Humans
don't do this affordably. Programs proven end-to-end correct using human provers and semi-automated proof assistants have required
extreme simplification of the program so that it could be verified at all (so no clever efficient algorithms), teams of graduate-student proof-monkeys working for
years, and even then the biggest programs we could prove that way were ~10KLOC. The greatest achievement of this semi-automated approach required years by highly-trained specialists to prove simplified programs that amount to 1/5th of jQuery.
We have some very interesting advances in formal verification, but that is probably not the way.