python,math,exponent , Difference between math.exp(2) and math.e**2 [duplicate]


Difference between math.exp(2) and math.e**2 [duplicate]

Question:

Tag: python,math,exponent

This question already has an answer here:

While programming I noticed a difference between the result of math.exp(2) and math.e**2. As you can see below, this difference does not arise when calculating e^1.

Not being an experienced programmer, I wondered why this differs? I assume it has something to do with rounding up. The python docs say that math.exp(x) return e**x, but this appears not to be precisely correct. So how come that the math.exp(x) operation differs from math.e**x?

>>> math.exp(1)
2.718281828459045
>>> math.e**1
2.718281828459045
>>> math.exp(1)==math.e**1
True
>>> math.exp(2)
7.38905609893065
>>> math.e**2
7.3890560989306495
>>> math.exp(2)==math.e**2
False
>>> math.exp(100)
2.6881171418161356e+43
>>> math.e**100
2.6881171418161212e+43
>>> math.exp(100)==math.e**100
False

Answer:

It's different due to differences in the implementation of the functions. Neither one is perfect due to the nature of floating point.

The ** operator is implemented in floatobject.c and contains a lot of special cases, but none of them are invoked here, so it ultimately reaches the standard C pow function. The math.pow function ultimately does the same thing.

The exp function is a straightforward wrapper around the C function of the same name, defined with a macro in mathmodule.c.

As it happens, exp is more precise (both of the results match, to within the precision allowed by floating point, high-precision answers I calculated in bc). Most likely it is because internally to pow, it is calculating the extended-precision logarithm of the double-precision e value you pass as the first argument, which is slightly smaller than 1.


The fundamental issue is that math.e, which is the number you're calculating the power of, is:

2.718281828459045 09079559829...

Whereas the real value of e is

2.718281828459045 23536028747...

And this error is compounded when you use pow or **, whereas it may not be (or may be using a higher precision value internally) if you're using exp due to the details of the algorithms it uses.


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