the forward-error rule of thumb

If you remember one inequality from this whole subject, make it this one: forward error is at most the condition number times the backward error. In plain words, how wrong your answer is (forward) is bounded by how hard the problem is (conditioning) multiplied by how good your algorithm is (backward error). It is the master accounting identity that ties the three central ideas together and lets you predict accuracy without ever computing the true answer.

To use it: estimate the backward error of your method (for a backward-stable algorithm this is roughly the unit roundoff, about 10^-16 in double precision), estimate the condition number kappa of the problem, and multiply. The product bounds the relative forward error. Example: a backward-stable solve of A x = b with kappa(A) ~ 10^8 gives a relative forward error around 10^8 times 10^-16 = 10^-8 — so about 8 correct significant digits out of 16. The condition number directly tells you how many of your digits survive.

Treat it honestly as a rule of thumb, not a strict theorem. It is an order-of-magnitude estimate: the true bound carries problem-dependent constants, the worst-case direction may not be hit, and conditioning is a first-order (linearized) notion that can mislead for large perturbations. Still, it is astonishingly useful in practice and it makes the central message unmistakable — to get an accurate answer you need BOTH a well-conditioned problem AND a stable algorithm. Either one failing is enough to wreck the result.