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Section: New Results

Floating-Point and Numerical Programs

Correct Average of Decimal Floating-Point Numbers

Some modern processors include decimal floating-point units, with a conforming implementation of the IEEE-754 2008 standard. Unfortunately, many algorithms from the computer arithmetic literature are not correct anymore when computations are done in radix 10. This is in particular the case for the computation of the average of two floating-point numbers. S. Boldo, F. Faissole and V. Tourneur developed a new radix-10 algorithm that computes the correctly-rounded average, with a Coq formal proof of its correctness, that takes gradual underflow into account [17].

Optimal Inverse Projection of Floating-Point Addition

In a setting where we have intervals for the values of floating-point variables $x$, $a$, and $b$, we are interested in improving these intervals when the floating-point equality $x\oplus a=b$ holds. This problem is common in constraint propagation, and called the inverse projection of the addition. It also appears in abstract interpretation for the analysis of programs containing IEEE 754 operations. D. Gallois-Wong, S. Boldo and P. Cuoq proposed floating-point theorems that provide optimal bounds for all the intervals. Fast loop-free algorithms compute these optimal bounds using only floating-point computations at the target precision [34].

Handbook of Floating-point Arithmetic

Initially published in 2010, the Handbook of Floating-Point Arithmetic has been heavily updated. G. Melquiond contributed to the second edition [28].

Error analysis of finite precision digital filters and controllers

The effort to provide accurate and reliable error analysis of fixed-point implementations of Signal Processing and Control algorithms was continued (see also the formalization effort above). A. Volkova, M. Istoan, F. de Dinechin and T. Hilaire (Citi Lyon, INSA Lyon) created an automatic code generator for FPGAs and dedicated roundoff analysis in order to minimize the bit-widths used for the intern computations while guaranteeing a bound on the output error [16]. The global workflow for the rigorous design of reliable Fixed-Point filters has been studied by A. Volkova, T. Hilaire and C. Lauter and submitted to a journal  [36] : it concerns the rigorous determination of the Most Significant Bit of each variable, to guaranty that no overflow will ever occur, also taking into account the roundoff error propagation.