Sharp ULP rounding error bound for the hypotenuse function

The hypotenuse function, $z=\sqrt{x^2+y^2}$, is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that $|\overline z-z|<1.222\, ulp(z)$ and $|\overline z-z|<1.222\, ulp(\overline z)$. Numerical experiments indicate that this bound is sharp and cannot be improved.