How accurate should numerical routines be?

J. H. Wilkinson, in justifying backward error analyses, advanced the proposition that a numerical routine could not be blamed if the computed answer was the exact answer to the problem with slightly perturbed input. Such a model was established by Wilkinson (1968) for the solution of linear equations, ns by (I) Gaussian elimination and (II) triangular decomposition using double precision accumulation of inner products. In the latter it is usual that inputs are perturbed only by a small multiple of a unit rounding error (see formula (175)).