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Annotation of OpenXM_contrib/gmp/mpfr/pi.c, Revision 1.1.1.1

1.1       maekawa     1: /* mpfr_pi -- compute Pi
                      2:
                      3: Copyright (C) 1999 PolKA project, Inria Lorraine and Loria
                      4:
                      5: This file is part of the MPFR Library.
                      6:
                      7: The MPFR Library is free software; you can redistribute it and/or modify
                      8: it under the terms of the GNU Library General Public License as published by
                      9: the Free Software Foundation; either version 2 of the License, or (at your
                     10: option) any later version.
                     11:
                     12: The MPFR Library is distributed in the hope that it will be useful, but
                     13: WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
                     14: or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Library General Public
                     15: License for more details.
                     16:
                     17: You should have received a copy of the GNU Library General Public License
                     18: along with the MPFR Library; see the file COPYING.LIB.  If not, write to
                     19: the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
                     20: MA 02111-1307, USA. */
                     21:
                     22: #include <stdio.h>
                     23: #include <math.h>
                     24: #include "gmp.h"
                     25: #include "gmp-impl.h"
                     26: #include "longlong.h"
                     27: #include "mpfr.h"
                     28:
                     29: /*
                     30: Set x to the value of Pi to precision PREC(x) rounded to direction rnd_mode.
                     31: Use the formula giving the binary representation of Pi found by Simon Plouffe
                     32: and the Borwein's brothers:
                     33:
                     34:                    infinity    4         2         1         1
                     35:                     -----   ------- - ------- - ------- - -------
                     36:                      \      8 n + 1   8 n + 4   8 n + 5   8 n + 6
                     37:               Pi =    )     -------------------------------------
                     38:                      /                         n
                     39:                     -----                    16
                     40:                     n = 0
                     41:
                     42: i.e. Pi*16^N = S(N) + R(N) where
                     43: S(N) = sum(16^(N-n)*(4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6)), n=0..N-1)
                     44: R(N) = sum((4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6))/16^(n-N), n=N..infinity)
                     45:
                     46: Let f(n) = 4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6), we can show easily that
                     47: f(n) < 15/(64*n^2), so R(N) < sum(15/(64*n^2)/16^(n-N), n=N..infinity)
                     48:                             < 15/64/N^2*sum(1/16^(n-N), n=N..infinity)
                     49:                            = 1/4/N^2
                     50:
                     51: Now let S'(N) = sum(floor(16^(N-n)*(120*n^2+151*n+47),
                     52:   (512*n^4+1024*n^3+712*n^2+194*n+15)), n=0..N-1)
                     53:
                     54: S(N)-S'(N) <= sum(1, n=0..N-1) = N
                     55:
                     56: so Pi*16^N-S'(N) <= N+1 (as 1/4/N^2 < 1)
                     57: */
                     58:
                     59: mpfr_t __mpfr_pi; /* stored value of Pi */
                     60: int __mpfr_pi_prec=0; /* precision of stored value */
                     61: char __mpfr_pi_rnd; /* rounding mode of stored value */
                     62:
                     63: void
                     64: #if __STDC__
                     65: mpfr_pi(mpfr_ptr x, unsigned char rnd_mode)
                     66: #else
                     67: mpfr_pi(x, rnd_mode)
                     68:      mpfr_ptr x;
                     69:      unsigned char rnd_mode;
                     70: #endif
                     71: {
                     72:   int N, oldN, n, prec; mpz_t pi, num, den, d3, d2, tmp; mpfr_t y;
                     73:
                     74:   prec=PREC(x);
                     75:
                     76:   /* has stored value enough precision ? */
                     77:   if ((prec==__mpfr_pi_prec && rnd_mode==__mpfr_pi_rnd) ||
                     78:       (prec<=__mpfr_pi_prec &&
                     79:       mpfr_can_round(__mpfr_pi, __mpfr_pi_prec, __mpfr_pi_rnd, rnd_mode, prec)))
                     80:     {
                     81:       mpfr_set(x, __mpfr_pi, rnd_mode); return;
                     82:     }
                     83:
                     84:   /* need to recompute */
                     85:   N=1;
                     86:   do {
                     87:     oldN = N;
                     88:     N = (prec+3)/4 + (int)ceil(log((double)N+1.0)/log(2.0));
                     89:   } while (N != oldN);
                     90:   mpz_init(pi); mpz_init(num); mpz_init(den); mpz_init(d3); mpz_init(d2);
                     91:   mpz_init(tmp);
                     92:   mpz_set_ui(pi, 0);
                     93:   mpz_set_ui(num, 16); /* num(-1) */
                     94:   mpz_set_ui(den, 21); /* den(-1) */
                     95:   mpz_set_si(d3, -2454);
                     96:   mpz_set_ui(d2, 14736);
                     97:   /* invariants: num=120*n^2+151*n+47, den=512*n^4+1024*n^3+712*n^2+194*n+15
                     98:                  d3 = 2048*n^3+400*n-6, d2 = 6144*n^2-6144*n+2448
                     99:    */
                    100:   for (n=0; n<N; n++) {
                    101:     /* num(n)-num(n-1) = 240*n+31 */
                    102:     mpz_add_ui(num, num, 240*n+31); /* no overflow up to PREC=71M */
                    103:     /* d2(n) - d2(n-1) = 12288*(n-1) */
                    104:     if (n>0) mpz_add_ui(d2, d2, 12288*(n-1));
                    105:     else mpz_sub_ui(d2, d2, 12288);
                    106:     /* d3(n) - d3(n-1) = d2 */
                    107:     mpz_add(d3, d3, d2);
                    108:     /* den(n)-den(n-1) = 2048*n^3 + 400n - 6 = d3 */
                    109:     mpz_add(den, den, d3);
                    110:     mpz_mul_2exp(tmp, num, 4*(N-n));
                    111:     mpz_fdiv_q(tmp, tmp, den);
                    112:     mpz_add(pi, pi, tmp);
                    113:   }
                    114:   mpfr_set_z(x, pi, rnd_mode);
                    115:   mpfr_init2(y, mpfr_get_prec(x));
                    116:   mpz_add_ui(pi, pi, N+1);
                    117:   mpfr_set_z(y, pi, rnd_mode);
                    118:   if (mpfr_cmp(x, y) != 0) {
                    119:     fprintf(stderr, "does not converge\n"); exit(1);
                    120:   }
                    121:   EXP(x) -= 4*N;
                    122:   mpz_clear(pi); mpz_clear(num); mpz_clear(den); mpz_clear(d3); mpz_clear(d2);
                    123:   mpz_clear(tmp); mpfr_clear(y);
                    124:
                    125:   /* store computed value */
                    126:   if (__mpfr_pi_prec==0) mpfr_init2(__mpfr_pi, prec);
                    127:   else mpfr_set_prec(__mpfr_pi, prec);
                    128:   mpfr_set(__mpfr_pi, x, rnd_mode);
                    129:   __mpfr_pi_prec=prec;
                    130:   __mpfr_pi_rnd=rnd_mode;
                    131: }