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 }

.

(scl 40)
 
(de pi ()
   (let 
      (A 1.0  N 1.0  Z 0.25
         G (/ (* 1.0 1.0) (sqrt 2.0 1.0)) )
      (use (X1 X2 V)
         (do 18
            (setq 
               X1 (/ (* (+ A G) 0.5) 1.0)
               X2 (sqrt (* A G))
               V (- X1 A)
               Z (- Z (/ (* (/ (* V V) 1.0) N) 1.0))
               N (+ N N)
               A X1
               G X2 ) ) )
      (round (/ (* A A) Z) 40)) )
 
(println (pi))
 
(bye)