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 }

.

/// calculate pi with algebraic/geometric mean
pub fn pi(n: usize) -> f64 {
    let mut a : f64 = 1.0;
    let two : f64= 2.0;
    let mut g = 1.0 / two.sqrt();
    let mut s = 0.0;
    let mut k = 1;
    while k<=n  {
 
        let a1 = (a+g)/two;
        let g1 = (a*g).sqrt();
        a = a1;
        g = g1;
        s += (a.powi(2)-g.powi(2)) * two.powi((k+1) as i32);
        k += 1;
 
 
    }
 
    4.0 * a.powi(2) / (1.0-s)
}
 