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Loop Transformations Chapter 14

Loop Transformations Chapter 14. Mooly Sagiv. Outline. Induction-Variable Optimizations Strength Reduction Live Variable Analysis Optimizing array bound checking. Induction Variable Optimizations.

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Loop Transformations Chapter 14

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  1. Loop Transformations Chapter 14 Mooly Sagiv

  2. Outline • Induction-Variable Optimizations • Strength Reduction • Live Variable Analysis • Optimizing array bound checking

  3. Induction Variable Optimizations • Variables that are updated in the “same way” every loop iteration are called “induction-variables” • May be used to decrease the cost of computations in loops

  4. A Simple Fortran Example t1 = 202 do i=1,100 t1 = t1 - 2 a(i) = t1 enddo do i=1,100 a(i) = 202 - 2 * i enddo

  5. A Simple ICAN Example x := init y := initz for ... a := x := x {a} if a z then y := y  {a} od x is induction variable z is loop invariant y is loop invariant x := init for ... a := ... x := x {a} y := x z od

  6. Identifying Induction Variables • Basic induction variables - modified by the same constant during each iteration • Dependent induction variables - depends on other induction variables

  7. t1  202 i  1 t3  addr(a) t4  t3 - 4 L1: t2  i >100 if t2 goto L2 t1  t1 - 2 t5  4 * i t6  t4 + t5 *t6  t1 i  i + 1 goto L1 L2: t1  202 i  1 L1: t2  i >100 if t2 goto L2 t1  t1 - 2 t3  addr(a) t4  t3 - 4 t5  4 * i t6  t4 + t5 *t6  t1 i  i + 1 goto L1 L2: t1 = 202 do i=1,100 t1 = t1 - 2 a(i) = t1 enddo

  8. t1  202; i  1 t3  addr(a) t4  t3 - 4 t5  4 t6  t4 L1: t2  i >100 if t2 goto L2 t1  t1 - 2 t6  t4 + t5 *t6  t1 i  i + 1 goto L1 L2: t1  202; i  1 t3  addr(a) t4  t3 - 4 t5  4 t6  t4 t7  t3 - 396 L1: t2  t6 > t7 if t2 goto L2 t1  t1 - 2 t6  t4 + t5 *t6  t1 goto L1 L2: t1  202 i  1 t3  addr(a) t4  t3 - 4 L1: t2  i >100 if t2 goto L2 t1  t1 - 2 t5  4 * i t6  t4 + t5 *t6  t1 i  i + 1 goto L1 L2:

  9. Algorithm for Identifying Induction variables • Identify loops • Identify loop invariants and constants • Identify basic induction vatiablesbiv = biv + c • Inductively identify variables j with a unique assignmentj = b * biv + c • Split multiple assignments into the same induction variables into different induction variables

  10. Strength Reduction • Replacej = b * biv + c in loop byj = j + c1 and appropriate initialization • The general idea “finite-differening” applied to computer programs 0 3 6 9 … 3 * i 3 3 3 3 … 3 s0=0, si+1 = si +3 0 1 4 9 16 … i2 1 3 5 7 … 2i+1

  11. Strength Reduction Algorithm • For every induction variablej = b * biv + c • Allocate a new temporary variable tj and replace the (single) assignment to j by:j  tj • After assignments biv  biv + c0insert an assignment tj  tj + b * c0 • Put an assignment tj b * biv + c in the preheader • Replace j by tj

  12. Eliminating Induction Variables • Induction variables can be eliminated if: • there computation is useless--- induction variable is not live • Loop tests can be replaced by the linear function

  13. Live Variable Analysis • A variable is live at a program point if it may be used before set in a path from this point • Local information • USE - variables that may be used before set in a block • DEF - variables that must be assigned in the block • Iterative solution LVout(EXIT) =  LVout(B) = B’  Succ(B) LVin(B’) LVin(B) = USE(B)  (LVout(B)-DEF(B))

  14. Other Usage of Live Information • Global register allocation • Find software defects • Garbage collection

  15. Unnecessary Bound Checking Elimination • Many bugs arise due to references beyond array bound • But checking array bounds at run-time is expensive • Compiler can optimize a way or most of the array checks in “reasonable” programs • Compiler can issue warning of inserted checks

  16. Pascal Example L1: ... if 1 > i trap 6 if i > 100 trap 6 t3  addr(b) t4  t3 - 4 t5  4 * i t6  t4 + t5 t1  *t6 goto L1 L2: var b: array[1…100] of integer; … for i := 1 to n do …. b[i] ….

  17. Eliminating Checks in Loops • Assume that we need to show that lo  e  hi • e is loop invariant  move the check to preheader block • if e= i where i is a basic induction variable such that i= i + 1 is the increment code replace the check by checking that lo  init and fin < hi in the preheader • Can be generalized (e.g.,, using partial redundecy elimination Kolte & Wolf 1995)

  18. Conclusions • Two optimizations for loops • Reduction in strength • Elimination of array bound checks

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