Optimizing floating-point arithmetic is vital because it is ubiquitous, costly, and used in compute-heavy workloads. Implementing precise optimizations correctly, however, is difficult, since developers must account for all the esoteric properties of floating-point arithmetic to ensure that their transformations do not alter the output of a program. Manual reasoning is error prone and stifles incorporation of new optimizations. We present an approach to automate reasoning about floating-point optimizations using satisfiability modulo theories (SMT) solvers. We implement the approach in LifeJacket, a system for automatically verifying precise floating-point optimizations for the LLVM assembly language. We have used LifeJacket to verify 43 LLVM optimizations and to discover eight incorrect ones, including three previously unreported problems. LifeJacket is an open source extension of the Alive system for optimization verification.
Research on compiler correctness has addressed floating-point and floating-point optimizations. CompCert, a formally-verified compiler, supports IEEE 754-2008 floating-point types and implements two floating-point optimizations @cite_10 . In CompCert, developers use Coq to prove optimizations correct, while proves optimization correctness automatically.
@cite_10
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Floating-point arithmetic is known to be tricky: roundings, formats, exceptional values. The IEEE-754 standard was a push towards straightening the field and made formal reasoning about floating-point computations easier and flourishing. Unfortunately, this is not sufficient to guarantee the final result of a program, as several other actors are involved: programming language, compiler, and architecture. The CompCert formally-verified compiler provides a solution to this problem: this compiler comes with a mathematical specification of the semantics of its source language (a large subset of ISO C99) and target platforms (ARM, PowerPC, x86-SSE2), and with a proof that compilation preserves semantics. In this paper, we report on our recent success in formally specifying and proving correct CompCert's compilation of floating-point arithmetic. Since CompCert is verified using the Coq proof assistant, this effort required a suitable Coq formalization of the IEEE-754 standard; we extended the Flocq library for this purpose. As a result, we obtain the first formally verified compiler that provably preserves the semantics of floating-point programs.