Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

Interval arithmetic underwent a standardization effort starting in 2008 by the IEEE P1788 working group. The structure of the proposed standard is presented: the mathematical level is distinguished from both the implementation and representation levels. The main definitions are introduced: what is an interval? How are mathematical functions, such as arithmetic operations or trigonometric functions, extended over intervals? What are the comparison relations? How are set operations, such as the union or the intersection, extended over intervals? While developing this standard, some topics led to hot debates: these topics correspond to points most delicate and difficult to define. Such a hot topic is the handling of exceptions. Eventually, the system of decorations was adopted. A decoration is a piece of information that is attached to each interval. Rules for the propagation of decorations have also been defined by the standardization group. Another hot topic is the mathematical model used for interval arithmetic. Historically, the model introduced by R. Moore in the 60s covered only non-empty and bounded intervals. The set-based model includes the empty set and unbounded intervals as well. Tenants of Kaucher arithmetic also insisted on offering "reverse" intervals. It was eventually decided that an implementation must provide at least one of these flavors of interval arithmetic. The standard provides hooks for these different flavors. Finally, a tentative list of missing items is given. As the preparation of the draft should end at the beginning of 2014, no chapter is missing. However, a reference implementation would be welcome to validate the choices made during the development of the standard for interval arithmetic.