375 lines
10 KiB
C
375 lines
10 KiB
C
/* mpn_tdiv_qr -- Divide the numerator (np,nn) by the denominator (dp,dn) and
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write the nn-dn+1 quotient limbs at qp and the dn remainder limbs at rp. If
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qxn is non-zero, generate that many fraction limbs and append them after the
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other quotient limbs, and update the remainder accordingly. The input
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operands are unaffected.
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Preconditions:
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1. The most significant limb of of the divisor must be non-zero.
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2. nn >= dn, even if qxn is non-zero. (??? relax this ???)
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The time complexity of this is O(qn*qn+M(dn,qn)), where M(m,n) is the time
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complexity of multiplication.
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Copyright 1997, 2000, 2001, 2002, 2005, 2009 Free Software Foundation, Inc.
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Copyright 2010 William Hart (modified to work with MPIR functions).
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This file is part of the GNU MP Library.
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The GNU MP 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 MP 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 MP Library. If not, see http://www.gnu.org/licenses/. */
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#include "mpir.h"
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#include "gmp-impl.h"
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#include "longlong.h"
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void
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mpn_tdiv_qr (mp_ptr qp, mp_ptr rp, mp_size_t qxn,
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mp_srcptr np, mp_size_t nn, mp_srcptr dp, mp_size_t dn)
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{
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ASSERT_ALWAYS (qxn == 0);
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ASSERT (nn >= 0);
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ASSERT (dn >= 0);
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ASSERT (dn == 0 || dp[dn - 1] != 0);
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ASSERT (! MPN_OVERLAP_P (qp, nn - dn + 1 + qxn, np, nn));
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ASSERT (! MPN_OVERLAP_P (qp, nn - dn + 1 + qxn, dp, dn));
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switch (dn)
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{
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case 0:
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DIVIDE_BY_ZERO;
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case 1:
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{
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rp[0] = mpn_divrem_1 (qp, (mp_size_t) 0, np, nn, dp[0]);
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return;
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}
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case 2:
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{
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mp_ptr n2p, d2p;
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mp_limb_t qhl, cy;
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TMP_DECL;
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TMP_MARK;
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if ((dp[1] & GMP_NUMB_HIGHBIT) == 0)
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{
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int cnt;
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mp_limb_t dtmp[2];
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count_leading_zeros (cnt, dp[1]);
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cnt -= GMP_NAIL_BITS;
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d2p = dtmp;
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d2p[1] = (dp[1] << cnt) | (dp[0] >> (GMP_NUMB_BITS - cnt));
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d2p[0] = (dp[0] << cnt) & GMP_NUMB_MASK;
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n2p = TMP_ALLOC_LIMBS (nn + 1);
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cy = mpn_lshift (n2p, np, nn, cnt);
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n2p[nn] = cy;
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qhl = mpn_divrem_2 (qp, 0L, n2p, nn + (cy != 0), d2p);
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if (cy == 0)
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qp[nn - 2] = qhl; /* always store nn-2+1 quotient limbs */
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rp[0] = (n2p[0] >> cnt)
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| ((n2p[1] << (GMP_NUMB_BITS - cnt)) & GMP_NUMB_MASK);
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rp[1] = (n2p[1] >> cnt);
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}
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else
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{
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d2p = (mp_ptr) dp;
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n2p = TMP_ALLOC_LIMBS (nn);
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MPN_COPY (n2p, np, nn);
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qhl = mpn_divrem_2 (qp, 0L, n2p, nn, d2p);
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qp[nn - 2] = qhl; /* always store nn-2+1 quotient limbs */
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rp[0] = n2p[0];
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rp[1] = n2p[1];
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}
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TMP_FREE;
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return;
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}
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default:
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{
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int adjust;
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mp_limb_t dinv, d1inv;
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TMP_DECL;
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TMP_MARK;
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adjust = np[nn - 1] >= dp[dn - 1]; /* conservative tests for quotient size */
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if (nn + adjust >= 2 * dn)
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{
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mp_ptr n2p, d2p;
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mp_limb_t cy;
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int cnt;
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qp[nn - dn] = 0; /* zero high quotient limb */
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if ((dp[dn - 1] & GMP_NUMB_HIGHBIT) == 0) /* normalize divisor */
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{
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count_leading_zeros (cnt, dp[dn - 1]);
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cnt -= GMP_NAIL_BITS;
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d2p = TMP_ALLOC_LIMBS (dn);
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mpn_lshift (d2p, dp, dn, cnt);
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n2p = TMP_ALLOC_LIMBS (nn + 1);
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cy = mpn_lshift (n2p, np, nn, cnt);
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n2p[nn] = cy;
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nn += adjust;
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}
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else
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{
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cnt = 0;
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d2p = (mp_ptr) dp;
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n2p = TMP_ALLOC_LIMBS (nn + 1);
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MPN_COPY (n2p, np, nn);
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n2p[nn] = 0;
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nn += adjust;
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}
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mpir_invert_pi2 (dinv, d1inv, d2p[dn - 1], d2p[dn - 2]);
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if (BELOW_THRESHOLD (dn, DC_DIV_QR_THRESHOLD))
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ASSERT_NOCARRY(mpn_sb_div_qr (qp, n2p, nn, d2p, dn, dinv, d1inv));
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else if (BELOW_THRESHOLD (dn, INV_DIV_QR_THRESHOLD) ||
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BELOW_THRESHOLD (nn, 2 * INV_DIV_QR_THRESHOLD))
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ASSERT_NOCARRY(mpn_dc_div_qr (qp, n2p, nn, d2p, dn, dinv, d1inv));
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else
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{
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mp_ptr dinv2 = TMP_ALLOC_LIMBS(dn);
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mpn_invert(dinv2, d2p, dn);
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ASSERT_NOCARRY(mpn_inv_div_qr (qp, n2p, nn, d2p, dn, dinv2));
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}
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if (cnt != 0)
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mpn_rshift (rp, n2p, dn, cnt);
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else
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MPN_COPY (rp, n2p, dn);
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TMP_FREE;
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return;
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}
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/* When we come here, the numerator/partial remainder is less
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than twice the size of the denominator. */
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{
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/* Problem:
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Divide a numerator N with nn limbs by a denominator D with dn
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limbs forming a quotient of qn=nn-dn+1 limbs. When qn is small
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compared to dn, conventional division algorithms perform poorly.
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We want an algorithm that has an expected running time that is
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dependent only on qn.
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Algorithm (very informally stated):
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1) Divide the 2 x qn most significant limbs from the numerator
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by the qn most significant limbs from the denominator. Call
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the result qest. This is either the correct quotient, but
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might be 1 or 2 too large. Compute the remainder from the
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division. (This step is implemented by a mpn_divrem call.)
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2) Is the most significant limb from the remainder < p, where p
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is the product of the most significant limb from the quotient
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and the next(d)? (Next(d) denotes the next ignored limb from
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the denominator.) If it is, decrement qest, and adjust the
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remainder accordingly.
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3) Is the remainder >= qest? If it is, qest is the desired
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quotient. The algorithm terminates.
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4) Subtract qest x next(d) from the remainder. If there is
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borrow out, decrement qest, and adjust the remainder
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accordingly.
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5) Skip one word from the denominator (i.e., let next(d) denote
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the next less significant limb. */
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mp_size_t qn;
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mp_ptr n2p, d2p;
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mp_ptr tp;
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mp_limb_t cy;
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mp_size_t in, rn;
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mp_limb_t quotient_too_large;
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unsigned int cnt;
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qn = nn - dn;
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qp[qn] = 0; /* zero high quotient limb */
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qn += adjust; /* qn cannot become bigger */
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if (qn == 0)
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{
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MPN_COPY (rp, np, dn);
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TMP_FREE;
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return;
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}
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in = dn - qn; /* (at least partially) ignored # of limbs in ops */
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/* Normalize denominator by shifting it to the left such that its
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most significant bit is set. Then shift the numerator the same
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amount, to mathematically preserve quotient. */
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if ((dp[dn - 1] & GMP_NUMB_HIGHBIT) == 0)
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{
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count_leading_zeros (cnt, dp[dn - 1]);
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cnt -= GMP_NAIL_BITS;
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d2p = TMP_ALLOC_LIMBS (qn);
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mpn_lshift (d2p, dp + in, qn, cnt);
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d2p[0] |= dp[in - 1] >> (GMP_NUMB_BITS - cnt);
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n2p = TMP_ALLOC_LIMBS (2 * qn + 1);
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cy = mpn_lshift (n2p, np + nn - 2 * qn, 2 * qn, cnt);
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if (adjust)
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{
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n2p[2 * qn] = cy;
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n2p++;
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}
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else
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{
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n2p[0] |= np[nn - 2 * qn - 1] >> (GMP_NUMB_BITS - cnt);
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}
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}
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else
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{
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cnt = 0;
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d2p = (mp_ptr) dp + in;
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n2p = TMP_ALLOC_LIMBS (2 * qn + 1);
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MPN_COPY (n2p, np + nn - 2 * qn, 2 * qn);
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if (adjust)
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{
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n2p[2 * qn] = 0;
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n2p++;
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}
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}
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/* Get an approximate quotient using the extracted operands. */
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if (qn == 1)
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{
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mp_limb_t q0, r0;
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udiv_qrnnd (q0, r0, n2p[1], n2p[0] << GMP_NAIL_BITS, d2p[0] << GMP_NAIL_BITS);
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n2p[0] = r0 >> GMP_NAIL_BITS;
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qp[0] = q0;
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}
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else if (qn == 2)
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mpn_divrem_2 (qp, 0L, n2p, 4L, d2p); /* FIXME: obsolete function */
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else
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{
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mpir_invert_pi2 (dinv, d1inv, d2p[qn - 1], d2p[qn - 2]);
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if (BELOW_THRESHOLD (qn, DC_DIV_QR_THRESHOLD))
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ASSERT_NOCARRY(mpn_sb_div_qr (qp, n2p, 2 * qn, d2p, qn, dinv, d1inv));
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else if (BELOW_THRESHOLD (qn, INV_DIV_QR_THRESHOLD))
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{
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mp_ptr temp = TMP_ALLOC_LIMBS(DC_DIVAPPR_Q_N_ITCH(qn));
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ASSERT_NOCARRY(mpn_dc_div_qr_n (qp, n2p, d2p, qn, dinv, d1inv, temp));
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} else
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{
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mp_ptr dinv2 = TMP_ALLOC_LIMBS(qn);
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mpn_invert(dinv2, d2p, qn);
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ASSERT_NOCARRY(mpn_inv_div_qr_n (qp, n2p, d2p, qn, dinv2));
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}
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}
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rn = qn;
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/* Multiply the first ignored divisor limb by the most significant
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quotient limb. If that product is > the partial remainder's
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most significant limb, we know the quotient is too large. This
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test quickly catches most cases where the quotient is too large;
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it catches all cases where the quotient is 2 too large. */
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{
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mp_limb_t dl, x;
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mp_limb_t h, dummy;
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if (in - 2 < 0)
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dl = 0;
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else
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dl = dp[in - 2];
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#if GMP_NAIL_BITS == 0
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x = (dp[in - 1] << cnt) | ((dl >> 1) >> ((~cnt) % GMP_LIMB_BITS));
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#else
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x = (dp[in - 1] << cnt) & GMP_NUMB_MASK;
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if (cnt != 0)
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x |= dl >> (GMP_NUMB_BITS - cnt);
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#endif
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umul_ppmm (h, dummy, x, qp[qn - 1] << GMP_NAIL_BITS);
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if (n2p[qn - 1] < h)
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{
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mp_limb_t cy;
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mpn_decr_u (qp, (mp_limb_t) 1);
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cy = mpn_add_n (n2p, n2p, d2p, qn);
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if (cy)
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{
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/* The partial remainder is safely large. */
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n2p[qn] = cy;
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++rn;
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}
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}
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}
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quotient_too_large = 0;
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if (cnt != 0)
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{
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mp_limb_t cy1, cy2;
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/* Append partially used numerator limb to partial remainder. */
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cy1 = mpn_lshift (n2p, n2p, rn, GMP_NUMB_BITS - cnt);
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n2p[0] |= np[in - 1] & (GMP_NUMB_MASK >> cnt);
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/* Update partial remainder with partially used divisor limb. */
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cy2 = mpn_submul_1 (n2p, qp, qn, dp[in - 1] & (GMP_NUMB_MASK >> cnt));
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if (qn != rn)
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{
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ASSERT_ALWAYS (n2p[qn] >= cy2);
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n2p[qn] -= cy2;
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}
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else
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{
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n2p[qn] = cy1 - cy2; /* & GMP_NUMB_MASK; */
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quotient_too_large = (cy1 < cy2);
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++rn;
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}
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--in;
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}
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/* True: partial remainder now is neutral, i.e., it is not shifted up. */
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tp = TMP_ALLOC_LIMBS (dn);
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if (in < qn)
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{
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if (in == 0)
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{
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MPN_COPY (rp, n2p, rn);
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ASSERT_ALWAYS (rn == dn);
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goto foo;
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}
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mpn_mul (tp, qp, qn, dp, in);
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}
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else
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mpn_mul (tp, dp, in, qp, qn);
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cy = mpn_sub (n2p, n2p, rn, tp + in, qn);
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MPN_COPY (rp + in, n2p, dn - in);
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quotient_too_large |= cy;
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cy = mpn_sub_n (rp, np, tp, in);
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cy = mpn_sub_1 (rp + in, rp + in, rn, cy);
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quotient_too_large |= cy;
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foo:
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if (quotient_too_large)
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{
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mpn_decr_u (qp, (mp_limb_t) 1);
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mpn_add_n (rp, rp, dp, dn);
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}
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}
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TMP_FREE;
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return;
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}
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}
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}
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