Consider the polynomial . Evaluate it at the RealBox object with center and radius . We first use simple evaluation using the regular arithmetic operators.
The radius of the result is smaller if we first convert the polynomial to Horner form.
However, this is still a severe overestimation of the radius: the minimal value on this interval is about and the maximal value of about is achieved at . We verify these values numerically and graphically below.
So ideally we would like the result to have a center of about and a radius of about . We don't quite achieve that with EvaluatePolynomial, but we get much closer than with the other options above.