We present a method to verify the correctness of parallel programs that perform complex numerical computations, including computations involving floating-point arithmetic. The method requires that a sequential version of the program be provided, to serve as the specification for the parallel one. The key idea is to use model checking, together with symbolic execution, to establish the equivalence of the two programs.

We discuss the characteristic properties of ASTRÉE, an automatic static analyzer for proving the absence of runtime errors in safety-critical real-time synchronous controlcommand C programs, and compare it with a variety of other program analysis tools. 1

We present a method to verify the correctness of parallel programs that perform complex numerical computations, including computations involving floating-point arithmetic. This method requires that a sequential version of the program be provided, to serve as the specification for the parallel one. The key idea is to use model checking, together with symbolic execution, to establish the equivalence of the two programs. In this approach the path condition from symbolic execution of the sequential program is used to constrain the search through the parallel program. To handle floating-point operations, three different types of equivalence are supported. Several examples are presented, demonstrating the approach and actual errors that were found. Limitations and directions for future research are also described.

In this article, we describe a new library for computing guaranteed bounds of the solutions of Initial Value Problems (IVP). Given an initial value problem and an end point, our library computes a sequence of approximation points together with a sequence of approximation errors such that the distance to the true solution of the IVP is below these error terms at each approximation point. These sequences are computed using a classical Runge-Kutta method for which truncation and roundoff errors may be over-approximated. We also compute the propagation of local errors to obtain an enclosure of the global error at each computation step. These techniques are implemented in a C++ library which provides an easy-to-use framework for the rigorous approximation of IVP. This library implements an error control technique based on step size reduction in order to reach a certain tolerance on local errors. 1.

Abstract. Floating-point arithmetic is an important source of errors in programs because of the loss of precision arising during a computation. Unfortunately, this arithmetic is not intuitive (e.g. many elementary operations are not associative, inversible, etc.) making the debugging phase very difficult and empiric. This article introduces a new kind of program transformation in order to automatically improve the accuracy of floating-point computations. We use P. Cousot and R. Cousot’s framework for semantics program transformation and we propose an offline transformation. This technique was implemented, and the first experimental results are presented. 1

Modern computing has adopted the floating point type as a default way to describe computations with real numbers. Thanks to dedicated hardware support, such computations are efficient on modern architectures, even in double precision. However, rigorous reasoning about the resulting programs remains difficult. This is in part due to a large gap between the finite floating point representation and the infiniteprecision real-number semantics that serves as the developers’ mental model. Because programming languages do not provide support for estimating errors, some computations in practice are performed more and some less precisely than needed. We present a library solution for rigorous arithmetic computation. Our numerical data type library tracks a (double) floating point value, but also a guaranteed upper bound on the error between this value and the ideal value that would be computed in the real-value semantics. Our implementation involves a set of linear approximations based on an extension of affine arithmetic. The derived approximations cover most of the standard mathematical operations, including trigonometric functions, and are more comprehensive than any publicly available ones. Moreover, while interval arithmetic rapidly yields overly pessimistic estimates, our approach remains precise for several computational tasks of interest. We evaluate the library on a number of examples from numerical analysis and physical simulations. We found it to be a useful tool for gaining confidence in the correctness of the computation.

Digital computers permeate our physical world. This phenomenon creates a pressing need for tools that help us understand a priori how digital computers can affect their physical environment. In principle, simulation can be a powerful tool for animating models of the world. Today, however, there is not a single simulation environment that comes with a guarantee that the results of the simulation are determined purely by a real-valued model and not by artifacts of the digitized implementation. As such, simulation with guaranteed fidelity does not yet exist. Towards addressing this problem, we offer an expository account of what is known about exact real arithmetic. We argue that this technology, which has roots that are over 200 years old, bears significant promise as offering exactly the right technology to build simulation environments with guaranteed fidelity. And while it has only been sparsely studied in this large span of time, there are reasons to believe that the time is right to accelerate research in this direction. ∗ This research was sponsored by the NSF under Award 0439017,