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 }

.

let limit = 10000 and n = 2800
let x = Array.make (n+1) 2000
 
let rec g j sum =
  if j < 1 then sum else
    let sum = sum * j + limit * x.(j) in
    x.(j) <- sum mod (j * 2 - 1);
    g (j - 1) (sum / (j * 2 - 1))
 
let rec f i carry =
  if i = 0 then () else
    let sum = g i 0 in
    Printf.printf "%04d" (carry + sum / limit);
    f (i - 14) (sum mod limit)
 
let () =
  f n 0;
  print_newline()
 