Inverse trigonometric functions are difficult to compute. Their use is often avoided like the plague. If, however, the value is absolutely necessary, it will probably be solved by approximations or, if possible, replacing them with easier functions by clever use of trig identities.

If you want to see how the experts approach this problem, then Sébastien Lagarde, a senior developer of the Frostbite engine at DICE, goes into detail with a blog post. By detail, I mean you will see some GPU assembly being stepped through by the end of it. What makes this particularly interesting is the diagrams at the end, showing what each method outputs as represented by the shading of a sphere.

If you are feeling brave, take a look.