This article presents advances on the subject of correctly rounded elementary functions since the publication of the libultim mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the worst-case overhead was still a factor 1000 or more. It is shown here that, with current processor technology, this worst-case overhead can be kept within a factor of 2 to 10 of current best libms. This low overhead has very positive consequences on the techniques for implementing and proving correctly rounded functions, which are also studied. These results lift the last technical obstacles to a generalisation of (at least some) correctly rounded double precision elementary functions.