  Discovering "good" algorithms for an operation is often considered an art
best left to experts. What if there is a simple methodology, an algorithm, for
systematically deriving a family of algorithms as well as their cost analyses,
so that the best algorithm can be chosen? We discuss such an approach for
deriving loop-based algorithms. The example used to illustrate this
methodology, evaluation of a polynomial, is itself simple yet the best
algorithm that results is surprising to a non-expert: Horner's rule. We finish
by discussing recent advances that make this approach highly practical for the
domain of high-performance linear algebra software libraries.
