Cost-optimal code motion

We generalize Knoop et al.'s Lazy Code Motion (LCM) algorithm for partial redundancy elimination so that the generalized version also performs strength reduction. Although Knoop et al. have themselves extended LCM to strength reduction with their Lazy Strength Reduction algorithm, our approach differs substantially from theirs and results in a broader class of candidate expressions, stronger safety guarantees, and the elimination of the potential for performance loss instead of gain. Also, our general framework is not limited to traditional strength reduction, but rather can also handle a wide variety of optimizations in which data-flow information enables the replacement of a computation with a less expensive one. As a simple example, computations can be hoisted to points where they are constant foldable. Another example we sketch is the hoisting of polymorphic operations to points where type analysis provides leverage for optimization. Our general approach consists of placing computations so as to minimize their cost, rather than merely their number. So long as the cost differences between flowgraph nodes obey a certain natural constraint, a cost-optimal code motion transformation that does not unnecessarily prolong the lifetime of temporary varibles can be found using techniques completely analogous to LCM. Specifically, the cost differences can be discovered using a wide variety of forward data-flow analyses in a manner which we describe.