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Example of Constructing the DAG (1)t 1 := 4 * iStep (1):create node 4 and i 0 Step (2):create node Step (3):attach identifier t 1 (2)t 2 := a[t 1 ]Step (1):create nodes labeled [], a Step (2):find previously node(t 1 ) Step (3):attach label (3)t 3 := 4 * i Here we determine that: node (4) was created node (i) was created node (*) was created just attach t 3 to *. * * t1t1 i0i0 4 * t 1,t 3 i0i0 4 [] t2t2 a0a0

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Common Subexpression Elimination Detection: Common subexpressions can be detected by noticing, as a new node m is about to be added, whether there is an existing node n with the same children, in the same order, and with the same operator. if so, n computes the same value as m and may be used in its place. Note: The expressions b+c in (1) and (3) are not confused. Example: (1)a:= b + c (2)b:= a – d (3)c:= b + c (4)d:= a - d a:= b + c b:= a – d c:= b + c d:= b + + - b0b0 c0c0 d0d0 c b,d a

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Dead Code Elimination Transfomation: if x is dead (i.e., not live) – never subsequently used – after a point where statement x := y + z appears in a basic block. Then, this statement may be safely removed. Method: Delete from the DAG any root (node with no ancestors) that has no live variable. Weaker form: delete (when multiple) dead variable label. Repeat until we remove all nodes (labels) that correspond to dead code. a:= b+c b:=b-d c:=c+d e:=b+c + +-+ + + - b0b0 c0c0 d0d0 c b,d a b is not used at the exit of the basic block Examples: a:=b + c d:=a - d c:=d + c e abc c0c0 d0d0 b0b0

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Renaming Temporary Variables Normal Form: Each statement that defines a temporary, defines a new temporary. - A basic block can always be transformed into an equivalent block which is in normal form Example: (1) t := b + c(1) u := b + c rename + t b0b0 c0c0 + u b0b0 c0c0 Change (rename) label

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Interchange of Statements Example: t 1 := b + c t 2 := x + y Observation: We can interchange the statements without affecting the value of the block if and only if neither x nor y is t 1 and neither b nor c is t 2, i.e. we have two trees. Note: Normal-form basic blocks permit all statement interchanges that are possible. + t1t1 b0b0 c0c0 + t2t2 x0x0 y0y0

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Algebraic Transformations Arithmetic Identities: x + 0 = 0 + x = x x – 0 = x x + 1 = 1 + x = x x / 1 = x - Replace left-hand side with simples right hand side. Associative/Commutative laws x + (y + z) = (x + y) + z x + y = y + x Reduction in strength: x ** 2 = x * x 2.0 * x = x + x x / 2 = x + 0.5 - Replace an expensive operator with a cheaper one. Constant folding 2 * 3.14 = 6.28 -Evaluate constant expression at compile time` Methodology: Before we create a node of the DAG we check whether such a node exists modulo the above identities.

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Global Data Flow Analysis To do code generation and optimization a compiler needs to collect information about the program as a whole and to distribute this information to each block in the flow-graph. Data flow analysis: the process by which the compiler collects data-flow information. - by setting up and solving systems of equations that relate information at various points in the program. Example (generic): out[S] = gen[S] U (in[S] – kill[S]) Information flowing after the execution of S Generated by execution of S Information flowing at the beginning of S Information killed by the exec. of S Note: Sometimes information flows backwards in[S] = … out[S] …

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Points and Paths Points between two adjacent statements within a block before the first statement after the last statement Paths Consider all points in all the blocks. A path from p 1 to p n is a sequence of points p 1, …, p n such that for each i [1, n-1]: - pi is the point immediately preceding a statement and p i+1 is the point immediately following that statement in the same block, or - p i is the end of some block and p i+1 is the beginning of a successor block. d1: i:= m-1 d2: j:= n flow-graph B d1d1 d2d2 i:= m-1 j:= n State machine

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Example of DFA Problem: Reaching Definitions Definition of variable x: is a statement that assigns or may assign a value to x. - unambiguous definitions: assignment to x or read & store a value in x from an I/O device. - ambiguous definitions: may define the value of x: procedure call with x as a parameter or global variable. assignment through a pointer. Definition d reaches a point p if there is a path from the point immediately following d to p such that d is not killed along that path. Killing a definition of a variable x happens if between two points along the path there is a definition of x.

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Reaching Definition (Cont) Inaccuracies: When defining reaching definition we sometimes allow inaccuracies. Undecidability: to decide in general whether each path in a flow graph can be taken is an undecidable problem. Conservative inaccuracy: an inaccuracy is conservative if it never leads to a change in what the program computes. - it is normally conservative to assume that a definition can reach a point even if it may not. - we allow definitions to pass through ambiguous definitions.

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DF Analysis of Strongly Structured Programs gen[s] = {d} kill[s] = D a – {d} out[s] = gen[s] (in[s] \ kill[s]) gen[s] = gen[s 2 ] (gen[s1] – kill[s 2 ]) kill[s] = kill[s 2 ] (kill[s1] – gen[s 2 ]) in[s 1 ] = in[s], in[s 2 ] = out[s 1 ] out[s] = out[s 2 ] gen[s] = gen[s 1 ] gen[s 2 ] kill[s] = kill[s 1 ] kill[s 2 ] in[s 1 ] = in[s], in[s 2 ] = in[s] out[s] = out[s 1 ] U out[s 2 ] Sd: a:= b + c S S1S1 S2S2 SS1S1 S2S2 (a) (b) (c)

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DFA of Strongly Structured Programs (Cont) Syntax directed definitions: the equations are syntax directed. - synthesized attributes: gen, kill, (out – depends on in) - inherited attributes: in ( in backward dir – out) Note: out[s] gen[s] (gen – definitions reaching end of S without following paths outside S) Loops & Fixpoints: Given gen[s], kill[s], in[s] we cannot simply use in[s] = in[s1].I = J O in[s1] = in[s] out[s1]O = G ( I – K) out[s] = out[s1]Take O 0 = out[s1] = gen[s1] (in[s1] – kill[s1])I 1 = J 1, O 1 = G ( J – K) I 2 = J G, O 2 = O 1 gen[s] = gen[s1] kill[s] = kill[s1] in[s1] = in[s] gen[s1] out[s] = out[s1] S (d) S1S1

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Code Optimization Data Flow Analysis. Data Flow Analysis (DFA) General framework Can be used for various optimization goals Some terms Basic block.

Code Optimization Data Flow Analysis. Data Flow Analysis (DFA) General framework Can be used for various optimization goals Some terms Basic block.

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