Error Reduction of the Taylor Centered Form by Half and an Inner Estimation of the Range

We show that the overestimation error of the interval Horner method for univariate polynomials on a centered interval is reduced at least by half if the interval is split at its midpoint zero and the interval Horner method is applied to both halves separately. This observation is used to reduce the overestimation error of the Taylor centered form at least by half. Further, it can be used to compute error bounds for the Taylor centered form and an inner range estimation.