When can forward stable algorithms be composed stably?

Abstract We state some widely satisfied hypotheses, depending only on two functions $g$ and $h$, under which the composition of a forward stable algorithm for $g$ and a forward stable algorithm for $h$ is a forward stable algorithm for the composition $g \circ h$. We show that the failure of these conditions can potentially lead to unstable algorithms. Finally, we list a number of examples to illustrate the new concepts and the usability of the results.