Solving a cubic equation

I've looked at the Wikipedia article and your program.

I also solved the equation using Wolfram Alpha and the results there don't match what you get.

I'd just go through your program at each step, use a lot of print statements, and get each intermediate result. Then go through with a calculator and do it yourself.

I can't find what's happening, but where your hand calculations and the program diverge is a good place to look.

Wikipedia's notation (rho^(1/3), theta/3) does not mean that rho^(1/3) is the real part and theta/3 is the imaginary part. Rather, this is in polar coordinates. Thus, if you want the real part, you would take rho^(1/3) * cos(theta/3).

I made these changes to your code and it worked for me:

theta = arccos(r/rho)
s_real = rho**(1./3.) * cos( theta/3)
t_real = rho**(1./3.) * cos(-theta/3)

(Of course, s_real = t_real here because cos is even.)