tdiv_qr.c
上传用户:qaz666999
上传日期:2022-08-06
资源大小:2570k
文件大小:11k
- /* mpn_tdiv_qr -- Divide the numerator (np,nn) by the denominator (dp,dn) and
- write the nn-dn+1 quotient limbs at qp and the dn remainder limbs at rp. If
- qxn is non-zero, generate that many fraction limbs and append them after the
- other quotient limbs, and update the remainder accordingly. The input
- operands are unaffected.
- Preconditions:
- 1. The most significant limb of of the divisor must be non-zero.
- 2. nn >= dn, even if qxn is non-zero. (??? relax this ???)
- The time complexity of this is O(qn*qn+M(dn,qn)), where M(m,n) is the time
- complexity of multiplication.
- Copyright 1997, 2000, 2001, 2002, 2005, 2009 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 3 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. If not, see http://www.gnu.org/licenses/. */
- #include "gmp.h"
- #include "gmp-impl.h"
- #include "longlong.h"
- void
- mpn_tdiv_qr (mp_ptr qp, mp_ptr rp, mp_size_t qxn,
- mp_srcptr np, mp_size_t nn, mp_srcptr dp, mp_size_t dn)
- {
- ASSERT_ALWAYS (qxn == 0);
- ASSERT (nn >= 0);
- ASSERT (dn >= 0);
- ASSERT (dn == 0 || dp[dn - 1] != 0);
- ASSERT (! MPN_OVERLAP_P (qp, nn - dn + 1 + qxn, np, nn));
- ASSERT (! MPN_OVERLAP_P (qp, nn - dn + 1 + qxn, dp, dn));
- switch (dn)
- {
- case 0:
- DIVIDE_BY_ZERO;
- case 1:
- {
- rp[0] = mpn_divrem_1 (qp, (mp_size_t) 0, np, nn, dp[0]);
- return;
- }
- case 2:
- {
- mp_ptr n2p, d2p;
- mp_limb_t qhl, cy;
- TMP_DECL;
- TMP_MARK;
- if ((dp[1] & GMP_NUMB_HIGHBIT) == 0)
- {
- int cnt;
- mp_limb_t dtmp[2];
- count_leading_zeros (cnt, dp[1]);
- cnt -= GMP_NAIL_BITS;
- d2p = dtmp;
- d2p[1] = (dp[1] << cnt) | (dp[0] >> (GMP_NUMB_BITS - cnt));
- d2p[0] = (dp[0] << cnt) & GMP_NUMB_MASK;
- n2p = TMP_ALLOC_LIMBS (nn + 1);
- cy = mpn_lshift (n2p, np, nn, cnt);
- n2p[nn] = cy;
- qhl = mpn_divrem_2 (qp, 0L, n2p, nn + (cy != 0), d2p);
- if (cy == 0)
- qp[nn - 2] = qhl; /* always store nn-2+1 quotient limbs */
- rp[0] = (n2p[0] >> cnt)
- | ((n2p[1] << (GMP_NUMB_BITS - cnt)) & GMP_NUMB_MASK);
- rp[1] = (n2p[1] >> cnt);
- }
- else
- {
- d2p = (mp_ptr) dp;
- n2p = TMP_ALLOC_LIMBS (nn);
- MPN_COPY (n2p, np, nn);
- qhl = mpn_divrem_2 (qp, 0L, n2p, nn, d2p);
- qp[nn - 2] = qhl; /* always store nn-2+1 quotient limbs */
- rp[0] = n2p[0];
- rp[1] = n2p[1];
- }
- TMP_FREE;
- return;
- }
- default:
- {
- int adjust;
- gmp_pi1_t dinv;
- TMP_DECL;
- TMP_MARK;
- adjust = np[nn - 1] >= dp[dn - 1]; /* conservative tests for quotient size */
- if (nn + adjust >= 2 * dn)
- {
- mp_ptr n2p, d2p;
- mp_limb_t cy;
- int cnt;
- qp[nn - dn] = 0; /* zero high quotient limb */
- if ((dp[dn - 1] & GMP_NUMB_HIGHBIT) == 0) /* normalize divisor */
- {
- count_leading_zeros (cnt, dp[dn - 1]);
- cnt -= GMP_NAIL_BITS;
- d2p = TMP_ALLOC_LIMBS (dn);
- mpn_lshift (d2p, dp, dn, cnt);
- n2p = TMP_ALLOC_LIMBS (nn + 1);
- cy = mpn_lshift (n2p, np, nn, cnt);
- n2p[nn] = cy;
- nn += adjust;
- }
- else
- {
- cnt = 0;
- d2p = (mp_ptr) dp;
- n2p = TMP_ALLOC_LIMBS (nn + 1);
- MPN_COPY (n2p, np, nn);
- n2p[nn] = 0;
- nn += adjust;
- }
- invert_pi1 (dinv, d2p[dn - 1], d2p[dn - 2]);
- if (BELOW_THRESHOLD (dn, DC_DIV_QR_THRESHOLD))
- mpn_sbpi1_div_qr (qp, n2p, nn, d2p, dn, dinv.inv32);
- else if (BELOW_THRESHOLD (dn, MUPI_DIV_QR_THRESHOLD) || /* fast condition */
- BELOW_THRESHOLD (nn, 2 * MU_DIV_QR_THRESHOLD) || /* fast condition */
- (double) (2 * (MU_DIV_QR_THRESHOLD - MUPI_DIV_QR_THRESHOLD)) * dn /* slow... */
- + (double) MUPI_DIV_QR_THRESHOLD * nn > (double) dn * nn) /* ...condition */
- mpn_dcpi1_div_qr (qp, n2p, nn, d2p, dn, &dinv);
- else
- {
- mp_size_t itch = mpn_mu_div_qr_itch (nn, dn, 0);
- mp_ptr scratch = TMP_ALLOC_LIMBS (itch);
- mpn_mu_div_qr (qp, rp, n2p, nn, d2p, dn, scratch);
- n2p = rp;
- }
- if (cnt != 0)
- mpn_rshift (rp, n2p, dn, cnt);
- else
- MPN_COPY (rp, n2p, dn);
- TMP_FREE;
- return;
- }
- /* When we come here, the numerator/partial remainder is less
- than twice the size of the denominator. */
- {
- /* Problem:
- Divide a numerator N with nn limbs by a denominator D with dn
- limbs forming a quotient of qn=nn-dn+1 limbs. When qn is small
- compared to dn, conventional division algorithms perform poorly.
- We want an algorithm that has an expected running time that is
- dependent only on qn.
- Algorithm (very informally stated):
- 1) Divide the 2 x qn most significant limbs from the numerator
- by the qn most significant limbs from the denominator. Call
- the result qest. This is either the correct quotient, but
- might be 1 or 2 too large. Compute the remainder from the
- division. (This step is implemented by a mpn_divrem call.)
- 2) Is the most significant limb from the remainder < p, where p
- is the product of the most significant limb from the quotient
- and the next(d)? (Next(d) denotes the next ignored limb from
- the denominator.) If it is, decrement qest, and adjust the
- remainder accordingly.
- 3) Is the remainder >= qest? If it is, qest is the desired
- quotient. The algorithm terminates.
- 4) Subtract qest x next(d) from the remainder. If there is
- borrow out, decrement qest, and adjust the remainder
- accordingly.
- 5) Skip one word from the denominator (i.e., let next(d) denote
- the next less significant limb. */
- mp_size_t qn;
- mp_ptr n2p, d2p;
- mp_ptr tp;
- mp_limb_t cy;
- mp_size_t in, rn;
- mp_limb_t quotient_too_large;
- unsigned int cnt;
- qn = nn - dn;
- qp[qn] = 0; /* zero high quotient limb */
- qn += adjust; /* qn cannot become bigger */
- if (qn == 0)
- {
- MPN_COPY (rp, np, dn);
- TMP_FREE;
- return;
- }
- in = dn - qn; /* (at least partially) ignored # of limbs in ops */
- /* Normalize denominator by shifting it to the left such that its
- most significant bit is set. Then shift the numerator the same
- amount, to mathematically preserve quotient. */
- if ((dp[dn - 1] & GMP_NUMB_HIGHBIT) == 0)
- {
- count_leading_zeros (cnt, dp[dn - 1]);
- cnt -= GMP_NAIL_BITS;
- d2p = TMP_ALLOC_LIMBS (qn);
- mpn_lshift (d2p, dp + in, qn, cnt);
- d2p[0] |= dp[in - 1] >> (GMP_NUMB_BITS - cnt);
- n2p = TMP_ALLOC_LIMBS (2 * qn + 1);
- cy = mpn_lshift (n2p, np + nn - 2 * qn, 2 * qn, cnt);
- if (adjust)
- {
- n2p[2 * qn] = cy;
- n2p++;
- }
- else
- {
- n2p[0] |= np[nn - 2 * qn - 1] >> (GMP_NUMB_BITS - cnt);
- }
- }
- else
- {
- cnt = 0;
- d2p = (mp_ptr) dp + in;
- n2p = TMP_ALLOC_LIMBS (2 * qn + 1);
- MPN_COPY (n2p, np + nn - 2 * qn, 2 * qn);
- if (adjust)
- {
- n2p[2 * qn] = 0;
- n2p++;
- }
- }
- /* Get an approximate quotient using the extracted operands. */
- if (qn == 1)
- {
- mp_limb_t q0, r0;
- udiv_qrnnd (q0, r0, n2p[1], n2p[0] << GMP_NAIL_BITS, d2p[0] << GMP_NAIL_BITS);
- n2p[0] = r0 >> GMP_NAIL_BITS;
- qp[0] = q0;
- }
- else if (qn == 2)
- mpn_divrem_2 (qp, 0L, n2p, 4L, d2p); /* FIXME: obsolete function */
- else
- {
- invert_pi1 (dinv, d2p[qn - 1], d2p[qn - 2]);
- if (BELOW_THRESHOLD (qn, DC_DIV_QR_THRESHOLD))
- mpn_sbpi1_div_qr (qp, n2p, 2 * qn, d2p, qn, dinv.inv32);
- else if (BELOW_THRESHOLD (qn, MU_DIV_QR_THRESHOLD))
- mpn_dcpi1_div_qr (qp, n2p, 2 * qn, d2p, qn, &dinv);
- else
- {
- mp_size_t itch = mpn_mu_div_qr_itch (2 * qn, qn, 0);
- mp_ptr scratch = TMP_ALLOC_LIMBS (itch);
- mp_ptr r2p = rp;
- if (np == r2p) /* If N and R share space, put ... */
- r2p += nn - qn; /* intermediate remainder at N's upper end. */
- mpn_mu_div_qr (qp, r2p, n2p, 2 * qn, d2p, qn, scratch);
- MPN_COPY (n2p, r2p, qn);
- }
- }
- rn = qn;
- /* Multiply the first ignored divisor limb by the most significant
- quotient limb. If that product is > the partial remainder's
- most significant limb, we know the quotient is too large. This
- test quickly catches most cases where the quotient is too large;
- it catches all cases where the quotient is 2 too large. */
- {
- mp_limb_t dl, x;
- mp_limb_t h, dummy;
- if (in - 2 < 0)
- dl = 0;
- else
- dl = dp[in - 2];
- #if GMP_NAIL_BITS == 0
- x = (dp[in - 1] << cnt) | ((dl >> 1) >> ((~cnt) % GMP_LIMB_BITS));
- #else
- x = (dp[in - 1] << cnt) & GMP_NUMB_MASK;
- if (cnt != 0)
- x |= dl >> (GMP_NUMB_BITS - cnt);
- #endif
- umul_ppmm (h, dummy, x, qp[qn - 1] << GMP_NAIL_BITS);
- if (n2p[qn - 1] < h)
- {
- mp_limb_t cy;
- mpn_decr_u (qp, (mp_limb_t) 1);
- cy = mpn_add_n (n2p, n2p, d2p, qn);
- if (cy)
- {
- /* The partial remainder is safely large. */
- n2p[qn] = cy;
- ++rn;
- }
- }
- }
- quotient_too_large = 0;
- if (cnt != 0)
- {
- mp_limb_t cy1, cy2;
- /* Append partially used numerator limb to partial remainder. */
- cy1 = mpn_lshift (n2p, n2p, rn, GMP_NUMB_BITS - cnt);
- n2p[0] |= np[in - 1] & (GMP_NUMB_MASK >> cnt);
- /* Update partial remainder with partially used divisor limb. */
- cy2 = mpn_submul_1 (n2p, qp, qn, dp[in - 1] & (GMP_NUMB_MASK >> cnt));
- if (qn != rn)
- {
- ASSERT_ALWAYS (n2p[qn] >= cy2);
- n2p[qn] -= cy2;
- }
- else
- {
- n2p[qn] = cy1 - cy2; /* & GMP_NUMB_MASK; */
- quotient_too_large = (cy1 < cy2);
- ++rn;
- }
- --in;
- }
- /* True: partial remainder now is neutral, i.e., it is not shifted up. */
- tp = TMP_ALLOC_LIMBS (dn);
- if (in < qn)
- {
- if (in == 0)
- {
- MPN_COPY (rp, n2p, rn);
- ASSERT_ALWAYS (rn == dn);
- goto foo;
- }
- mpn_mul (tp, qp, qn, dp, in);
- }
- else
- mpn_mul (tp, dp, in, qp, qn);
- cy = mpn_sub (n2p, n2p, rn, tp + in, qn);
- MPN_COPY (rp + in, n2p, dn - in);
- quotient_too_large |= cy;
- cy = mpn_sub_n (rp, np, tp, in);
- cy = mpn_sub_1 (rp + in, rp + in, rn, cy);
- quotient_too_large |= cy;
- foo:
- if (quotient_too_large)
- {
- mpn_decr_u (qp, (mp_limb_t) 1);
- mpn_add_n (rp, rp, dp, dn);
- }
- }
- TMP_FREE;
- return;
- }
- }
- }