/* mpz_n_pow_ui -- mpn raised to ulong. Copyright 2001, 2002 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 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., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */ #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_n 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_N(dst, alloc, src, size) \ do { \ ASSERT (2*(size) <= (alloc)); \ mpn_sqr_n (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; unsigned long 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 (marker); 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), ABSIZ(r), bp, 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 (marker); 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)); MPZ_REALLOC (r, ralloc + rtwos_limbs); rp = PTR(r); /* 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, 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_N (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_N (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_N (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 (marker); /* 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); }

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