Almkvist-Giullera formula for pi: Difference between revisions

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; the nth integer term

(defun integral (n)

~~(/r~~ (*~~r 32~~ (!r (*r 6 n)) (+r (*r 532 n n) (*r 126 n) 9~~)) (*r 3 (expt-r (!r n) 6))~~))

(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

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; the nth term of the series

(defun a-g (n)

(defun almkvist-giullera (n)

(/r (integral n) (expt-r 10 (abs (power n)))))

; the sum of the first n terms

(defun a-g-sigma (n)

(defun almkvist-giullera-sigma (n)

(let ((s 0))

(loop for i from 0 to n doing (setq s (+r s (a-g i))))

(loop for i from 0 to n doing (setq s (+r s (almkvist-giullera i))))

s))

; the approximation to pi after n terms

(defun a-g-pi (n)

(defun almkvist-giullera-pi (n)

(sqrt-r (/r 1 (a-g-sigma 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")

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(format t "~&~a. ~44d ~3d " i (integral i) (power i))

(finish-output *standard-output*)

(print-r (a-g i) 50 nil))

(print-r (almkvist-giullera i) 50 nil))

(format t "~%~%Pi after ~a iterations: " *iterations*)

(print-r (a-g-pi *iterations*) *print-prec*)~~</lang>~~

(print-r (almkvist-giullera-pi *iterations*) *print-prec*)

</lang>