Use Monte Carlo Simulation to approximate pi by considering the percentage of po

Question

Use Monte Carlo Simulation to approximate pi by considering the percentage of points with coordinates (x,y) generated randomly by choosing x and y both between 0 and 1, inside the unit quarted circle

Q: X^2 + y^2 = 1, x>=0 y>=0

when the quarter circle is taken to be inside the square

S: 0<=x<=1 and 0<=y<=1

Use approximation pi/4 = area Q / area S.

Hint: Do the simulation 10000 times and then 100000. (meaning choose the value of x and y randomly 10000 times and then 100000 times all between 0 and 1. find out how many times the pair of (x,y) falls inside the Q and divide that by the total number of trials. Does the zccuracy of pi increase as the number of trials increase?)

Explanation / Answer

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

double pi(double tolerance)
{
          double x, y, val, error;
          unsigned long sampled = 0, hit = 0, i;

          do {
                                        for (i = 1000000; i; i--, sampled++) {
                               x = rand() / (RAND_MAX + 1.0);
                               y = rand() / (RAND_MAX + 1.0);
                               if (x * x + y * y < 1) hit ++;
                    }

                    val = (double) hit / sampled;
                    error = sqrt(val * (1 - val) / sampled) * 4;
                    val *= 4;
                     fprintf(stderr, "Pi = %f +/- %5.3e at %ldM samples. ",
                    val, error, sampled/1000000);
          } while (!hit || error > tolerance);

          return val;
}

int main()
{
          printf("Pi is %f ", pi(3e-4));

          return 0;
}

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