machin.py

#!/usr/bin/env python3
#
#   machin.py  -  by Don Cross
#
#   Test Machin's Formula: pi = 4*(4*arctan(1/5) - arctan(1/239)).
#

import math

def ArctanDenom(d):
    # Calculates arctan(1/d) = 1/d - 1/(3*d^3) + 1/(5*d^5) - 1/(7*d^7) + ...
    tolerance = 1.0e-20     # how small the residue needs to be before we stop iterating
    total = term = 1.0 / d
    n = 0
    while abs(term) > tolerance:
        n += 1
        term /= -d*d
        total += term / (2*n + 1)
    print('ArctanDenom({}) = {:0.15f}; took {} iterations.'.format(d, total, n))
    return total

if __name__ == '__main__':
    calc_pi = 4 * (4*ArctanDenom(5) - ArctanDenom(239))
    print('pi = {:0.15f}, error = {:0.4g}'.format(calc_pi, math.pi - calc_pi))