the intermediate rational.
(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 again.
* (time (machin-pi 10000)) Evaluation took: 0.662 seconds of real time 0.634038 seconds of total run time (0.495454 user, 0.138584 system) [ Run times consist of 0.233 seconds GC time, and 0.402 seconds non-GC time. ] 95.77% CPU 1,491,387,858 processor cycles 109,530,592 bytes consed 31415926535897932384626433832795028841971693993751058209749445923078164062862089 ...
Algorithmic optimizations would take us much further. For example the Gauss–Legendre or Salamin–Brent formula.
Then there is the fastest known(at the turn of the millenium), Chudnovsky’s formula :