A Very Rapidly Convergent Product Expansion For

There has recently been considerable interest in an essentially quadratic method for computing n. The algorithm, first suggested by Salamin [6], is based upon an identity known to Gauss [-3, p. 377], [4]. This iteration has been used by two Japanese researchers, Y. Tamura and Y. Kanada, to compute 223 decimal digits of n in under 7 hours. They have now successfully computed 224 digits (more than 16.7 million places). This is reported in the January 1983 Scientific American and the February 1983 Discover Magazine. In the process of surveying this and related fast methods of computing elementary functions ([2], [5]), the authors discovered a new quadratically convergent product expansion for n [1]. Our algorithm, like Salamin's, is intimately related to the Gaussian arithmetic-geometric mean iteration. However, it requires considerably less elliptic function theory to establish. The iteration is initialized by