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 and have been formally verified. We hope this work will motivate the next IEEE 754 revision committee to require correct rounding for mathematical functions.