Abstract: Floating-point numbers use finite precision to represent real numbers,and their inherent rounding errors can accumulate during calculations,potentially leading to serious errors that jeopardize program safety and reliability.Theoretically,the most precise method for detecting floating-point errors is exhaustive search of all possible floating-point inputs to determine the maximum error between actual computation results and theoretical values.However,the search space is enormous.Effectively and efficiently detecting maximum floating-point errors has been a challenge.Based on the study on condition numbers,a tool for floating-point expression error detection,CNFED,has been designed and implemented.CNFED divides the input interval into multiple sub-intervals,conducts random sampling and evaluation for each sub-interval to quickly locate multiple hotspot sub-intervals.It then hierarchically applies global and local search algorithms to these hotspot sub-intervals,using corresponding evaluation functions for filtering,ultimately identifying potential maximum floating-point errors and reporting the corresponding input values.Experiment selects 26 expressions from the FPBench standard test suite as test cases and compare CNFED with advanced detection tools ATOMU and HSED.The experimental results indicate that CNFED outperforms ATOMU in 96.15% of cases(25/26).Compared to the detection tool HSED for floating-point expressions,CNFED surpasses HSED in 34.62% of cases(9/26),while the average time taken by HSED is 4.8 times that of CNFED.

Key words: Error detection, Floating-point arithmetic, Condition number, Dynamic analysis, Hierarchical search