Section: New Results

Verification of Computer Arithmetic

Formal Verification of a State-of-the-Art 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 fixed-point arithmetic algorithm. This is the case of the function from the GMP library that computes the square root of a 64-bit 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. Rieu-Helft have verified this algorithm using the Why3 tool and automated solvers such as Gappa [28].

Round-off error and exceptional behavior analysis of explicit Runge-Kutta methods

S. Boldo, F. Faissole, and A. Chapoutot developed a new fine-grained analysis of round-off errors in explicit Runge-Kutta 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 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. D. Gallois-Wong, S. Boldo, and P. Cuoq proposed floating-point theorems that provide optimal bounds for all the intervals [13].

Emulating round-to-nearest-ties-to-zero "augmented" floating- point operations using round-to-nearest-ties-to-even arithmetic

The 2019 version of the IEEE 754 Standard for Floating-Point Arithmetic recommends that new “augmented” operations should be provided for the binary formats. These operations use a new “rounding direction”: round to nearest ties-to-zero. S. Boldo, C. Lauter, and J.-M. Muller show how they can be implemented using the currently available operations, using round-to-nearest ties-to-even 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 time-invariant) numerical filters: a word-length optimization problem under accuracy constraints [26] by T. Hilaire, H. Ouzia, and B. Lopez, and a tight worst-case error analysis [16] by A. Volkova, T. Hilaire, and C. Lauter.