Lazy Strength Reduction

We present a bit-vector algorithm that uniformly combines code motion and strength reduction, avoids superruous register pressure due to unnecessary code motion, and is as eecient as standard unidirectional analyses. The point of this algorithm is to combine the concept of lazy code motion of 1] with the concept of unifying code motion and strength reduction of 2, 3, 4]. This results in an algorithm for lazy strength reduction, which consists of a sequence of unidirectional analyses, and is unique in its transformational power. 1 Motivation Code motion improves the runtime eeciency of a program by avoiding unnecessary recom-putations of a value at runtime. Strength reduction improves runtime eeciency by reducing \expensive" recomputations to less expensive ones, e.g., by reducing computations involving multiplication to computations involving only addition. Common to both techniques is replacing the original computations of a program by auxiliary variables (registers) that are initialized at suitable program points. In the case of strength reduction, they are additionally updated at certain points. The similarity between strength reduction and code motion suggests the combination of both techniques, an idea which was rst realized by an algorithm of Joshi and Dhamdhere 2, 3] that enhances the code motion algorithm of Morel and Renvoise 5] to capture strength reduction. In this paper we combine Joshi and Dhamdhere's approach with the idea of lazy code motion presented in 1]. This results in an algorithm for lazy strength reduction, which uniformly combines code motion and strength reduction, and avoids any unnecessary register pressure. In fact, in contrast to previous algorithms, it does not insert multiplications and additions on the same path, minimizes the lifetimes of moved computations, and limits the insertion of multiple additions on a path to a minimum. Moreover, as our algorithm is composed of a sequence of unidirectional bit-vector analyses, it is as eecient as the standard unidirectional analyses (cf. The illustration of the essential new features of our algorithm requires a rather complex program structure. In the example of Figure 1, which is a slight modiication of an example of 3], our lazy strength reduction algorithm is unique in yielding the result shown in Figure 2.