Code Optimization More Optimization Techniques

More Optimization Techniques  Loop optimization  Code motion  Strength reduction for induction variables  Loop unrolling  Function call optimization  Function in-lining  Tail recursion elimination  Alias analysis  Data flow analysis for pointers

Code Motion  Concept  Find loop invariants  Code that will not change its results in each iteration  Move them outside the loop whenever possible  Method  First identify the loop  Then perform reachability analysis  Find the define-use links  For each variable used, find where it is defined  If x’s definitions are all outside the loop, then x is a loop invariant  Identify the loop invariant statements  Exam the movability of the statements  The process continues till no more code to be moved

Code Motion  First identify the loop and the entry node  Then compute the In and Out sets 1: read m 2: i := 1 3: n := 2 4: if (i>m) goto 10 5: k := m + 2 6: j := 3 * k 7: n := j + n 8: i := i + 1 9: goto 4 10: print n B1 B2 B4 B3 In[B2] = {} Out[B2] = {d1, d2, d3} In[B2] = {d1, d2, d3, d5, d6, d7, d8} Out[B2] = {d1, d2, d3, d5, d6, d7, d8} In[B3] = {d1, d2, d3, d5, d6, d7, d8} Out[B3] = {d1, d5, d6, d7, d8} In[B4] = {d1, d2, d3, d5, d6, d7, d8} Out[B4] = {d1, d2, d3, d5, d6, d7, d8}

Code Motion  Now compute define-use links  For each use, find the corresponding definitions in the In set  If a variable has more than one possible source for its definition, then all links should be given 1: read m 2: i := 1 3: n := 2 4: if (i>m) goto 10 5: k := m + 2 6: j := 3 * k 7: n := j + n 8: i := i + 1 9: goto 4 10: print n B1 B2 B4 B3 In[B2]: {d1, d2, d3, d5, d6, d7, d8} In[B3]: {d1, d2, d3, d5, d6, d7, d8} m: d1 i: d2, d8 n: d3, d7 k: d5 j: d6

Code Motion  Now find invariant code  Statements whose operands are not defined within the loop  Computation does not depend on any value computed in the loop 1: read m 2: i := 1 3: n := 2 4: if (i>m) goto 10 5: k := m + 2 6: j := 3 * k 7: n := j + n 8: i := i + 1 9: goto 4 10: print n B1 B2 B4 B3 Statement 5: m: defined outside the loop 2: constant  Is a loop invariant Move 5 out of the loop No need to worry about the dependency to external definitions Statement 6: k: defined inside the loop 3: constant  Not a loop invariant

Code Motion  Create a block before the entry block (if not done so yet)  Move the invariant code into the new block  If there are other back edges pointing to the same entry block  If it is an outer loop then Point to the new block  May have to perform code motion for the other loops 1: read m 2: read n 3: i := 1 4: if (i>m) goto 10 5: k := m + 2 6: j := 3 * k 7: n := j + n 8: i := i + 1 9: goto 4 10: print n B1 B2 B4 B3 5: k := m + 2 Move 5 out of the loop Redirect the dependency link

Code Motion  Continue to find new invariant code 1: read m 2: read n 3: i := 1 4: if (i>m) goto 10 5: k := m + 2 6: j := 3 * k 7: n := j + n 8: i := i + 1 9: goto 4 10: print n B1 B2 B4 B3 5: k := m + 2 No more statements can be moved out. Done! 6: Now, 6 has no operands that are defined within the loop  Move 6 out of the loop 5: k := m + 2 6: j := 3 * k

Code Motion  Problem  Moving code causes the execution order to change i := 1 v := v – 1 If v > 0 goto B2 i := 2 u := u + 1 B1 B2 B4 B3 If u>v goto B4 j := i … B5 i := 2 is a loop invariant Move out of the loop? What would i be? Both i := 1 and i := 2 reaches B5 Moving i := 2 out would cause i := 1 “dead”  Move only if the block (containing the statement to be moved out) dominates all exit blocks

Code Motion  Problem  Moving code causes the execution order to change i := 1 v := v – 1 If v > 0 goto B2 i := 2 u := u + 1 B1 B2 B4 B3 i := 3 If u>v goto B4 j := i … B5 Should i := 3 be moved out of the loop? B2 does dominate the only exit block What would i be? Moving i := 3 out would cause i=2 reaches the end of B2  Move only if there is no other definition of i (the variable being defined) in the loop

Code Motion  Problem  Moving code causes the execution order to change i := 1 v := v – 1 If v > 0 goto B2 i := 2 u := u + 1 B1 B2 B4 B3 k := i j := i … B5 Should i := 2 be moved out of the loop? It does dominate the exit. It is the only definition of i in the loop. What would i be? B2 uses i, and i := 2 as well as i := 1 reaches B2  Move only if all the use of i (the variable being defined) in the loop can only be reached by i := 2 (the statement to be moved out)

Code Motion  Problem  Moving code causes the execution order to change  Algorithm  Detect loop invariant code  s: invariant statement, a candidate to be moved out  B: the block containing s  d: the definition produced by s  x: the variable defined in d  Check  B dominates all exits  No other definition of x in the loop  All use of x in the loop is from d  Only if all three conditions are met, move the code  To the block immediately before the loop entry

Code Motion  Check  B dominates all exit nodes  What is an exit node? Entry node? Node of the back edge?  Any node in the loop with an outgoing edge to a node that is not in the loop  Check dominator tree for this condition  No other definition of x in the loop  Simply check all the definitions  All uses of x in the loop are from d  Check the In set of each block after the exit block

Induction Variables  Finding basic induction variables in loop L  Scanning the statements in L and for each variable, say x  Within L, x is only defined once  Definition of x is of the form x := x  b, b is a constant  Could be x := b  x also  Perform constant propagation and constant folding to allow better recognition of b

Induction Variables  Finding other induction variables y, y is defined as a linear function of x  Scan the statements in L and find all variables, say y  y is only defined once in L  Definition of y is equivalent to  y := c * x + d  x is a basic induction variable  c and d are constants  Then: y is in the family of x, expressed as (x, c, d)  x is expressed as (x, 1, 0)  y := 4 * x, then y is (x, 4, 0)

Strength Reduction for Induction Variables  x is the basic IV  Definition of x is: x := x + b  For each y in the family of x, expressed as (x, c, d)  Create a new variable sy outside the loop  Immediately before the loop entry  If not created yet  Initialize sy outside the loop  sy := c * x + d  Assignment to sy in the loop  Add sy := sy + c * b immediately after x’s definition  Replacing definition of y  Replaced by y := sy

Strength Reduction for Induction Variables i := i + 1 t2 := 4 * i t3 := a[t2] if t3 < v … i := m – 1 t1 := 4 * n v := a[t1] i := i + 1 s2 := s2 + 4 t2 := s2; t3 := a[t2] if t3 < v … s2 := 4 * i i := m – 1 t1 := 4 * n v := a[t1] Add s2 and initialize s2 i is a basic IV t2 is an IV in i’s family (i, 4, 0) Add: s2 := s2 + 4 Replace: t2 := 4*i by t2 := s2 Why not just replace t2 := 4 * i by t2 := t2 + 4 and initialize t2 := 4 * i

Strength Reduction for Induction Variables use t2 i := i + 1 t2 := t2 + 4 t3 := a[t2] if t3 < v … t2’s use is now from an incorrect def t2 := 4 * i i := m – 1 t1 := 4 * n v := a[t1] t2 := t1 + v use t2 i := i + 1 t2 := 4 * i t3 := a[t2] if t3 < v … i := m – 1 t1 := 4 * n v := a[t1] t2 := t1 + v use t2 i := i + 1 s2 := s2 + 4 t2 := s2 t3 := a[t2] if t3 < v … s2 := 4 * i i := m – 1 t1 := 4 * n v := a[t1] t2 := t1 + v Why can’t we just replace t2 := 4 * i by s2 := s2 + 4 and t2 := s2 (on the same spot)

Strength Reduction for Induction Variables s2 := s2 + 4 t2 := s2 use t2 i := i + 1 s2 := s2 + 4 t3 := a[t2] if t3 < v … s2 := 4 * i i := m – 1 t1 := 4 * n v := a[t1] The update has to be done after i’s def t2 := s2 has to be where t2’s def is change s2 and so t2 at an incorrect time t2 := 4 * i use t2 i := i + 1 t3 := a[t2] if t3 < v … i := m – 1 t1 := 4 * n v := a[t1] t2 := s2; use t2 i := i + 1 s2 := s2 + 4 t3 := a[t2] if t3 < v … s2 := 4 * i i := m – 1 t1 := 4 * n v := a[t1] t2 := s2 is in t2’s original location Def of s2 is right after i’s def

Finding More Induction Variables  Consider y being in the family of a basic IV x and z is defined as a linear function of y  Scan the statements in L and find all variables, say z  z is only defined once in L  Definition of z is equivalent to  z := c’ * y + d’  y is an induction variable, and y is in the family of x  c’ and d’ are constants  No assignment to x between assignments to y and z  No assignment to y outside L reaches the use of z  Then: z is in the family of x, expressed as (x, c’*c, c’*d+d’)

Loop Unrolling  Advantages  Saves loop tests  Facilitate the application of optimization techniques  Increases instruction level parallelism fac := 1; for (int i:=5; i>1; i--) { fac := (fac * i) % Max; } fac := 1; fac := (fac * 5) % Max; fac := (fac * 4) % Max; fac := (fac * 3) % Max; fac := (fac * 2) % Max; do constant propagation can achieve great optimization

Optimization for Function Calls  Tail recursion elimination  A function is tail-recursive if the last stmt is a call to itself list_type last (lst: list_type) { if (lst.next = null) then return lst; else return last (lst.next); }  Tail recursive call can be replaced with list_type last (lst: list_type) { start: if (lst.next == null) then return lst; else { lst := lst.next; go to start } }

Optimization for Function Calls  Tail recursion elimination  Replacement rule  Formal param of the func := Actual param in the tail-recursive call  Apply to multiple parameters, as long as they have one to one correspondence  Go back to the start of the function  Advantages of tail recursion elimination  The assignments and a jump were anyway needed by a function call  Function call requires additional copies and jumps  Saves stack space  Recursive call: require linear stack usage  Converted code: only constant stack usage

Optimization for Function Calls  Function Inlining  Advantages  Enable additional optimizations oDue to the revealing of the actual parameters, a lot of optimizations may be possible  Save copying and jump required due to a function call  Disadvantages  Code size increases

Function Inlining – Return and Exit  Simple case  Replace return stmt by assigning returned value to the target variable  Assume that there is returned value and the returned value is assigned to a target variable in the caller  Some issues  The function may have multiple exits  Use a label to serve as a single exit point  The function may not assign return values on all paths  Have to be truthful to original code  Create a temporary variable to keep the returned result

Function Inlining – Return and Exit  General rules  Replace return statements by  Assigning returned value to the temporary variable (if there is returned value)  Followed by goto to exit-label  Assigning the temporary variable to the target variable at the exit label  Optimizations  If the function has a single return statement and the caller stmt is an assignment stmt  Use the simple case rules  If the function has no return statement  Simply use the exit label  Apply all data flow analysis and optimization techniques

Function Inlining – Parameter Passing  Simple case  Replace formal parameters by actual parameters  If an actual parameter is an expression, evaluated it once  Assign evaluation result to a temporary variable  Replace formal parameter with temporary variable  Actual parameter is A[exp]  Mix of the above two cases  Problems  Array size mismatch, subtyping, etc.  Global variables and static  Recognize global variables and preserve its scope  Rename for matching names  Retain the static status of static variables

Function Inlining  int Max := 10**6; … int factorial (int x) { int fac := 1; for (int i:=x; i>1; i--) { fac := (fac * i) % Max; } return fac; } … int val1 := factorial(10); … int val2 := factorial(5); … int Max := 10**6; … int val1; { int fac := 1; for (int i:=10; i>1; i--) { fac := (fac * i) % Max; } val1 := fac; } … int val2; { int fac := 1; for (int i:=5; i>1; i--) { fac := (fac * i) % Max; val2 := fac; } … single return point global var preserve scope

Function Inlining  int Max := 10**6; int overflow := -2; … int factorial (int x) { if (x >= 1) { int fac := 1; for (int i:=x; i>1; i--) { fac := (fac * i); if (fac > Max) return overflow; } return fac; } … int val := factorial (x*y+2); … Multiple return points Undefined return path for x < 1 actual parameter is an expression int Max := 10**6; int overflow := -2; … int t1 := x*y+2; int ret; { if (t1 >= 1) { int fac := 1; for (int i:=t1; i>1; i--) { fac := (fac * i); if (fac > Max) { ret := overflow; goto L1; } } { ret := fac; goto L1; } } L1: int val := ret … define temp ret var define exit label define temp var evaluate exp same as original code ret undefined on some paths

A Headache in Data Flow Analysis - Alias  Pointer aliases  E.g., *p := y may write to any memory location  E.g., x := *p may read from any memory location  If p is dynamically computed  Problems  Example: Live variable analysis  …, x := *p  all variables in all prior blocks may be alive  Example: Constant folding  a := 1; b := 2; *p := 0; c := a + b;  c := 3 only if *p is not an alias for a or b!  All data flow analysis techniques can be messed up by aliases  Need to perform alias analysis when possible

Alias Analysis  Ptr(v)  For each variable v that may hold an address, computer Ptr(v)  Ptr(v) includes all variables v may point to  Ptr(v) may include variables allocated in stack and heap  Can be represented as a graph G, G  2 V  V  V: the set of all variables in the program  E.g., Ptr(x) = {y}, Ptr(y) = {u, v}  Anderson’s graphical representation x y u v x y u v a b

Alias Flow Analysis  Data flow analysis  For p := &q  Add (p,q) to Gen[B]  Add (p,x) to Kill[B], for all x in V (the entire set of variables) oBesides the ones in Gen[B]  For p := q  Add (p,x) to Gen[B], for all x, x in Ptr(q)  Add (p,x) to Kill[B], for all x in V (the entire set of variables) oBesides the ones in Gen[B]  For p := *q  Add (p,x) to Gen[B], for all x, x in Ptr(Ptr(q))  Add (p,x) to Kill[B], for all x in V (the entire set of variables) p q … p q … p q ? …

Alias Flow Analysis  Data flow analysis  For *p := q  Add (x,y) to Gen[B], for all x in Ptr(p) and all y in Ptr(q)  Add (x,y) to Kill[B], for all x in Ptr(p) and all y in V oBesides those in Gen[B]  For *p := *q  Add (x,y) to Gen[B], for all x in Ptr(p) and all y in Ptr(Ptr(q))  Add (x,y) to Kill[B], for all x in Ptr(p) and all y in V oBesides those in Gen[B]  When you have *…* or &…&  Resolve layer by layer p q ? … ? q … p ?

Alias Flow Analysis  Data flow analysis  Out[B] = Gen[B]  (In[B] – Kill[B])  If (x,y) is in the In set and also in the Kill set, then it is killed  In[B] =  Out[P]  As long as a definition of a pointer definition pair (x,y) reaches B, the pointer x could have value y  Assumption  No pointer arithmetic  If there is, then need to use offset, instead of actual variables, for data flow analysis

Alias Flow Analysis  Example x := &a y := &b c := &i if (i) *x := c x := y x a i b y c Out: (x,a), (y,b), (c,i) In: (x,a), (y,b), (c,i) Killed: (x,a) Out: (x,b), (y,b), (c,i) In: (x,a), (x,b), (y,b), (c,i) Killed: (a,*), (b,*) -- none Out: (x,a), (x,b), (y,b), (c,i), (a,i), (b,i) x points to a and b a and b are now possible to point to whatever c points to

Data Flow Analysis with Alias  Apply the reachability information to perform analysis  E.g.  a := 1; b := 2; *p := 0; c := a + b;  *p := 0 defines x := 0, for all (p,x) that reaches *p := 0  If (p,a) or (p,b) reaches *p := 0, then c is no longer a constant  If only (p,a) reaches *p := 0 does not imply p = a, p could also be undefined  Use the reachable Ptr set to help with data flow analysis

Data Flow Analysis with Alias  Define-use  For c := *p op b  Only p is of pointer type  All x, x in Ptr(p) are added to the use set oOr, all x, (p,x) reaches here, are added to the use set oIn the example, for c:= *x op b, a, b are added to the use set  p is also added to the use set (need to use p to get *p)  Of course, b in the use set, c in the define set  For *p := b op c  Only p is of pointer type  All x, x in Ptr(p) are added to the def set oOr, all x, (p,x) reaches here, are added to the def set oIn the example, a := b op c, b := b op c, are added to the def set  p is also in the use set (need to use p to get *p)  Of course, b and c are in the use set x a i b y c

Data Flow Analysis with Alias  Define-use  For p := …  Only p is defined  For p := …q…  Only p is defined  Only q is used (and other variables appear in the right side), not the objects q points to

Code Optimization -- Summary  Read Chapter 9  Sections 9.2, 9.4, 9.5, 9.6  Optimization of intermediate code  Overview of optimization techniques  Construct CFG and loop recognition  Data flow analysis based on CFG  Constant propagation and constant folding  Copy propagation  Code motion  Induction variable identification and Strength reduction  Dead code elimination

Code Optimization -- Summary  Optimization of intermediate code  Loop optimization (9.1)  Function call optimization (2.5.4)  Alias analysis (12.4)