Centered Taylor model with derivative-interval remainder

Let $f$ be a real function that is $n+1$ times continuously differentiable on an open interval containing the finite real interval $X=[a,b]$. Here $n\ge 0$ is an integer, and continuous differentiability means that the derivatives through order $n+1$ exist there and are continuous. Let $c\in X$ be a dyadic point. Put

$$P_n(t)=\sum _{j=0}^n\frac{f^{(j)}(c)}{j!}(t-c{)}^j\mathrm{.}$$

Supply dyadic intervals $C_j$ containing $f^{(j)}(c)\mathrm{/}j!$ for $0\le j\le n$ and a dyadic interval $D$ containing $f^{(n+1)}(X)$. Put $M=\max (\mathrm{\mid }\inf D\mathrm{\mid },\mathrm{\mid }\sup D\mathrm{\mid })$ and $R=\max (\mathrm{\mid }a-c\mathrm{\mid },\mathrm{\mid }b-c\mathrm{\mid })$. The centered Taylor-model enclosure is ${\mathcal{T}}_n(f,X,c,(C_j{)}_{j=0}^n,D)={\sum }_{j=0}^n{C}_j(X-c{)}^j+[-\frac{M{R}^{n+1}}{(n+1)!},\frac{M{R}^{n+1}}{(n+1)!}],$ where products, powers, sums, and the final sum are evaluated by outward-rounded interval operations. The first term carries input variation through the polynomial; the symmetric second interval is the explicit derivative-based remainder. If a required coefficient or derivative enclosure is unavailable, or any operation or endpoint conversion returns a non-finite status, no finite Taylor-model certificate is returned.