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Reduction in Strength

Reduction in Strength. CS 480. Our sample calculation. for i := 1 to n for j := 1 to m c [i, j] := 0 for k := 1 to p c[i, j] := c[i, j] + a[i, k] * b[k, j]. Graph rep of our program. Node 1 t1 <- n i <- 1 (implicit goto node 2)

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Reduction in Strength

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  1. Reduction in Strength CS 480

  2. Our sample calculation for i := 1 to n for j := 1 to m c [i, j] := 0 for k := 1 to p c[i, j] := c[i, j] + a[i, k] * b[k, j]

  3. Graph rep of our program Node 1 t1 <- n i <- 1 (implicit goto node 2) Node 2 If i > t1 goto node 10 Node 3 t2 <- m j <- 1 Node 4 if j > t2 goto node 9 Node 5 (c + ((i*4-4) * m + (j*4-4))) <- 0 t3 <- p k <- 1 Node 6 If k > t3 goto node 8 Node 7 t4 <- i*4-4 t5 <- j*4-4 t7 <- t4 * m + t5 t8 <- k*4-4 (c + t7) <- @(c+t7) + @(a + (t4 * p + t8)) * @(b + (t8 * m + t5)) k <- k + 1 goto node 6 Node 8 j <- j + 1 goto Node 4 Node 9 i <- i + 1 goto Node 2 Node 10 (procedure exit)

  4. Innermost loop 7 *, 12 + Node 5 (c + ((i*4-4) * m + (j*4-4))) <- 0 t3 <- p k <- 1 Node 6 If k > t3 goto node 8 Node 7 t4 <- i*4-4 t5 <- j*4-4 t7 <- t4 * m + t5 t8 <- k*4-4 (c + t7) <- @(c+t7) + @(a + (t4 * p + t8)) * @(b + (t8 * m + t5)) k <- k + 1 goto node 6

  5. Loop Invariant Code Removal • Last time we looked for expressions that did not change within a loop, and moved them out • Called Loop Invariant Code removal • We were able to reduce the operations to 8 additions and 3 multiplications

  6. After loop invariant code removal Node 5 t4 <- i * 4 – 4 t5 <- j * 4 – 4 t7 <- t4 * m + t5 t6 <- t4 * p (c + t7) <- 0 t3 <- p k <- 1 Node 6 If k > t3 goto node 8 Node 7 t8 <- k*4-4 (c + t7) <- @(c+t7) + @(a + (t6 + t8)) * @(b + (8 * m + t5)) k <- k + 1 goto node 6

  7. So, are we done? • Originally had +/-: 12 and */%: 7 in innermost loop • Now have +/-: 8 and */%: 3 • Are we done? Can we do better? • Sure we can.. • Next optimization

  8. Reduction in Strength • Reduce a costly (“strong”) operation to a less costly (“weak”) operation • Typically replace multiplications by additions, which can be executed much faster

  9. Reduction in Strength algorithm • Step 1 within a loop, look for variables that are changing by the addition or subtraction of a constant • Called Induction Variables • Step 2 look for expressions of the form (iv + c) * c + c (these come up a lot in subscript calculations) Step 3. Replace these with a temporary variable

  10. Justification • Think about the variable k in the innermost loop • How is it changing? • 1,2,3,……. • Then how does the expression k * 4 – 4 change?

  11. Finding induction variables • Same type of analysis as before, only now we need to know not only is a variable being changed, but NOW it is being changed • Look for iv <- iv + c • Most commonly from for statements, but can actually come from anyplace (or even multiple assignments)

  12. Finding expressions • Look for expressions of the form (iv + c) * c + c (again, these are common in subscripts) Create a temporary that tracks these expressions Every time our induction variable changes, change the temporary (by adding a constant)

  13. See what the constant is • Suppose we have iv = iv + c1 • Suppose we have (iv + c2) * c3 + c4 • New expression is (iv + c1 + c2) * c3 + c4 • Which is (iv + c2) * c3 + c4 + c1 * c2 • Which is old expression + c1 * c2 • So new expression is just a constant change from old expression

  14. Lets try it Node 5 t4 <- i * 4 – 4 t5 <- j * 4 – 4 t7 <- t4 * m + t5 t6 <- t4 * p (c + t7) <- 0 t3 <- p k <- 1 t8 <- 0 (this is k * 4 – 4 at this point) Node 6 If k > t3 goto node 8 Node 7 t8 <- k*4-4 (NO LONGER NEEDED) (c + t7) <- @(c+t7) + @(a + (t6 + t8)) * @(b + (t8 * m + t5)) k <- k + 1 t8 <- t8 + 4 goto node 6

  15. Are we done yet? • Nope. • As aways, after we have done one optimization, we have a different program • Need to look once more since there might now be further optimizations that were not possible previously

  16. New induction variable • T8 is now an induction variable • It changes only by a constant amount (namely, m) each time through the loop • Can replace with another temporary

  17. Lets try it Node 5 t4 <- i * 4 – 4 t5 <- j * 4 – 4 t7 <- t4 * m + t5 t6 <- t4 * p (c + t7) <- 0 t3 <- p k <- 1 t8 <- 0 t11 <- 0 (this is t8* m at this point) Node 6 If k > t3 goto node 8 Node 7 (c + t7) <- @(c+t7) + @(a + t6 + t8) * @(b + t11+ t5) k <- k + 1 t8 <- t8 + 4 t11 <- t11 + 4m goto node 6

  18. Operation counts • Original 7 * 12 + • After loop invariant code removal 3 * 8 + • After reduction in strength 1 * 9 + • Still other types of optizations that could be done, but this is pretty good

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