this is not as cool as i had hoped because it just hardcodes the first 14 terms of the continued fraction for pi, i.e. 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1
(i did implement a generalized continued fraction which isn't hardcoded, but (A) it converges much more slowly and (B) it doesn't actually find the "best" convergents (e.g. 355/113) so it just didn't seem very useful.)
i should have put "useful" in scare-quotes