Optimizing Compilers CISC 673 Spring 2011 Static Single Assignment II

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Optimizing CompilersCISC 673

Spring 2011Static Single Assignment II

John Cavazos

University of Delaware


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What is SSA?

• Many data-flow problems have been formulated

• To limit number of analyses, use single analysis to perform multiple transformations SSA

• Compiler optimization algorithms enhanced or enabled by SSA: Constant propagation Dead code elimination Global value numbering Partial redundancy elimination Register allocation


3

SSA Construction Algorithm (High-level sketch)

1. Insert -functions

2. Rename values


4

SSA Construction Algorithm (Less high-level sketch)

• Insert -functions at every join for every name

• Solve reaching definitions

• Rename each use to def that reaches it (will be unique)

What’s wrong with this approach

• Too many -functions!

To do better, we need a more complex approach

Builds maximal SSA


5

1. Insert -functions

a.) calculate dominance frontiers

b.) find global namesfor each name, build list of blocks that

define it

c.) insert -functions



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Insert -functions

global name n

worklist ← Block(n) // blocks in which n is assigned block b ∈ worklist

block d in b’s dominance frontierinsert a -function for n in dadd d to worklist


7

Computing Dominance Frontiers

•Only join points are in DF(n) for some n

•Simple algorithm for computing dominance frontiers

For each join point x (i.e., |preds(x)| > 1) For each CFG predecessor of xRun up to IDOM(x ) in dominator tree,

add x to DF(n) for each n between x and IDOM(x )


8

Dominance Frontiers

n B0 B1 B2 B3 B4 B5 B6 B7

DOM(n) 0 0,1 0,1,2 0,1,3 0,1,3,4

0,1,3,5

0,1,3,6

0,1,7

IDOM(n) -- 0 1 1 3 3 3 1

DF(n) -- -- 7 7 6 6 7 1

For each join point x

For each CFG pred of x

Run to IDOM(x ) in dom tree,

add x to DF(n) for each n

between x and IDOM(x )

B1

B2 B3

B4 B5

B6

B7

B0

B1

B2 B3

B4 B5

B6

B7

B0

Flow Graph Dominance Tree


Dominance Frontiers & -Function Insertion

Dominance Frontiers

• A definition at n forces a -function at m iff n DOM(m) but n DOM(p) for some p preds(m)

• DF(n ) is fringe just beyond region n dominates

B1

B2 B3

B4 B5

B6

B7

B0

0 1 2 3 4 5 6 7DOM 0 0,1 0,1,2 0,1,3 0,1,3,4 0,1,3,5 0,1,3,6 0,1,7DF – – 7 7 6 6 7 1

in 1 forces -function in DF(1) = Ø (halt )

x ...

x (...)• DF(4) is {6}, so in 4 forces -function

in 6x (...) in 6 forces -function in DF(6) = {7}

x (...)

in 7 forces -function in DF(7) = {1}

For each assignment, we insert the -functions

a (a,a)b (b,b)c (c,c)d (d,d)y a+bz c+di i+1

B7

i > 100i ...B0

b ...c ...d ...

B2

a (a,a)b (b,b)c (c,c)d (d,d)i (i,i) a ...c ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

With all the -functions

• Lots of new ops

• Renaming is next

Assume a, b, c, & d defined before B0

Excluding local names avoids ’s for y & z


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2. Rename variables in a pre-order walk over dominator tree(uses counter and a stack per global name)

Staring with the root block, b

a.) generate unique names for each -functionand push them on the appropriate stacks



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2. Rename variables (cont’d)

b.) rewrite each operation in the blocki. Rewrite uses of global names with the current version (from the stack)ii. Rewrite definition by creating & pushing new name

c.) fill in -function parameters of successor blocks

d.) recurse on b’s children in the dominator tree

e.) <on exit from block b > pop names generated in b from stacks



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SSA Construction Algorithm

for each global name icounter[i] 0stack[i]

call Rename(n0)

NewName(n)i counter[n]counter[n] counter[n] + 1push ni onto stack[n]

return ni

Rename(b)for each -function in b, x (…)

rename x as NewName(x)

for each operation “x y op z” in brewrite y as top(stack[y])rewrite z as top(stack[z])rewrite x as NewName(x)

for each successor of b in the CFGrewrite appropriate parameters

for each successor s of b in dom. treeRename(s)

for each operation “x y op z” in bpop(stack[x])

Adding all the details ...


B7

i > 100i ...B0

b ...c ...d ...

B2

a (a,a)b (b,b)c (c,c)d (d,d)i (i,i) a ...c ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

CountersStacks

1 1 1 1 0

a

a0 b0 c0 d0

Before processing B0

b c d i

Assume a, b, c, & d defined before B0

i has not been defined

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B7

i > 100i0 ...B0

b ...c ...d ...

B2

a (a0,a)b (b0,b)c (c0,c)d (d0,d)i (i0,i)

a ...c ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

CountersStacks

1 1 1 1 1

a b c d i

a0 b0 c0 d0

End of B0

i0

21


B7

i > 100i0 ...B0

b ...c ...d ...

B2

a1 (a0,a)b1 (b0,b)c1 (c0,c)d1 (d0,d)i1 (i0,i)

a2 ...c2 ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

CountersStacks

3 2 3 2 2

a b c d i

a0 b0 c0 d0

End of B1

i0

a1 b1 c1 d1 i1

a2 c2

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a (a2,a)

b (b2,b)c (c3,c)

d (d2,d)y a+bz c+di i+1

B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

CountersStacks

3 3 4 3 2

a b c d i

a0 b0 c0 d0

End of B2

i0

a1 b1 c1 d1 i1

a2 c2b2 d2

c3

23

a (a2,a)

b (b2,b)c (c3,c)


B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a ...d ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

i ≤ 100

CountersStacks

3 3 4 3 2

a b c d i

a0 b0 c0 d0

Before starting B3

i0

a1 b1 c1 d1 i1

a2 c2

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a (a2,a)

b (b2,b)c (c3,c)


B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d ...B4 c ...

B5

d (d,d)c (c,c)b ...B6

i > 100

CountersStacks

4 3 4 4 2

a b c d i

a0 b0 c0 d0

End of B3

i0

a1 b1 c1 d1 i1

a2 c2

a3

d3

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a (a2,a)

b (b2,b)c (c3,c)


B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c ...

B5

d (d4,d)c (c2,c)b ...B6

i > 100

CountersStacks

4 3 4 5 2

a b c d i

a0 b0 c0 d0

End of B4

i0

a1 b1 c1 d1 i1

a2 c2

a3

d3

d4

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a (a2,a)

b (b2,b)c (c3,c)


B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c4 ...

B5

d (d4,d3)c (c2,c4)

b ...B6

i > 100

CountersStacks

4 3 5 5 2

a b c d i

a0 b0 c0 d0

End of B5

i0

a1 b1 c1 d1 i1

a2 c2

a3

d3

c4

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a (a2,a3)

b (b2,b3)c (c3,c5)

d (d2,d5)y a+bz c+di i+1

B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c4 ...

B5

d5 (d4,d3)c5 (c2,c4)

b3 ...B6

i > 100

CountersStacks

4 4 6 6 2

a b c d i

a0 b0 c0 d0

End of B6

i0

a1 b1 c1 d1 i1

a2 c2

a3

d3

c5 d5

b3

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a (a2,a3)

b (b2,b3)c (c3,c5)

d (d2,d5)y a+bz c+di i+1

B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a)

b1 (b0,b)c1 (c0,c)

d1 (d0,d)

i1 (i0,i) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c4 ...

B5

d5 (d4,d3)c5 (c2,c4)

b3 ...B6

i > 100

CountersStacks

4 4 6 6 2

a b c d i

a0 b0 c0 d0

Before B7

i0

a1 b1 c1 d1 i1

a2 c2

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a4 (a2,a3)

b4 (b2,b3)c6 (c3,c5)

d6 (d2,d5)y a4+b4

z c6+d6

i2 i1+1

B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a4)

b1 (b0,b4)c1 (c0,c6)

d1 (d0,d6)

i1 (i0,i2) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c4 ...

B5

d5 (d4,d3)c5 (c2,c4)

b3 ...B6

i > 100

CountersStacks

5 5 7 7 3

a b c d i

a0 b0 c0 d0

End of B7

i0

a1 b1 c1 d1 i1

a2 c2

a4

b4

c6

d6 i2

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a4 (a2,a3)

b4 (b2,b3)c6 (c3,c5)

d6 (d2,d5)y a4+b4

z c6+d6

i2 i1+1

B7

i > 100i0 ...B0

b2 ...c3 ...d2 ...

B2

a1 (a0,a4)

b1 (b0,b4)c1 (c0,c6)

d1 (d0,d6)

i1 (i0,i2) a2 ...c2 ...

B1

a3 ...d3 ...

B3

d4 ...B4 c4 ...

B5

d5 (d4,d3)c5 (c2,c4)

b3 ...B6

i > 100

After renaming

• Semi-pruned SSA form

• We’re done …

Semi-pruned only names live in 2 or more blocks are “global names”.

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26

SSA Construction Algorithm (Pruned SSA)

What’s this “pruned SSA” stuff?

• Minimal SSA still contains extraneous -functions

• Inserts some -functions where they are dead

• Would like to avoid inserting them


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SSA Construction Algorithm (Two Ideas)

• Semi-pruned SSA: discard names used in only one block Significant reduction in total number of -functions Needs only local Live information (cheap to

compute)

• Pruned SSA: only insert -functions where their value is live Inserts even fewer -functions, but costs more to do Requires global Live variable analysis (more expensive)

In practice, both are simple modifications to step 1.


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SSA Deconstruction

At some point, we need executable code

• Few machines implement operations

• Need to fix up the flow of values

Basic idea

• Insert copies -function pred’s

• Simple algorithm Works in most cases

• Adds lots of copies Many of them coalesce away

X17 (x10,x11)

... x17

... ...

... x17

X17 x10 X17 x11

Optimizing Compilers CISC 673 Spring 2011 Static Single Assignment II

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