Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.

Quantitative semantics of Signal Temporal Logic (STL) play an important role in both the falsification and control synthesis for dynamical systems by assigning numerical quantities to truth values. Recently, several different quantitative semantics have been proposed, offering better performance in many cases. Yet a general, systematic understanding of the structure and properties of quantitative semantics is missing. In this paper, we develop a general framework to model quantitative semantics. We focus mainly on soundness, which requires that the quantitative semantics of a statement is positive when the statement is true, and negative when the statement is false. This ensures that counterexamples will not be missed during verification. We derive simple, necessary conditions in our framework for soundness. We show how several recently proposed quantitative semantics fit in our framework, and how others do not, typically because they do not strictly satisfy soundness. We implement various quantitative semantics, including existing semantics from literature, in our framework and compare their effectiveness as objective functions for optimization-based falsification on both novel and existing benchmarks.