In this article, we propose an algorithmic approach to determine the integer points located near a transcendental curve. This approach is 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. From a practical point of view, we focus on an instance of our general problem, called the Table Maker's Dilemma, whose solving makes it possible to evaluate a given function with correct rounding. Our experiments show a significant speedup. 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.

翻译：在本文中，我们提出了一种算法方法，用于确定位于超越曲线附近的整数点。该方法与Bombieri和Pila的著名工作以及所谓的Coppersmith方法密切相关。我们建立了相关的理论基础，证明了算法的正确性，研究了其复杂性，并进行了实际实验；同时，我们将我们的方法与先前存在的方法进行了比较。从实践角度来看，我们关注于一个称为“制表者困境”的一般问题实例，解决该问题能够以正确舍入的方式评估给定函数。我们的实验表明，计算速度显著提升。特别地，我们的结果表明，现在以比先前方法更低的成本开发二进制128格式的正确舍入数学库是可行的。