153 lines
4.2 KiB
C
153 lines
4.2 KiB
C
/* mpn_sb_divappr_q - schoolbook approximate quotient.
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THE FUNCTIONS IN THIS FILE ARE INTERNAL FUNCTIONS WITH MUTABLE
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INTERFACES. IT IS ONLY SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES.
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IN FACT, IT IS ALMOST GUARANTEED THAT THEY'LL CHANGE OR DISAPPEAR IN A
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FUTURE GNU MP RELEASE.
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Copyright 2009 William Hart.
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This file is part of the MPIR Library.
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The MPIR 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 2.1 of the License, or (at your
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option) any later version.
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The MPIR 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 MPIR Library; see the file COPYING.LIB. If not, write to
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the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
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MA 02110-1301, USA. */
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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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/*
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Given n = {np, nn} and d = {dp, dn} and a 2 limb inverse
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x = {dip, 2} (with implicit top bit), satisfying
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x*d0 < B^4 <= (x+1)*d0 where d0 = {dp + dn - 2, 2} is the
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top two limbs of the denominator, returns an approximate
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quotient q = {qp, nn - dn + 1} such that d*q + r = n for
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some remainder r with -d < r < d.
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Requires d = {dp, dn} to be normalised, i.e. the most
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significant bit of the most significant limb must be set.
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n = {np, nn} is destroyed.
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*/
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mp_limb_t
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mpn_sb_divappr_q (mp_ptr qp, mp_ptr np, mp_size_t nn,
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mp_srcptr dp, mp_size_t dn, mp_srcptr dip)
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{
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/*
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In order to make use of the the two limb inverse we
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use the following theorem of Torbjorn Granlund and
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Peter Montgomery from their paper, "Division by
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invariant integers using multiplication" (restated
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here for clarity):
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Lemma 8.1: Let d be normalised, d < B^2 (i.e.
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fits in two words), and suppose that
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m*d < B^4 <= (m+1)*d.
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Let 0 <= n <= B^2*d - 1. Write
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n = n2*B^2 + n1*B^2/2 + n0
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with n1 = 0 or 1 and n0 < B^2/2.
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Suppose
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q1*B^2 + q0 = n2*B^2 + (n2 + n1)*(m-B^2)
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+ n1*(d-B^2/2) + n0
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and 0 <= q0 < B^2.
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Then 0 <= q1 < B^2 and 0 <= n - q1*d < 2d.
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We apply the theorem as follows. Note that
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n0 and n1*(d-B^2/2) are both less than B^2/2.
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Also note that n1*(m-B^2) < B^2. Thus the sum
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of all these terms contributes at most 1 to q1.
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We are left with n2*B^2 + n2*(m-B^2). But note
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that (m-B^2) is precisely our precomputed inverse
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without the implied leading bit. If we write
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q1*B^2 + q0 = n2*B^2 + n2*(m-B^2), then from the
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theorem, we have 0 <= n-q1*d < 3d.
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*/
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mp_limb_t ret, di0, di1, p1, p2, p3, p4, q, q0, n21, n20, cy;
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mp_size_t qn = nn - dn + 1;
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mp_size_t i;
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/*
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We only need to use the top qn limbs of the
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denominator and the same applies for the
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numerator.
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*/
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if (qn < dn)
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{
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dp += (dn - qn);
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dn = qn;
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}
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if (qn < nn)
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{
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np += (nn - qn);
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nn = qn;
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}
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/*
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It may be that the top limbs of the numerator
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are bigger than the denominator, in which case
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we return the high top limb of the quotient as
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1 instead of 0.
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*/
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if (mpn_cmp(np + nn - dn, dp, dn) >= 0)
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{
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ret = CNST_LIMB(1);
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mpn_sub_n(np + nn - dn, np + nn - dn, dp, nn);
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} else
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ret = CNST_LIMB(0);
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di1 = dip[1];
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di0 = dip[0];
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for (i = qn - 2; i >= 0; i--)
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{
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/*
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Compute n2 + top two limbs of n2*di, but
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caring only about the top limb q, which we
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allow to be off by up to 1.
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*/
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n21 = np[nn - 1];
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n20 = np[nn - 2];
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umul_ppmm(p2, p1, di0, n21);
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umul_ppmm(p4, p3, di1, n20);
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add_ssaaaa(q, q0, n21, p2, CNST_LIMB(0), p4);
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umul_ppmm(p1, p2, di1, n21);
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add_ssaaaa(q, q0, q, q0, p1, p2);
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add_ssaaaa(q, q0, q, q0, CNST_LIMB(0), n20);
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cy = mpn_submul_1(np + nn - dn - 1, dp, dn, q);
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/* Either q was correct or too small by 1 */
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if ((np[nn-1] > cy) || (mpn_cmp(np + nn - dn - 1, dp, dn) >= 0))
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{
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q++; /* q can't overflow */
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mpn_sub_n(np + nn - dn - 1, np + nn - dn - 1, dp, dn);
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}
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qp[i] = q;
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dn--;
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nn--;
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}
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return ret;
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}
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