Chudnovsky algorithm

The Chudnovsky algorithm is a fast method for calculating the digits of π. It was published by the Chudnovsky brothers in 1989,^[1] and was used in the world record calculations of 2.7 trillion digits of π in December 2009,^[2] 5 trillion digits of π in August 2010,^[3] 10 trillion digits of π in October 2011,^[4][5] and 12.1 trillion digits in December 2013.^[6]

The algorithm is based on the negated Heegner number d = −163, the j-function j(1+√−163/2) = −7005640320000000000♠640320³, and on the following rapidly convergent generalized hypergeometric series:^[2]

For high performance iterative implementation, this can be simplified to -

There are 3 big integer terms (multinomial term Mk, linear term Lk and exponential term Xk) that make up the series and π is just the constant (C) divided by the sum of the series, as below.

, where -

C = 426880√10005

Mk+1 = Mk * (12k+2) * (12k+6) * (12k+10) / (k+1)^3 and M0 = 1

Lk+1 = Lk + 545,140,134 and L0 = 13,591,409

Xk+1 = Xk * -262,537,412,640,768,000 and X0 = 1

Mk can be optimized further -

Kk+1 = Kk + 12 and K0 = 6

Mk+1 = Mk * (K^3 - (K<<4)) / (k+1)^3 and M0 = 1

Note that 7008545140134000000♠545140134 = 163 × 7006334441800000000♠3344418 and,

and

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

Example: Python Implementation

π can be computed to any precision using the above algorithm in any environment that supports Big Integer and configurable precision arithmetic (such as Python, Ruby, Perl, PHP, C#, Java, C++ and so many other languages, libraries, frameworks and runtimes). As an example, here is a Python implementation.

#!/usr/bin/env python -i

from decimal import getcontext, Decimal as Dec

def PI(maxK=70, prec=1008, disp=1007): # parameter defaults chosen to gain 1000+ digits within a few seconds
    getcontext().prec = prec
    k12_6,M,L,X,S = 6, 1, 13591409, 1, 13591409 # At k=0: 12k+6=6, (6k)!/(3k)!(k!)^3=1, 545140134k+13591409, (-640320^3)^k=1, 0+Dec(M*L)/X=L
    for k in xrange(1,maxK+1):
        M = (k12_6**3 - (k12_6<<4))*M / k**3; k12_6 += 12
        L += 545140134
        X *= -262537412640768000 # -640320**3;
        S += Dec(M*L) / X
    pi = 426880 * 10005**Dec(".5") / S
    pi = Dec(("%s"%pi)[:disp]) # drop few digits of precision for accuracy
    print "PI(maxK=%d iterations, getcontext().prec=%d, disp=%d digits) =\n%s" % (maxK, prec, disp, pi)
    return pi

Pi = PI()
print "\nFor greater precision and more digits (takes a few extra seconds) - Try"
print "Pi = PI(317,4501,4500)"
print "Pi = PI(353,5022,5020)"

See also

• Numerical approximations of π
• Ramanujan–Sato series

References