Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation

Abstract

More than 200 million decimal places of pi were calculated using an arithmetic geometric mean formula independently discovered by E. Salamin and R.P. Brent in 1976. Correctness of the calculation was verified through Borwein's quartic convergent formula developed in 1983. The computation took CPU times of 5 hours 57 minutes for the main calculation and 7 hours 30 minutes for the verification calculation on the HITAC S-820 model 80 supercomputer. Two programs generated values up to 3*2/sup 26/, about 201 million. The two results agreed except for the last 21 digits. The results also agree with the 133,554,000-place calculation of pi that was done by the author in January 1987. Compared to the record in 1987, 50% more decimal digits were calculated with about 1/6 of CPU time. The computation was performed with a real-arithmetic-based vectorized fast Fourier transform (FFT) multiplier and vectorized multiple-precision add, subtract, and (single-word) constant multiplication programs. Vectorizations for the later cases were realized through first order linear recurrence vector instruction on the S-820.<>