The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan’s π formulae. It was published by the Chudnovsky brothers in 1988 and was used in the world record calculations of 2.7 trillion digits of π in December 2009, 10 trillion digits in October 2011, 22.4 trillion digits in November 2016, 31.4 trillion digits in September 2018–January 2019, 50 trillion digits on January 29, 2020, and 62.8 trillion digits on August 14, 2021.
A detailed proof of this formula can be found here:
For a high performance iterative implementation, this can be simplified to
There are 3 big integer terms (the multinomial term Mq, the linear term Lq, and the exponential term Xq) that make up the series and π equals the constant C divided by the sum of the series, as below:
- , where:
The terms Mq, Lq, and Xq satisfy the following recurrences and can be computed as such:
The computation of Mq can be further optimized by introducing an additional term Kq as follows:
- Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to ramanujan, Ramanujan revisited: proceedings of the centenary conference
- Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009), "Ramanujan's series for 1/π: a survey", American Mathematical Monthly, 116 (7): 567–587, doi:10.4169/193009709X458555, JSTOR 40391165, MR 2549375
- Yee, Alexander; Kondo, Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer Science Department, University of Illinois, hdl:2142/28348
- Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
- "22.4 Trillion Digits of Pi". www.numberworld.org.
- "Google Cloud Topples the Pi Record". www.numberworld.org/.
- "The Pi Record Returns to the Personal Computer". www.numberworld.org/.
- "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17.
- Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533
- "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.