Global optimizations

Global optimizations

2

Local analysis framework

• Define an analysis of a basic block as a quadruple (D, V, F, I) where– D is a direction (forwards or backwards)– V is a set of values the program can have at any

point– F is a family of transfer functions defining the

meaning of any expression as a function f : V V– I is the initial information at the top (or bottom)

of a basic block

3

Available Expressions

• Direction: Forward• Values: Sets of expressions assigned to variables• Transfer functions: Given a set of variable

assignments V and statement a = b + c:– Remove from V any expression containing a as a

subexpression– Add to V the expression a = b + c– Formally: Vout = (Vin \ {e | e contains a}) {a = b + c}

• Initial value: Empty set of expressions

4

Available expressions analysis

a = b;

c = b;

d = a + b;

e = a + b;

d = b;

f = a + b;

sIN[s] OUT[s]

{a=b, c=b, d=b, e=a+b, f=a+b}

{a=b, c=b, d=b, e=a+b}

{a=b, c=b, d=a+b, e=a+b}

{a=b, c=b, d=a+b}

{a=b, c=b}

{a=b}

entry: {} Initial value

5

Optimizing from available expressions

• Common sub-expression elimination– If {… t = y op z … } x = y op z– Can transform statement into x = t

• Copy propagation– If {… y = t … } x = y op z– Can transform statement into x = t op z

• Note: same for x=y and x=op y

6

Common sub-expression elimination

a = b;

c = b;

d = a + b;

e = a + b;

d = b;

f = a + b; {a=b, c=b, d=b, e=a+b, f=a+b}

{a=b, c=b, d=b, e=a+b}

{a=b, c=b, d=a+b, e=a+b}

{a=b, c=b, d=a+b}

{a=b, c=b}

{a=b}

entry: {}

e = d;

f = e;

7

Liveness Analysis• Direction: Backward• Values: Sets of variables• Transfer functions: Given a set of variable assignments V

and statement a = b + c:– Remove a from V (any previous value of a is now dead)– Add b and c to V (any previous value of b or c is now live)– Formally: Vin = (Vout \ {a}) {b,c}

• Initial value: Depends on semantics of language– E.g., function arguments and return values (pushes)– Result of local analysis of other blocks as part of a global analysis

Liveness analysis

8

sIN[s] OUT[s]a = b;

c = a;

d = a + b;

e = d;

d = a;

f = e;

{ b, d } exit:

{ b, d, e }

{ a, b, e }

{ a, b, d }

{ a, b }

{ a, b }

{ b }

Initial value

9

Optimizing from liveness analysis

• Dead code elimination– If x = y op z {v1,…,vk}– And x {v1,…,vk}– We can eliminate x = y op z

• Note: same for x=y and x=op y

Dead code elimination

10

a = b;

c = a;

d = a + b;

e = d;

d = a;

f = e;

{ b, d } exit:

{ a, b}

{ b }

Zero value Analysis• Direction: Forward• Values: Sets of valuations (values assigned to variables)• Transfer functions:

– Given a set of variable assignments V and statement a = b op c: update valuation of a at V according to values of b, c and operator op.

– Formally: Vout = (Vin \ {a = value’}) {a = value}

• Initial value: Depends on semantics of language– E.g., function arguments and return values (pushes)– Result of local analysis of other blocks as part of a global

analysis

12

Zero value Analysis

a = b;

c = a + b;

d = a / c;

c = b - b;

a = d - b;

f = d / c;

sIN[s] OUT[s]

{b=, a=?, c=0, d=, f=?}

{b=, a=?, c=0, d=}

{b=, a=, c=0, d=}

{b=, a=, c=, d=}

{b=, a=, c=}

{b=, a=}

entry: {b=} Initial value

13

Optimizing from zero value analysis

• Safe division– If x = y / z {v1,…,vk}– And z {v1,…,z=0,…,vk} or {v1,…,z=?,…,vk}

– We can update x = y safe div z

14

Safe division

a = b;

c = a + b;

d = a / c;

c = b - b;

a = d - b;

f = d / c; {b=, a=?, c=0, d=, f=?}

{b=, a=?, c=0, d=}

{b=, a=, c=0, d=}

{b=, a=, c=, d=}

{b=, a=, c=}

{b=, a=}

entry: {b=}

f = d safe div c;

Global Optimizations

15

16

Global dead code elimination

• Local dead code elimination needed to know what variables were live on exit from a basic block

• This information can only be computed as part of a global analysis

• How do we modify our liveness analysis to handle a CFG?

17

CFGs without loops

Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

b = c + d;e = c + d;Entry

18

CFGs without loops

Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;


{x, y}

{x, y}

{a, b, c, d}

{a, b, c, d} {a, b, c, d}

{a, b, c, d}{b, c, d}

{a, b, c, d}

{a, c, d}

?

Which variables may be live on some execution path?

19

CFGs without loops

Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;


{x, y}

{x, y}

{a, b, c, d}

{a, b, c, d} {a, b, c, d}

{a, b, c, d}{b, c, d}

{b, c, d}

{c, d}Need to combine currently-computed value with new value

Need to combine currently-computed value with new value

20

CFGs without loops

Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;


{x, y}

{x, y}

{a, b, c, d}

{a, b, c, d} {a, b, c, d}

{a, b, c, d}{b, c, d}

{a, b, c, d}

{a, c, d}

21

CFGs without loops

Exit

x = a + b;y = c + d;

a = b + c;

b = c + d;Entry

22

CFGs without loops

Exit

x = a + b;y = c + d;

a = b + c;

b = c + d;Entry

23

CFGs with loops

Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

b = c + d;c = c + d;IfZ ...

Entry

{a}

?

24

CFGs with loops - initialization

Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

b = c + d;c = c + d;Entry

{a}

{}{}

{}

{}

25

CFGs with loops - iteration

Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{}

{}

{}

{a}

26


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{}

{}

{a, b, c}

{a}

27


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{}

{}

{a, b, c}

{a}

{a, b, c}

28


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{b, c}

{}

{a, b, c}

{a}

{a, b, c}

29


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{b, c}

{}

{a, b, c}

{a}

{a, b, c}

{b, c}

30


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{}{b, c}

{c, d}

{a, b, c}

{a}

{a, b, c}

{b, c}

{a, b, c}

31


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a}

{a, b, c}

{b, c}

{a, b, c}

32


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a}

{a, b, c}

{b, c}

{a, b, c}

33


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{b, c}

{a, b, c}

34


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{b, c}

{a, b, c}

35


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{b, c}

{a, b, c}

36


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{b, c}

{a, b, c}

37


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{a, b, c}

{a, b, c}

38


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{a, c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{a, b, c}

{a, b, c}

39


Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;


{a}

{a, b}{b, c}

{a, c, d}

{a, b, c}

{a, c, d}

{a, b, c}

{a, b, c}

{a, b, c}

40

Join semilattices• A join semilattice is a ordering defined on a set of elements• Any two elements have some join that is the smallest

element larger than both elements• There is a unique bottom element, which is smaller than all

other elements• Intuitively:

– The join of two elements represents combining information from two elements by an overapproximation

• The bottom element represents “no information yet” or “the least conservative possible answer”

41

Formal definitions

• A join semilattice is a pair (V, ), where• V is a domain of elements• is a join operator that is– commutative: x y = y x– associative: (x y) z = x (y z)– idempotent: x x = x

• If x y = z, we say that z is the joinor (least upper bound) of x and y

• Every join semilattice has a bottom element denoted such that x = x for all x

42

A general framework

• A global analysis is a tuple (D, V, , F, I), where– D is a direction (forward or backward)

• The order to visit statements within a basic block, not the order in which to visit the basic blocks

– V is a set of values– is a join operator over those values– F is a set of transfer functions f : V V– I is an initial value

• The only difference from local analysis is the introduction of the join operator

43

A semilattice for zero value analysis• One possible semilattice for this analysis is

shown here (for each variable):

Undefined

0𝟎− 𝟎+¿¿

?

44

Global constant propagation

exit x = -5;Undefined

z = y / x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

x = 6;Undefined

entryUndefined

x=Undefinedy=Undefinedz=Undefinedw=Undefined

45



z = y / x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

x = 6;Undefined

entryUndefined

46



z = y / x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;Undefined

entryUndefined

47



z = y / x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

48



z = y / x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

49



z = y / x;Undefined

w = x;Undefined

x= y = x;Undefined

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

50



z = y / x;Undefined

w = x;Undefined

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

51



z = y / x;Undefined

w = x;Undefined

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

y= y=Undefined gives what?

52



z = y / x;Undefined

x= ,y= w = x;Undefined

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

53



z = y / x;Undefined

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

54



z = y / x;Undefined

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

55



x=y=w= z = y / x;Undefined

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

56



x=y=w=z = y / x;x=y=w=z=

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

57



x=y=w= z = y / x;x=y=w=z=

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

58


exitx=y=w=z= x = -5;Undefined


x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

59


exitx=y=w=z=x = -5;x=, y=w=z=


x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

60


exitx=y=w=z= x = -5;x=, y=w=z=


x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

z = y;Undefined

Undefinedx = 6;x =

entryUndefined

61




x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

x = z = y;Undefined

Undefinedx = 6;x =

entryUndefined

62




x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

63




x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

x= x= gives what?

64



y=w= ,x=?z = y / x;x=y=w=z=

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

65



y=w=,x=?z = y / x;y=w=,x=z=?

x=,y=w = x;x=y=w=

x=y = x;x=,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

66


exity=w=,x=z=? x = -5;x=, y=w=z=

y=w=,x=?z = y / x;y=w=,x=z=?

x=,y=w = x;x=y=w=

x=y = x;x=,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

67


exity=w=,x=z=? x = -5;y=w=,x=z=?

y=w=,x=?z = y / x;y=w=,x=z=?

x= ,y= w = x;x=y=w=

x= y = x;x= ,y=

x = z = y;x =

Undefinedx = 6;x =

entryUndefined

Global analysisreached fixpoint

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Global optimizations