Sharp ULP rounding error bound for the hypotenuse function

The hypotenuse function, z = √ x2 + y2, 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 |z − z| < 1.222 ulp(z) and |z − z| < 1.222 ulp(z). Numerical experiments indicate that this bound is sharp and cannot be improved.