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Section: New Results

Floating-Point and Numerical Programs

  • T. Nguyen and C. Marché have worked on how to prove floating-point programs while taking into account architecture- and compiler-dependent features such as the use of the x87 stack in Intel micro-processors. 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 floating-point algorithms for computing the exact and approximated errors of the FMA (fused multiply-and-add) [16] .

  • S. Boldo and G. Melquiond have developed in Coq a comprehensive formalization of floating-point arithmetic: core definitions, axiomatic and computational rounding operations, high-level 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 floating-point 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 Paris-Rocquencourt) 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 floating-point 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 Nancy-Lorraine), has designed new methods for computing guaranteed enclosures of the Masser-Gramain constant, a two-dimensional analogue of the Euler-Mascheroni constant  [86] .

  • G. Melquiond, in collaboration with J-M. Muller (CNRS, Arénaire, LIP, ÉNS Lyon) and E. Martin-Dorel (Arénaire, LIP, ÉNS Lyon), has worked on weakening the assumptions floating-point error-free transformations rely on [39] .