/* mpz_n_pow_ui -- mpn raised to ulong.
THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY. THEY'RE ALMOST
CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN
FUTURE GNU MP RELEASES.
Copyright 2001, 2002, 2005, 2012 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.h"
#include "gmp-impl.h"
#include "longlong.h"
/* Change this to "#define TRACE(x) x" for some traces. */
#define TRACE(x)
/* Use this to test the mul_2 code on a CPU without a native version of that
routine. */
#if 0
#define mpn_mul_2 refmpn_mul_2
#define HAVE_NATIVE_mpn_mul_2 1
#endif
/* mpz_pow_ui and mpz_ui_pow_ui want to share almost all of this code.
ui_pow_ui doesn't need the mpn_mul based powering loop or the tests on
bsize==2 or >2, but separating that isn't easy because there's shared
code both before and after (the size calculations and the powers of 2
handling).
Alternatives:
It would work to just use the mpn_mul powering loop for 1 and 2 limb
bases, but the current separate loop allows mul_1 and mul_2 to be done
in-place, which might help cache locality a bit. If mpn_mul was relaxed
to allow source==dest when vn==1 or 2 then some pointer twiddling might
let us get the same effect in one loop.
The initial powering for bsize==1 into blimb or blimb:blimb_low doesn't
form the biggest possible power of b that fits, only the biggest power of
2 power, ie. b^(2^n). It'd be possible to choose a bigger power, perhaps
using mp_bases[b].big_base for small b, and thereby get better value
from mpn_mul_1 or mpn_mul_2 in the bignum powering. It's felt that doing
so would be more complicated than it's worth, and could well end up being
a slowdown for small e. For big e on the other hand the algorithm is
dominated by mpn_sqr so there wouldn't much of a saving. The current
code can be viewed as simply doing the first few steps of the powering in
a single or double limb where possible.
If r==b, and blow_twos==0, and r must be realloc'ed, then the temporary
copy made of b is unnecessary. We could just use the old alloc'ed block
and free it at the end. But arranging this seems like a lot more trouble
than it's worth. */
/* floor(sqrt(GMP_NUMB_MAX)), ie. the biggest value that can be squared in
a limb without overflowing.
FIXME: This formula is an underestimate when GMP_NUMB_BITS is odd. */
#define GMP_NUMB_HALFMAX (((mp_limb_t) 1 << GMP_NUMB_BITS/2) - 1)
/* The following are for convenience, they update the size and check the
alloc. */
#define MPN_SQR(dst, alloc, src, size) \
do { \
ASSERT (2*(size) <= (alloc)); \
mpn_sqr (dst, src, size); \
(size) *= 2; \
(size) -= ((dst)[(size)-1] == 0); \
} while (0)
#define MPN_MUL(dst, alloc, src, size, src2, size2) \
do { \
mp_limb_t cy; \
ASSERT ((size) + (size2) <= (alloc)); \
cy = mpn_mul (dst, src, size, src2, size2); \
(size) += (size2) - (cy == 0); \
} while (0)
#define MPN_MUL_2(ptr, size, alloc, mult) \
do { \
mp_limb_t cy; \
ASSERT ((size)+2 <= (alloc)); \
cy = mpn_mul_2 (ptr, ptr, size, mult); \
(size)++; \
(ptr)[(size)] = cy; \
(size) += (cy != 0); \
} while (0)
#define MPN_MUL_1(ptr, size, alloc, limb) \
do { \
mp_limb_t cy; \
ASSERT ((size)+1 <= (alloc)); \
cy = mpn_mul_1 (ptr, ptr, size, limb); \
(ptr)[size] = cy; \
(size) += (cy != 0); \
} while (0)
#define MPN_LSHIFT(ptr, size, alloc, shift) \
do { \
mp_limb_t cy; \
ASSERT ((size)+1 <= (alloc)); \
cy = mpn_lshift (ptr, ptr, size, shift); \
(ptr)[size] = cy; \
(size) += (cy != 0); \
} while (0)
#define MPN_RSHIFT_OR_COPY(dst, src, size, shift) \
do { \
if ((shift) == 0) \
MPN_COPY (dst, src, size); \
else \
{ \
mpn_rshift (dst, src, size, shift); \
(size) -= ((dst)[(size)-1] == 0); \
} \
} while (0)
/* ralloc and talloc are only wanted for ASSERTs, after the initial space
allocations. Avoid writing values to them in a normal build, to ensure
the compiler lets them go dead. gcc already figures this out itself
actually. */
#define SWAP_RP_TP \
do { \
MP_PTR_SWAP (rp, tp); \
ASSERT_CODE (MP_SIZE_T_SWAP (ralloc, talloc)); \
} while (0)
void
mpz_n_pow_ui (mpz_ptr r, mp_srcptr bp, mp_size_t bsize, unsigned long int e)
{
mp_ptr rp;
mp_size_t rtwos_limbs, ralloc, rsize;
int rneg, i, cnt, btwos, r_bp_overlap;
mp_limb_t blimb, rl;
mp_bitcnt_t rtwos_bits;
#if HAVE_NATIVE_mpn_mul_2
mp_limb_t blimb_low, rl_high;
#else
mp_limb_t b_twolimbs[2];
#endif
TMP_DECL;
TRACE (printf ("mpz_n_pow_ui rp=0x%lX bp=0x%lX bsize=%ld e=%lu (0x%lX)\n",
PTR(r), bp, bsize, e, e);
mpn_trace ("b", bp, bsize));
ASSERT (bsize == 0 || bp[ABS(bsize)-1] != 0);
ASSERT (MPN_SAME_OR_SEPARATE2_P (PTR(r), ALLOC(r), bp, ABS(bsize)));
/* b^0 == 1, including 0^0 == 1 */
if (e == 0)
{
PTR(r)[0] = 1;
SIZ(r) = 1;
return;
}
/* 0^e == 0 apart from 0^0 above */
if (bsize == 0)
{
SIZ(r) = 0;
return;
}
/* Sign of the final result. */
rneg = (bsize < 0 && (e & 1) != 0);
bsize = ABS (bsize);
TRACE (printf ("rneg %d\n", rneg));
r_bp_overlap = (PTR(r) == bp);
/* Strip low zero limbs from b. */
rtwos_limbs = 0;
for (blimb = *bp; blimb == 0; blimb = *++bp)
{
rtwos_limbs += e;
bsize--; ASSERT (bsize >= 1);
}
TRACE (printf ("trailing zero rtwos_limbs=%ld\n", rtwos_limbs));
/* Strip low zero bits from b. */
count_trailing_zeros (btwos, blimb);
blimb >>= btwos;
rtwos_bits = e * btwos;
rtwos_limbs += rtwos_bits / GMP_NUMB_BITS;
rtwos_bits %= GMP_NUMB_BITS;
TRACE (printf ("trailing zero btwos=%d rtwos_limbs=%ld rtwos_bits=%lu\n",
btwos, rtwos_limbs, rtwos_bits));
TMP_MARK;
rl = 1;
#if HAVE_NATIVE_mpn_mul_2
rl_high = 0;
#endif
if (bsize == 1)
{
bsize_1:
/* Power up as far as possible within blimb. We start here with e!=0,
but if e is small then we might reach e==0 and the whole b^e in rl.
Notice this code works when blimb==1 too, reaching e==0. */
while (blimb <= GMP_NUMB_HALFMAX)
{
TRACE (printf ("small e=0x%lX blimb=0x%lX rl=0x%lX\n",
e, blimb, rl));
ASSERT (e != 0);
if ((e & 1) != 0)
rl *= blimb;
e >>= 1;
if (e == 0)
goto got_rl;
blimb *= blimb;
}
#if HAVE_NATIVE_mpn_mul_2
TRACE (printf ("single power, e=0x%lX b=0x%lX rl=0x%lX\n",
e, blimb, rl));
/* Can power b once more into blimb:blimb_low */
bsize = 2;
ASSERT (e != 0);
if ((e & 1) != 0)
{
umul_ppmm (rl_high, rl, rl, blimb << GMP_NAIL_BITS);
rl >>= GMP_NAIL_BITS;
}
e >>= 1;
umul_ppmm (blimb, blimb_low, blimb, blimb << GMP_NAIL_BITS);
blimb_low >>= GMP_NAIL_BITS;
got_rl:
TRACE (printf ("double power e=0x%lX blimb=0x%lX:0x%lX rl=0x%lX:%lX\n",
e, blimb, blimb_low, rl_high, rl));
/* Combine left-over rtwos_bits into rl_high:rl to be handled by the
final mul_1 or mul_2 rather than a separate lshift.
- rl_high:rl mustn't be 1 (since then there's no final mul)
- rl_high mustn't overflow
- rl_high mustn't change to non-zero, since mul_1+lshift is
probably faster than mul_2 (FIXME: is this true?) */
if (rtwos_bits != 0
&& ! (rl_high == 0 && rl == 1)
&& (rl_high >> (GMP_NUMB_BITS-rtwos_bits)) == 0)
{
mp_limb_t new_rl_high = (rl_high << rtwos_bits)
| (rl >> (GMP_NUMB_BITS-rtwos_bits));
if (! (rl_high == 0 && new_rl_high != 0))
{
rl_high = new_rl_high;
rl <<= rtwos_bits;
rtwos_bits = 0;
TRACE (printf ("merged rtwos_bits, rl=0x%lX:%lX\n",
rl_high, rl));
}
}
#else
got_rl:
TRACE (printf ("small power e=0x%lX blimb=0x%lX rl=0x%lX\n",
e, blimb, rl));
/* Combine left-over rtwos_bits into rl to be handled by the final
mul_1 rather than a separate lshift.
- rl mustn't be 1 (since then there's no final mul)
- rl mustn't overflow */
if (rtwos_bits != 0
&& rl != 1
&& (rl >> (GMP_NUMB_BITS-rtwos_bits)) == 0)
{
rl <<= rtwos_bits;
rtwos_bits = 0;
TRACE (printf ("merged rtwos_bits, rl=0x%lX\n", rl));
}
#endif
}
else if (bsize == 2)
{
mp_limb_t bsecond = bp[1];
if (btwos != 0)
blimb |= (bsecond << (GMP_NUMB_BITS - btwos)) & GMP_NUMB_MASK;
bsecond >>= btwos;
if (bsecond == 0)
{
/* Two limbs became one after rshift. */
bsize = 1;
goto bsize_1;
}
TRACE (printf ("bsize==2 using b=0x%lX:%lX", bsecond, blimb));
#if HAVE_NATIVE_mpn_mul_2
blimb_low = blimb;
#else
bp = b_twolimbs;
b_twolimbs[0] = blimb;
b_twolimbs[1] = bsecond;
#endif
blimb = bsecond;
}
else
{
if (r_bp_overlap || btwos != 0)
{
mp_ptr tp = TMP_ALLOC_LIMBS (bsize);
MPN_RSHIFT_OR_COPY (tp, bp, bsize, btwos);
bp = tp;
TRACE (printf ("rshift or copy bp,bsize, new bsize=%ld\n", bsize));
}
#if HAVE_NATIVE_mpn_mul_2
/* in case 3 limbs rshift to 2 and hence use the mul_2 loop below */
blimb_low = bp[0];
#endif
blimb = bp[bsize-1];
TRACE (printf ("big bsize=%ld ", bsize);
mpn_trace ("b", bp, bsize));
}
/* At this point blimb is the most significant limb of the base to use.
Each factor of b takes (bsize*BPML-cnt) bits and there's e of them; +1
limb to round up the division; +1 for multiplies all using an extra
limb over the true size; +2 for rl at the end; +1 for lshift at the
end.
The size calculation here is reasonably accurate. The base is at least
half a limb, so in 32 bits the worst case is 2^16+1 treated as 17 bits
when it will power up as just over 16, an overestimate of 17/16 =
6.25%. For a 64-bit limb it's half that.
If e==0 then blimb won't be anything useful (though it will be
non-zero), but that doesn't matter since we just end up with ralloc==5,
and that's fine for 2 limbs of rl and 1 of lshift. */
ASSERT (blimb != 0);
count_leading_zeros (cnt, blimb);
ralloc = (bsize*GMP_NUMB_BITS - cnt + GMP_NAIL_BITS) * e / GMP_NUMB_BITS + 5;
TRACE (printf ("ralloc %ld, from bsize=%ld blimb=0x%lX cnt=%d\n",
ralloc, bsize, blimb, cnt));
rp = MPZ_REALLOC (r, ralloc + rtwos_limbs);
/* Low zero limbs resulting from powers of 2. */
MPN_ZERO (rp, rtwos_limbs);
rp += rtwos_limbs;
if (e == 0)
{
/* Any e==0 other than via bsize==1 or bsize==2 is covered at the
start. */
rp[0] = rl;
rsize = 1;
#if HAVE_NATIVE_mpn_mul_2
rp[1] = rl_high;
rsize += (rl_high != 0);
#endif
ASSERT (rp[rsize-1] != 0);
}
else
{
mp_ptr tp;
mp_size_t talloc;
/* In the mpn_mul_1 or mpn_mul_2 loops or in the mpn_mul loop when the
low bit of e is zero, tp only has to hold the second last power
step, which is half the size of the final result. There's no need
to round up the divide by 2, since ralloc includes a +2 for rl
which not needed by tp. In the mpn_mul loop when the low bit of e
is 1, tp must hold nearly the full result, so just size it the same
as rp. */
talloc = ralloc;
#if HAVE_NATIVE_mpn_mul_2
if (bsize <= 2 || (e & 1) == 0)
talloc /= 2;
#else
if (bsize <= 1 || (e & 1) == 0)
talloc /= 2;
#endif
TRACE (printf ("talloc %ld\n", talloc));
tp = TMP_ALLOC_LIMBS (talloc);
/* Go from high to low over the bits of e, starting with i pointing at
the bit below the highest 1 (which will mean i==-1 if e==1). */
count_leading_zeros (cnt, (mp_limb_t) e);
i = GMP_LIMB_BITS - cnt - 2;
#if HAVE_NATIVE_mpn_mul_2
if (bsize <= 2)
{
mp_limb_t mult[2];
/* Any bsize==1 will have been powered above to be two limbs. */
ASSERT (bsize == 2);
ASSERT (blimb != 0);
/* Arrange the final result ends up in r, not in the temp space */
if ((i & 1) == 0)
SWAP_RP_TP;
rp[0] = blimb_low;
rp[1] = blimb;
rsize = 2;
mult[0] = blimb_low;
mult[1] = blimb;
for ( ; i >= 0; i--)
{
TRACE (printf ("mul_2 loop i=%d e=0x%lX, rsize=%ld ralloc=%ld talloc=%ld\n",
i, e, rsize, ralloc, talloc);
mpn_trace ("r", rp, rsize));
MPN_SQR (tp, talloc, rp, rsize);
SWAP_RP_TP;
if ((e & (1L << i)) != 0)
MPN_MUL_2 (rp, rsize, ralloc, mult);
}
TRACE (mpn_trace ("mul_2 before rl, r", rp, rsize));
if (rl_high != 0)
{
mult[0] = rl;
mult[1] = rl_high;
MPN_MUL_2 (rp, rsize, ralloc, mult);
}
else if (rl != 1)
MPN_MUL_1 (rp, rsize, ralloc, rl);
}
#else
if (bsize == 1)
{
/* Arrange the final result ends up in r, not in the temp space */
if ((i & 1) == 0)
SWAP_RP_TP;
rp[0] = blimb;
rsize = 1;
for ( ; i >= 0; i--)
{
TRACE (printf ("mul_1 loop i=%d e=0x%lX, rsize=%ld ralloc=%ld talloc=%ld\n",
i, e, rsize, ralloc, talloc);
mpn_trace ("r", rp, rsize));
MPN_SQR (tp, talloc, rp, rsize);
SWAP_RP_TP;
if ((e & (1L << i)) != 0)
MPN_MUL_1 (rp, rsize, ralloc, blimb);
}
TRACE (mpn_trace ("mul_1 before rl, r", rp, rsize));
if (rl != 1)
MPN_MUL_1 (rp, rsize, ralloc, rl);
}
#endif
else
{
int parity;
/* Arrange the final result ends up in r, not in the temp space */
ULONG_PARITY (parity, e);
if (((parity ^ i) & 1) != 0)
SWAP_RP_TP;
MPN_COPY (rp, bp, bsize);
rsize = bsize;
for ( ; i >= 0; i--)
{
TRACE (printf ("mul loop i=%d e=0x%lX, rsize=%ld ralloc=%ld talloc=%ld\n",
i, e, rsize, ralloc, talloc);
mpn_trace ("r", rp, rsize));
MPN_SQR (tp, talloc, rp, rsize);
SWAP_RP_TP;
if ((e & (1L << i)) != 0)
{
MPN_MUL (tp, talloc, rp, rsize, bp, bsize);
SWAP_RP_TP;
}
}
}
}
ASSERT (rp == PTR(r) + rtwos_limbs);
TRACE (mpn_trace ("end loop r", rp, rsize));
TMP_FREE;
/* Apply any partial limb factors of 2. */
if (rtwos_bits != 0)
{
MPN_LSHIFT (rp, rsize, ralloc, (unsigned) rtwos_bits);
TRACE (mpn_trace ("lshift r", rp, rsize));
}
rsize += rtwos_limbs;
SIZ(r) = (rneg ? -rsize : rsize);
}