Interval operations involving NaNs

Ten years ago IEEE standard 1] for oating-point arithmetic became oocial. Each IEEE oating-point format supports: its own set of nite real numbers, 1, two distinguished values +0 and ?0 and a set of special values called NaNs (Not-a-Number). Arithmetic operations include operations on numeric, non-numeric or mixed operands in four rounding modes. A number of exceptional situations may arise during numerical computations. Every exception, when it occurs must raise a ag that a program may subsequently sense and/or take a trap intended to handle the detected exceptional condition. The mandatory default response to the exceptional situations is not to trap on them, but to compute and deliver to the destination a default result, speciied for each possible exception. Now, an increasing number of computers and software feature IEEE arithmetic. Contrary to the programs written before IEEE Standard became oocial, programs which are written to be used under IEEE arithmetic should be prepared to expect any exceptions that can arise and deal with them properly. Some recent works 3] show that algorithms working uniformly and robustly across rather diierent systems and languages are a lot easier to design and usually more eecient if they rely on non-trapping exception handling paradigm. Most recent interval arithmetic implementations 8] are in Standard conforming environment. Recently, some speciications of Basic Interval Arithmetic Subroutines (BIAS) appeared 2, 5] showing a movement toward standardization of the user interfaces for interval arithmetic software. Although IEEE Standard has been intended to facilitate, between other things, the implementation of interval arithmetic 6] nowadays there are no general implementation requirements for interval arithmetic under IEEE systems. Since there is no meaning of the arithmetic operations on intervals involving NaNs their implementation is up to the implementor's option. Moreover, no agreement exists about how to deal with the exceptions arising on interval operations and no default interval response has been proposed. The emphasis in computing was traditionally on speed but we have to develop also credible and accurate programs. For a program to be credible, the result it produces must never be misleading.