Unexpected promotion to 'complex' type in Python math expression

Question

While implementing a numerical optimization algorithm I came across some rather strange behavior in Python: after applying the Six-hump cambel back function on some values form the function's domain, some results evaluate to 'complex' numbers.

The code for evaluating a number is as simple as it can get:

def f(x1, x2):   
    return ((4 - 2.1 * x1 ** 2 + x1 ** (4 / 3)) * x1 ** 2 + x1 * x2 +
            (-4 + 4 * x2 ** 2) * x2 ** 2)

Here are some numbers for which the above function returns a complex number:

x1                    x2                   Result
-1.30423635697717    -1.750915750915751    27.13078459548355-2.099189473463638j
-0.28677817116347226  0.3413919413919415  -0.20277691810793963-0.013469483248785653j
-1.105725796606031   -0.5816849816849816   0.8001125335594826-1.2106597512385535j

And some numbers for which the above functions returns a normal float:

x1   x2  Result
1    2   52.9
1.4 -2   48.04232245707135

Why exactly is this happening and what can I do to fix it?

The code is run under Python 3.3.2.

Answer 1

The expression x1 ** (4 / 3) is taking the variable x1 to the power of 4/3. When x1 is negative, the result of the expression has 3 roots; 2 of them are complex. Python is choosing the "principal root" for you, which is complex in this case.

See this question for ideas.