CMP.8:4.4 - Include finite arithmetic and the sensitivity of the question

An algorithm implemented with finite representations performs approximations beyond those in its ideal construction. Identify consequential rounding, overflow, underflow, cancellation and approximate comparisons. Use integer or rational operations, higher precision, a stable reformulation or directed bounds where they improve the required answer affordably.

Conditioning describes how the mathematical answer responds to changes in the problem data. Forward error compares the returned value with the requested answer. Backward error asks how much the data would need to change to make the returned value an answer to the changed problem. Fix the permitted data changes and the measure of their size; those choices determine the claim.

A small backward error supports a small forward error only with suitable sensitivity control. A stable calculation cannot recover a distinction already absent from uncertain input data. Conversely, a well-conditioned question can be computed poorly by an unstable expression; :5.2 demonstrates a local repair.

Use a cheap bound or estimate of sensitivity when a stronger computation would not change the decision. When even the error bound is computed approximately, preserve the direction needed for its use.