Calculating the digits of pi
10th March 2026
One of the examples I wanted to include for my Simple infinite precision arithmetic package is a function to calculate an arbitrary number of digits of π. I finally got it working, so here it is.
Digits of π
The formula is based on the following equation found by John Machin, an 18th century astronomer and mathematician:
If you're wondering where the values 5 and 239 come from, see the Wikipedia article [1].
The two arctangents can be calculated using the arctangent series:
Machin's formula converges extremely rapidly, making it a widely used method of calculating π to a large number of digits.
Here's the function arctan that iteratively calculates the arctangent of 1/x with a scale factor of n:
(defun arctan (n x)
(let* ((i 0)
(xn (div n x))
(x2 (mul x x))
(term xn)
result)
(loop
(setq result
(if (evenp i) (add result term)
(sub result term)))
(incf i)
(setq xn (div xn x2))
(setq term (div xn (bign (1+ (* 2 i)))))
(when (null term) (return result)))))
So to calculate arctan 1/5 to 8 digits evaluate:
> (pri (arctan (big 100000000) (big 5))) 19739557
Compare this with the calculation using floating-point arithmetic:
> (atan (/ 1 5)) 0.19739557
Finally, the function machin calculates π to d digits using Machin's formula:
(defun machin (d)
(let ((n (iex 10 d)))
(mul (big 4) (sub (mul (big 4) (arctan n (big 5))) (arctan n (big 239))))))
Here's π to 80 digits:
> (pri (machin 80)) 314159265358979323846264338327950288419716939937510582097494459230781640628620948
- ^ Machin-like formula on Wikipedia.
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