The problem of computing a desired function value to within a prescribed tolerance can be formulated in the following two distinct ways: Formulation I: Given x and ∈ > 0, compute f (x) to within ∈. Formulation II: Given only that x is in a closed interval X, compute a subinterval of the image, f (X) = { f (x) : x ∈ X}. The first formulation is applicable when x is known to arbitrary accuracy. The second formulation is applicable when x is known only to a limited accuracy, in which case the tolerance is prescribed albeit indirectly by the interval X, and one must be satisfied with all or part of the set f (X) of possible function values. Elsewhere the author has presented an efficient solution to Formulation I for any rational f and many nonrational f . B. A. Chartres has presented an efficient solution to Formulation II for a very restricted class of rational f and for a few nonrational f . In this paper a solution to Formulation II for the arbitrary nonconstant rational f is presented. By bounding df/dx away from zero over some subset of X, it is shown how to reduce Formulation II to Formulation I, yielding the solution given here. In generalizing to vector-valued functions f, Chartres has solved Formulation II only for rational f which satisfy a linear system of equations, while this paper presents a solution for arbitrary non-degenerate rational vector-valued f.

Finite-precision interval arithmetic evaluation of a function ƒ of n variables at an n -dimensional rectangle T which is the Cartesian product of intervals yields an interval which is denoted by F ( T ). Correspondingly, finite-precision real arithmetic evaluation of ƒ at the midpoint m ( T ) of T yields a number which is denoted by f ( m ( T )) Ε F ( T ). Often, f ( m ( T )) is surprisingly close to m ( F ( T )). The purpose of this note is to provide some insight into this phenomenon by examining the case of infinite precision and rational functions. It is shown that if the gradient of ƒ is nonzero at a fixed point t Ε T , then as the maximum edge length w ( T ) of T approaches zero, [ m ( F ( T )) - ƒ( m ( T ))]/ w ( F ( T )) = O ( w ( T )), where F ( T ) and ƒ( m ( T )) denote the infinite-precision results corresponding to F ( T ) and f ( m ( T )), respectively. More precise results are derived when ƒ is one of +, -, ×, or /.