If we want to evaluate $$f(x)=\frac{e^x-1-x}{x^2}$$ then we have to observe its large relative error as $x\to 0$. My question is that how can we find a method so that we can compute $f(x)$ to full machine precision for all $|x|<1$?
So far, I have been able to use a Taylor series to show that $$f(x)=\frac{1}{2!}+\frac{x}{3!}+\frac{x^2}{4!}+\ldots$$ And now I suspect that we have to find the remainder, right? How would I do this?
