/*
 * estimate pi using numerical integration
 * input is number of steps
 */
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
#include <math.h>
/* copied from not-strictly-standard part of math.h */
#define M_PI		3.14159265358979323846

double estimate_pi(int steps);

int main(void) {
	int steps;
	printf("how many steps?  maximum %d\n", INT_MAX);
	if ((scanf("%d", &steps) != 1) || (steps <= 0)) {
		printf("invalid input\n");
		return EXIT_FAILURE;
	}
	double pi = estimate_pi(steps);

	printf("results with %d steps:\n", steps);
	printf("computed pi = %g  (%17.15f)\n",pi, pi);
	printf("difference between computed pi and math.h M_PI = %g\n", 
			fabs(pi - M_PI));
	return EXIT_SUCCESS;
}

/* returns result of computation with "steps" steps */
double estimate_pi(int steps) {
	double sum = 0.0;
	double width = 1.0 / (double) steps;
	/* 
	 * we want to add up areas of rectangles:
	 * width of each is given above
	 * height of each is function value at midpoint
	 * we can factor out width and multiply after computing sum of heights
	 */
	for (int i = 0 ; i < steps; ++i) {
		/* midpoint */
		double x = (i+0.5)*width;
		sum += 4.0/(1.0+x*x);
	}
	return width * sum;
}