DELA: A Novel Approach for Detecting Errors Induced by Large Atomic Condition Numbers

Numerical programs form the foundation of modern science and engineering, providing essential solutions to complex mathematical problems. Therefore, errors in numerical results would lead to harmful consequences, especially in safety-critical applications. Since only a few inputs may lead to substantial errors for numerical programs, it is essential to determine whether a given input could result in a significant error. Existing researchers tend to use the results of high-precision programs to assess whether there is a substantial error, which introduces three main challenges: difficulty of implementation, existence of potential faults in the detection of numerical errors, and long execution time. To address these limitations, we propose a novel approach named DELA. Our approach is based on the observation that most numerical errors stem from large condition numbers in atomic operations (such as subtraction), which then propagate and accumulate. DELA injects small perturbations into the results of individual atomic operations within the program and compares the outcomes of the original program with the perturbed version to detect errors. We evaluate DELA with datasets from ATOMU and HSED, as well as data from a complex linear system-solving program. Experimental results demonstrate that we can detect all the significant errors that were reported by prior research. DELA shows strong alignment with high-precision programs of ATOMU and HSED, with average Pearson and Spearman correlations of 0.86 and 0.61. Additionally, DELA effectively detects significant errors in complex programs, achieving correlation scores of 0.9763 and 0.8993. More importantly, in experiments with ATOMU and HSED, DELA's perturbed programs run within only 0.13% of the time needed by high-precision versions; while for the linear system-solving programs, DELA is 73.46 times faster than the high-precision programs.