We design algorithms for the correct rounding of the power function $x^y$ in the binary64 IEEE 754 format, for all rounding modes, modulo the knowledge of hardest-to- round cases. Our implementation of these algorithms largely outperforms previous correctly-rounded implementations and is not far from the efficiency of current mathematical libraries, which are not correctly-rounded. Still, we expect our algorithms can be further improved for speed. The proofs of correctness are fully detailed in an extended version of this paper, with the goal to enable a formal proof of these algorithms. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.