Abstract. We present a new method for solving the fixed point equations that appear in the static analysis of programs by abstract interpretation. We introduce and analyze a policy iteration algorithm for monotone self-maps of complete lattices. We apply this algorithm to the particular case of lattices arising in the interval abstraction of values of variables. We demonstrate the improvements in terms of speed and precision over existing techniques based on Kleene iteration, including traditional widening/narrowing acceleration mecanisms.

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Abstract. Exact computations being in general not tractable for computers, they are approximated by floating-point computations. This is the source of many errors in numerical programs. Because the floatingpoint arithmetic is not intuitive, these errors are very difficult to detect and to correct by hand and we consider the problem of automatically synthesizing accurate formulas. We consider that a program would return an exact result if the computations were carried out using real numbers. In practice, roundoff errors arise during the execution and these errors are closely related to the way formulas are written. Our approach is based on abstract interpretation. We introduce Abstract Program Equivalence Graphs (APEGs) to represent in polynomial size an exponential number of mathematically equivalent expressions. The concretization of an APEG yields expressions of very different shapes and accuracies. Then, we extract optimized expressions from APEGs by searching the most accurate concrete expressions among the set of represented expressions.

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We introduce a concrete semantics for floating-point operations which describes the propagation of roundoff errors throughout a computation. This semantics is used to assert the correctness of an abstract interpretation which can be straightforwardly derived from it. In our model, every elementary operation introduces a new first order error term, which is later combined with other error terms, yielding higher order error terms. The semantics is parameterized by the maximal order of error to be examined and verifies whether higher order errors actually are negligible. We consider also coarser semantics computing the contribution, to the final error, of the errors due to some intermediate computations.

We introduce FPBench, a standard benchmark format for validation and optimization of numerical accuracy in floating-point computations. FPBench is a first step toward addressing an increasing need in our community for comparisons and combinations of tools from different application domains. To this end, FPBench provides a basic floatingpoint benchmark format and accuracy measures for comparing different tools. The FPBench format and measures allow comparing and composing different floating-point tools. We describe the FPBench format and measures and show that FPBench expresses benchmarks from recent papers in the literature, by building an initial benchmark suite drawn from these papers. We intend for FPBench to grow into a standard benchmark suite for the members of the floating-point tools research community. 3 We are using |S| to denote the number of elements in a set S 4 Where n is the number of arguments. 5 Where N is the number of valid inputs, and n is the number of arguments.

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