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/* mpfr_const_pi -- compute Pi
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#include "mpfr-impl.h"
/* Declare the cache */
#ifndef MPFR_USE_LOGGING
MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_pi, mpfr_const_pi_internal);
#else
MPFR_DECL_INIT_CACHE(__gmpfr_normal_pi, mpfr_const_pi_internal);
MPFR_DECL_INIT_CACHE(__gmpfr_logging_pi, mpfr_const_pi_internal);
mpfr_cache_ptr MPFR_THREAD_ATTR __gmpfr_cache_const_pi = __gmpfr_normal_pi;
#endif
/* Set User Interface */
#undef mpfr_const_pi
int
mpfr_const_pi (mpfr_ptr x, mpfr_rnd_t rnd_mode) {
return mpfr_cache (x, __gmpfr_cache_const_pi, rnd_mode);
}
/* Don't need to save/restore exponent range: the cache does it */
int
mpfr_const_pi_internal (mpfr_ptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t a, A, B, D, S;
mpfr_prec_t px, p, cancel, k, kmax;
MPFR_ZIV_DECL (loop);
int inex;
MPFR_LOG_FUNC
(("rnd_mode=%d", rnd_mode),
("x[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(x), mpfr_log_prec, x, inex));
px = MPFR_PREC (x);
/* we need 9*2^kmax - 4 >= px+2*kmax+8 */
for (kmax = 2; ((px + 2 * kmax + 12) / 9) >> kmax; kmax ++);
p = px + 3 * kmax + 14; /* guarantees no recomputation for px <= 10000 */
mpfr_init2 (a, p);
mpfr_init2 (A, p);
mpfr_init2 (B, p);
mpfr_init2 (D, p);
mpfr_init2 (S, p);
MPFR_ZIV_INIT (loop, p);
for (;;) {
mpfr_set_ui (a, 1, MPFR_RNDN); /* a = 1 */
mpfr_set_ui (A, 1, MPFR_RNDN); /* A = a^2 = 1 */
mpfr_set_ui_2exp (B, 1, -1, MPFR_RNDN); /* B = b^2 = 1/2 */
mpfr_set_ui_2exp (D, 1, -2, MPFR_RNDN); /* D = 1/4 */
#define b B
#define ap a
#define Ap A
#define Bp B
for (k = 0; ; k++)
{
/* invariant: 1/2 <= B <= A <= a < 1 */
mpfr_add (S, A, B, MPFR_RNDN); /* 1 <= S <= 2 */
mpfr_div_2ui (S, S, 2, MPFR_RNDN); /* exact, 1/4 <= S <= 1/2 */
mpfr_sqrt (b, B, MPFR_RNDN); /* 1/2 <= b <= 1 */
mpfr_add (ap, a, b, MPFR_RNDN); /* 1 <= ap <= 2 */
mpfr_div_2ui (ap, ap, 1, MPFR_RNDN); /* exact, 1/2 <= ap <= 1 */
mpfr_mul (Ap, ap, ap, MPFR_RNDN); /* 1/4 <= Ap <= 1 */
mpfr_sub (Bp, Ap, S, MPFR_RNDN); /* -1/4 <= Bp <= 3/4 */
mpfr_mul_2ui (Bp, Bp, 1, MPFR_RNDN); /* -1/2 <= Bp <= 3/2 */
mpfr_sub (S, Ap, Bp, MPFR_RNDN);
MPFR_ASSERTN (mpfr_cmp_ui (S, 1) < 0);
cancel = mpfr_cmp_ui (S, 0) ? (mpfr_uexp_t) -mpfr_get_exp(S) : p;
/* MPFR_ASSERTN (cancel >= px || cancel >= 9 * (1 << k) - 4); */
mpfr_mul_2ui (S, S, k, MPFR_RNDN);
mpfr_sub (D, D, S, MPFR_RNDN);
/* stop when |A_k - B_k| <= 2^(k-p) i.e. cancel >= p-k */
if (cancel + k >= p)
break;
}
#undef b
#undef ap
#undef Ap
#undef Bp
mpfr_div (A, B, D, MPFR_RNDN);
/* MPFR_ASSERTN(p >= 2 * k + 8); */
if (MPFR_LIKELY (MPFR_CAN_ROUND (A, p - 2 * k - 8, px, rnd_mode)))
break;
p += kmax;
MPFR_ZIV_NEXT (loop, p);
mpfr_set_prec (a, p);
mpfr_set_prec (A, p);
mpfr_set_prec (B, p);
mpfr_set_prec (D, p);
mpfr_set_prec (S, p);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (x, A, rnd_mode);
mpfr_clear (a);
mpfr_clear (A);
mpfr_clear (B);
mpfr_clear (D);
mpfr_clear (S);
return inex;
}
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