Section: New Results
FloatingPoint and Numerical Programs

T. Nguyen and C. Marché have worked on how to prove floatingpoint programs while taking into account architecture and compilerdependent features such as the use of the x87 stack in Intel microprocessors. This is done by analyzing the assembly code generated by the compiler [40] , [28]

S. Boldo and C. Marché published a survey article on the proofs of numerical C programs using both automatic provers and Coq [15] .

S. Boldo and T. Nguyen have worked on how to prove numerical programs on multiple architectures and compilers [17] . More precisely, it covers all the compiler choices about the use of extended registers, FMA, and reorganization of additions.

S. Boldo and J.M. Muller (CNRS, Arénaire, LIP, ÉNS Lyon) have worked on new floatingpoint algorithms for computing the exact and approximated errors of the FMA (fused multiplyandadd) [16] .

S. Boldo and G. Melquiond have developed in Coq a comprehensive formalization of floatingpoint arithmetic: core definitions, axiomatic and computational rounding operations, highlevel properties [23] . It provides a framework for developers to formally certify numerical applications.

G. Melquiond, in collaboration with F. de Dinechin (Arénaire, LIP, ÉNS Lyon) and C. Lauter (Intel Hillsboro), has improved the methodology for formally proving floatingpoint mathematical functions when their correctness depends on relative errors [19] .

S. Boldo, J.C. Filliâtre and G. Melquiond, in collaboration with F. Clément (Estime, INRIA ParisRocquencourt) and M. Mayero (University Paris 13) have finished a full formal proof of a program solving a partial differential equation (the wave equation) using a finite difference scheme [36] . This proof includes both the mathematical convergence proof (method error) [57] , a tricky floatingpoint proof [56] and proofs of the absence of runtime errors.

C. Lelay, under the supervision of S. Boldo and G. Melquiond, has worked on differentiability in Coq. The goal was to prove the existence of a solution to the wave equation thanks to D'Alembert's formula and to automatize the process as much as possible [44] [77] .

G. Melquiond, in collaboration with W. G. Nowak (Institute of Mathmatics, Austria) and P. Zimmermann (Caramel, INRIA NancyLorraine), has designed new methods for computing guaranteed enclosures of the MasserGramain constant, a twodimensional analogue of the EulerMascheroni constant [86] .

G. Melquiond, in collaboration with JM. Muller (CNRS, Arénaire, LIP, ÉNS Lyon) and E. MartinDorel (Arénaire, LIP, ÉNS Lyon), has worked on weakening the assumptions floatingpoint errorfree transformations rely on [39] .