Efficient and reliable methods for rounded-interval arithmetic

We present an efficient and reliable method for computing the unit–in–the–last–place (ulp) of a double precision floating–point number, taking advantage of the standard binary representation for floating– point numbers defined by IEEE Std 754–1985. The ulp is necessary to perform software rounding for robust rounded interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling.