MIT 6.00 Lecture 8 - Stirling's approximation

Approximating large factorials:

import math

def stirling(n):
    return math.sqrt(2*math.pi*n) * (n/math.e)**n

if __name__ == '__main__':
    for i in range(1,20):
        s = stirling(i)
        f = math.factorial(i)
        print "stirling: %21.2f  factorial: %21d  error: %3.2f%%" % (s, f, abs(((f-s)/s)*100.0))

Output:

stirling:                  0.92  factorial:                     1  error: 8.44%
stirling:                  1.92  factorial:                     2  error: 4.22%
stirling:                  5.84  factorial:                     6  error: 2.81%
stirling:                 23.51  factorial:                    24  error: 2.10%
stirling:                118.02  factorial:                   120  error: 1.68%
stirling:                710.08  factorial:                   720  error: 1.40%
stirling:               4980.40  factorial:                  5040  error: 1.20%
stirling:              39902.40  factorial:                 40320  error: 1.05%
stirling:             359536.87  factorial:                362880  error: 0.93%
stirling:            3598695.62  factorial:               3628800  error: 0.84%
stirling:           39615625.05  factorial:              39916800  error: 0.76%
stirling:          475687486.47  factorial:             479001600  error: 0.70%
stirling:         6187239475.19  factorial:            6227020800  error: 0.64%
stirling:        86661001740.60  factorial:           87178291200  error: 0.60%
stirling:      1300430722199.47  factorial:         1307674368000  error: 0.56%
stirling:     20814114415223.14  factorial:        20922789888000  error: 0.52%
stirling:    353948328666101.12  factorial:       355687428096000  error: 0.49%
stirling:   6372804626194313.00  factorial:      6402373705728000  error: 0.46%
stirling: 121112786592294176.00  factorial:    121645100408832000  error: 0.44%

As n increases the error decreases:

n=165 error: 0.0505177446%
n=166 error: 0.0502133452%
n=167 error: 0.0499125922%
n=168 error: 0.0496154204%
n=169 error: 0.0493217663%