Well, most calculators find the square root of X as e^(0.5*ln(X)), where e and ln are found using a Taylor series expansion or some other efficient algorithm. Intel processors speed up trig functions by using a look-up table for a close initial guess so the series converges much more rapidly.

I don't know of any processor or compiler that would use a random number generator for a common math functions because random numbers are somewhat expensive to generate (unless done in hardware) and using them would make execution times harder to predict.