What does this equation mean?

The difficulty is that a simulation is not one operation; it is billions of them, composed. A relative error of u per operation does not by itself say anything about the error after a million timesteps, because errors can compound, cancel, or amplify depending entirely on the structure of the calculation. This is the subject of rounding error analysis , and it produces two genuinely different kinds of statement that are often conflated in casual discussion. A forward error bound asks how far the computed answer can be from the true answer. A backward error bound asks, instead, how large a perturbation of the original input would have been needed to produce the computed output exactly — and it is frequently the more tractable and more informative quantity, because it separates the algorithm’s behavior from the problem’s own sensitivity. Higham’s systematic treatment of this distinction, and of the perturbation theory needed to convert a backward bound into a usable forward bound, is the standard reference for how these bounds are actually derived and what they do and do not promise [ 6 ] .