1 /* mpfr_const_pi -- compute Pi 2 3 Copyright 1999-2020 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramba projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 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. 12 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. 17 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 https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23 #include "mpfr-impl.h" 24 25 /* Declare the cache */ 26 #ifndef MPFR_USE_LOGGING 27 MPFR_DECL_INIT_CACHE (__gmpfr_cache_const_pi, mpfr_const_pi_internal) 28 #else 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_THREAD_VAR (mpfr_cache_ptr, __gmpfr_cache_const_pi, __gmpfr_normal_pi) 32 #endif 33 34 /* Set User Interface */ 35 #undef mpfr_const_pi 36 int 37 mpfr_const_pi (mpfr_ptr x, mpfr_rnd_t rnd_mode) { 38 return mpfr_cache (x, __gmpfr_cache_const_pi, rnd_mode); 39 } 40 41 /* The algorithm used here is taken from Section 8.2.5 of the book 42 "Fast Algorithms: A Multitape Turing Machine Implementation" 43 by A. Schönhage, A. F. W. Grotefeld and E. Vetter, 1994. 44 It is a clever form of Brent-Salamin formula. */ 45 46 /* Don't need to save/restore exponent range: the cache does it */ 47 int 48 mpfr_const_pi_internal (mpfr_ptr x, mpfr_rnd_t rnd_mode) 49 { 50 mpfr_t a, A, B, D, S; 51 mpfr_prec_t px, p, cancel, k, kmax; 52 MPFR_GROUP_DECL (group); 53 MPFR_ZIV_DECL (loop); 54 int inex; 55 56 MPFR_LOG_FUNC 57 (("rnd_mode=%d", rnd_mode), 58 ("x[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(x), mpfr_log_prec, x, inex)); 59 60 px = MPFR_PREC (x); 61 62 /* we need 9*2^kmax - 4 >= px+2*kmax+8 */ 63 for (kmax = 2; ((px + 2 * kmax + 12) / 9) >> kmax; kmax ++); 64 65 p = px + 3 * kmax + 14; /* guarantees no recomputation for px <= 10000 */ 66 67 MPFR_GROUP_INIT_5 (group, p, a, A, B, D, S); 68 69 MPFR_ZIV_INIT (loop, p); 70 for (;;) { 71 mpfr_set_ui (a, 1, MPFR_RNDN); /* a = 1 */ 72 mpfr_set_ui (A, 1, MPFR_RNDN); /* A = a^2 = 1 */ 73 mpfr_set_ui_2exp (B, 1, -1, MPFR_RNDN); /* B = b^2 = 1/2 */ 74 mpfr_set_ui_2exp (D, 1, -2, MPFR_RNDN); /* D = 1/4 */ 75 76 #define b B 77 #define ap a 78 #define Ap A 79 #define Bp B 80 for (k = 0; ; k++) 81 { 82 /* invariant: 1/2 <= B <= A <= a < 1 */ 83 mpfr_add (S, A, B, MPFR_RNDN); /* 1 <= S <= 2 */ 84 mpfr_div_2ui (S, S, 2, MPFR_RNDN); /* exact, 1/4 <= S <= 1/2 */ 85 mpfr_sqrt (b, B, MPFR_RNDN); /* 1/2 <= b <= 1 */ 86 mpfr_add (ap, a, b, MPFR_RNDN); /* 1 <= ap <= 2 */ 87 mpfr_div_2ui (ap, ap, 1, MPFR_RNDN); /* exact, 1/2 <= ap <= 1 */ 88 mpfr_sqr (Ap, ap, MPFR_RNDN); /* 1/4 <= Ap <= 1 */ 89 mpfr_sub (Bp, Ap, S, MPFR_RNDN); /* -1/4 <= Bp <= 3/4 */ 90 mpfr_mul_2ui (Bp, Bp, 1, MPFR_RNDN); /* -1/2 <= Bp <= 3/2 */ 91 mpfr_sub (S, Ap, Bp, MPFR_RNDN); 92 MPFR_ASSERTD (mpfr_cmp_ui (S, 1) < 0); 93 cancel = MPFR_NOTZERO (S) ? (mpfr_uexp_t) -mpfr_get_exp(S) : p; 94 /* MPFR_ASSERTN (cancel >= px || cancel >= 9 * (1 << k) - 4); */ 95 mpfr_mul_2ui (S, S, k, MPFR_RNDN); 96 mpfr_sub (D, D, S, MPFR_RNDN); 97 /* stop when |A_k - B_k| <= 2^(k-p) i.e. cancel >= p-k */ 98 if (cancel >= p - k) 99 break; 100 } 101 #undef b 102 #undef ap 103 #undef Ap 104 #undef Bp 105 106 mpfr_div (A, B, D, MPFR_RNDN); 107 108 /* MPFR_ASSERTN(p >= 2 * k + 8); */ 109 if (MPFR_LIKELY (MPFR_CAN_ROUND (A, p - 2 * k - 8, px, rnd_mode))) 110 break; 111 112 p += kmax; 113 MPFR_ZIV_NEXT (loop, p); 114 MPFR_GROUP_REPREC_5 (group, p, a, A, B, D, S); 115 } 116 MPFR_ZIV_FREE (loop); 117 inex = mpfr_set (x, A, rnd_mode); 118 119 MPFR_GROUP_CLEAR (group); 120 121 return inex; 122 } 123