Now a very trivial question about this book Ralston: A first course in numerical analysis the Hermite formula example, how it follows $$\frac{-1}{32768}<E(0.60)<\frac{-1}{2^{23}}$$ from this: $p_n(x),p_r(x)$ and the tables of $\ln(x)$ and $1/x$ at the points $0.4,0.5,0.7$ and $0.8$ ?
The error is, per the cited formula and $n=r=4$, $$ E(0.6)=\frac{f^{(8)}(\xi)}{8!}(0.1)^4(0.2)^4 $$ and $f^{(8)}(\xi)=-7!\xi^{-8}$, $\xi\in[0.4,0.8]$. This gives a bound to the error as $$ E(0.6)=-\frac{1}{8\cdot \xi^8}\frac{2^4}{10^8}\in -\frac1{2^{15}}[2^{-8},1], $$ which is the claim.
