For a real variable x in the closed interval X=[-2,2], both occurrences of x in xx take the same value. The square range is therefore square(X)=[0,4]. Interval multiplication instead encloses products of independently chosen operands from X: XX=[-4,4]. For example, choosing -2 and 2 gives -4, although a real square cannot be negative. Thus interval evaluation of x*x can overestimate the actual square range by losing the dependence between repeated occurrences of the variable. The product interval still contains every actual square value.

Conditions and limits:

• These exact mathematical ranges include their endpoints.
• This example establishes overestimation for this interval and expression; it does not quantify the effect for other expressions.
• Computational containment depends on exact arithmetic or suitable directed rounding and supported operations.
• Containment alone establishes neither the narrowest enclosure nor statistical confidence.

Source:

• Boost Interval Arithmetic Library — Introduction and Overview