Strength Reduction

In compiler construction, strength reduction is a compiler optimization where expensive operations are replaced with equivalent but less expensive operations. The classic example of strength reduction converts ''strong'' multiplications inside a loop into ''weaker'' additions – something that frequently occurs in array addressing.

Examples of strength reduction include replacing a multiplication within a loop with an addition and replacing exponentiation within a loop with a multiplication.

Code analysis

Most of a program's execution time is typically spent in a small section of code (called a hot spot), and that code is often inside a loop that is executed over and over.

A compiler uses methods to identify loops and recognize the characteristics of register values within those loops. For strength reduction, the compiler is interested in:
*Loop invariants: the values which do not change within the body of a loop.
*Induction variables: the values which are being iterated each time through the loop.

Loop invariants are essentially constants within a loop, but their value may change outside of the loop. Induction variables are changing by known amounts. The terms are relative to a particular loop. When loops are nested, an induction variable in the outer loop can be a loop invariant in the inner loop.

Strength reduction looks for expressions involving a loop invariant and an induction variable. Some of those expressions can be simplified. For example, the multiplication of loop invariant c and induction variable i

c = 7;
for (i = 0; i < N; i++)

can be replaced with successive weaker additions

c = 7;
k = 0;
for (i = 0; i < N; i++)

Strength reduction example

Below is an example that will strength-reduce all the loop multiplications that arose from array indexing address calculations.

Imagine a simple loop that sets an array to the identity matrix.

for (i = 0; i < n; i++)

Intermediate code

The compiler will view this code as

0010 ; for (i = 0, i < n; i++)
0020 ;
0410 G0001:

This expresses 2-dimensional array ''A'' as a 1-dimensional array of n*n size, so that whenever the high-level code expresses A, yit will internally be A x*n)+yfor any given valid indices x and y.

Many optimizations

The compiler will start doing many optimizations – not just strength reduction. Expressions that are constant (invariant) within a loop will be hoisted out of the loop. Constants can be loaded outside of both loops—such as floating point registers fr3 and fr4. Recognition that some variables don't change allows registers to be merged; n is constant, so r2, r4, r7, r12 can be hoisted and collapsed. The common value i*n is computed in (the hoisted) r8 and r13, so they collapse. The innermost loop (0120-0260) has been reduced from 11 to 7 intermediate instructions. The only multiply that remains in the innermost loop is line 0210's multiply by 8.

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

There are more optimizations to do. Register r3 is the main variable in the innermost loop (0140-0260); it gets incremented by 1 each time through the loop. Register r8 (which is invariant in the innermost loop) is added to r3. Instead of using r3, the compiler can eliminate r3 and use r9. The loop, instead of being controlled by r3 = 0 to n-1, can be controlled by r9=r8+0 to r8+n-1. That adds four instructions and kills four instructions, but there's one fewer instruction inside the loop.

0110 ; r3 = #0 killed ; j = 0
0115 r9 = r8 ; new assignment
0117 r20 = r8 + r2 ; new limit
0120 G0002:
0140 ; cmp r3, r2 killed ; j < n
0145 cmp r9, r20 ; r8 + j < r8 + n = r20
0150 bge G0003
0160
0170 ; A ,j= 0.0;
0200 ; r9 = r8 + r3 killed ; calculate subscript i * n + j
0210 r10 = r9 * #8 ; calculate byte address
0230 fstore fr3, A 100240
0250 ; r3 = r3 + #1 killed ; j++
0255 r9 = r9 + #1 ; new loop variable
0260 br G0002

Now r9 is the loop variable, but it interacts with the multiply by 8. Here we get to do some strength reduction. The multiply by 8 can be reduced to some successive additions of 8. Now there are no multiplications inside the loop.

0115 r9 = r8 ; new assignment
0117 r20 = r8 + r2 ; new limit
0118 r10 = r8 * #8 ; initial value of r10
0120 G0002:
0145 cmp r9, r20 ; r8 + j < r8 + n = r20
0150 bge G0003
0160
0170 ; A ,j= 0.0;
0210 ; r10 = r9 * #8 killed ; calculate byte address
0230 fstore fr3, A 100240
0245 r10 = r10 + #8 ; strength reduced multiply
0255 r9 = r9 + #1 ; loop variable
0260 br G0002

Registers r9 and r10 (= 8*r9) aren't both needed; r9 can be eliminated in the loop. The loop is now 5 instructions.

0115 ; r9 = r8 killed
0117 r20 = r8 + r2 ; limit
0118 r10 = r8 * #8 ; initial value of r10
0119 r22 = r20 * #8 ; new limit
0120 G0002:
0145 ; cmp r9, r20 killed ; r8 + j < r8 + n = r20
0147 cmp r10, r22 ; r10 = 8*(r8 + j) < 8*(r8 + n) = r22
0150 bge G0003
0160
0170 ; A ,j= 0.0;
0230 fstore fr3, A 100240
0245 r10 = r10 + #8 ; strength reduced multiply
0255 ; r9 = r9 + #1 killed ; loop variable
0260 br G0002

Outer loop

Back to the whole picture:

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

There are now four multiplications within the outer loop that increments r1. Register r8 = r1*r2 at 0190 can be strength reduced by setting it before entering the loop (0055) and incrementing it by r2 at the bottom of the loop (0385).

The value r8*8 (at 0118) can be strength reduced by initializing it (0056) and adding 8*r2 to it when r8 gets incremented (0386).

Register r20 is being incremented by the invariant/constant r2 each time through the loop at 0117. After being incremented, it is multiplied by 8 to create r22 at 0119. That multiplication can be strength reduced by adding 8*r2 each time through the loop.

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

The last multiply

That leaves the two loops with only one multiplication operation (at 0330) within the outer loop and no multiplications within the inner loop.

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

At line 0320, r14 is the sum of r8 and r1, and r8 and r1 are being incremented in the loop. Register r8 is being bumped by r2 (=n) and r1 is being bumped by 1. Consequently, r14 is being bumped by n+1 each time through the loop. The last loop multiply at 0330 can be strength reduced by adding (r2+1)*8 each time through the loop.

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

There's still more to go. Constant folding will recognize that r1=0 in the preamble, so several instructions will clean up. Register r8 isn't used in the loop, so it can disappear.

Furthermore, r1 is only being used to control the loop, so r1 can be replaced by a different induction variable such as r40. Where i went 0 <= i < n, register r40 goes 0 <= r40 < 8 * n * n.

0010 ; for (i = 0, i < n; i++)
0020
0410 G0001:

Other strength reduction operations

Operator strength reduction uses mathematical identities to replace slow math operations with faster operations. The benefits depend on the target CPU and sometimes on the surrounding code (which can affect the availability of other functional units within the CPU).

* replacing integer division or multiplication by a power of 2 with an arithmetic shift or logical shift
* replacing integer multiplication by a constant with a combination of shifts, adds or subtracts
* replacing integer division by a constant with a multiplication, taking advantage of the limited range of machine integers. This method also works if divisor is a non-integer sufficiently greater than 1, e.g. √2 or π.

Induction variable (orphan)

Induction variable or recursive strength reduction replaces a function of some systematically changing variable with a simpler calculation using previous values of the function. In a procedural programming language this would apply to an expression involving a loop variable and in a declarative language it would apply to the argument of a recursive function. For example,

f x = ... (3 ** x) ... (f (x + 1)) ...

becomes

f x = f' x 1
where
f' x z = ... z ... (f' (x + 1) (3 * z)) ...

Here modified recursive function takes a second parameter z = 3 ** x, allowing the expensive computation (3 ** x) to be replaced by the cheaper (3 * z).

See also

* Fast inverse square root
* Hacker's Delight
* Partial evaluation

Notes

References

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{{DEFAULTSORT:Strength Reduction
Compiler optimizations