Rosa Abbasi

• Floating-point numbers often represent real numbers in computer systems. They are applicable in many domains, including embedded systems, machine learning, and scientific computing. Despite their widespread use, they pose some difficulties. Floating-point numbers and operations typically suffer from roundoff errors, making computations over floating-points inaccurate with respect to a real-valued specification. Moreover, the IEEE 754 floating-point standard, a fundamental element in formalizing floating-point arithmetic for today’s computers, presents additional challenges due to special values and resulting unintuitive behaviors. This thesis has three main contributions that address existing gaps in automated reasoning about floating-point arithmetic, making it easier for developers and researchers to understand, verify, and trust the floating-point computations in their programs. First, we introduce the first floating-point support in a deductive verifier for the Java programming language. Our support in the KeY verifier automatically handles floating-point arithmetic and transcendental functions. We achieve this with a combination of delegation to external SMT solvers on one hand, and rule-based reasoning within KeY on the other, exploiting the complementary strengths of both approaches. As a result, this approach can prove functional floating-point properties for realistic programs. Second, inspired by KeY’s treatment of method calls and the need for a scalable roundoff error analysis, we present the first modular optimization-based roundoff error analysis for non-recursive procedural floating-point programs. Our key idea is to achieve modularity while maintaining reasonable accuracy by automatically computing procedure summaries that are a function of the input parameters. Technically, we extend an existing optimization-based roundoff error analysis and show how to effectively use first-order Taylor approximations to compute precise procedure summaries, and how to integrate those to obtain end-to-end roundoff error bounds. Third, our experience using SMT solvers to discharge KeY’s floating-point verification conditions revealed unexpected performance behavior, motivating a systematic study of floating-point reasoning in SMT solvers. We propose a metamorphic testing approach that uses semantics-preserving rewrite rules, focusing on floating-point special values, to uncover unexpected performance behavior in SMT solvers’ handling of floating-point formulas, such as an increase in solving time when the SMT queries are simplified. Using real-world test inputs, our approach can identify such performance bugs for every SMT solver tested.