On relative errors of floating-point operations: Optimal bounds and applications

Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t − fl(t)|/|t| and E 2 (t) = |t − fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We provide explicit and attainable bounds on E 1 (t), which are all less than or equal to u/(1 + u) and, therefore, smaller than u. For E 2 (t) the bound u is attainable whenever t = x ± y or t = xy or, in base β > 2, t = x/y with x, y two floating-point numbers. However, for division in base 2 as well as for square root, smaller bounds are derived, which are also shown to be attainable. This set of sharp bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain either much shorter proofs of the best-known error bounds for such algorithms, or improvements on these bounds themselves.