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297 lines
10 KiB
C
297 lines
10 KiB
C
/* Mulders' short product, square and division.
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Copyright 2005-2025 Free Software Foundation, Inc.
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Contributed by the Pascaline and Caramba projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER.
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If not, see <https://www.gnu.org/licenses/>. */
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/* References:
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[1] Short Division of Long Integers, David Harvey and Paul Zimmermann,
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Proceedings of the 20th Symposium on Computer Arithmetic (ARITH-20),
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July 25-27, 2011, pages 7-14.
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*/
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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/* Don't use MPFR_MULHIGH_SIZE since it is handled by tuneup */
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#ifdef MPFR_MULHIGH_TAB_SIZE
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static short mulhigh_ktab[MPFR_MULHIGH_TAB_SIZE];
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#else
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static short mulhigh_ktab[] = {MPFR_MULHIGH_TAB};
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#define MPFR_MULHIGH_TAB_SIZE (numberof_const (mulhigh_ktab))
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#endif
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/* Put in rp[n..2n-1] an approximation of the n high limbs
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of {up, n} * {vp, n}. The error is less than n ulps of rp[n] (and the
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approximation is always less or equal to the truncated full product).
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Assume 2n limbs are allocated at rp.
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Implements Algorithm ShortMulNaive from [1].
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*/
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static void
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mpfr_mulhigh_n_basecase (mpfr_limb_ptr rp, mpfr_limb_srcptr up,
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mpfr_limb_srcptr vp, mp_size_t n)
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{
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mp_size_t i;
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rp += n - 1;
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umul_ppmm (rp[1], rp[0], up[n-1], vp[0]); /* we neglect up[0..n-2]*vp[0],
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which is less than B^n */
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for (i = 1 ; i < n ; i++)
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/* here, we neglect up[0..n-i-2] * vp[i], which is less than B^n too */
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rp[i + 1] = mpn_addmul_1 (rp, up + (n - i - 1), i + 1, vp[i]);
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/* in total, we neglect less than n*B^n, i.e., n ulps of rp[n]. */
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}
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/* Put in rp[n..2n-1] an approximation of the n high limbs
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of {np, n} * {mp, n}. The error is less than n ulps of rp[n] (and the
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approximation is always less or equal to the truncated full product).
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Implements Algorithm ShortMul from [1].
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*/
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void
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mpfr_mulhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mpfr_limb_srcptr mp,
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mp_size_t n)
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{
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mp_size_t k;
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MPFR_STAT_STATIC_ASSERT (MPFR_MULHIGH_TAB_SIZE >= 8); /* so that 3*(n/4) > n/2 */
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k = MPFR_LIKELY (n < MPFR_MULHIGH_TAB_SIZE) ? mulhigh_ktab[n] : 3*(n/4);
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/* Algorithm ShortMul from [1] requires k >= (n+3)/2, which translates
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into k >= (n+4)/2 in the C language. */
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MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n));
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if (k < 0)
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mpn_mul_basecase (rp, np, n, mp, n); /* result is exact, no error */
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else if (k == 0)
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mpfr_mulhigh_n_basecase (rp, np, mp, n); /* basecase error < n ulps */
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else if (n > MUL_FFT_THRESHOLD)
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mpn_mul_n (rp, np, mp, n); /* result is exact, no error */
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else
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{
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mp_size_t l = n - k;
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mp_limb_t cy;
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mpn_mul_n (rp + 2 * l, np + l, mp + l, k); /* fills rp[2l..2n-1] */
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mpfr_mulhigh_n (rp, np + k, mp, l); /* fills rp[l-1..2l-1] */
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cy = mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
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mpfr_mulhigh_n (rp, np, mp + k, l); /* fills rp[l-1..2l-1] */
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cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
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mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */
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}
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}
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#ifdef MPFR_SQRHIGH_TAB_SIZE
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static short sqrhigh_ktab[MPFR_SQRHIGH_TAB_SIZE];
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#else
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static short sqrhigh_ktab[] = {MPFR_SQRHIGH_TAB};
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#define MPFR_SQRHIGH_TAB_SIZE (numberof_const (sqrhigh_ktab))
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#endif
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/* Put in rp[n..2n-1] an approximation of the n high limbs
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of {np, n}^2. The error is less than n ulps of rp[n]. */
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void
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mpfr_sqrhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mp_size_t n)
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{
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mp_size_t k;
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MPFR_STAT_STATIC_ASSERT (MPFR_SQRHIGH_TAB_SIZE > 2); /* ensures k < n */
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k = MPFR_LIKELY (n < MPFR_SQRHIGH_TAB_SIZE) ? sqrhigh_ktab[n]
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: (n+4)/2; /* ensures that k >= (n+3)/2 */
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MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n));
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if (k < 0)
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/* we can't use mpn_sqr_basecase here, since it requires
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n <= SQR_KARATSUBA_THRESHOLD, where SQR_KARATSUBA_THRESHOLD
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is not exported by GMP */
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mpn_sqr (rp, np, n);
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else if (k == 0)
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mpfr_mulhigh_n_basecase (rp, np, np, n);
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else
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{
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mp_size_t l = n - k;
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mp_limb_t cy;
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mpn_sqr (rp + 2 * l, np + l, k); /* fills rp[2l..2n-1] */
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mpfr_mulhigh_n (rp, np, np + k, l); /* fills rp[l-1..2l-1] */
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/* {rp+n-1,l+1} += 2 * {rp+l-1,l+1} */
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cy = mpn_lshift (rp + l - 1, rp + l - 1, l + 1, 1);
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cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1);
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mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */
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}
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}
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#ifdef MPFR_DIVHIGH_TAB_SIZE
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static short divhigh_ktab[MPFR_DIVHIGH_TAB_SIZE];
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#else
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static short divhigh_ktab[] = {MPFR_DIVHIGH_TAB};
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#define MPFR_DIVHIGH_TAB_SIZE (numberof_const (divhigh_ktab))
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#endif
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/* Put in Q={qp, n} an approximation of N={np, 2*n} divided by D={dp, n},
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with the most significant limb of the quotient as return value (0 or 1).
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Assumes the most significant bit of D is set. Clobbers N.
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The approximate quotient Q satisfies - 2(n-1) < N/D - Q <= 4.
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Assumes n >= 2.
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*/
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static mp_limb_t
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mpfr_divhigh_n_basecase (mpfr_limb_ptr qp, mpfr_limb_ptr np,
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mpfr_limb_srcptr dp, mp_size_t n)
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{
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mp_limb_t qh, d1, d0, q2, q1, q0;
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mpfr_pi1_t dinv2;
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MPFR_ASSERTD(n >= 2);
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np += n;
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if ((qh = (mpn_cmp (np, dp, n) >= 0)))
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mpn_sub_n (np, np, dp, n);
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/* now {np, n} is less than D={dp, n}, which implies np[n-1] <= dp[n-1] */
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d1 = dp[n - 1];
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/* we assumed n >= 2 */
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d0 = dp[n - 2];
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invert_pi1 (dinv2, d1, d0);
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/* dinv2.inv32 = floor ((B^3 - 1) / (d0 + d1 B)) - B */
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while (n > 1)
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{
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/* Invariant: it remains to reduce n limbs from N (in addition to the
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initial low n limbs).
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Since n >= 2 here, necessarily we had n >= 2 initially, which means
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that in addition to the limb np[n-1] to reduce, we have at least 2
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extra limbs, thus accessing np[n-3] is valid. */
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/* Warning: we can have np[n-1]>d1 or (np[n-1]=d1 and np[n-2]>=d0) here,
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since we truncate the divisor at each step, but since {np,n} < D
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originally, the largest possible partial quotient is B-1. */
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if (MPFR_UNLIKELY(np[n-1] > d1 || (np[n-1] == d1 && np[n-2] >= d0)))
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q2 = MPFR_LIMB_MAX;
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else
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udiv_qr_3by2 (q2, q1, q0, np[n - 1], np[n - 2], np[n - 3],
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d1, d0, dinv2.inv32);
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/* since q2 = floor((np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0)),
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we have q2 <= (np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0),
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thus np[n-1]*B^2+np[n-2]*B+np[n-3] >= q2*(d1*B+d0)
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and {np-1, n} >= q2*D - q2*B^(n-2) >= q2*D - B^(n-1)
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thus {np-1, n} - (q2-1)*D >= D - B^(n-1) >= 0
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which proves that at most one correction is needed */
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q0 = mpn_submul_1 (np - 1, dp, n, q2);
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if (MPFR_UNLIKELY(q0 > np[n - 1]))
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{
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mpn_add_n (np - 1, np - 1, dp, n);
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q2 --;
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}
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qp[--n] = q2;
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dp ++;
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}
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/* we have B+dinv2 = floor((B^3-1)/(d1*B+d0)) < B^2/d1
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q1 = floor(np[0]*(B+dinv2)/B) <= floor(np[0]*B/d1)
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<= floor((np[0]*B+np[1])/d1)
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thus q1 is not larger than the true quotient.
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q1 > np[0]*(B+dinv2)/B - 1 > np[0]*(B^3-1)/(d1*B+d0)/B - 2
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For d1*B+d0 <> B^2/2, we have B+dinv2 = floor(B^3/(d1*B+d0))
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thus q1 > np[0]*B^2/(d1*B+d0) - 2, i.e.,
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(d1*B+d0)*q1 > np[0]*B^2 - 2*(d1*B+d0)
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d1*B*q1 > np[0]*B^2 - 2*d1*B - 2*d0 - d0*q1 >= np[0]*B^2 - 2*d1*B - B^2
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thus q1 > np[0]*B/d1 - 2 - B/d1 > np[0]*B/d1 - 4.
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For d1*B+d0 = B^2/2, dinv2 = B-1 thus q1 > np[0]*(2B-1)/B - 1 >
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np[0]*B/d1 - 2.
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In all cases, if q = floor((np[0]*B+np[1])/d1), we have:
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q - 4 <= q1 <= q
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*/
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umul_ppmm (q1, q0, np[0], dinv2.inv32);
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qp[0] = np[0] + q1;
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return qh;
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}
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/* Put in {qp, n} an approximation of N={np, 2*n} divided by D={dp, n},
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with the most significant limb of the quotient as return value (0 or 1).
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Assumes the most significant bit of D is set. Clobbers N.
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This implements the ShortDiv algorithm from reference [1].
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Assumes n >= 2 (which should be fulfilled also in the recursive calls).
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*/
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mp_limb_t
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mpfr_divhigh_n (mpfr_limb_ptr qp, mpfr_limb_ptr np, mpfr_limb_ptr dp,
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mp_size_t n)
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{
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mp_size_t k, l;
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mp_limb_t qh, cy;
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mpfr_limb_ptr tp;
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MPFR_TMP_DECL(marker);
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MPFR_STAT_STATIC_ASSERT (MPFR_DIVHIGH_TAB_SIZE >= 15); /* so that 2*(n/3) >= (n+4)/2 */
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MPFR_ASSERTD(n >= 2);
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k = MPFR_LIKELY (n < MPFR_DIVHIGH_TAB_SIZE) ? divhigh_ktab[n] : 2*(n/3);
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if (k == 0)
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{
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#if defined(WANT_GMP_INTERNALS) && defined(HAVE___GMPN_SBPI1_DIVAPPR_Q)
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mpfr_pi1_t dinv2;
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invert_pi1 (dinv2, dp[n - 1], dp[n - 2]);
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if (n > 2) /* sbpi1_divappr_q wants n > 2 */
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return __gmpn_sbpi1_divappr_q (qp, np, n + n, dp, n, dinv2.inv32);
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else
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return mpfr_divhigh_n_basecase (qp, np, dp, n);
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#else /* use our own code for base-case short division */
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return mpfr_divhigh_n_basecase (qp, np, dp, n);
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#endif
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}
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/* Check the bounds from [1]. In addition, we forbid k=n-1, which would
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give l=1 in the recursive call. It follows n >= 5. */
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MPFR_ASSERTD ((n+4)/2 <= k && k < n-1);
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MPFR_TMP_MARK (marker);
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l = n - k;
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/* first divide the most significant 2k limbs from N by the most significant
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k limbs of D */
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qh = mpn_divrem (qp + l, 0, np + 2 * l, 2 * k, dp + l, k); /* exact */
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/* it remains {np,2l+k} = {np,n+l} as remainder */
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/* now we have to subtract high(Q1)*D0 where Q1=qh*B^k+{qp+l,k} and
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D0={dp,l} */
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tp = MPFR_TMP_LIMBS_ALLOC (2 * l);
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mpfr_mulhigh_n (tp, qp + k, dp, l);
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/* we are only interested in the upper l limbs from {tp,2l} */
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cy = mpn_sub_n (np + n, np + n, tp + l, l);
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if (qh)
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cy += mpn_sub_n (np + n, np + n, dp, l);
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while (cy > 0) /* Q1 was too large: subtract 1 to Q1 and add D to np+l */
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{
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qh -= mpn_sub_1 (qp + l, qp + l, k, MPFR_LIMB_ONE);
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cy -= mpn_add_n (np + l, np + l, dp, n);
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}
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/* now it remains {np,n+l} to divide by D */
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cy = mpfr_divhigh_n (qp, np + k, dp + k, l);
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qh += mpn_add_1 (qp + l, qp + l, k, cy);
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MPFR_TMP_FREE(marker);
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return qh;
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}
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