Cubic root from negative number in Python

Quite recently I’ve encountered a weird issue. I tried to calculate a cubic root from a negative number in Python:

    $ python3
    Python 3.5.3 (default, May 10 2017, 15:05:55) 
    [GCC 6.3.1 20161221 (Red Hat 6.3.1-1)] on linux
    Type "help", "copyright", "credits" or "license" for more information.
    >>> import math
    >>> math.pow(8, 1/3)
    2.0
    >>> math.pow(-8, 1/3)
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    ValueError: math domain error
 

But it’s kind of obvious that \(-8 = (-2)^{1/3} = (-2) * (-2) * (-2)\). So why is this error?

Next thing I tried to the same in JavaScript (it’s easy to find a console in browser…):

    console.log(Math.pow(8, 1/3)); // 2
    console.log(Math.pow(-8, 1/3)); // NaN

It seems that it worked the same, clearly the issue was in the underlying implementation. But first quick glance on the workaround solution:

Workaround solution

I needed to calculate that cubic root and I knew that there has to be at least one real solution, so I used this workaround:

    
    def cubic_pow(n):
        if n < 0:
            return -(-n) ** (1.0 / 3.0)
        else:
            return n ** (1.0 / 3.0)

That’s because it’s \(-(-n) ^ \frac{1}{3} = n ^ \frac{1}{3}\) for negative numbers.

Note that there are usually \(3\) roots of cubic root: two complex ones and one real.

Reason of issues with cubic root

I was wondering about the reason behind that behaviour. To find out I decided to check the documentation of Python and source code of Python implementation and see where it can lead me next.

Power function in Python documentation and source code

In python doc it’s clearly written:

If both x and y are finite, `x` is negative, and `y` is not an integer then `pow(x, y)` is undefined, and raises `ValueError`.

The source code of cpython only confirmed that mathmodule.c.h behaviour. But there is no explicit reason why mentioned. It’s also written:

Exceptional cases follow Annex ‘F’ of the C99 standard as far as possible.

It seemed that C99 standard is a good lead for next steps.

Annex F of the C99 standard

Here is a link to the standard: C99 standard In F.9.4.4 it’s clearly written:

pow(x, y) returns a NaN and raises the ‘‘invalid’’ floating-point exception for
finite x < 0 and finite non-integer y.

So it seems that when designing floating-point arithmetic they just made a decision to raise exception when it’s not obvious when the result of power function will be complex or not. I think it might be good decision - the C99 standard is already quite complicated and adding more corner cases there could create a nightmare for people who wants to implement math functions using that standard. And finding a workaround is really easy in cases when this is obvious that real roots exist.

This is also complaint with IEEE Standard 754.

What about pow in C++?

From C++ reference math pow:

If base is finite and negative and exp is finite and non-integer, a domain 
    error occurs and a range error may occur.
(...)
If the implementation supports IEEE floating-point arithmetic (IEC 60559)

Actually in C++11 and later ones there is a special function for cubic roots cbrt, which can handle the negative argument, but they didn’t decide to use that one in cpython implementation of power function.