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99d8158635
git-svn-id: file:///srv/svn/repos/haiku/buildtools/trunk@29042 a95241bf-73f2-0310-859d-f6bbb57e9c96
225 lines
6.9 KiB
C
225 lines
6.9 KiB
C
/* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
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Copyright 2005, 2006, 2007 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the MPFR Library.
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The 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 2.1 of the License, or (at your
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option) any later version.
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The 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 MPFR Library; see the file COPYING.LIB. If not, write to
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the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
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MA 02110-1301, USA. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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int
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mpfr_zeta_ui (mpfr_ptr z, unsigned long m, mp_rnd_t r)
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{
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MPFR_ZIV_DECL (loop);
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if (m == 0)
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{
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mpfr_set_ui (z, 1, r);
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mpfr_div_2ui (z, z, 1, r);
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MPFR_CHANGE_SIGN (z);
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MPFR_RET (0);
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}
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else if (m == 1)
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{
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MPFR_SET_INF (z);
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MPFR_SET_POS (z);
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return 0;
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}
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else /* m >= 2 */
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{
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mp_prec_t p = MPFR_PREC(z);
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unsigned long n, k, err, kbits;
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mpz_t d, t, s, q;
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mpfr_t y;
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int inex;
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if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
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2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
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i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
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{
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if (m == 2) /* necessarily p=2 */
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return mpfr_set_ui_2exp (z, 13, -3, r);
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else if (r == GMP_RNDZ || r == GMP_RNDD || (r == GMP_RNDN && m > p))
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{
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mpfr_set_ui (z, 1, r);
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return -1;
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}
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else
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{
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mpfr_set_ui (z, 1, r);
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mpfr_nextabove (z);
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return 1;
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}
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}
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/* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
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and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
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mpfr_init2 (y, 31);
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if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
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{
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/* the following is a lower bound for log(3)/log(2) */
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mpfr_set_str_binary (y, "1.100101011100000000011010001110");
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mpfr_mul_ui (y, y, m, GMP_RNDZ); /* lower bound for log2(3^m) */
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if (mpfr_cmp_ui (y, p + 2) >= 0)
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{
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mpfr_clear (y);
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mpfr_set_ui (z, 1, GMP_RNDZ);
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mpfr_div_2ui (z, z, m, GMP_RNDZ);
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mpfr_add_ui (z, z, 1, GMP_RNDZ);
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if (r != GMP_RNDU)
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return -1;
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mpfr_nextabove (z);
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return 1;
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}
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}
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mpz_init (s);
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mpz_init (d);
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mpz_init (t);
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mpz_init (q);
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p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
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p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
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MPFR_ZIV_INIT (loop, p);
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for(;;)
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{
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/* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
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n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
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err = n + 4;
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mpfr_set_prec (y, p);
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/* computation of the d[k] */
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mpz_set_ui (s, 0);
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mpz_set_ui (t, 1);
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mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
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mpz_set (d, t);
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for (k = n; k > 0; k--)
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{
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count_leading_zeros (kbits, k);
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kbits = BITS_PER_MP_LIMB - kbits;
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/* if k^m is too large, use mpz_tdiv_q */
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if (m * kbits > 2 * BITS_PER_MP_LIMB)
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{
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/* if we know in advance that k^m > d, then floor(d/k^m) will
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be zero below, so there is no need to compute k^m */
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kbits = (kbits - 1) * m + 1;
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/* k^m has at least kbits bits */
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if (kbits > mpz_sizeinbase (d, 2))
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mpz_set_ui (q, 0);
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else
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{
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mpz_ui_pow_ui (q, k, m);
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mpz_tdiv_q (q, d, q);
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}
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}
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else /* use several mpz_tdiv_q_ui calls */
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{
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unsigned long km = k, mm = m - 1;
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while (mm > 0 && km < ULONG_MAX / k)
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{
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km *= k;
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mm --;
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}
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mpz_tdiv_q_ui (q, d, km);
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while (mm > 0)
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{
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km = k;
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mm --;
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while (mm > 0 && km < ULONG_MAX / k)
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{
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km *= k;
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mm --;
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}
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mpz_tdiv_q_ui (q, q, km);
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}
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}
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if (k % 2)
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mpz_add (s, s, q);
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else
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mpz_sub (s, s, q);
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/* we have d[k] = sum(t[i], i=k+1..n)
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with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
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t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
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#if (BITS_PER_MP_LIMB == 32)
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#define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
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#elif (BITS_PER_MP_LIMB == 64)
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#define KMAX 3037000500
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#endif
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#ifdef KMAX
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if (k <= KMAX)
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mpz_mul_ui (t, t, k * (2 * k - 1));
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else
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#endif
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{
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mpz_mul_ui (t, t, k);
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mpz_mul_ui (t, t, 2 * k - 1);
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}
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mpz_div_2exp (t, t, 1);
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if (n < 1UL << (BITS_PER_MP_LIMB / 2))
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/* (n - k + 1) * (n + k - 1) < n^2 */
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mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
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else
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{
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mpz_divexact_ui (t, t, n - k + 1);
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mpz_divexact_ui (t, t, n + k - 1);
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}
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mpz_add (d, d, t);
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}
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/* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
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mpz_div_2exp (t, s, m - 1);
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do
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{
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err ++;
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mpz_add (s, s, t);
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mpz_div_2exp (t, t, m - 1);
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}
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while (mpz_cmp_ui (t, 0) > 0);
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/* divide by d[n] */
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mpz_mul_2exp (s, s, p);
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mpz_tdiv_q (s, s, d);
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mpfr_set_z (y, s, GMP_RNDN);
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mpfr_div_2ui (y, y, p, GMP_RNDN);
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err = MPFR_INT_CEIL_LOG2 (err);
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if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
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break;
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MPFR_ZIV_NEXT (loop, p);
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}
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MPFR_ZIV_FREE (loop);
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mpz_clear (d);
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mpz_clear (t);
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mpz_clear (q);
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mpz_clear (s);
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inex = mpfr_set (z, y, r);
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mpfr_clear (y);
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return inex;
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}
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}
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