Almkvist Berndt 1988 begins with an investigation of why the agm is such an efficient algorithm, and proves that it converges quadratically. This is an efficient method to calculate 



π


{\displaystyle \pi }

.

With the same notations used in Arithmetic-geometric mean, we can summarize the paper by writing:





π
=



4


a
g
m

(
1
,
1

/



2



)

2




1
−

∑

n
=
1


∞



2

n
+
1


(

a

n


2


−

g

n


2


)





{\displaystyle \pi ={\frac {4\;\mathrm {agm} (1,1/{\sqrt {2}})^{2}}{1-\sum \limits _{n=1}^{\infty }2^{n+1}(a_{n}^{2}-g_{n}^{2})}}}



This allows you to make the approximation, for any large   N:





π
≈



4


a

N


2




1
−

∑

k
=
1


N



2

k
+
1


(

a

k


2


−

g

k


2


)





{\displaystyle \pi \approx {\frac {4\;a_{N}^{2}}{1-\sum \limits _{k=1}^{N}2^{k+1}(a_{k}^{2}-g_{k}^{2})}}}



The purpose of this task is to demonstrate how to use this approximation in order to compute a large number of decimals of 



π


{\displaystyle \pi }

.

#include "gmp.h"
 
void agm (const mpf_t in1, const mpf_t in2, mpf_t out1, mpf_t out2) {
	mpf_add (out1, in1, in2);
	mpf_div_ui (out1, out1, 2);
	mpf_mul (out2, in1, in2);
	mpf_sqrt (out2, out2);
}
 
int main (void) {
	mpf_set_default_prec (300000);
	mpf_t x0, y0, resA, resB, Z, var;
 
	mpf_init_set_ui (x0, 1);
	mpf_init_set_d (y0, 0.5);
	mpf_sqrt (y0, y0);
	mpf_init (resA);
	mpf_init (resB);
	mpf_init_set_d (Z, 0.25);
	mpf_init (var);
 
	int n = 1;
        int i;
	for(i=0; i<8; i++){
		agm(x0, y0, resA, resB);
		mpf_sub(var, resA, x0);
		mpf_mul(var, var, var);
		mpf_mul_ui(var, var, n);
		mpf_sub(Z, Z, var);
		n += n;
		agm(resA, resB, x0, y0);
		mpf_sub(var, x0, resA);
		mpf_mul(var, var, var);
		mpf_mul_ui(var, var, n);
		mpf_sub(Z, Z, var);
		n += n;
	}
	mpf_mul(x0, x0, x0);
	mpf_div(x0, x0, Z);
	gmp_printf ("%.100000Ff\n", x0);
	return 0;
}