e to the pi gives 1.0 and I am unable to comprehend the why behind it. Need help in understanding the underlying science

Question

I created a simple function to solve e to the power pi explained here

def e_to_power_pi(number):
    return (1 + (1/number)) ** (number * math.pi)

from the look of it, clearly simple piece of code. But look at the output difference of these two values:

Example one:

e_to_power_pi(1000000000000000)

output:

32.71613881872869

Example Two:

e_to_power_pi(10000000000000000)

output:

1.0

upon tear down of the code, I learnt that 1.0 is coming from this portion 1 + (1/number) of the code above.

When I tore it down further, I learnt that 1/10000000000000000 outputs correct answer as it should 0.00000000000000001.

But when I add 1 to result it returns 1.0 instead of 1.00000000000000001.

I presumed that it must be default round off in python that may be changing the value.

I decided to use round(<float>, 64) # where <float> is any computation taking place in code above to try and get 64 digits post decimal result. But still I got stuck with the same result when addition was performed i.e. 1.0.

Can someone guide me or point me to the direction where I can learn or further read about it?

Answer 1

You are using double-precision binary floating-point format, with 53-bit significand precision, which is not quite enough to represent your fraction:

10000000000000001/10000000000000000 = 1.0000000000000001

See IEEE 754 double-precision binary floating-point format: binary64

Mathematica can operate in precisions higher than the architecturally imposed machine precision.

See Wolfram Language: MachinePrecision

The Mathematica screenshot below shows you would need a significand precision higher than 53-bit to obtain a result other than 1.

N numericises the fractional result to the requested precision. Machine precision is the default; higher precision calculations are done in software.