efee5258bc
The MPFR library is a C library for multiple-precision floating-point computations with exact rounding (also called correct rounding). It is based on the GMP multiple-precision library and should replace the MPF class in further releases of GMP. GCC >= 4.2 requires MPFR.
153 lines
4.2 KiB
C
153 lines
4.2 KiB
C
/* mpfr_const_catalan -- compute Catalan's constant.
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Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 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 GNU MPFR Library.
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The GNU 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 3 of the License, or (at your
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option) any later version.
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The GNU 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, 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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/* Declare the cache */
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MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_catalan, mpfr_const_catalan_internal);
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/* Set User Interface */
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#undef mpfr_const_catalan
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int
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mpfr_const_catalan (mpfr_ptr x, mpfr_rnd_t rnd_mode) {
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return mpfr_cache (x, __gmpfr_cache_const_catalan, rnd_mode);
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}
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/* return T, Q such that T/Q = sum(k!^2/(2k)!/(2k+1)^2, k=n1..n2-1) */
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static void
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S (mpz_t T, mpz_t P, mpz_t Q, unsigned long n1, unsigned long n2)
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{
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if (n2 == n1 + 1)
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{
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if (n1 == 0)
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{
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mpz_set_ui (P, 1);
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mpz_set_ui (Q, 1);
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}
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else
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{
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mpz_set_ui (P, 2 * n1 - 1);
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mpz_mul_ui (P, P, n1);
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mpz_ui_pow_ui (Q, 2 * n1 + 1, 2);
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mpz_mul_2exp (Q, Q, 1);
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}
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mpz_set (T, P);
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}
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else
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{
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unsigned long m = (n1 + n2) / 2;
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mpz_t T2, P2, Q2;
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S (T, P, Q, n1, m);
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mpz_init (T2);
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mpz_init (P2);
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mpz_init (Q2);
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S (T2, P2, Q2, m, n2);
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mpz_mul (T, T, Q2);
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mpz_mul (T2, T2, P);
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mpz_add (T, T, T2);
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mpz_mul (P, P, P2);
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mpz_mul (Q, Q, Q2);
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mpz_clear (T2);
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mpz_clear (P2);
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mpz_clear (Q2);
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}
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}
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/* Don't need to save/restore exponent range: the cache does it.
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Catalan's constant is G = sum((-1)^k/(2*k+1)^2, k=0..infinity).
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We compute it using formula (31) of Victor Adamchik's page
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"33 representations for Catalan's constant"
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http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
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G = Pi/8*log(2+sqrt(3)) + 3/8*sum(k!^2/(2k)!/(2k+1)^2,k=0..infinity)
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*/
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int
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mpfr_const_catalan_internal (mpfr_ptr g, mpfr_rnd_t rnd_mode)
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{
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mpfr_t x, y, z;
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mpz_t T, P, Q;
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mpfr_prec_t pg, p;
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int inex;
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MPFR_ZIV_DECL (loop);
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MPFR_GROUP_DECL (group);
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MPFR_LOG_FUNC (("rnd_mode=%d", rnd_mode), ("g[%#R]=%R inex=%d", g, g, inex));
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/* Here are the WC (max prec = 100.000.000)
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Once we have found a chain of 11, we only look for bigger chain.
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Found 3 '1' at 0
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Found 5 '1' at 9
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Found 6 '0' at 34
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Found 9 '1' at 176
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Found 11 '1' at 705
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Found 12 '0' at 913
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Found 14 '1' at 12762
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Found 15 '1' at 152561
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Found 16 '0' at 171725
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Found 18 '0' at 525355
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Found 20 '0' at 529245
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Found 21 '1' at 6390133
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Found 22 '0' at 7806417
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Found 25 '1' at 11936239
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Found 27 '1' at 51752950
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*/
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pg = MPFR_PREC (g);
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p = pg + MPFR_INT_CEIL_LOG2 (pg) + 7;
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MPFR_GROUP_INIT_3 (group, p, x, y, z);
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mpz_init (T);
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mpz_init (P);
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mpz_init (Q);
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MPFR_ZIV_INIT (loop, p);
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for (;;) {
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mpfr_sqrt_ui (x, 3, MPFR_RNDU);
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mpfr_add_ui (x, x, 2, MPFR_RNDU);
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mpfr_log (x, x, MPFR_RNDU);
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mpfr_const_pi (y, MPFR_RNDU);
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mpfr_mul (x, x, y, MPFR_RNDN);
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S (T, P, Q, 0, (p - 1) / 2);
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mpz_mul_ui (T, T, 3);
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mpfr_set_z (y, T, MPFR_RNDU);
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mpfr_set_z (z, Q, MPFR_RNDD);
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mpfr_div (y, y, z, MPFR_RNDN);
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mpfr_add (x, x, y, MPFR_RNDN);
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mpfr_div_2ui (x, x, 3, MPFR_RNDN);
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if (MPFR_LIKELY (MPFR_CAN_ROUND (x, p - 5, pg, rnd_mode)))
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break;
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MPFR_ZIV_NEXT (loop, p);
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MPFR_GROUP_REPREC_3 (group, p, x, y, z);
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}
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MPFR_ZIV_FREE (loop);
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inex = mpfr_set (g, x, rnd_mode);
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MPFR_GROUP_CLEAR (group);
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mpz_clear (T);
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mpz_clear (P);
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mpz_clear (Q);
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return inex;
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}
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