Section: New Results
Verification of Computer Arithmetic
 Formal Verification of a StateoftheArt Integer Square Root

Even though some programs only use integer operations, the best way to understand and verify them might be to view them as fixedpoint arithmetic algorithm. This is the case of the function from the GMP library that computes the square root of a 64bit integer. The C code is short but intricate, as it implements Newton's method and it relies on magic constants and intentional arithmetic overflows. G. Melquiond and R. RieuHelft have verified this algorithm using the Why3 tool and automated solvers such as Gappa [28].
 Roundoff error and exceptional behavior analysis of explicit RungeKutta methods

S. Boldo, F. Faissole, and A. Chapoutot developed a new finegrained analysis of roundoff errors in explicit RungeKutta integration methods, taking into account exceptional behaviors, such as underflow and overflow [12]. First steps towards the formalization has been done by F. Faissole [34].
 Optimal Inverse Projection of FloatingPoint Addition

In a setting where we have intervals for the values of floatingpoint variables $x$, $a$, and $b$, we are interested in improving these intervals when the floatingpoint equality $x\oplus a=b$ holds. This problem is common in constraint propagation, and called the inverse projection of the addition. D. GalloisWong, S. Boldo, and P. Cuoq proposed floatingpoint theorems that provide optimal bounds for all the intervals [13].
 Emulating roundtonearesttiestozero "augmented" floating point operations using roundtonearesttiestoeven arithmetic

The 2019 version of the IEEE 754 Standard for FloatingPoint Arithmetic recommends that new “augmented” operations should be provided for the binary formats. These operations use a new “rounding direction”: round to nearest tiestozero. S. Boldo, C. Lauter, and J.M. Muller show how they can be implemented using the currently available operations, using roundtonearest tiestoeven with a partial formal proof of correctness [42].
 LTI filters

Several developments were made towards the efficency and accuracy of the implementation of LTI (linear timeinvariant) numerical filters: a wordlength optimization problem under accuracy constraints [26] by T. Hilaire, H. Ouzia, and B. Lopez, and a tight worstcase error analysis [16] by A. Volkova, T. Hilaire, and C. Lauter.