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.
The main difference it that humans have to learn to do math, and with computers that is an innate part of their function. If a computer is to think like a human, it needs to be bad in math, as all this mathematical precision is expensive in energy terms. Humans don't usually do e^(0.5*ln(X)) when trying to figure out the square root of some number. One way is to pick a number between 0 and X and multiply that number by itself, if that number is greater than X, then that number is the upper bound of the next number you pick and 0 is the lower bound, otherwise that number picked is the lower bound of the next number picked and X is the upper. If one keeps following that algorithm one gets fairly close to the actual square root fairly quickly. The brain does a lot of its thinking by guesses or random numbers, and often times a quick decision is better than a mathematically precise answer which is what computers are used for.