp, ws, vn);
MPN_COPY (prodp + vn, ws + vn, un);
mpn_incr_u (prodp + vn, cy);
TMP_FREE;
}
else
mpn_fft_mul (prodp, up, un, vp, vn);
}
return prodp[un + vn - 1]; /* historic */
}
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