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Gal’s accurate tables algorithm aims at providing an ef-ficient implementation of mathematical functions with cor-rect rounding as often as possible. This method requires an expensive pre-computation of the values taken by the function – or by several related functions – at some dis-tinguished 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 ef-ficiency and accuracy of the function implementation, and on the other hand we give an algorithm which drastically decreases the cost of the pre-computation. These improve-ments are related to the worst cases for the correct rounding of mathematical functions and to the algorithms for finding them. We demonstrate how the whole method can be turne