How to resolve the algorithm Almkvist-Giullera formula for pi step by step in the dc programming language

Published on 12 May 2024 09:40 PM
#Dc

How to resolve the algorithm Almkvist-Giullera formula for pi step by step in the dc programming language

Table of Contents

Problem Statement

The Almkvist-Giullera formula for calculating   1/π2   is based on the Calabi-Yau differential equations of order 4 and 5,   which were originally used to describe certain manifolds in string theory.

The formula is:

This formula can be used to calculate the constant   π-2,   and thus to calculate   π. Note that, because the product of all terms but the power of 1000 can be calculated as an integer, the terms in the series can be separated into a large integer term: multiplied by a negative integer power of 10:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Almkvist-Giullera formula for pi step by step in the dc programming language

Source code in the dc programming language

[* factorial *]sz
[ 1 Sp [ d lp * sp 1 - d 1 

[* nth integral term *]sz
[ sn 32 6 ln * lFx 532 ln * ln * 126 ln * + 9 + * * 3 ln lFx 6 ^ * / ]sI

[* nth exponent of 10 *]sz
[ 1 + 6 * 3 r - ]sE

[* nth term in series *]sz
[ d lIx r 10 r lEx _1 * ^ / ]sA

[* sum of the first n terms *]sz
[ [li lAx ls + ss li 1 - d si 0 r !

[* approximation of pi after n terms *]sz
[ lSx 1 r / v ]sP

[* count digits in a number *]sz
[sn 0 sd lCx ld]sD
[ld 1 + sd ln 10 0k / d sn 0 !=C]sC

[* print a number in a given column width *]sz
[sw d lDx si lw li 
[[ ]n li 1 + si lw li 

[* main loop: print values for first 10 terms *]sz
[N. Integral part of Nth term .................. × 10^ =Actual value of Nth term]p
0 sj
[
  lj n [. ]n
  lj lIx 0k 1 / 44 lWx [ ]n
  lj lEx 4 lWx [ ]n
  lj 99k lAx 50k 1 / p
  lj 1 + d sj 10 >M
] sM
lMx

[]p

[* print resulting value of pi to 70 places *]sz
[Pi after ]n 52n [ iterations:]p
99k 52 lPx 70k 1 / p

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