This work studies the connection between the problem of analyzing floating-point code and that of function minimization. It formalizes this connection as a reduction theory, where the semantics of a floating-point program is measured as a generalized metric, called weak distance, which faithfully captures any given analysis objective. It is theoretically guaranteed that minimizing the weak distance (e.g., via mathematical optimization) solves the underlying problem. This reduction theory provides a general framework for analyzing numerical code. Two important separate analyses from the literature, branch-coverage-based testing and quantifier-free floating-point satisfiability, are its instances.

To further demonstrate our reduction theory’s generality and power, we develop three additional applications, including boundary value analysis, path reachability and overflow detection. Critically, these analyses do not rely on the modeling or abstraction of floating-point semantics; rather, they explore a program’s input space guided by runtime computation and minimization of the weak distance. This design, combined with the aforementioned theoretical guarantee, enables the application of the reduction theory to real-world floating-point code. In our experiments, our boundary value analysis is able to find all reachable boundary conditions of the GNU sin function, which is complex with several hundred lines of code, and our floating-point overflow detection detects a range of overflows and inconsistencies in the widely-used numerical library GSL, including two latent bugs that developers have already confirmed.