For a real function f defined throughout a closed interval X, an interval evaluation F(X) satisfies inclusion when f(x) belongs to F(X) for every x in X.

For finite real intervals [a,b] and [c,d], outward-rounded addition gives [down(a+c),up(b+d)]. Here down returns a representable lower bound on the exact sum, and up returns a representable upper bound. Guaranteed containment requires exact arithmetic or suitable directed rounding, together with supported operations and correct implementation of their bounds; merely storing two floating-point endpoints is insufficient.

Conditions and limits:

• An enclosure need not have the smallest possible width and does not represent statistical confidence.
• Using a no-rounding policy with inexact floating-point arithmetic removes the containment guarantee.
• Input conversions, rounding-state management, exceptional values, exponent limits, and support for particular functions require appropriate handling.
• Correct scalar directed rounding supports endpoint bounds but does not by itself establish a complete interval algorithm.

Sources:

• Boost Interval Arithmetic Library — Introduction and Overview
• Boost Interval Arithmetic Library — Rounding Policies
• GNU MPFR — Directed rounding