Building Better Bit-Blasting for Floating-Point Problems

Abstract

An effective approach to handling the theory of floatingpoint is to reduce it to the theory of bit-vectors. Implementing the required encodings is complex, error prone and requires a deep understanding of floating-point hardware. This paper presents SymFPU, a library of encodings that can be included in solvers. It also includes a verification argument for its correctness, and experimental results showing that its use in CVC4 out-performs all previous tools. As well as a significantly improved performance and correctness, it is hoped this will give a simple route to add support for the theory of floating-point.

Counterexample- and Simulation-Guided Floating-Point Loop Invariant Synthesis

“…We note that it is in principle possible to encode the queries using the floating-point theory, and thus to encode the semantics of the loop body, including roundoff errors, exactly. However, despite the recent advances in floating-point decision procedures [8], we have observed that their performance is still prohibitively slow for our purpose (CVC4's state-of-the-art floating-point procedure [8] was several orders of magnitude slower than Z3's real-valued procedure [24]).…”

A Two-Phase Approach for Conditional Floating-Point Verification

“…Similarly, bounded model checking [52] is limited by the performance of the underlying SAT/SMT solvers. While the floating-point support in today's SMT solvers [17,16] has improved significantly in recent years, it is still limited to relatively few arithmetic expressions.…”

Invertibility Conditions for Floating-Point Formulas

“…While the extension was optimized for this task, we stress that our techniques are theory-agnostic and can be used for synthesis problems over any finite domain. Our approach builds upon the SyGuS capabilities of the SMT solver CVC4 [5,29], which has recently been extended to support reasoning about the theory of floating-points [11]. We use the invertibility condition for floatingpoint addition with equality here as a running example.…”

“…Problems that combine universal quantification over floating-points are rare-experience to date has suggested they are hard for solvers and would-be users should either give up or develop their own incomplete techniques. However, progress in theory solvers for floating-point [11] and the use of expression synthesis for handling universal quantifiers [27,29] suggest that these problems may not be entirely out of reach after all, which could potentially impact a number of interesting applications.…”

“…To our knowledge, no previous work specifically discusses techniques for solving universally quantified floating-point formulas. Brain et al [11] provide a comprehensive review of decision procedures for quantifier-free bitexact floating-point using both SMT-based as well as other approaches. They identify four groups of techniques: bit-blasting approaches that use floating-point circuits to generate bit-vector formulas [13,16,20,33], interval techniques that use partitioning and interval propagation [10,22,23,31], optimization and numerical approaches that work with complete valuations [4,7,18,21], and axiomatic techniques that use partial or total axiomatizations of the theory of floating-point numbers in other theories such as real arithmetic [14,15].…”