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


void menu_method();
void menu_sigdig();
double power (int sigdig);
double sqroot(double num);
double Gauss_Legendre(double digit);
double BBP(double digit);
double Area(double digit);


int main void()
{
	int method, sigdig;     //declare variables 
	double pi, digit;


	while(1)
	{ 
		do{
		menu_method();
		scanf("%d", &method);
		fflush(stdin);
	}while(method<1||method>4)(method!=(int)method));

	if (method==4) {
		printf(Good Bye!!!\n");
		exit(1);
	}

	do{			//choose significant digits 
		menu_sigdig();	
		scanf("%d", &sigdig);
		fflush(stdin);
	}while (sigdig<0||sigdig>99);
	digit= power(sigdig);

	swith(method)
	{
	case 1: pi=Gauss_Legendre(digit); break;
	case 2: pi=BBP(digit); break;
	case 3: pi= Area(digit); break;
	}
	printf("\nthe approximation of pi is << %e >>\n", pi);  
	}

	return 0;
}

void menu_method() 
{
	
	double pi=1, temp, diff, a_temp, a=(double)1, b=(double)1/sqroot(2), c, t=(double)1/4, p=(double)1;
	int flag=0, i=2;
	pi= ((a+b)*(a+b))/(4*t);
	printf("\ni=1>> pi = %e\n", pi);
	do{

		a_temp=(a+b)/2;
		b=sqroot(a*b);
		t=t-p*(a-a_temp)*(a-a_temp);
		p=2*p;
		a=a_temp;
		temp=((a+b)*(a+b))/(4*t);
		diff = temp-pi;
		if(diff < 0)
		diff = diff*-1;
		if(diff<=digit)
		flag = 1;
		pi=temp;
		printf("i = %d >> pi = %e\n", i++, pi);
	}while (flag==0);
	return pi;
}

double BBP(double digit)
{
	int i, k=1, flag=0;
	double a=1.00, b, pi, temp;
	pi= 4-((double)1/2)-((double)1/5)-((double)1/6);
	printf("\nk = 0 >> pi = %e\n", pi);
	do{
		for (i=o; i<k; i++)
			a *= (double)16;
		b = 1/a; 
		temp = ((double)4/(8*k+1))-((double)2/(8*k+4))-((double)1/(8*k+5))-((double)1/(8*k+6));
		temp = temp*b;
		if(temp<= digit)
			flag =1;
		pi= pi + temp;
		printf("k = %d >> pi = %e\n", k, pi);
		k++;
	}while(flag==0); 

}

double Area(double digit)
{
	double pi=-1, temp, x, y, diff;
	int r=10, i, area, flag=0;
	do{
		area = 0;
		for (i=0; i<(2*r+1)*(2*r+1); i++)
		{
			x = ((double)rand()/RAND_MAX)*2*r - r;
			y = ((double)rand()/RAND_MAX)*2*r - r;
			if(((x*x)+(y*y))<= (r*r))
			area++;
		}

		temp=(double)area/(r*r);
		diff = temp-pi;
		if(diff < 0)
			diff = diff*-1;
		if(diff<=digit)
			flag= 1;
		pi= temp;
		printf("r = %d, area = %d, >> pi = %e, diff = %e\n", r, area, pi, diff);
		r++;
		}while (flag == 0);

return pi;