/** ***************************************************************************
 * @remark approximate Pi adding n terms from an arithmetic series            * 
 *                                                                            *
 *   pi is the sum of terns Tn   for i = 0, 1, . . .                          *
 *      where t0 = T1 = 1 and T(n) = 2nT(n-1)/(2(2n+1))                       *
 *                                                                            *
 *     for 0 <= i <= numTerms, this sum is evaluated                          *
 *         from largest term to smallest, and                                 *
 *         from smallest term to largest                                      *
 *                                                                            *
 * @author Henry M. Walker                                                    *
 * @file  pi-approx.c                                                         *
 * @date  Summer, 2023                                                        *
 *                                                                            *
 * @remark References                                                         *
 * @remark Consequences of Data Representation on Programming                 *
  https://blue.cs.sonoma.edu/~hwalker/courses/415-sonoma.sp23/readings/data-rep-consequences.html
 * @remark Wilkpedia article on Leibniz formula for π                         *
 *   combining terms pairwise from a Taylor's series
  https://en.wikipedia.org/wiki/Leibniz_formula_for_π
  * @remark Series that converge to π quickly                                 *
 https://math.stackexchange.com/questions/14113/series-that-converge-to-pi-quickly
 *                                                                            *
 *****************************************************************************/

#include <stdio.h>

int main ( ) {
  printf ("approximating Pi, based on a Taylor series and algebra\n");
 
  double actualPi = 3.1415926535897932384626433;
  int numTerms;
  printf ("Enter number of terms: ");
  scanf ("%d", &numTerms);

  // Set up array for terms of series
  double  T[numTerms +1];
  T[0] = 2.0;
  T[1] = 2.0/3.0;
  int i;
  for (i = 2; i <= numTerms; i++) {
    T[i] = 2.0 * i * T[i-1] / (2.0 * (2.0*i+1.0));
  }

  printf ("first two terms computed:  T[0]:  %15.12lf,  T[1]:   %15.12lf,\n", T[0], T[1]);
  printf (" last two terms computed:  T[%d]: %15.12lf,  T[%d]:  %15.12lf,\n\n",
          numTerms-1, T[numTerms-1], numTerms, T[numTerms]); 
  
 //print table headers
  printf ("                   approximation    actual value (to 16 places)    error\n\n");

  // adding terms with i from 0 to numTerms
  double sum = 0;
  for (i = 0 ; i < numTerms ; i++) {
    sum += T[i];
   }
  printf ("number terms:  %d\n", numTerms+1);
  printf (" i goes 0 to %d: %18.16lf", numTerms, sum); 
  printf (" %20.16lf", actualPi);
  printf (" %20.16lf\n", actualPi - sum);


  sum = 0;
  for (i = numTerms-1 ; i >= 0 ; i--) {
    sum += T[i];
   }
  printf (" i goes %d to 0: %18.16lf", numTerms, sum); 
  printf (" %20.16lf", actualPi);
  printf (" %20.16lf\n\n", actualPi - sum);  
  
  return 0;
}