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 ...```