Computing E becomes chaotic at limits 10 ** 12 in Python?

Question

I wrote a program that computes the base of natural logarithms (known as e in mathematics) using the following well-known formula:

e = (1 + 1.0/n) ** n

The code is:

def e_formula(lim):
    n = lim
    e = (1 + 1.0/n) **n
    return e

I set up a test that iterates from 101 to 10100:

if __name__ == "__main__":
    for i in range(1,100):
        print e_formula(10**i)

However the following results around 10**11 blow up.

Actual results from shell:

2.5937424601

2.70481382942

2.71692393224

2.71814592682

2.71826823719

2.7182804691

2.71828169413

2.71828179835

2.71828205201

2.71828205323

2.71828205336

2.71852349604

2.71611003409

2.71611003409

3.03503520655

1.0

I am looking for a reason for this, either to do with the result exceeding the floating-point limit in a 32-bit machine or because of the way Python itself computes floating point numbers. I am not looking for a better solution; I just want to understand why it blows up.

Answer 1

This is simply due to the limited precision of floating point numbers. You get about 15 significant digits.

You are taking (1 + very_small_number). Most of the digits of very_small_number are truncated at this stage.

The **n just multiplies this error