posted on 2025-05-09, 08:45authored byJ. M. Borwein, P. B. Borwein
We present a quadratically converging algorithm for π based on a formula of Legendre's for complete elliptic integrals of modulus sin(π/12) and the arithmetic-geometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent large-scale calculations of π.