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455 lines
14 KiB
C
455 lines
14 KiB
C
/* mpfr_zeta -- compute the Riemann Zeta function
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Copyright 2003, 2004, 2005, 2006, 2007 Free Software Foundation, Inc.
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Contributed by Jean-Luc Re'my and the Spaces project, INRIA Lorraine.
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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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/*
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Parameters:
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s - the input floating-point number
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n, p - parameters from the algorithm
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tc - an array of p floating-point numbers tc[1]..tc[p]
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Output:
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b is the result, i.e.
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sum(tc[i]*product((s+2j)*(s+2j-1)/n^2,j=1..i-1), i=1..p)*s*n^(-s-1)
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*/
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static void
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mpfr_zeta_part_b (mpfr_t b, mpfr_srcptr s, int n, int p, mpfr_t *tc)
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{
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mpfr_t s1, d, u;
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unsigned long n2;
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int l, t;
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MPFR_GROUP_DECL (group);
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if (p == 0)
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{
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MPFR_SET_ZERO (b);
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MPFR_SET_POS (b);
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return;
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}
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n2 = n * n;
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MPFR_GROUP_INIT_3 (group, MPFR_PREC (b), s1, d, u);
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/* t equals 2p-2, 2p-3, ... ; s1 equals s+t */
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t = 2 * p - 2;
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mpfr_set (d, tc[p], GMP_RNDN);
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for (l = 1; l < p; l++)
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{
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mpfr_add_ui (s1, s, t, GMP_RNDN); /* s + (2p-2l) */
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mpfr_mul (d, d, s1, GMP_RNDN);
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t = t - 1;
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mpfr_add_ui (s1, s, t, GMP_RNDN); /* s + (2p-2l-1) */
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mpfr_mul (d, d, s1, GMP_RNDN);
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t = t - 1;
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mpfr_div_ui (d, d, n2, GMP_RNDN);
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mpfr_add (d, d, tc[p-l], GMP_RNDN);
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/* since s is positive and the tc[i] have alternate signs,
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the following is unlikely */
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if (MPFR_UNLIKELY (mpfr_cmpabs (d, tc[p-l]) > 0))
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mpfr_set (d, tc[p-l], GMP_RNDN);
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}
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mpfr_mul (d, d, s, GMP_RNDN);
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mpfr_add (s1, s, __gmpfr_one, GMP_RNDN);
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mpfr_neg (s1, s1, GMP_RNDN);
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mpfr_ui_pow (u, n, s1, GMP_RNDN);
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mpfr_mul (b, d, u, GMP_RNDN);
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MPFR_GROUP_CLEAR (group);
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}
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/* Input: p - an integer
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Output: fills tc[1..p], tc[i] = bernoulli(2i)/(2i)!
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tc[1]=1/12, tc[2]=-1/720, tc[3]=1/30240, ...
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*/
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static void
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mpfr_zeta_c (int p, mpfr_t *tc)
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{
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mpfr_t d;
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int k, l;
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if (p > 0)
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{
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mpfr_init2 (d, MPFR_PREC (tc[1]));
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mpfr_div_ui (tc[1], __gmpfr_one, 12, GMP_RNDN);
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for (k = 2; k <= p; k++)
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{
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mpfr_set_ui (d, k-1, GMP_RNDN);
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mpfr_div_ui (d, d, 12*k+6, GMP_RNDN);
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for (l=2; l < k; l++)
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{
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mpfr_div_ui (d, d, 4*(2*k-2*l+3)*(2*k-2*l+2), GMP_RNDN);
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mpfr_add (d, d, tc[l], GMP_RNDN);
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}
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mpfr_div_ui (tc[k], d, 24, GMP_RNDN);
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MPFR_CHANGE_SIGN (tc[k]);
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}
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mpfr_clear (d);
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}
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}
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/* Input: s - a floating-point number
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n - an integer
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Output: sum - a floating-point number approximating sum(1/i^s, i=1..n-1) */
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static void
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mpfr_zeta_part_a (mpfr_t sum, mpfr_srcptr s, int n)
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{
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mpfr_t u, s1;
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int i;
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MPFR_GROUP_DECL (group);
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MPFR_GROUP_INIT_2 (group, MPFR_PREC (sum), u, s1);
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mpfr_neg (s1, s, GMP_RNDN);
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mpfr_ui_pow (u, n, s1, GMP_RNDN);
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mpfr_div_2ui (u, u, 1, GMP_RNDN);
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mpfr_set (sum, u, GMP_RNDN);
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for (i=n-1; i>1; i--)
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{
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mpfr_ui_pow (u, i, s1, GMP_RNDN);
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mpfr_add (sum, sum, u, GMP_RNDN);
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}
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mpfr_add (sum, sum, __gmpfr_one, GMP_RNDN);
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MPFR_GROUP_CLEAR (group);
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}
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/* Input: s - a floating-point number >= 1/2.
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rnd_mode - a rounding mode.
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Assumes s is neither NaN nor Infinite.
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Output: z - Zeta(s) rounded to the precision of z with direction rnd_mode
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*/
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static int
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mpfr_zeta_pos (mpfr_t z, mpfr_srcptr s, mp_rnd_t rnd_mode)
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{
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mpfr_t b, c, z_pre, f, s1;
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double beta, sd, dnep;
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mpfr_t *tc1;
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mp_prec_t precz, precs, d, dint;
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int p, n, l, add;
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int inex;
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MPFR_GROUP_DECL (group);
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MPFR_ZIV_DECL (loop);
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MPFR_ASSERTD (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0);
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precz = MPFR_PREC (z);
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precs = MPFR_PREC (s);
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/* Zeta(x) = 1+1/2^x+1/3^x+1/4^x+1/5^x+O(1/6^x)
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so with 2^(EXP(x)-1) <= x < 2^EXP(x)
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So for x > 2^3, k^x > k^8, so 2/k^x < 2/k^8
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Zeta(x) = 1 + 1/2^x*(1+(2/3)^x+(2/4)^x+...)
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= 1 + 1/2^x*(1+sum((2/k)^x,k=3..infinity))
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<= 1 + 1/2^x*(1+sum((2/k)^8,k=3..infinity))
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And sum((2/k)^8,k=3..infinity) = -257+128*Pi^8/4725 ~= 0.0438035
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So Zeta(x) <= 1 + 1/2^x*2 for x >= 8
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The error is < 2^(-x+1) <= 2^(-2^(EXP(x)-1)+1) */
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if (MPFR_GET_EXP (s) > 3)
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{
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mp_exp_t err;
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err = MPFR_GET_EXP (s) - 1;
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if (err > (mp_exp_t) (sizeof (mp_exp_t)*CHAR_BIT-2))
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err = MPFR_EMAX_MAX;
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else
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err = ((mp_exp_t)1) << err;
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err = 1 - (-err+1); /* GET_EXP(one) - (-err+1) = err :) */
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MPFR_FAST_COMPUTE_IF_SMALL_INPUT (z, __gmpfr_one, err, 0, 1,
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rnd_mode, {});
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}
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d = precz + MPFR_INT_CEIL_LOG2(precz) + 10;
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/* we want that s1 = s-1 is exact, i.e. we should have PREC(s1) >= EXP(s) */
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dint = (mpfr_uexp_t) MPFR_GET_EXP (s);
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mpfr_init2 (s1, MAX (precs, dint));
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inex = mpfr_sub (s1, s, __gmpfr_one, GMP_RNDN);
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MPFR_ASSERTD (inex == 0);
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/* case s=1 */
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if (MPFR_IS_ZERO (s1))
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{
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MPFR_SET_INF (z);
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MPFR_SET_POS (z);
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MPFR_ASSERTD (inex == 0);
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goto clear_and_return;
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}
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MPFR_GROUP_INIT_4 (group, MPFR_PREC_MIN, b, c, z_pre, f);
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MPFR_ZIV_INIT (loop, d);
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for (;;)
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{
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/* Principal loop: we compute, in z_pre,
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an approximation of Zeta(s), that we send to can_round */
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if (MPFR_GET_EXP (s1) <= -(mp_exp_t) ((mpfr_prec_t) (d-3)/2))
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/* Branch 1: when s-1 is very small, one
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uses the approximation Zeta(s)=1/(s-1)+gamma,
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where gamma is Euler's constant */
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{
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dint = MAX (d + 3, precs);
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MPFR_TRACE (printf ("branch 1\ninternal precision=%d\n", dint));
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MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f);
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mpfr_div (z_pre, __gmpfr_one, s1, GMP_RNDN);
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mpfr_const_euler (f, GMP_RNDN);
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mpfr_add (z_pre, z_pre, f, GMP_RNDN);
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}
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else /* Branch 2 */
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{
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size_t size;
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MPFR_TRACE (printf ("branch 2\n"));
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/* Computation of parameters n, p and working precision */
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dnep = (double) d * LOG2;
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sd = mpfr_get_d (s, GMP_RNDN);
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/* beta = dnep + 0.61 + sd * log (6.2832 / sd);
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but a larger value is ok */
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#define LOG6dot2832 1.83787940484160805532
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beta = dnep + 0.61 + sd * (LOG6dot2832 - LOG2 *
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__gmpfr_floor_log2 (sd));
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if (beta <= 0.0)
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{
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p = 0;
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/* n = 1 + (int) (exp ((dnep - LOG2) / sd)); */
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n = 1 + (int) __gmpfr_ceil_exp2 ((d - 1.0) / sd);
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}
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else
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{
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p = 1 + (int) beta / 2;
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n = 1 + (int) ((sd + 2.0 * (double) p - 1.0) / 6.2832);
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}
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MPFR_TRACE (printf ("\nn=%d\np=%d\n",n,p));
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/* add = 4 + floor(1.5 * log(d) / log (2)).
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We should have add >= 10, which is always fulfilled since
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d = precz + 11 >= 12, thus ceil(log2(d)) >= 4 */
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add = 4 + (3 * MPFR_INT_CEIL_LOG2 (d)) / 2;
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MPFR_ASSERTD(add >= 10);
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dint = d + add;
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if (dint < precs)
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dint = precs;
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MPFR_TRACE (printf("internal precision=%d\n",dint));
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size = (p + 1) * sizeof(mpfr_t);
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tc1 = (mpfr_t*) (*__gmp_allocate_func) (size);
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for (l=1; l<=p; l++)
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mpfr_init2 (tc1[l], dint);
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MPFR_GROUP_REPREC_4 (group, dint, b, c, z_pre, f);
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MPFR_TRACE (printf ("precision of z =%d\n", precz));
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/* Computation of the coefficients c_k */
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mpfr_zeta_c (p, tc1);
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/* Computation of the 3 parts of the fonction Zeta. */
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mpfr_zeta_part_a (z_pre, s, n);
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mpfr_zeta_part_b (b, s, n, p, tc1);
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/* s1 = s-1 is already computed above */
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mpfr_div (c, __gmpfr_one, s1, GMP_RNDN);
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mpfr_ui_pow (f, n, s1, GMP_RNDN);
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mpfr_div (c, c, f, GMP_RNDN);
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MPFR_TRACE (MPFR_DUMP (c));
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mpfr_add (z_pre, z_pre, c, GMP_RNDN);
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mpfr_add (z_pre, z_pre, b, GMP_RNDN);
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for (l=1; l<=p; l++)
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mpfr_clear (tc1[l]);
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(*__gmp_free_func) (tc1, size);
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/* End branch 2 */
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}
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MPFR_TRACE (MPFR_DUMP (z_pre));
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if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, d-3, precz, rnd_mode)))
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break;
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MPFR_ZIV_NEXT (loop, d);
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}
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MPFR_ZIV_FREE (loop);
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inex = mpfr_set (z, z_pre, rnd_mode);
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MPFR_GROUP_CLEAR (group);
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clear_and_return:
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mpfr_clear (s1);
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return inex;
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}
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int
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mpfr_zeta (mpfr_t z, mpfr_srcptr s, mp_rnd_t rnd_mode)
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{
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mpfr_t z_pre, s1, y, p;
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double sd, eps, m1, c;
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long add;
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mp_prec_t precz, prec1, precs, precs1;
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int inex;
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MPFR_GROUP_DECL (group);
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MPFR_ZIV_DECL (loop);
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_LOG_FUNC (("s[%#R]=%R rnd=%d", s, s, rnd_mode),
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("z[%#R]=%R inexact=%d", z, z, inex));
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/* Zero, Nan or Inf ? */
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if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (s)))
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{
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if (MPFR_IS_NAN (s))
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{
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MPFR_SET_NAN (z);
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MPFR_RET_NAN;
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}
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else if (MPFR_IS_INF (s))
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{
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if (MPFR_IS_POS (s))
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return mpfr_set_ui (z, 1, GMP_RNDN); /* Zeta(+Inf) = 1 */
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MPFR_SET_NAN (z); /* Zeta(-Inf) = NaN */
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MPFR_RET_NAN;
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}
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else /* s iz zero */
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{
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MPFR_ASSERTD (MPFR_IS_ZERO (s));
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mpfr_set_ui (z, 1, rnd_mode);
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mpfr_div_2ui (z, z, 1, rnd_mode);
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MPFR_CHANGE_SIGN (z);
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MPFR_RET (0);
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}
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}
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/* s is neither Nan, nor Inf, nor Zero */
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/* check tiny s: we have zeta(s) = -1/2 - 1/2 log(2 Pi) s + ... around s=0,
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and for |s| <= 0.074, we have |zeta(s) + 1/2| <= |s|.
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Thus if |s| <= 1/4*ulp(1/2), we can deduce the correct rounding
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(the 1/4 covers the case where |zeta(s)| < 1/2 and rounding to nearest).
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A sufficient condition is that EXP(s) + 1 < -PREC(z). */
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if (MPFR_EXP(s) + 1 < - (mp_exp_t) MPFR_PREC(z))
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{
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int signs = MPFR_SIGN(s);
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mpfr_set_si_2exp (z, -1, -1, rnd_mode); /* -1/2 */
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if ((rnd_mode == GMP_RNDU || rnd_mode == GMP_RNDZ) && signs < 0)
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{
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mpfr_nextabove (z); /* z = -1/2 + epsilon */
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inex = 1;
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}
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else if (rnd_mode == GMP_RNDD && signs > 0)
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{
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mpfr_nextbelow (z); /* z = -1/2 - epsilon */
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inex = -1;
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}
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else
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{
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if (rnd_mode == GMP_RNDU) /* s > 0: z = -1/2 */
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inex = 1;
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else if (rnd_mode == GMP_RNDD)
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inex = -1; /* s < 0: z = -1/2 */
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else /* (GMP_RNDZ and s > 0) or GMP_RNDN: z = -1/2 */
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inex = (signs > 0) ? 1 : -1;
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}
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return mpfr_check_range (z, inex, rnd_mode);
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}
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/* Check for case s= -2n */
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if (MPFR_IS_NEG (s))
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{
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mpfr_t tmp;
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tmp[0] = *s;
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MPFR_EXP (tmp) = MPFR_EXP (s) - 1;
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if (mpfr_integer_p (tmp))
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{
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MPFR_SET_ZERO (z);
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MPFR_SET_POS (z);
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MPFR_RET (0);
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}
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}
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MPFR_SAVE_EXPO_MARK (expo);
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/* Compute Zeta */
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if (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0) /* Case s >= 1/2 */
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inex = mpfr_zeta_pos (z, s, rnd_mode);
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else /* use reflection formula
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zeta(s) = 2^s*Pi^(s-1)*sin(Pi*s/2)*gamma(1-s)*zeta(1-s) */
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{
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precz = MPFR_PREC (z);
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precs = MPFR_PREC (s);
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/* Precision precs1 needed to represent 1 - s, and s + 2,
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without any truncation */
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precs1 = precs + 2 + MAX (0, - MPFR_GET_EXP (s));
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sd = mpfr_get_d (s, GMP_RNDN) - 1.0;
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if (sd < 0.0)
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sd = -sd; /* now sd = abs(s-1.0) */
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/* Precision prec1 is the precision on elementary computations;
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it ensures a final precision prec1 - add for zeta(s) */
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/* eps = pow (2.0, - (double) precz - 14.0); */
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eps = __gmpfr_ceil_exp2 (- (double) precz - 14.0);
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m1 = 1.0 + MAX(1.0 / eps, 2.0 * sd) * (1.0 + eps);
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c = (1.0 + eps) * (1.0 + eps * MAX(8.0, m1));
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/* add = 1 + floor(log(c*c*c*(13 + m1))/log(2)); */
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add = __gmpfr_ceil_log2 (c * c * c * (13.0 + m1));
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prec1 = precz + add;
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prec1 = MAX (prec1, precs1) + 10;
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MPFR_GROUP_INIT_4 (group, prec1, z_pre, s1, y, p);
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MPFR_ZIV_INIT (loop, prec1);
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for (;;)
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{
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mpfr_sub (s1, __gmpfr_one, s, GMP_RNDN);/* s1 = 1-s */
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|
mpfr_zeta_pos (z_pre, s1, GMP_RNDN); /* zeta(1-s) */
|
|
mpfr_gamma (y, s1, GMP_RNDN); /* gamma(1-s) */
|
|
if (MPFR_IS_INF (y)) /* Zeta(s) < 0 for -4k-2 < s < -4k,
|
|
Zeta(s) > 0 for -4k < s < -4k+2 */
|
|
{
|
|
MPFR_SET_INF (z_pre);
|
|
mpfr_div_2ui (s1, s, 2, GMP_RNDN); /* s/4, exact */
|
|
mpfr_frac (s1, s1, GMP_RNDN); /* exact, -1 < s1 < 0 */
|
|
if (mpfr_cmp_si_2exp (s1, -1, -1) > 0)
|
|
MPFR_SET_NEG (z_pre);
|
|
else
|
|
MPFR_SET_POS (z_pre);
|
|
break;
|
|
}
|
|
mpfr_mul (z_pre, z_pre, y, GMP_RNDN); /* gamma(1-s)*zeta(1-s) */
|
|
mpfr_const_pi (p, GMP_RNDD);
|
|
mpfr_mul (y, s, p, GMP_RNDN);
|
|
mpfr_div_2ui (y, y, 1, GMP_RNDN); /* s*Pi/2 */
|
|
mpfr_sin (y, y, GMP_RNDN); /* sin(Pi*s/2) */
|
|
mpfr_mul (z_pre, z_pre, y, GMP_RNDN);
|
|
mpfr_mul_2ui (y, p, 1, GMP_RNDN); /* 2*Pi */
|
|
mpfr_neg (s1, s1, GMP_RNDN); /* s-1 */
|
|
mpfr_pow (y, y, s1, GMP_RNDN); /* (2*Pi)^(s-1) */
|
|
mpfr_mul (z_pre, z_pre, y, GMP_RNDN);
|
|
mpfr_mul_2ui (z_pre, z_pre, 1, GMP_RNDN);
|
|
|
|
if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz,
|
|
rnd_mode)))
|
|
break;
|
|
|
|
MPFR_ZIV_NEXT (loop, prec1);
|
|
MPFR_GROUP_REPREC_4 (group, prec1, z_pre, s1, y, p);
|
|
}
|
|
MPFR_ZIV_FREE (loop);
|
|
inex = mpfr_set (z, z_pre, rnd_mode);
|
|
MPFR_GROUP_CLEAR (group);
|
|
}
|
|
|
|
MPFR_SAVE_EXPO_FREE (expo);
|
|
return mpfr_check_range (z, inex, rnd_mode);
|
|
}
|