Plouffe, Simon | Engineering Papers

NASA NTRS1996

The Quest for Pi

This article gives a brief history of the analysis and computation of the mathematical constant Pi=3.14159 ..., including a number of the formulas that have been used to compute Pi through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of Pi to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of Pi to be computed.

Bailey, David H.↗

NASA NTRS1996

On the Rapid Computation of Various Polylogarithmic Constants

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or pi on a modest workstation in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of pi, the billionth hexadecimal digits of pi-squared, log(2) and log-squared(2), and the ten billionth decimal digit of log(9/10). These calculations rest on the observation that very special types of identities exist for certain numbers like pi, pi-squared, log(2) and log-squared(2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example a critical identity for pi.

Bailey, David H.↗