Exploiting Binary Floating-Point Representations for Constraint Propagation

Abstract

Floating-point computations are quickly finding their way in the design of safety-and mission-critical systems, despite the fact that designing floating-point algorithms is significantly more difficult than designing integer algorithms. For this reason, verification and validation of floating-point computations is a hot research topic. An important verification technique, especially in some industrial sectors, is testing. However, generating test data for floating-point intensive programs proved to be a challenging problem. Existing approaches usually resort to random or search-based test data generation, but without symbolic reasoning it is almost impossible to generate test inputs that execute complex paths controlled by floating-point computations.Moreover, as constraint solvers over the reals or the rationals do not natively support the handling of rounding errors, the need arises for efficient constraint solvers over floating-point domains. In this paper, we present and fully justify improved algorithms for the propagation of arithmetic IEEE 754 binary floating-point constraints. The key point of these algorithms is a generalization of an idea by B. Marre and C. Michel that exploits a property of the representation of floating-point numbers.

A Practical Approach to Verification of Floating-Point C/C++ Programs with math.h/cmath Functions

“…• algorithms realizing optimal direct projections as well as correct and precise indirect projections: the result is similar to the projections defined in [53], but the ECLAIR algorithms never require working with precision greater than the operation data type; • algorithms that exploit properties of the binary floating-point representations in order to obtain enhanced precision [5]; • dynamic linear relaxation techniques [9,25] using the PPL to enhance constraint propagation with the relational information provided by convex polyhedra.…”

“…Propagators for basic floating-point operations are already present in the literature [3,5,13], and are out of the scope of this paper. Moreover, integer variables can also be treated with this approach, by employing appropriate propagators.…”

“…We indicate with d ℓ and i ℓ the direct and the indirect projections for the constraint at line ℓ, respectively. The related propagators are either those described in Section 5, or in [3,5].…”

“…Interval consistency techniques have been extended to floating-point computations in [13], with the purpose of symbolic execution. More advanced techniques for refining arithmetic floating-point operations have been proposed in [5]. All such works only deal with basic arithmetic operations, and do not provide any technique to tackle math.h/cmath mathematical functions.…”

“…We do not have a precise specification for the library functions that are assumed in the code of Figure 1. Its sources refer to GNU libc and to Newlib, 5 but no specific versions are mentioned. We can probably assume a POSIX-compliant behavior with respect to special values.…”