270 lines
7.7 KiB
C
270 lines
7.7 KiB
C
/* mpn_mulhigh_n
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Copyright 2009 Jason Moxham
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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
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by the Free Software Foundation; either version 2.1 of the License, or (at
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your 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
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to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
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Boston, MA 02110-1301, USA.
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*/
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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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Let X = sum over 0 <= i < n of x[i]B^i
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Let Y = sum over 0 <= i < n of y[i]B^i
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Define the usual multiplication as
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XY = sum over 0 <= i < n, 0 <= j < n, x[i]y[j]B^(i + j)
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Define short product as
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XY_k = sum over i + j >= k, x[i]y[j]B^(i + j)
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and approx short product as a superset of short product and subset of usual product
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Now consider the usual product XY
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XY = sum over {0 <= i < n, 0 <= j < n} x[i]y[j]B^(i+j)
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from now we just show the sum bounds with these implicit limits on i and j
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= {0 <= i < n, 0 <= j < n}
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split into four pieces (requires 0 <= m <= n)
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= {i < n - m, j < n - m} {i >= n-m, j >= n - m}
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{i < n - m, j >= n - m} {i >= n - m, j < n - m}
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split last two pieces again (requires n - m <= m - 1)
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= {i < n - m, j < n - m} {i >= n - m, j >= n - m} {i < n - m, n - m <= j < m}
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{i < n - m, m <= j} {n - m <= i < m, j < n - m} {m <= i, j < n - m}
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rearrange
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= {i < n - m, j < n - m} {i >= n - m, j >= n - m} {i < n - m, m <= j}
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{m <= i, j < n - m} {i < n - m, n - m <= j < m} { n - m <= i < m, j < n - m}
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split last two again (requires n - m <= m - 2)
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= {i < n - m, j < n - m} {i >= n - m, j >= n - m} {i < n - m, m <= j}
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{m <= i, j < n - m} {i < n - m, n - m <= j <= m - 2} {i < n - m, m - 2 < j < m}
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{n - m <= i <= m-2, j < n - m} {m - 2 < i < m, j < n - m}
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rearrange
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= {i < n - m, j < n - m} {i >= n - m, j >= n - m} {i < n - m, m <= j}
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{m <= i, j < n - m} {i < n - m, n - m <= j <= m - 2} {n - m <= i <= m - 2, j < n - m} {i<n-m,j=m-1} {i=m-1,j<n-m}
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split last two again
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= {i < n - m, j < n - m} {i >= n - m, j >= n - m}
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{i < n - m, m <= j} {m <= i, j < n - m} {i < n - m, n - m <= j <= m - 2}
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{n - m <= i <= m - 2, j < n - m} {i < n - m - 1, j = m - 1}
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{i = n - m - 1, j = m - 1} {i = m - 1, j < n - m - 1} {i = m - 1, j = n - m - 1}
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Now choose any m such that n + 2 <= 2m, m <= n
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so n - m <= m - 2 so our requirements above are satisfied
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Now consider the short product with k = n - 2, so we discard those
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with i + j < k = n - 2
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= {i < n - m, j < n - m}, i + j <= 2(n - m) - 2
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as n + 2 <= 2m, so n < 2m so 2n < 2m + n so 2n - 2m < n so i + j < n - 2 = k
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so empty
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{i >= n - m, j >= n - m}, i + j >= 2(n - m) keep most
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{i < n - m, m <= j}, keep some
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{m <= i, j < n - m}, keep some
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{i < n - m, n - m <= j <= m - 2}, i + j <= n - m - 1 + m - 2 = n - 3 < n - 2 = k, empty
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{n - m <= i <= m - 2, j < n - m}, i + j <= n - m - 1 + m - 2 = n - 3 < n - 2 = k, empty
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{i < n - m - 1, j = m - 1}, i + j <= n - m - 2 + m - 1 = n - 3 < n - 2 = k, empty
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{i = n - m - 1, j = m - 1}, i + j = n - m - 1 + m - 1 = n - 2 = k, keep all
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{i = m - 1, j < n - m - 1}, i + j <= m - 1 + n - m - 2 = n - 3 < n - 2 = k, empty
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{i = m - 1, j = n - m - 1}, i + j = m - 1 + n - m - 1 = n - 2 = k, keep all
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so the approx short product XY_k is
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{i >= n - m, j >= n - m} {i < n - m, m <= j}
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{m <= i, j < n - m} {i = n - m - 1, j = m - 1} {i = m - 1, j = n - m - 1}
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Now for {i < n - m, m <= j} with i + j > = k = n - 2, let u = i, v = j - m
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so we have {0 <= u < n - m, 0 <= v < n - m} with u + v >= n - m - 2
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which is the same short product
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Summary
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-----------
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Given n digit xp and yp,
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define mulshort_n(xp,yp,n) to be sum
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{i + j >= n - 2, and perhaps some i + j < n - 2} xp[i]yp[i]B^(i+j)
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choose m such that n+2 <= 2m and m < n then from above
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mulshort_n(xp, yp, n) = mul(xp + n - m, yp + n - m, m)B^(2n - 2m)
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+ mulshort_n(xp + m,yp, n - m)B^m
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+ mulshort_n(xp, yp + m, n - m)B^m
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+ xp[n - m - 1]yp[m - 1]B^(n - 2)
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+ xp[m - 1]yp[n - m - 1]B^(n - 2)
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and clearly when summing the above we can ignore any products from i + j < n - 2
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Theorem
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Let (zp, 2n) = mulshort_n(xp, yp, n)
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if zp[n - 1] + n - 2 < B then mulhigh_n(xp, yp, n) = (zp, 2n)
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*/
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/* (rp, 2n) = (xp, n)*(yp, n) / B^n */
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inline static void
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mpn_mulshort_n_basecase(mp_ptr rp, mp_srcptr xp, mp_srcptr yp, mp_size_t n)
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{
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mp_size_t i, k;
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#if GMP_NAIL_BITS==0
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mp_limb_t t1, t2, t3;
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#endif
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ASSERT(n >= 3); /* this restriction doesn't make a lot of sense in general */
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ASSERT_MPN(xp, n);
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ASSERT_MPN(yp, n);
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ASSERT(!MPN_OVERLAP_P (rp, 2 * n, xp, n));
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ASSERT(!MPN_OVERLAP_P (rp, 2 * n, yp, n));
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k = n - 2; /* so want short product sum_(i + j >= k) x[i]y[j]B^(i + j) */
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#if GMP_NAIL_BITS!=0
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rp[n] = mpn_mul_1(rp + k, xp + k, 2, yp[0]);
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#else
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umul_ppmm(t1, rp[k], xp[k], yp[0]);
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umul_ppmm(t3, t2, xp[k + 1], yp[0]);
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add_ssaaaa(rp[n], rp[k + 1], t3, t2, 0, t1);
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#endif
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for (i = 1; i <= n - 2; i++)
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rp[n + i] = mpn_addmul_1 (rp + k, xp + k - i, 2 + i, yp[i]);
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rp[n + n - 1] = mpn_addmul_1 (rp + n - 1, xp, n, yp[n - 1]);
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return;
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}
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/* (rp, 2n) = (xp, n)*(yp, n) */
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static void
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mpn_mulshort_n(mp_ptr rp, mp_srcptr xp, mp_srcptr yp, mp_size_t n)
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{
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mp_size_t m;
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mp_limb_t t;
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mp_ptr rpn2;
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ASSERT(n >= 1);
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ASSERT_MPN(xp, n);
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ASSERT_MPN(yp, n);
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ASSERT(!MPN_OVERLAP_P (rp, 2 * n, xp, n));
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ASSERT(!MPN_OVERLAP_P (rp, 2 * n, yp, n));
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if (BELOW_THRESHOLD(n, MULHIGH_BASECASE_THRESHOLD))
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{
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mpn_mul_basecase(rp, xp, n, yp, n);
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return;
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}
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if (BELOW_THRESHOLD (n, MULHIGH_DC_THRESHOLD))
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{
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mpn_mulshort_n_basecase(rp, xp, yp, n);
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return;
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}
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/* choose optimal m s.t. n + 2 <= 2m, m < n */
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ASSERT (n >= 4);
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m = 87 * n / 128;
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if (2 * m < n + 2)
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m = (n + 1) / 2 + 1;
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if (m >= n)
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m = n - 1;
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ASSERT (n + 2 <= 2 * m);
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ASSERT (m < n);
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rpn2 = rp + n - 2;
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mpn_mul_n (rp + n - m + n - m, xp + n - m, yp + n - m, m);
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mpn_mulshort_n (rp, xp, yp + m, n - m);
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ASSERT_NOCARRY (mpn_add (rpn2, rpn2, n + 2, rpn2 - m, n - m + 2));
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mpn_mulshort_n (rp, xp + m, yp, n - m);
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ASSERT_NOCARRY (mpn_add (rpn2, rpn2, n + 2, rpn2 - m, n - m + 2));
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umul_ppmm (rp[1], t, xp[m - 1], yp[n - m - 1] << GMP_NAIL_BITS);
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rp[0] = t >> GMP_NAIL_BITS;
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ASSERT_NOCARRY (mpn_add (rpn2, rpn2, n + 2, rp, 2));
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umul_ppmm (rp[1], t, xp[n - m - 1], yp[m - 1] << GMP_NAIL_BITS);
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rp[0] = t >> GMP_NAIL_BITS;
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ASSERT_NOCARRY (mpn_add (rpn2, rpn2, n + 2, rp, 2));
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return;
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}
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/* (rp, 2n) = (xp, n)*(yp, n) */
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void
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mpn_mulhigh_n (mp_ptr rp, mp_srcptr xp, mp_srcptr yp, mp_size_t n)
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{
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mp_limb_t t;
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ASSERT(n > 0);
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ASSERT_MPN(xp, n);
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ASSERT_MPN(yp, n);
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ASSERT(!MPN_OVERLAP_P(rp, 2 * n, xp, n));
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ASSERT(!MPN_OVERLAP_P(rp, 2 * n, yp, n));
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if (BELOW_THRESHOLD(n, MULHIGH_BASECASE_THRESHOLD))
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{
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mpn_mul_basecase(rp, xp, n, yp, n);
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return;
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}
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if (ABOVE_THRESHOLD (n, MULHIGH_MUL_THRESHOLD))
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{
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mpn_mul_n(rp, xp, yp, n);
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return;
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}
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mpn_mulshort_n(rp, xp, yp, n);
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t = rp[n - 1] + n - 2;
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if (UNLIKELY(t < n - 2))
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mpn_mul_n(rp, xp, yp, n);
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return;
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}
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