  We provide tools to help automate the error analysis of algorithms that
evaluate simple functions over the floating-point numbers. The aim is to obtain
tight relative error bounds for these algorithms, expressed as a function of
the unit round-off. Due to the discrete nature of the set of floating-point
numbers, the largest errors are often intrinsically "arithmetic" in the sense
that their appearance may depend on specific bit patterns in the binary
representations of intermediate variables, which may be present only for some
precisions. We focus on generic (i.e., parameterized by the precision) and
analytic over-estimations that still capture the correlations between the
errors made at each step of the algorithms. Using methods from computer
algebra, which we adapt to the particular structure of the polynomial systems
that encode the errors, we obtain bounds with a linear term in the unit
round-off that is sharp in manycases. An explicit quadratic bound is given,
rather than the $O()$-estimate that is more common in this area. This is
particularly important when using low precision formats, which are increasingly
common in modern processors. Using this approach, we compare five algorithms
for computing the hypotenuse function, ranging from elementary to quite
challenging.
