Numerical software is used in a wide variety of applications including safety-critical systems, which have stringent correctness requirements, and whose failures have catastrophic consequences that endanger human life. Numerical bugs are known to be particularly difficult to diagnose and fix, largely due to the use of approximate representations of numbers such as floating point. Understanding the characteristics of numerical bugs is the first step to combat them more effectively. In this paper, we present the first comprehensive study of real-world numerical bugs. Specifically, we identify and carefully examine 269 numerical bugs from five widely-used numerical software libraries: NumPy, SciPy, LAPACK, GNU Scientific Library, and Elemental. We propose a categorization of numerical bugs, and discuss their frequency, symptoms and fixes. Our study opens new directions in the areas of program analysis, testing, and automated program repair of numerical software, and provides a collection of real-world numerical bugs.

The original flocking model proposed and simulated by Reynolds [1987] and recently analytically verified to be convergent in a modified form by Tanner et. al. demands tracking and combining the movement parameters of a potentially large number of neighboring agents. We investigate an alternative simplification of the verified model wherein only two nearest neighbors are taken into account in a way that minimally preserves the convergence properties. We find that this choice also simplifies the geometry of the converged state from an amorphous distribution to a line. We speculate that this method might serve as the basis for forming more structured equilibrium distributions.