A catalogue of mathematical formulas involving \(\pi\), with analysis

David H. Bailey

doi:10.33993/jnaat502-1259

Authors

David H. Bailey Lawrence Berkeley National Laboratory, USA

DOI:

https://doi.org/10.33993/jnaat502-1259

Keywords:

Pi, computation of pi, history of pi

Abstract views: 307

Abstract

This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant \(\pi\), ranging from Archimedes' 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs.

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References

Victor Adamchik and Stan Wagon, A simple formula for pi, American Mathematical Monthly, vol. 104 (Nov. 1997), pg. 852-855, https://doi.org/10.2307/2975292

Gert Almkvist and Jesus Guillera, Ramanujan-like series for 1/π2 and string theory, 27 Sept. 2010, https://doi.org/10.1080/10586458.2012.656059

David H. Bailey, A compendium of BBP-type formulas for mathematical constants, updated 29 April 2013, http://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf.

David H. Bailey, Simple proofs: Archimedes calculation of pi, Math Scholar, 9 Feb 2019, https://mathscholar.org/2019/02/simple-proofs-archimedes-calculation-of-pi/.

David H. Bailey, MPFUN2015: A thread-safe arbitrary precision package (full documentation), manuscript, updated 7 Feb 2020, https://www.davidhbailey.com/dhbpapers/mpfun2015.pdf.

David H. Bailey and Jonathan M. Borwein, Ancient Indian square roots: An exercise in forensic paleomathematics, American Mathematical Monthly, vol. 119, no. 8 (Oct 2012), pg. 646-657, http://doi.org/10.4169/amer.math.monthly.119.08.646

David H. Bailey and Jonathan M. Borwein, Hand-to-hand combat with thousand-digit integrals, Journal of Computational Science, vol. 3 (2012), pg. 77-86, preprint at https://doi.org/10.1016/j.jocs.2010.12.004

David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, Integrals of the Ising class, Journal of Physics A: Mathematical and General, vol. 39 (2006), pg. 12271-12302, https://doi.org/10.1088/0305-4470/39/40/001

David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, Box integrals, Journal of Computational and Applied Mathematics, vol. 206 (2007), pg. 196–208, https://doi.org/10.1016/j.cam.2006.06.010

David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, Advances in the theory of box integrals, Mathematics of Computation, vol. 79, no. 271 (Jul. 2010), pg. 1839-1866, https://doi.org/10.1090/S0025-5718-10-02338-0

David H. Bailey, Jonathan M. Borwein, Andrew Mattingly and Glenn Wightwick, The computation of previously inaccessible digits of π and Catalan’s constant, Notices of the American Mathematical Society, vol. 60 (2013), no. 7, pg. 844-854, https://doi.org/10.1090/noti1015

David H. Bailey, Peter B. Borwein and Simon Plouffe, On the rapid computation of various polylogarithmic constants, Mathematics of Computation, vol. 66, no. 218 (Apr. 1997), pg. 903-913, https://doi.org/10.1090/S0025-5718-97-00856-9

David H. Bailey and David J. Broadhurst, Parallel integer relation detection: Techniques and applications, Mathematics of Computation, vol. 70, no. 236 (Oct. 2000), pg. 1719–1736, https://doi.org/10.1090/S0025-5718-00-01278-3

David H. Bailey, Xiaoye S. Li and Karthik Jeyabalan, A comparison of three high-precision quadrature schemes, Experimental Mathematics, vol. 14 (2005), no. 3, pg. 317-329, http://projecteuclid.org/euclid.em/1128371757

Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, AK Peters, Natick, MA, 2008.

Jonathan M. Borwein, David H. Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, AK Peters, Natick, MA 2004.

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, CMS Series of monographs and Advanced Books in Mathematics, John Wiley, Hoboken, NJ, 1987.

David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe, The quest for Pi, Mathematical Intelligencer, vol. 19, no. 1 (Jan 1997), pg. 50–57, https://doi.org/10.1007/BF03024340

David J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, manuscript, 1998, http://www.arxiv.org/abs/hep-th/9803091.

David and Gregory Chudnovsky, Listing of Ramanujan-type formulas, copy in author’s possession, 2000.

Helaman R. P. Ferguson, David H. Bailey and Stephen Arno, Analysis of PSLQ, an integer relation finding algorithm, Mathematics of Computation, vol. 68, no. 225 (Jan. 1999), pg. 351-369, https://doi.org/10.1090/S0025-5718-99-00995-3

Laurent Fousse, Guillaume Hanrot, Vincent Lefevre, Patrick Pelissier and Paul Zimmermann, MPFR:A multiple-precision binary floating-point library with correct rounding, ACM Transactions on Mathematical Software, vol. 33, no. 2 (June 2007), https://doi.org/10.1145/1236463.1236468.

Jesus Guillera, Some binomial series obtained by the WZ-method, Advances in Applied Mathematics, vol. 29 (2002), pg. 599-603, https://doi.org/10.1016/S0196-8858(02)00034-9

MPFR research team, Comparison of multiple-precision floating-point software, accessed 12 Mar 2020, https://www.mpfr.org/mpfr-4.0.1/timings.html.

Alexander J. Yee, y-cruncher - A multi-threaded pi-program,” updated 12 Mar 2020, http://www.numberworld.org/y-cruncher/.