Sharp ULP rounding error bound for the hypotenuse function

Source abstract

The hypotenuse function, z = x 2 + y 2 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 | z ¯ − z | > 1.222 u l p ( z ) |\overline z-z|>1.222\, ulp(z) and | z ¯ − z | > 1.222 u l p ( z ¯ ) |\overline z-z|>1.222\, ulp(\overline z) . Numerical experiments indicate that this bound is sharp and cannot be improved.