Analysis of floating-point programs is a topic that received an increasing attention the past few years. However, only very few works have been done regarding their termination analysis. We address that problem in this paper. We present a technique that takes advantage of the already existing works on termination analysis of rational programs. Our approach consists in translating the floating-point programs into rational ones by means of sound approximations. We approximate the floating-point expressions using piecewise linear functions. Our approximation differs from the already existing ones in the sense that it can be as precise as needed.

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Termination of loops can be inferred from the existence of linear ranking functions. We already know that the existence of these functions is PTIME decidable for simple rational loops. Since very recently, we know that the problem is coNP-complete for simple integer loops. We continue along this path by investigating programs dealing with floating-point computations. First, we show that the problem is at least in coNP for simple floating-point loops. Then, in order to work around that theoretical limitation we present an algorithm which remains polynomial by sacrificing completeness. The algorithm, based on the Podelski-Rybalchenko algorithm, can also synthesize in polynomial time the linear ranking functions it detects. To our knowledge, our work is the first adaptation of this well-known algorithm to floating-points.

Floating-point numbers are used in a wide variety of programs, from numerical analysis programs to control command programs. However floating-point computations are affected by rounding errors that render them hard to be verified efficiently. We address in this paper termination proving of an important class of programs that manipulate floating-point numbers: the simple floating-point loops. Our main contribution is an optimal approximation to the rationals that allows us to efficiently analyze their termination.