/* 
 * Program to approximate pi using numerical integration, starting with initial
 * number of slices and repeating with increasing number of slices until 
 * "convergence" (previous and current values "close").
 */ 
#include <stdio.h>
#include <math.h>

/* best known value of pi (surprisingly, no standard M_PI defined!) */
#define M_PI acos(-1.0)

double pi_approx(long nslices);

int main(void) {

	long num_slices;
	double threshold;
	printf("enter initial number of slices, threshold\n");
	if (scanf("%ld %lf", &num_slices, &threshold) != 2) {
		printf("wrong format\n");
		return 1;
	}
	double prev;
	double current = 0.0;
	do {
		prev = current;
		current = pi_approx(num_slices);
		printf("with %ld slices approximation is %.15f\n", num_slices, current);
		num_slices *= 2;
	} while (fabs(current-prev) > threshold);
	printf("difference from best available constant %g\n", 
            fabs(current - M_PI));

	return 0;
}
double pi_approx(long nslices) {

	double width = 1.0 / nslices;
	double sum = 0.0;

	/* compute heights of rectangles and add */
	for ( int i = 0; i < nslices; ++i) {
		double mid = i*width + 0.5*width;
		double f_at_mid = 4.0/(1 + mid*mid);
		sum += f_at_mid;
	}
	/* area is sum of heights times width */
	return sum * width;
}