Numerical libraries constitute the building blocks for software applications that perform numerical calculations. Thus, it is paramount that such libraries provide accurate and consistent results. To that end, this paper addresses the problem of finding discrepancies between synonymous functions in different numerical libraries as a means of identifying incorrect behavior. Our approach automatically finds such synonymous functions, synthesizes testing drivers, and executes differential tests to discover meaningful discrepancies across numerical libraries. We implement our approach in a tool named FPDiff, and provide an evaluation on four popular numerical libraries: GNU Scientific Library (GSL), SciPy, mpmath, and jmat. FPDiff finds a total of 126 equivalence classes with a 95.8% precision and 79.0% recall, and discovers 655 instances in which an input produces a set of disagreeing outputs between function synonyms, 150 of which we found to represent 125 unique bugs. We have reported all bugs to library maintainers; so far, 30 bugs have been fixed, 9 have been found to be previously known, and 25 more have been acknowledged by developers.