Update 2009-07-23 : Faster version in CL and a Haskell version.

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A trivial approximation using the Leibniz formula.

(defun leibniz-pi() (labels ((local-pi(sum n) (if (< n (expt 10 6)) (local-pi (+ sum (/ (expt -1 n) (+ (* 2 n) 1))) (+ n 1)) sum))) (* 4 (local-pi 0f0 0f0))))

And here’s a longer version(but faster and more precise) using Machin’s formula with fixed point arithmetic to x digits.

(defun machin-pi (digits) "Calculates PI digits using fixed point arithmetic and Machin's formula with double recursion" (labels ((arccot-minus (xsq n xpower) (let ((term (floor (/ xpower n)))) (if (= term 0) 0 (- (arccot-plus xsq (+ n 2) (floor (/ xpower xsq))) term)))) (arccot-plus (xsq n xpower) (let ((term (floor (/ xpower n)))) (if (= term 0) 0 (+ (arccot-minus xsq (+ n 2) (floor (/ xpower xsq))) term)))) (arccot (x unity) (let ((xpower (floor (/ unity x)))) (arccot-plus (* x x) 1 xpower)))) (let* ((unity (expt 10 (+ digits 10))) (thispi (* 4 (- (* 4 (arccot 5 unity)) (arccot 239 unity))))) (floor (/ thispi (expt 10 10))))))

The first 10000 digits.

* (time (pidigits 10000)) Evaluation took: 2.496 seconds of real time 1.149998 seconds of total run time (0.974647 user, 0.175351 system) [ Run times consist of 0.325 seconds GC time, and 0.825 seconds non-GC time. ] 46.07% CPU 5,626,335,988 processor cycles 217,893,792 bytes consed 31415926535897932384626433832795028841971693993751058209749445923078164062862089 ...