I played with this myself. Imperito's formula of '4 arctan (1)', given appropriate precision, gives results accurate to, at least, ten digits.


Using a FOR loop, and the formula of:

4 * [1 + ((-1)^n)/((2*n) + 1)]

(start the iterations at '1', not '0'.), I ran 20,000,000,000 iterations (3.75 hours) and didn't achieve the same precision. Close, but no cigar. (Low boredom threshold. )

Just a little boost for built-in functions.