/*
Approximating Pi

Pi ~= 2^n * sin(pi/2^n); //this uses pi in definition, so we have to calculate sin(pi/2^n)

sin(pi/2^n)                          2^n
sqrt(2)/2                            4
sqrt(2-sqrt(2))/2                    8
sqrt(2-sqrt(2+sqrt(2)))/2            16
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2    32

--recursive part is how many times +sqrt(2) is added
*/


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

#define N 20     //limited because of the double data type and that we use 2^N

double getapprox(int n)
{
      double ret;
      if(n==4) return 0;      
      return sqrt(2+getapprox(n/2));
}

double approxPI(int n)
{
      double val = pow(2.0, n);
      double res = getapprox(val);
      return val*0.5*sqrt(2.0-res);
}

int main()
{
    double x = approxPI(N);
    printf("pi at %d approximation is %.10lf\n", N, x);
    system("pause");
}