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.[1]

It was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 10 trillion digits in October 2011,[3][4] 22.4 trillion digits in November 2016,[5] 31.4 trillion digits in September 2018–January 2019,[6] 50 trillion digits on January 29, 2020,[7] 62.8 trillion digits on August 14, 2021,[8] and 100 trillion digits on March 21, 2022.[9]

Algorithm

The algorithm is based on the negated Heegner number , the j-function , and on the following rapidly convergent generalized hypergeometric series:[2]

A detailed proof of this formula can be found here: [10]

This identity is similar to some of Ramanujan's formulas involving π,[2] and is an example of a Ramanujan–Sato series.

The time complexity of the algorithm is .[11]

Optimizations

The optimization technique used for the world record computations is called binary splitting.[12]

Binary splitting

A factor of can be taken out of the sum and simplified to


Let , and substitute that into the sum.


can be simplified to , so

from the original definition of , so

This definition of isn't defined for , so compute the first term of the sum and use the new definition of

Let and , so

Let and

can never be computed, so instead compute and as approaches , the approximation will get better.

From the original definition of ,

Recursively computing the functions

Consider a value such that

Base case for recursion

Consider

Python code

import decimal


def binary_split(a, b):
    if b == a + 1:
        Pab = -(6*a - 5)*(2*a - 1)*(6*a - 1)
        Qab = 10939058860032000 * a**3
        Rab = Pab * (545140134*a + 13591409)
    else:
        m = (a + b) // 2
        Pam, Qam, Ram = binary_split(a, m)
        Pmb, Qmb, Rmb = binary_split(m, b)
        
        Pab = Pam * Pmb
        Qab = Qam * Qmb
        Rab = Qmb * Ram + Pam * Rmb
    return Pab, Qab, Rab


def chudnovsky(n):
    P1n, Q1n, R1n = binary_split(1, n)
    return (426880 * decimal.Decimal(10005).sqrt() * Q1n) / (13591409*Q1n + R1n)


print(chudnovsky(2))  # 3.141592653589793238462643384

decimal.getcontext().prec = 100
for n in range(2,10):
    print(f"{n=} {chudnovsky(n)}")  # 3.14159265358979323846264338...

Notes

See also

References

  1. Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to Ramanujan, Ramanujan revisited: proceedings of the centenary conference
  2. 1 2 3 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
  3. 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
  4. Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
  5. "22.4 Trillion Digits of Pi". www.numberworld.org.
  6. "Google Cloud Topples the Pi Record". www.numberworld.org/.
  7. "The Pi Record Returns to the Personal Computer". www.numberworld.org/.
  8. "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17.
  9. "Calculating 100 trillion digits of pi on Google Cloud". cloud.google.com. Retrieved 2022-06-10.
  10. Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533
  11. "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.
  12. Rayton, Joshua (Sep 2023), How is π calculated to trillions of digits?, YouTube
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