michael@0: /*
michael@0:  * pi.c
michael@0:  *
michael@0:  * Compute pi to an arbitrary number of digits.  Uses Machin's formula,
michael@0:  * like everyone else on the planet:
michael@0:  * 
michael@0:  *    pi = 16 * arctan(1/5) - 4 * arctan(1/239)
michael@0:  *
michael@0:  * This is pretty effective for up to a few thousand digits, but it
michael@0:  * gets pretty slow after that.
michael@0:  *
michael@0:  * This Source Code Form is subject to the terms of the Mozilla Public
michael@0:  * License, v. 2.0. If a copy of the MPL was not distributed with this
michael@0:  * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
michael@0: 
michael@0: #include <stdio.h>
michael@0: #include <stdlib.h>
michael@0: #include <string.h>
michael@0: #include <limits.h>
michael@0: #include <time.h>
michael@0: 
michael@0: #include "mpi.h"
michael@0: 
michael@0: mp_err arctan(mp_digit mul, mp_digit x, mp_digit prec, mp_int *sum);
michael@0: 
michael@0: int main(int argc, char *argv[])
michael@0: {
michael@0:   mp_err       res;
michael@0:   mp_digit     ndigits;
michael@0:   mp_int       sum1, sum2;
michael@0:   clock_t      start, stop;
michael@0:   int          out = 0;
michael@0: 
michael@0:   /* Make the user specify precision on the command line */
michael@0:   if(argc < 2) {
michael@0:     fprintf(stderr, "Usage: %s <num-digits>\n", argv[0]);
michael@0:     return 1;
michael@0:   }
michael@0: 
michael@0:   if((ndigits = abs(atoi(argv[1]))) == 0) {
michael@0:     fprintf(stderr, "%s: you must request at least 1 digit\n", argv[0]);
michael@0:     return 1;
michael@0:   }
michael@0: 
michael@0:   start = clock();
michael@0:   mp_init(&sum1); mp_init(&sum2);
michael@0: 
michael@0:   /* sum1 = 16 * arctan(1/5)  */
michael@0:   if((res = arctan(16, 5, ndigits, &sum1)) != MP_OKAY) {
michael@0:     fprintf(stderr, "%s: arctan: %s\n", argv[0], mp_strerror(res));
michael@0:     out = 1; goto CLEANUP;
michael@0:   }
michael@0: 
michael@0:   /* sum2 = 4 * arctan(1/239) */
michael@0:   if((res = arctan(4, 239, ndigits, &sum2)) != MP_OKAY) {
michael@0:     fprintf(stderr, "%s: arctan: %s\n", argv[0], mp_strerror(res));
michael@0:     out = 1; goto CLEANUP;
michael@0:   }
michael@0: 
michael@0:   /* pi = sum1 - sum2         */
michael@0:   if((res = mp_sub(&sum1, &sum2, &sum1)) != MP_OKAY) {
michael@0:     fprintf(stderr, "%s: mp_sub: %s\n", argv[0], mp_strerror(res));
michael@0:     out = 1; goto CLEANUP;
michael@0:   }
michael@0:   stop = clock();
michael@0: 
michael@0:   /* Write the output in decimal */
michael@0:   {
michael@0:     char  *buf = malloc(mp_radix_size(&sum1, 10));
michael@0: 
michael@0:     if(buf == NULL) {
michael@0:       fprintf(stderr, "%s: out of memory\n", argv[0]);
michael@0:       out = 1; goto CLEANUP;
michael@0:     }
michael@0:     mp_todecimal(&sum1, buf);
michael@0:     printf("%s\n", buf);
michael@0:     free(buf);
michael@0:   }
michael@0: 
michael@0:   fprintf(stderr, "Computation took %.2f sec.\n", 
michael@0: 	  (double)(stop - start) / CLOCKS_PER_SEC);
michael@0: 
michael@0:  CLEANUP:
michael@0:   mp_clear(&sum1);
michael@0:   mp_clear(&sum2);
michael@0: 
michael@0:   return out;
michael@0: 
michael@0: }
michael@0: 
michael@0: /* Compute sum := mul * arctan(1/x), to 'prec' digits of precision */
michael@0: mp_err arctan(mp_digit mul, mp_digit x, mp_digit prec, mp_int *sum)
michael@0: {
michael@0:   mp_int   t, v;
michael@0:   mp_digit q = 1, rd;
michael@0:   mp_err   res;
michael@0:   int      sign = 1;
michael@0: 
michael@0:   prec += 3;  /* push inaccuracies off the end */
michael@0: 
michael@0:   mp_init(&t); mp_set(&t, 10);
michael@0:   mp_init(&v); 
michael@0:   if((res = mp_expt_d(&t, prec, &t)) != MP_OKAY ||  /* get 10^prec    */
michael@0:      (res = mp_mul_d(&t, mul, &t)) != MP_OKAY ||    /* ... times mul  */
michael@0:      (res = mp_mul_d(&t, x, &t)) != MP_OKAY)        /* ... times x    */
michael@0:     goto CLEANUP;
michael@0: 
michael@0:   /*
michael@0:     The extra multiplication by x in the above takes care of what
michael@0:     would otherwise have to be a special case for 1 / x^1 during the
michael@0:     first loop iteration.  A little sneaky, but effective.
michael@0: 
michael@0:     We compute arctan(1/x) by the formula:
michael@0: 
michael@0:          1     1       1       1
michael@0: 	 - - ----- + ----- - ----- + ...
michael@0: 	 x   3 x^3   5 x^5   7 x^7
michael@0: 
michael@0:     We multiply through by 'mul' beforehand, which gives us a couple
michael@0:     more iterations and more precision
michael@0:    */
michael@0: 
michael@0:   x *= x; /* works as long as x < sqrt(RADIX), which it is here */
michael@0: 
michael@0:   mp_zero(sum);
michael@0: 
michael@0:   do {
michael@0:     if((res = mp_div_d(&t, x, &t, &rd)) != MP_OKAY)
michael@0:       goto CLEANUP;
michael@0: 
michael@0:     if(sign < 0 && rd != 0)
michael@0:       mp_add_d(&t, 1, &t);
michael@0: 
michael@0:     if((res = mp_div_d(&t, q, &v, &rd)) != MP_OKAY)
michael@0:       goto CLEANUP;
michael@0: 
michael@0:     if(sign < 0 && rd != 0)
michael@0:       mp_add_d(&v, 1, &v);
michael@0: 
michael@0:     if(sign > 0)
michael@0:       res = mp_add(sum, &v, sum);
michael@0:     else
michael@0:       res = mp_sub(sum, &v, sum);
michael@0: 
michael@0:     if(res != MP_OKAY)
michael@0:       goto CLEANUP;
michael@0: 
michael@0:     sign *= -1;
michael@0:     q += 2;
michael@0: 
michael@0:   } while(mp_cmp_z(&t) != 0);
michael@0: 
michael@0:   /* Chop off inaccurate low-order digits */
michael@0:   mp_div_d(sum, 1000, sum, NULL);
michael@0: 
michael@0:  CLEANUP:
michael@0:   mp_clear(&v);
michael@0:   mp_clear(&t);
michael@0: 
michael@0:   return res;
michael@0: }
michael@0: 
michael@0: /*------------------------------------------------------------------------*/
michael@0: /* HERE THERE BE DRAGONS                                                  */