1  /* mpfr_const_pi -- compute Pi
       2  
       3  Copyright 1999-2023 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[%Pd]=%.*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  }