Despite several significant advances over the last 30 years, guaranteeing the correctly rounded evaluation of elementary functions, such as $\cos, \exp, \sqrt[3]{\cdot}$ for instance, is still a difficult issue. This can be formulated as a Diophantine approximation problem, called the Table Maker’s Dilemma, which consists in determining points with integer coordinates that are close to a curve. In this article, we propose two algorithmic approaches to tackle this problem, closely related to a celebrated work by Bombieri and Pila and to the so-called Coppersmith's method. We establish the underlying theoretical foundations, prove the algorithms, study their complexity and present practical experiments; we also compare our approach with previously existing ones. In particular, our results show that the development of a correctly rounded mathematical library for the binary128 format is now possible at a much smaller cost than with previously existing approaches.