ECE540S Optimizing Compilers

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ECE540SOptimizing Compilers

• http//www.eecg.toronto.edu/voss/ece540/
• January 31, 2002

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Corrections and Clarifications
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Value Numbering Algorithm
Initialize AVAIL set to empty for each
instruction (res expr) in BB if
(instruction is of form res rhs ) rv
value number of rhs // give new value if
necessary associate res with rv
else if (instruction is of form res lhs op rhs)
lv value number of lhs // give
new value number if necessary rv value
number of rhs // give new value number if
necessary if ( (lv op rv) Î AVAIL )
v value number of (lv op rv)
replace (lhs op rhs) by a variable with value
number v associate with res the value
number v else v new value
number associate with res the value
number v associate with (lv op rv) the
value number v add (lv op rv) to AVAIL
else if (instruction is of form res
op rhs ) possibly cleanup AVAIL set
(remove elements that contain value
numbers
that are no longer active).
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Value Numbering Example
g x y h u v i x y i u v x u
v x k u g h u x y w g h
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Irreducible CFG

• BB1 dom BB3
• entry at BB2
• not natural
• BB2 dom BB4
• natural loop
• rare in real apps

BB1
BB2
BB3
BB4
Reducible! Since for all retreating edges h dom
t. BB3 -gt BB1 is therefore natural.
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Finding Back Edges

• Perform depth-first traversal of CFG (N,E).
• A retreating edge e (t,h) Î E is a back edge if
  h dom t.
• A retreating edge is always a back edge in a loop
  written using structured constructs (i.e., no
  gotos).
• The CFGs of such programs are said to be
  reducible graphs.
• One can use gotos to obtain irreducible flow
  graphs.
• We will only consider reducible CFGs , so we can
  just look for a retreating edge e (t,h) Î E.

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Algorithm for Finding the Nodes in the Loop
Stack S Empty Set Loop h insert_on_stack
(t) while (S is not empty) m pop (S)
for each pred(m)
insert_on_stack (p) insert_on_stack
(a) if (a Ï Loop) loop loop U
a push_on_stack (a)
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Natural Loops and Reducible Graphs

• If all retreating edges are back edges (h dom t),
  the graph is reducible
• if all back edges are removed, a directed acyclic
  graph remains where all nodes can be reached from
  Entry.
• back edges define natural loops according to the
  given algorithm.

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Redundancy Optimizations
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Some well known redundancy optimizations

• Value numbering
• Common subexpression elimination
• Forward substitution (reverse of CSE)
• Copy propagation
• Loop-invariant code motion
• Code Hoisting

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Common Subexpression Elimination

• An occurrence of an expression is a common
  subexpression if there is another occurrence that
  always precedes it in execution order and whose
  operands remain unchanged between these
  evaluations.
• i.e. the expression has been already computed and
  the result is still valid.
• Common Subexpression Elimination replaces the
  recomputations with a saved value.
• reduces the number of computations

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CSE Algorithm
Perform Available Expressions analysis for each b
Î N for each instruction Î b in the form z
x op y such that x op y is available at
entry of b 1. determine if x op y
is available at instruction 2.
determine evaluations of x op y that reach z
3. create a new variable t 4.
replace definitions w x op y found in step 2.
with t x op y
w t 5. replace z x op y with z
t
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AE - Data flow Equations

• AEin(b) Set of available expression at the
  beginning of b.
• AEout(b) Set of available that reach the end
  of b.

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CSE Example 1
Entry
c ab d ac e dd i 1
BB 1
f ab c c2 c gt d?
BB 2
d c g dd
BB 4
BB 3
g ac
i i1 i gt 10?
BB 5
Exit
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CSE Example 2
Entry
c ab d ac e dd i 1
BB 1
f ab c c2 c gt d?
BB 2
d c g dd
BB 4
BB 3
g ac
i i1 i gt 10?
BB 5
Exit
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CSE Example 3
Entry
c ab d ac e dd i 1
BB 1
f ab c c2 c gt d?
BB 2
d c g dd
BB 4
BB 3
g dd
i i1 i gt 10?
BB 5
Exit
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CSE Example 4
Entry
c ab d ac e dd i 1
BB 1
f ab c c2 c gt d?
BB 2
d c g dd
BB 4
BB 3
g dd
i dd i gt 10?
BB 5
Exit
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CSE Limitations

• Expressions must be textually identical
• sort operands
• global value numbering (see Muchnick)

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Forward Substitution

• Replace a copy by reevaluation of the expression
• Why?
• perhaps holds a register too long, causes spills
• See that you have a store of an expression to a
  temporary followed by an assignment to a
  variable. If the expression operands are not
  changed to point of substitution replace with
  expression.

t1 b 2 a t1
a b 2
c b 2 d a b
c t1 d a b
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Copy Propagation

• A copy instruction is an instruction in the form
  x y.
• Copy propagation replaces later uses of x with
  uses of y provided intervening instructions do
  not change the value of either x or y.
• Benefit saves computations, reduces space
  enables other transformations.

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Copy Propagation (cont)

• To propagate a copy statement in the form s x
  y, we must
• Determine all places where this definition of x
  is used.
• For each such use, u
• s must be the only definition of x reaching u
  and
• on every path from s to u, there are no
  assignments to y.

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Available Copy Expressions

• Copy expressions available at the input of basic
  blocks can be computed using a data-flow analysis
  similar to available expressions.

• Copy(b) Set of copy instructions u v in b
  such that u and v are not assigned to later in
  b (i.e. copy instructions available at end of
  b).
• Kill(b) Set of copy instructions that are
  killed by b. That is, the set of copy
  instructions in other blocks that have one or
  more of their operands are assigned to in
  b.
• CopyIn(b) Set of copy instructions available at
  the beginning of b.
• CopyOut(b) Set of copy instructions available
  at the end of b.

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Copy Propagation - Example
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Copy Propagation - Example

• the copy expression is the only definition that
  reaches the statement
• the copy expression is still valid, i.e. its RHS
  has not been modified.

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Copy Propagation - Example

• the copy expression is the only definition that
  reaches the statement
• the copy expression is still valid, i.e. its RHS
  has not been modified.

after dead-code elimination
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Loop-Invariant Code Motion

• A computation inside a loop is said to be
  loop-invariant if its execution produces the same
  value as long as control stays within the loop.
• Loop-invariant code motion moves such
  computations outside the loop (into the loop
  pre-header).
• Benefit eliminates redundant computations.

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Loop Invariants

• Appear mostly (but not always) because of array
  references in loops.

for (i0iltni) for (j0jltmj)
aij 10nj
for (i0iltni) for (j0jltmj)
t1 im t2 t1 j t3
10n (a t2) t3j
t3 10n for (i0iltni) t1 im
for (j0jltmj) t2 t1 j
(a t2) t3j
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Loop Invariants

• An operation in a BB that belongs to a loop is
  loop-invariant if for each operand of the
  operation
• the operand is constant, or
• all definitions that reach the operand are
  outside the loop.

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Loop Invariants (Contd)

• An operation in a BB that belongs to a loop is
  also loop-invariant if for each operand of the
  operation
• there is exactly one definition that reaches the
  operand and this definition is inside the loop
  and is itself loop-invariant.Why exactly one?

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Loop Invariants (Contd)

• An iterative algorithm is used to determine
  loop-invariants
• 1. Mark invariant instructions for which each
  operand
• is constant, or
• has all its reaching definitions outside loop.
• 2. Repeat until no more instructions are marked
  loop invariant Mark invariant each instruction
  not previously marked, whose
• operands are all marked as loop-invariants, or
• operands have exactly one reaching definition
  from inside the loop, and that definition is in
  an instruction which is marked invariant.

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Loop-Invariants - Example
Entry
b 2 i 1
i gt 100
a b 1 c 2 i2 0
d a d e 1 d
d -c f 1 a
i i 1 a lt 2
Exit
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Loop-Invariant Code Motion Algorithm
Perform Reaching Definitions analysis Build
ud-chains Determine Loop-Invariants in loop
L for each loop-invariant instruction s that
defines v
move loop-invariant
instruction to Ls pre-hearder
if ( 1. BB s dominates all exits of L
2. v is not defined elsewhere in L
3. all uses of v in L can only be
reached by the definition of v in
s )
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Code Motion - Condition 1
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Code Motion - Condition 2
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Code Motion - Condition 3
the instruction defining a variable v must be in
a BB the dominates all uses of v in the loop.
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Loop-Invariants - Example
Entry
b 2 i 1
i gt 100
a b 1 c 2 i2 0
d a d e 1 d
d -c f 1 a
i i 1 a lt 2
Exit
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Entry
b 2 i 1

i gt 100
a b 1 c 2 i2 0
d a d e 1 d
d -c f 1 a
i i 1 a lt 2
Exit
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Code Motion - BB Examining Order
y y d
a b 1 c 2 d2 0
d y d y 1 d
f 1 a y gt 20
d i 1 x a

• Breadth-first traversal visits siblings before
  children.
• Process the BBs in breadth first order.
• Move loop invariants into pre-header in the order
  they are discovered!

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Zero-Iteration Loops

• Move loop-invariant into pre-header and protect
  with a guard that tests whether a loop is entered
  or not.
• Invert the loop.

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Loop Invariants (Contd)

• An operation in a BB that belongs to a loop is
  also loop-invariant if for each operand of the
  operation
• there is exactly one definition that reaches the
  operand and this definition is inside the loop
  and is itself loop-invariant.Why exactly one?

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Example
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Code Hoisting

• Code Hoisting finds expressions that are always
  evaluated following a point p and moves them to
  the latest point beyond which they are always
  evaluated.
• reduces space occupied by the program
• may impact execution time positively, negatively
  or not at all
• Uses Very Busy Expressions

• VBEin(b) Set of very busy expressions at the
  beginning of b.
• VBEout(b) Set of very busy expressions at the
  end of b.

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Code Hoisting Algorithm
for each BB b for each exp in VBEout(b)
for each BB x dominated by b
replace 1st reachable exp in x with tmp
if replacement not yet inserted
insert tmp exp at end of b
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Code Hoisting Example
ENTRY
cd
a lt 10
cd
cd
ab,ac,cd
e c d d 2
f a c c d gt 0
ab,ac
?
ab,ac
ab,ac,ad,cd
i a b j a d
g a b h a c
?
ab,ac,cd
EXIT
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Code Hoisting Example
ENTRY
cd
a lt 10
cd
cd
ab,ac,cd
e c d d 2
f a c c d gt 0
?
ab,ac
ab,ac
ab,ac,ad,cd
i a b j a d
g a b h a c
?
ab,ac,cd
EXIT