1 /* mpfr_const_pi -- compute Pi
3 Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramel projects, INRIA.
6 This file is part of the GNU MPFR Library.
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
23 #include "mpfr-impl.h"
25 /* Declare the cache */
26 #ifndef MPFR_USE_LOGGING
27 MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_pi
, mpfr_const_pi_internal
);
29 MPFR_DECL_INIT_CACHE(__gmpfr_normal_pi
, mpfr_const_pi_internal
);
30 MPFR_DECL_INIT_CACHE(__gmpfr_logging_pi
, mpfr_const_pi_internal
);
31 mpfr_cache_ptr MPFR_THREAD_ATTR __gmpfr_cache_const_pi
= __gmpfr_normal_pi
;
34 /* Set User Interface */
37 mpfr_const_pi (mpfr_ptr x
, mpfr_rnd_t rnd_mode
) {
38 return mpfr_cache (x
, __gmpfr_cache_const_pi
, rnd_mode
);
41 /* Don't need to save/restore exponent range: the cache does it */
43 mpfr_const_pi_internal (mpfr_ptr x
, mpfr_rnd_t rnd_mode
)
46 mpfr_prec_t px
, p
, cancel
, k
, kmax
;
51 (("rnd_mode=%d", rnd_mode
),
52 ("x[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(x
), mpfr_log_prec
, x
, inex
));
56 /* we need 9*2^kmax - 4 >= px+2*kmax+8 */
57 for (kmax
= 2; ((px
+ 2 * kmax
+ 12) / 9) >> kmax
; kmax
++);
59 p
= px
+ 3 * kmax
+ 14; /* guarantees no recomputation for px <= 10000 */
67 MPFR_ZIV_INIT (loop
, p
);
69 mpfr_set_ui (a
, 1, MPFR_RNDN
); /* a = 1 */
70 mpfr_set_ui (A
, 1, MPFR_RNDN
); /* A = a^2 = 1 */
71 mpfr_set_ui_2exp (B
, 1, -1, MPFR_RNDN
); /* B = b^2 = 1/2 */
72 mpfr_set_ui_2exp (D
, 1, -2, MPFR_RNDN
); /* D = 1/4 */
80 /* invariant: 1/2 <= B <= A <= a < 1 */
81 mpfr_add (S
, A
, B
, MPFR_RNDN
); /* 1 <= S <= 2 */
82 mpfr_div_2ui (S
, S
, 2, MPFR_RNDN
); /* exact, 1/4 <= S <= 1/2 */
83 mpfr_sqrt (b
, B
, MPFR_RNDN
); /* 1/2 <= b <= 1 */
84 mpfr_add (ap
, a
, b
, MPFR_RNDN
); /* 1 <= ap <= 2 */
85 mpfr_div_2ui (ap
, ap
, 1, MPFR_RNDN
); /* exact, 1/2 <= ap <= 1 */
86 mpfr_mul (Ap
, ap
, ap
, MPFR_RNDN
); /* 1/4 <= Ap <= 1 */
87 mpfr_sub (Bp
, Ap
, S
, MPFR_RNDN
); /* -1/4 <= Bp <= 3/4 */
88 mpfr_mul_2ui (Bp
, Bp
, 1, MPFR_RNDN
); /* -1/2 <= Bp <= 3/2 */
89 mpfr_sub (S
, Ap
, Bp
, MPFR_RNDN
);
90 MPFR_ASSERTN (mpfr_cmp_ui (S
, 1) < 0);
91 cancel
= mpfr_cmp_ui (S
, 0) ? (mpfr_uexp_t
) -mpfr_get_exp(S
) : p
;
92 /* MPFR_ASSERTN (cancel >= px || cancel >= 9 * (1 << k) - 4); */
93 mpfr_mul_2ui (S
, S
, k
, MPFR_RNDN
);
94 mpfr_sub (D
, D
, S
, MPFR_RNDN
);
95 /* stop when |A_k - B_k| <= 2^(k-p) i.e. cancel >= p-k */
104 mpfr_div (A
, B
, D
, MPFR_RNDN
);
106 /* MPFR_ASSERTN(p >= 2 * k + 8); */
107 if (MPFR_LIKELY (MPFR_CAN_ROUND (A
, p
- 2 * k
- 8, px
, rnd_mode
)))
111 MPFR_ZIV_NEXT (loop
, p
);
112 mpfr_set_prec (a
, p
);
113 mpfr_set_prec (A
, p
);
114 mpfr_set_prec (B
, p
);
115 mpfr_set_prec (D
, p
);
116 mpfr_set_prec (S
, p
);
118 MPFR_ZIV_FREE (loop
);
119 inex
= mpfr_set (x
, A
, rnd_mode
);