$ \pi $ at the limits of computation

Abstract

Before 1976, all record-broken computation for the digits of $ \pi $ completely depended on arctangent-type identities; of them the most celebrated is John Machin's identity ${ \pi\over 4}=4\tan^{-1} ({1\over 5})\-tan^{-1}({1\over 239})$, discovered in 1706. But, in 1976, Eugene Salamin moved in a powerful heavy artillery. The method is an adaptation of an algorithm discovered by Gauss for the evaluation of elliptic integrals. Then, a new era comes. In 1983, Y.~Kanada, Y.~Tanura, S.~Yoshino and Y.~Ushiro used Gauss-Legendre-Brent-Salamin algorithm to calculate $ \pi $ to $2^{24}$ (16,777,216) decimal places on a HITAC M-280H supercomputer and used an FFT-based fast multiplication. In this article, we present an easy-to-understand explanation of this amazing method.