Computing Floating-Point Errors by Injecting Perturbations

Abstract

Floating-point programs form the foundation of modern science and engineering, providing the essential computational framework for a wide range of applications, such as safety-critical systems, aerospace engineering, and financial analysis. Floating-point errors can lead to severe consequences. Although floating-point errors widely exist, only a subset of inputs may trigger significant errors in floating-point programs. Therefore, it is crucial to determine whether a given input could produce such errors. Researchers tend to take the results of high-precision floating-point programs as oracles for detecting floating-point errors, which introduces two main limitations: (1) difficulty of implementation and (2) prolonged execution time. The two recent tools, ATOMU and FPCC, can partially address these issues. However, ATOMU suffers from false positives; while FPCC, though eliminating false positives, operates at a considerably slower speed. To address these two challenges, we propose a novel approach named PI-detector to computing floating-point errors effectively and efficiently. Our approach is based on the observation that floating-point errors stem from large condition numbers in atomic operations (such as addition and subtraction), which then propagate and accumulate. PI-detector injects small perturbations into the operands of individual atomic operations within the program and compares the outcomes of the original program with the perturbed version to compute floating-point errors. We evaluate PI-detector with datasets from ATOMU and HSED, as well as a complex linear system-solving program. Experimental results demonstrate that PI-detector can perform efficient and accurate floating-point error computation.