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IR Code Optimization
Jingke Li
Portland State University
Jingke Li (Portland State University) CS322 IR Code Optimization 1 / 26
IR Code Optimization
Optimizations transform a program into a functionally-equivalent program with better performance. They can be performed at various stages and levels.
Advantages of IR-Level Optimization:
- (^) All operations are explicit, so cost estimations can be accurate.
- Optimizations at this level are machine-independent, hence the results are portable across different target machines.
Scopes of Optimization:
- Local — Transforming code by analyzing a single basic block.
- Global — Transforming code by analyzing a whole subroutine.
- Inter-Procedural — Trans· · · by analyzing the whole program.
Concepts and Techniques:
- (^) Basic blocks & flow graphs
- Control-flow analysis & data-flow analysis
Central Theme: Redundancy Elimination
A common IR code optimization theme is removing redundant computations. The following are some concrete examples:
- (^) Common Subexpression Elimination — Based on lexical representation, applicable to global scope.
- Partial Redundancy Elimination — More powerful than CSE.
- Copy Propagation — A companion optimization to CSE.
- Value Numbering — Value based, very efficient, applies to a single block.
- Superlocal Value Numbering — Extends VN to multiple blocks.
- Loop Invariant Elimination — Removes a special form of redundancy.
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Common Subexpression Elimination
An occurrence of an expression E is a common subexpression if E was previously computed, and its value has not changed.
To eliminate a common subexpression, one can introduce a temp to hold the value of the subexpression when it is first evaluated.
Example:
t6 := 4i x := a[t6] t7 := 4i t8 := 4j t9 := a[t8] a[t7] := t t10 := 4j a[t10] := x goto B
t6 := 4i x := a[t6] t7 := t t8 := 4j t9 := a[t8] a[t7] := t t10 := t a[t10] := x goto B
B5 B
Before After The second occurrence of 4i in the above basic block is a common subexpression; so is the second occurrence of 4j.
A Global CSE Example (cont.)
i := m-1j := n t1 := 4*nv := a[t1]
i := i+1t2 := 4*i t3 := a[t2]if t3<v goto B
j := j-1t4 := 4*j t5 := a[t4]if t5>v goto B
if i>=j goto B x := t3a[t2] := t a[t4] := xgoto B
x := t3t14 := a[t1] a[t2] := t14a[t1] := x
B
B
B
B B5 B
After
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CSE Algorithm
First, the concept of available expressions — An expression x ⊕ y is available at node n if every path from the entry node to n evaluates the expression, and there are no definitions of x or y after the last evaluation.
Algorithm:
- Compute available expressions for all expressions.
- At each node n : w := x ⊕ y , where the expression x ⊕ y is available:
- Search backwards for the evaluations of x ⊕ y that reach n. Replace each evaluation v := x ⊕ y found in the search by t := x ⊕ y ; v := t. Replace n by w := t.
An Illustrating Example
i := j a := 4*i
i := i+ b := 4*i
c := 4i d := 4i
B
B
B
Before
i := j t2 := 4*i t1 := t a := t
i := i+ t2 := 4*i t1 := t b := t
c := t d := t
B
B
B
After
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An Improved CSE Algorithm
The previous CSE algorithm performs the expensive backward search and inserts a new temp for every use of a common subexpression. The following ideas can be be used to improve the algorithm:
- Reduce the number of new temps by assigning a unique name to each unique expression.
- Avoid backward search by a separate traversal of the CFG.
Algorithm:
- Compute available expressions for all expressions.
- Initialize an array Name[e] = φ for all expressions.
- At each node n : w := x ⊕ y , where the expression x ⊕ y (denoted e below) is available: - If Name[e] = φ, allocate new name t and set Name[e] = t. - Else let t = Name[e]. - Replace n by w := t.
- In a subsequent traversal of CFG, at each node of form v := e, if Name[e] 6 = φ, let t = Name[e]; replace the node by t := e; v := t.
Copy Propagation
Copy statement has the form f := g.
A large number of copy statements may be generated after performing a CSE optimization. Copy propagation eliminates copy statements by using g for f wherever possible.
Example:
t6 := 4i x := a[t6] t7 := t t8 := 4j t9 := a[t8] a[t7] := t t10 := t a[t10] := x goto B
t6 := 4i x := a[t6] t8 := 4j t9 := a[t8] a[t6] := t a[t8] := x goto B
B B
Before
After
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The Cascading Problem
CSE transformation may have a cascading effect — more rounds of CSE/Copy-propagation may be needed before reaching the final form:
x := b + c y := a + x u := b + c v := a + u
x := b + c y := a + x u := x v := a + u
x := b + c y := a + x v := a + x
x := b + c y := a + x v := y
Value Numbering
The Idea:
- (^) Each variable is assumed to have a unique initial value.
- (^) Each unique value is assigned a unique number.
- (^) An expression’s value is represented by a corresponding symbolic expression based on the operands’ numbers, e.g. expression “x + y ”’s value is “ 1 ”+“ 2 ”, if “ 1 ” and “ 2 ” are x and y ’s value numbers, respectively.
- (^) Each unique expression value is also assigned a unique number.
- (^) When a new variable or expression is encountered, check to see if it has been assigned a number, if so, use the number, otherwise assign it a new number.
- Use a hash table for efficient number lookup.
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A Value-Numbering Example
x := b + c y := a + x u := b + c v := a + u
statement var or expr asgnd# x := b + c b 1 c 2 b + c (“1”+“2”) 3 x 3 y := a + x a 4 a + x (“4”+“3”) 5 y 5 u := b + c u (“1”+“2”) 3 v := a + u v (“4”+“3”) 5
Value numbering uses a single round to calculate the effect of cascaded optimizations.
Loop Invariant Code Motion
If a loop contains a statement t ← a ⊕ b such that a and b have the same values each time around the loop, then t will also have the same value each time. We would like to hoist such loop-invariant statement out of the loop.
t <- 0
i <- i+ t <- a⊕b M[i] <- t if a<N
x <- t
B
B
B
t <- 0 t <- a⊕b
i <- i+ M[i] <- t if a<N
x <- t
B
B
B
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Loop Invariant Criteria
A statement S : t ← a 1 ⊕ a 2 is loop-invariant within loop L if, for each operand ai ,
- ai is a constant, or
- all the definitions of ai that reach S are outside the loop, or
- only one definition of ai reaches S, and that definition is loop-invariant.
An iterative algorithm can be used to find all loop-invariant statements.
Counter Examples
Not all loop-invariant statements can be hoisted out of a loop.
t <- 0
if a>=N
i <- i+1t <- a⊕b M[i] <- t
x <- t
B
B
B
B
t <- 0
i <- i+ t <- aM[i] <- t⊕b t <- 0M[j] <- t if a<N
x <- t
B
B
B
t <- 0
i <- i+ M[j] <- tt <- a⊕b M[i] <- tif a<N
x <- t
B
B
B
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PRE vs. Loop Invariant Code Motion
- A loop invariant expression is partially redundant.
- PRE hence can remove this redundancy.
t <- 0
i <- i+ t <- a⊕b M[i] <- t if a<N
x <- t
B
B
B
t <- 0 t <- a⊕b
i <- i+ M[i] <- t if a<N
x <- t
B
B
B
Induction-Variable Elimination (cont.)
i, j, t2, and t4 are four induction variables. Since the only use of i and j is the comparison “i>=j” in block B5. However, we have “i>=j” ⇔ “4i>=4j” ⇔ “t2>=t4”
After replacing “i>=j” by “t2>=t4”, both i and j become dead variables, hence can be eliminated.
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Final CFG for Quicksort
i := m-1j := n t1 := 4n v := a[t1] t2 := 4it4 := 4*j
t2 := t2+ t3 := a[t2]if t3<v goto B
t4 := t4-4t5 := a[t4] if t5>v goto B
if t2>=t4 goto B
x := t a[t2] := t5a[t4] := x goto B
x := t t14 := a[t1]a[t2] := t a[t1] := x
B
B
B
B
B5 B
