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 <gmpxx.h>
#include <chrono>
 
using namespace std;
using namespace chrono;
 
void agm(mpf_class& rop1, mpf_class& rop2, const mpf_class& op1,
        const mpf_class& op2)
{
    rop1 = (op1 + op2) / 2;
    rop2 = op1 * op2;
    mpf_sqrt(rop2.get_mpf_t(), rop2.get_mpf_t());
}
 
int main(void)
{
    auto st = steady_clock::now();
    mpf_set_default_prec(300000);
    mpf_class x0, y0, resA, resB, Z;
 
    x0 = 1;
    y0 = 0.5;
    Z  = 0.25;
    mpf_sqrt(y0.get_mpf_t(), y0.get_mpf_t());
 
    int n = 1;
    for (int i = 0; i < 8; i++) {
        agm(resA, resB, x0, y0);
        Z -= n * (resA - x0) * (resA - x0);
        n *= 2;
 
        agm(x0, y0, resA, resB);
        Z -= n * (x0 - resA) * (x0 - resA);
        n *= 2;
    }
 
    x0 = x0 * x0 / Z;
    printf("Took %f ms for computation.\n", duration<double>(steady_clock::now() - st).count() * 1000.0);
    st = steady_clock::now();
    gmp_printf ("%.89412Ff\n", x0.get_mpf_t());
    printf("Took %f ms for output.\n", duration<double>(steady_clock::now() - st).count() * 1000.0);
    return 0;
}