Strength Reduction

Strength Reduction (SR) [2] replaces expensive operations by equivalent cheaper ones from the target machine language. Some machine instructions are considerably cheaper than others and can often be used as special cases of more expensive operators. For instance, $x^2$ is invariably cheaper to implement as than as a call to an exponentiation routine. Moreover, an additive operator is generally cheaper than a multiplicative operator. Some loops have a variable $i$ that is incremented or decremented, and a variable $j$ that is set in the loop to where $c$ and are loop-invariant. Then we can calculate $j$'s value without reference to $i$; whenever $i$ is incremented by $a$ we can increment $j$ by $c * a$

Figure 5.3: Before and After Strength Reduction

     s = 0                               s = 0
     i = 0                               k = a
 L1: if i < n goto L2                    b = n * 4
     j = i * 4                           c = a + b
     k = j + d                       L1: if k < c goto L2
     x = M[k]                            x = M[k]
     s = s + x                           s = s + x
     i = i + 1                           k = k + 4
     goto L1                             goto L1

 L2                                  L2

     (a) Before                          (b) After

The program (a) from Figure 5.3 is a program to sum the elements of an array. Using induction variable analysis to find that $i$ and $j$ are related induction variables, strength reduction to replace a multiplication by 4 with an addition, then induction variable elimination to replace by $k < 4 * n + a$, we get the program (b). We have reviewed the key features of SR and its algorithm briefly. SR is a very powerful algorithm to transform computations involving multiplications and additions together into computations only using additions. Thus, it could be a very competitive candidate to optimize the current HPJava compiler because of the complexity of the associated terms in the subscript expression of a multiarray element access.

Figure 5.4: Translation of overall construct, applying Strength Reduction.

SOURCE:

TRANSLATION:

where:

Before applying PRE and SR to some scientific algorithms, we can apply this SR technique on the basic translation of overall construct in the subsection 4.5.5. The naive translation generates a local subscript, sub, and a global subscript, glb for the index i. We call these two variables, control subscripts for the index i. The computations for the control subscripts are obviously amenable to SR. Thus, Figure 5.4 shows the translation scheme for overall after applying Strength Reduction.