New Developments in Interval Arithmetic and Their Implications for Floating-Point Standardization

We consider the prospect of a processor that can perform interval arithmetic at the same speed as conventional floating-point arithmetic. This makes it possible for all arithmetic to be performed with the superior security of interval methods without any penalty in speed. In such a situation the IEEE floating-point standard needs to be compared with a version of floating-point arithmetic that is ideal for the purpose of interval arithmetic. Such a comparison requires a succinct and complete exposition of interval arithmetic according to its recent developments. We present such an exposition in this paper. We conclude that the directed roundings toward the infinities and the definition of division by the signed zeros are valuable features of the standard. Because the operations of interval arithmetic are always defined, exceptions do not arise. As a result neither Nans nor exceptions are needed. Of the status flags, only the inexact flag may be useful. Denormalized numbers seem to have no use for interval arithmetic; in the use of interval constraints, they are a handicap.