Alternatives to floating-point satisfiability

Floating-point satisfiability (FPSAT) is the problem of determining if a given set of constraints involving floating-point variables and operations is satisfiable. It is crucial for verifying the correctness of numerical algorithms and software, especially in safety-critical domains where floating-point inaccuracies can lead to significant errors.

At a glance

Executive summary

Floating-point satisfiability (FPSAT) is the problem of determining if there exists an assignment of floating-point numbers that satisfies a given set of constraints. It is a specialized area within constraint satisfaction and verification, particularly relevant for analyzing programs that heavily rely on floating-point arithmetic.

TL;DR

If you need to verify properties of floating-point computations use FPSAT; if you need to find optimal solutions to numerical problems use numerical optimization; if you need to solve general logical formulas with integer/real variables use SMT solving; if you need to minimize floating-point errors use ULP optimization.

Key points

Consider FPSAT when your primary concern is the exact satisfaction of constraints involving floating-point numbers, not just finding approximate solutions.
If the problem involves finding the best possible values for variables under numerical constraints, numerical optimization is likely more appropriate.
For general logical reasoning that may include floating-point variables but also complex logical structures, SMT solvers offer broader capabilities.
If the goal is to reduce the impact of rounding errors in floating-point computations, ULP optimization is the direct approach.
Evaluate whether your problem can be precisely formulated with floating-point arithmetic as the core constraint type, which is the domain of FPSAT.

Our Take

## Our Take In the realm of computational problem-solving, floating-point satisfiability (FPS) and Satisfiability Modulo Theories (SMT) represent two distinct yet interrelated paradigms. FPS focuses on determining the satisfiability of constraints involving floating-point numbers, while SMT extends this concept by integrating various theories, such as integers, arrays, and bit-vectors, into the decision-making process. Recent studies, such as those by de Moura and Bjørner (2008), highlight that SMT solvers can efficiently handle a broader range of problems, leveraging their ability to combine different logical theories. On the other hand, numerical optimization and ULP (Uncertainty in Floating-Point) optimization tackle the challenges of finding optimal solutions under constraints, particularly in environments where precision is critical. Research by Gurov et al. (2020) emphasizes that while numerical optimization focuses on minimizing or maximizing objective functions, ULP optimization specifically addresses the nuances of floating-point arithmetic, ensuring that solutions remain robust against rounding errors and other numerical inaccuracies. The intersection of these areas reveals a rich landscape of opportunities and challenges. FPS and SMT solvers can benefit from the insights gained in numerical optimization, particularly in refining their approaches to handle floating-point constraints. Conversely, ULP optimization can leverage the decision-making capabilities of SMT to improve the reliability of its solutions. In conclusion, while FPS and SMT solvers excel in different contexts, their integration with numerical and ULP optimization techniques can lead to more powerful and reliable computational tools. The ongoing research in these areas promises to enhance our ability to tackle complex problems in a world increasingly reliant on precise numerical computations.

Alternative	Difference	Papers (with floating-point satisfiability)	Avg viability
SMT solving	—	1	—
numerical optimization	—	1	—
ULP optimization	—	1	—