Context: The goal of this thesis is to contribute to proposing efficient methods and tools for controlling the numerical reliability of large-scale simulations. In mathematics and computer science, interval arithmetic is a method of computation that involves manipulating intervals, as opposed to numbers (e.g., integers or floating-point numbers), with the aim of obtaining rigorous results. This approach allows for the bounding of rounding errors and, consequently, the development of numerical methods that provide reliable results. In interval arithmetic, a real number x is represented by a pair of floating-point numbers (x_inf , x_sup). Stating that x is represented by this pair means that x belongs to the interval [x_inf , x_sup]. This is referred to as the "inf-sup" notation. There is another way to represent intervals, where an interval is defined by its center and radius. It is denoted as ?c, r? with r > 0, representing the interval [c ? r, c + r]. This is known as the "center-radius" notation.
Objectives: Currently, it is necessary to use two numbers to encode an interval. The objective of this PhD subject is to define a "compressed" format for storing intervals. To do this, we can draw inspiration from the FP-ANR format, which allows the representation of both a number and the associated rounding error without any additional memory overhead. The primary goal is to analyse the impact of this compressed format on the assessment of numerical quality in interval arithmetic. A new interval arithmetic library based on this compressed format will then be developed for single and double precision floating-point arithmetic. Its performance will be compared to that of conventional interval libraries that use two numbers to represent results. As it is sometimes necessary to work with more precision than double precision, we will extend this compressed format to multiple precision arithmetic. This idea is to use this compressed format to obtain a new library similar to the MPFI library. We also plan to use this new compressed format in INTLAB that is an interval library for MATLAB and Octave. This will make it possible to study self-validating methods in this new context. As the idea behind this compress format is closed to Posit arithmetic, we will also study the self-validating methods with Posit format.