advanced compiler techniques liu xianhua school of eecs, peking university partial redundancy...
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Advanced Compiler Techniques
LIU Xianhua
School of EECS, Peking University
Partial Redundancy Elimination
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REVIEW
Foundations Data Flow Framework Lattice-Theoretic Formulation Meet-Over-Paths Solution
Extensions Other DFA Methods
“Advanced Compiler Techniques”
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REVIEW
“Advanced Compiler Techniques”
Available Expressions Analysis
Live Variables Analysis
≤ is ⊆, ∧ is ∩ ≤ is ⊇, ∧ is ∪
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REVIEW
All possible assignments
All safe assignments
All fixed point solutions
Meet Over Paths Assignment
Maximum Fixed Point
Least Fixed Point
∀i , Ini = Outi = ⊤
∀i , Ini = Outi = ⊥
“Advanced Compiler Techniques”
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REVIEW
Entry B
Since f(x ∧ y) ≤ f(x) ∧ f(y), it is as if we addednonexistent paths, but we’re safe.
f
OUT = x
OUT = y
IN = x∧y
OUT = f(x∧y)
In MFP, Values x and yget combined too soon. f(x)
f(y)
MOP considers pathsindependently andand combines at the last possible moment.
OUT = f(x) ∧ f(y)
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Summary of DFA Methods
Method Speed Simple?
Structure
Both Way?
Graph Class
Iterative O(n2) Simple No Yes All
Interval O(n2) Middle Yes Yes Reducible
Balance Tree
O(nlogn)
Complicated
Yes No Reducible
Path Comp.
O(nlogn)
Middle Semi Yes Reducible
Node List O(nlogn)
Middle No Yes Reducible
Balance Path
O(nα(n,n))
Complicated
No ? Reducible
Grammar
n Middle Yes Yes L(Grammar)
High Level
n Simple Yes Yes Parse Trees
“Advanced Compiler Techniques”
“Advanced Compiler Techniques”
Outline
• Forms of redundancy– global common sub-expression– loop invariant– partial redundancy expression
• Lazy Code Motion Algorithm– A set of four analysis
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Role of PRE
• Goal: Minimize the number of expression evaluations.– Keep the value of evaluation in a
temporary variable and use it later.
• Sources of redundancy:– Global common sub-expressions– Loop-invariant computations– True partial redundancy: an expression
is sometimes available, sometimes not
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Partial redundancy elimination
One of the most complex dataflow analysis
Subsumes common sub-expression elimination and loop invariant code motion
Originally proposed in 1979 by Morel and Renvoise, Used a bi-directional dataflow analysis
Reformulated by Knoop, Rüthing and Steffen in 1992, Uses a backward dataflow analysis followed by a forward analysis
We will discuss this latter formulation“Advanced Compiler Techniques”
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Convention
• Throughout, assume that neither argument of an expression x+y is modified unless we explicitly assign to x or y.
• And of course, we assume x+y is the only expression anyone would ever want to compute.
• Can easily extend this to multiple expressions by using a bit vector lattice. “Advanced Compiler Techniques”
“Advanced Compiler Techniques”
Example: Global CSE
• A common expression may have different values on different paths.
• On every path reaching p,– expression x+y has been computed– x, y not overwritten after the expression
a = x+y b = x+y
c = x+y
t = x+ya = t
t = x+yb = t
c = t
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a=x+y
d=x+y
a=x+y
x=7
d=x+y
a=x+y x=7f=x+y
d=x+y
“Advanced Compiler Techniques”
Example:Loop-Invariant Code Motion
• Given an expression (x+y) inside a loop,– does the value of x+y change inside the loop?– is the code executed at least once?
t = x+ya = x+y
a = t
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Example: True Partial Redundancy
• Can we place calculations of x+y such that no path re-executes the same expression?
a = x+y
b = x+y
t = x+ya = t
t = x+y
b = t
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Modifying the Flow Graph
• We could:1. Add a new block along an edge.
Only necessary if the edge enters a block with several predecessors.
2. Duplicate blocks so an expression x+y is evaluated only along paths where it is needed.
“Advanced Compiler Techniques”
“Advanced Compiler Techniques”
Can All Redundancy Be Eliminated?
• Critical edges– source basic block has multiple successors– destination basic block has multiple
predecessors16
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Problem With Node-Splitting
• Can exponentiate the number of nodes.
• Our PRE algorithm needs to move code to new blocks along edges, but will not split blocks.
• Convention: All new instructions are either inserted at the beginning of a block or placed in a new block.
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Lazy Code Motion Problem
• Desired properties of a PRE algorithm– All redundant computations of
expressions that can be eliminated without code duplication are eliminated.
– The optimized program does not perform any computation that is not in the original program execution.
– Expressions are computed at the latest possible time.
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Full Redundancy
• Full redundancy at p: expression a+b redundant on all paths– cutset: nodes that separate entry from p– cutset contains calculation of a+b– a, b, not redefined
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Partial Redundancy
• Partial redundancy at p: redundant on some but not all paths– Add operations to create a cutset containing
a+b– Note: Moving operations up can eliminate
redundancy21
“Advanced Compiler Techniques”
The Plan
1. Determine for each expression the earliest place(s) it can be computed while still being sure that it will be used.
2. Postpone the expressions as long as possible without introducing redundancy.– We trade space for time --- an
expression can be computed in many places, but never if it is already computed.
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The Guarantee
• No expression is computed at a place where it its value might have been computed previously, and preserved instead.– Even along a subset of the possible
paths.
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The Plan – (2)
3. Determine those places where it is really necessary to store x+y in a temporary rather than compute it when needed.
Example: If x+y is computed in only one place.
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More about the plan
Don’t introduce/insert new operations didn’t exist originally:
Anticipate the range of code motion Eliminate as many redundant
calculations of an expression as possible, without duplicating code
Move it up as early as possible Delay computation as much as
possible to minimize register Lifetimesmove it down unless it creates redundancy (lazy code motion) Remove temporary assignment
“Advanced Compiler Techniques”
“Advanced Compiler Techniques”
More About the Plan
• We use four data-flow analysis, in succession, plus some set operations on the results of these analysis.– Anticipated Expressions– Available Expressions– Postponable Expressions– Used Expressions
• After the first, each analysis uses the results of the previous ones.
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Assumptions
• Assume every statement is a basic block– Only place statements at the beginning
of a basic block– Add a basic block for every edge that
leads to a basic block with multiple predecessors
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Anticipated Expressions
• Expression x+y is anticipated at a point if x+y is certain to be evaluated along any computation path, before any recomputation of x or y.
• Copies of an expression must be placed only at program points where the expression is anticipated.– The earlier an expression is placed, the
more redundancy can be removed
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Example: Anticipated Expressions
= x+y = x+y
= x+y
x+y is anticipatedhere and could becomputed now ratherthan later.
= x+y
x+y isanticipatedhere, but isalso available.No computa-tion is needed.
“Advanced Compiler Techniques”
“Advanced Compiler Techniques”
Computing Anticipated Expressions
• Use(B) = set of expressions x+y evaluated in B before any assignment to x or y.
• Def(B) = set of expressions one of whose arguments is assigned in B.
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Computing Anticipated Expressions
• Direction = backwards.• Join (or Meet) = intersection.• Boundary condition: IN[exit] = ∅.• Transfer function: IN[B] = (OUT[B] – Def(B)) ∪ Use(B)
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Example: Anticipated Expressions
= x+y
= x+y
Anticipated
Backwards; Intersection; IN[B] = (OUT[B] – Def(B)) ∪ Use(B)
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“Available” Expressions
• Modification of the usual AE.• x+y is “available” at a point if
either:1. It is available in the usual sense; i.e., it
has been computed and not killed, or2. It is anticipated; i.e., it could be
available if we chose to precompute it there.
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“Available” Expressions
• x+y is in Kill(B) if x or y is defined, and x+y is not recomputed later in B (same as previously).
• Direction = Forward• Meet = intersection.• Transfer function:
OUT[B] = (IN[B] ∪ INANTICIPATED[B]) – Kill(B)
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Earliest Placement
x+y is in Earliest[B] if it is anticipated at the beginning of B but not “available” there. That is: when we compute anticipated
expressions, x+y is in IN[B], but When we compute “available”
expressions, x+y is not in IN[B]. I.e., x+y is anticipated at B, but not
anticipated at OUT of some predecessor.
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Example: Available/Earliest
= x+y
= x+y
Anticipated
“Available”
Earliest = anticipatedbut not available
Forward; Intersection; OUT[B] = (IN[B] ∪ INANTICIPATED[B]) – Kill(B)“Advanced Compiler Techniques”
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Postponable Expressions
• Now, we need to delay the evaluation of expressions as long as possible, but …
1. Not past the use of the expression.2. Not so far that we wind up computing
an expression that is already evaluated.
• Note viewpoint: It is OK to use code space if we save register use.
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Postponable Expressions – (2)
x+y is postponable to a point p if on every path from the entry to p:
1. There is a block B for which x+y is in earliest[B], and
2. After that block, there is no use of x+y.
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Postponable Expressions – (3)
Computed like “available” expressions, with two differences:
1. In place of killing an expression (assigning to one of its arguments): Use(B), the set of expressions used in block B.
2. In place of INANTICIPATED[B]: earliest[B].
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Postponable Expressions – (4)
Direction = forward. Meet = intersection. Transfer function:OUT[B] = (IN[B] ∪ earliest[B]) – Use(B)
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Example: Postponable Expressions
= x+y = x+y
= x+y
Earliest
Postponable
Three places tocompute x+y
Forward; Intersection; OUT[B] = (IN[B] ∪ earliest[B]) – Use(B)“Advanced Compiler Techniques”
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Latest Placement
• We want to postpone as far as possible.
• How do we compute the “winners” – the blocks such that we can postpone no further?
• Remember – postponing stops at a use or at a block with another predecessor where x+y is not postponable.
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Latest[B]
• For x+y to be in latest[B]:1. x+y is either in earliest[B] or in
INPOSTPONABLE[B].• I.e., we can place the computation at B.
2. x+y is either used in B or there is some successor of B for which (1) does not hold.• I.e., we cannot postpone further along all
branches.
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Example: Latest
= x+y = x+y
= x+y
EarliestOr Postponableto beginning
UsedOr has a suc-cessor not red.
Latest =Blue and red.
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Final Touch – Used Expressions
We’re now ready to introduce a temporary t to hold the value of expression x+y everywhere.
But there is a small glitch: t may be totally unnecessary. E.g., x+y is computed in exactly one
place.
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Used Expressions
An expression is used at point p if There exists a path leading from p that
uses the expression before the value is reevaluated.
Essentially liveness analysis for expressions rather than for variables.
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Example: Used
= x+y = x+y
= x+y
Recall: Latest
Used
Backwards; Union; IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B)“Advanced Compiler Techniques”
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Used Expressions – (2)
• used[B] = expressions used along some path from the end of B.
• Direction = backward.• Meet = union.• Transfer function:
IN[B] = (OUT[B] ∪ e-used[B]) – Latest(B)– e-used = “expression is used in B.”
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Rules for Introducing Temporaries
1. If x+y is in both Latest[B] and OUTUSED[B], introduce t = x+y at the beginning of B.
2. If x+y is used in B, but either1. Is not in Latest[B] or2. Is in OUTUSED[B],
replace the use(s) of x+y by uses of t.
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Example:Where is a Temporary Used?
= x+y = x+y
= x+y
Recall: Latest
Recall OUTUSED
Create temp-orary here
Use it here
But not here ---x+y is in Latest andnot in OUTUSED
“Advanced Compiler Techniques”
“Advanced Compiler Techniques”
Summary
• Cannot execute any operations not executed originally– Pass 1: Anticipation: range of code motion
• Eliminate as many redundant calculations of an expression as possible, without duplicating code– Pass 2: Availability: move it up as early as
possible• Delay computation as much as
possible to minimize register lifetimes– Pass 3: Postponable: move it down unless it
creates redundancy (lazy code motion)• Pass 4: Remove temporary
assignment
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Algorithm of Lazy Code Motion
INPUT: A flow graph for which e_useB and e_killB have been computed for each block B.
OUTPUT: A modified flow graph satisfying the four lazy code motion conditions.
METHOD: Insert an empty block along all edges entering a
block with more than one predecessor Find anticipated[B] .in for all blocks B Find available[B] .in for all blocks B Compute the earliest placements for all blocks B Find postponable[B].in for all blocks B Compute the latest placements for all blocks B Find used[B] .out for all blocks B
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Algorithm of Lazy Code Motion
METHOD:8. For each expression, say x+y, computed by the
program, do the following: Create a new temporary, say t, for x + y. For all blocks B such that x + y is in latest[B]∩
used[B].out, add t = x +y at the beginning of B.
For all blocks B such that x + y is ine_use B ∩ (┐latest[B] ∪ used.out[B] )
replace every original x + y by t.
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More about PRE
Don’t need heuristic Dhamdhere, Drechsler-Stadel, Knoop,et.al. use restricted flow graph or allow edge placements.
Data flow can be separated into unidirectional passes Dhamdhere, Knoop, et. al.
Improvement still tied to accuracy of computational model Assumes performance depends only on the number of
computations along any path. Ignores resource constraint issues: register alloc., etc. Knoop, et.al. give “earliest” and “latest” placement
algorithms which begin to address this.
Further issues: more than one expression at once, strength reduction,
redundant assignments, redundant stores With GVN,SSA… “Advanced Compiler Techniques”