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

Published on 12 May 2024 09:40 PM
#Common lisp

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

Source code in the common programming language

(ql:quickload :computable-reals :silent t)
(use-package :computable-reals)
(setq *print-prec* 70)
(defparameter *iterations* 52)

; factorial using computable-reals multiplication op to keep precision
(defun !r (n)
  (let ((p 1))
    (loop for i from 2 to n doing (setq p (*r p i)))
    p))

; the nth integer term
(defun integral (n)
   (let* ((polynomial (+r (*r 532 n n) (*r 126 n) 9))
          (numer  (*r 32 (!r (*r 6 n)) polynomial))
          (denom  (*r 3 (expt-r (!r n) 6))))
    (/r  numer denom)))
      

; the exponent for 10 in the nth term of the series
(defun power (n) (- 3 (* 6 (1+ n))))

; the nth term of the series
(defun almkvist-giullera (n)
  (/r (integral n) (expt-r 10 (abs (power n)))))

; the sum of the first n terms
(defun almkvist-giullera-sigma (n)
  (let ((s 0)) 
    (loop for i from 0 to n doing (setq s (+r s (almkvist-giullera i))))
    s))

; the approximation to pi after n terms
(defun almkvist-giullera-pi (n)
  (sqrt-r (/r 1 (almkvist-giullera-sigma n))))

(format t "~A. ~44A~4A ~A~%" "N" "Integral part of Nth term" "×10^" "=Actual value of Nth term")
(loop for i from 0 to 9 doing
  (format t "~&~a. ~44d ~3d " i (integral i) (power i))
  (finish-output *standard-output*)
  (print-r (almkvist-giullera i) 50 nil))

(format t "~%~%Pi after ~a iterations: " *iterations*)
(print-r (almkvist-giullera-pi *iterations*) *print-prec*)

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