Gal's accurate tables algorithm aims at providing an efficient implementation of elementary functions with correct rounding as often as possible. This method requires an expensive pre-computation of a table made of the values taken by the function - or by several related functions - at some distinguished points. Our improvements of Gal's method are two-fold: on the one hand we describe what is the arguably best set of distinguished values and how it improves the efficiency and correctness of the implementation of the function, and on the other hand we give an algorithm which drastically decreases the cost of the pre-computation. These improvements are related to the worst cases for the correct rounding of mathematical functions and to the algorithms for finding them. We show that the whole method can be turned into practice by giving complete tables for $2^x$ and $\sin x$ for $x \in [\frac{1}{2},1[$, in double precision.