Why is my program stopping when doing a seemingly infinite loop?

Question

This must be really obvious but I am currently doing a little tutorial that features this code snippet:

n=0
a=1
while a>0:
    n=n+1
    a=(1.0+2.0**(-n))-1.0
    print (n)

And I've tried to run it but it keeps getting stuck at n=53. Why? I just assumed that while would always be true ...

Answer 1

If you change the last line to print(n, a) you can see what's happening more clearly:

n = 0
a = 1
while a > 0:
    n = n + 1
    a = (1.0 + 2.0 ** (-n)) - 1.0
    print(n, a)

Output:

1 0.5
2 0.25
3 0.125
4 0.0625
# ...
50 8.881784197001252e-16
51 4.440892098500626e-16
52 2.220446049250313e-16
53 0.0

As you can see, a is half the size each time through the loop. Eventually, 2.0 ** (-n) is so small that floating point math (which has limited precision) is unable to tell the difference between 1.0 and 1.0 + 2.0 ** (-n):

>>> 1.0 + 2.0 ** -51
1.0000000000000004
>>> 1.0 + 2.0 ** -52
1.0000000000000002
>>> 1.0 + 2.0 ** -53
1.0

… and when that happens, subtracting 1.0 from 1.0 gives you 0.0, and the while loop terminates.