Assuming that both $x$ and $z$ have no representation errors, and that $\vert z^2 \vert \ll \vert x^2 \vert$. There must exist an expression for $\sqrt{x^2 + z^2}$ that is the same in exact arithmetic but more precise in floating point arithmetic, according to my teacher.
I cannot possibly figure out how to find such an expression. If we assume some error $(1+\epsilon)$ to be applied to every operation in the above expression (e.g. $x \cdot x$ becomes $(x \cdot x)(1 + \epsilon)$) there cannot possibly be a better expression for this right? I have already tried rewriting the expression but everything I find seems to only add more errors, as the expression cannot be simplified further. Am I missing something here? Or did my teacher ask something impossible from me?