Machine-Independent Optimizations Ⅰ CS308 Compiler Theory1

Code optimization Elimination of unnecessary instructions Replacement of one sequence of instructions by a faster sequence of instructions Local optimization Global optimizations –based on data flow analyses CS308 Compiler Theory2

The Principal Sources of Optimization Optimization –Preserves the semantics of the original program –Applies relatively low-level semantic transformations CS308 Compiler Theory3

Causes of Redundancy Redundant operations are –at the source level –a side effect of having written the program in a high-level language Each of high-level data-structure accesses expands into a number of low-level arithmetic operations Programmers are not aware of these low-level operations and cannot eliminate the redundancies themselves. By having a compiler eliminate the redundancies –The programs are both efficient and easy to maintain. CS308 Compiler Theory4

A Running Example: Quicksort CS308 Compiler Theory5

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Semantics-Preserving Transformations A number of ways in which a compiler can improve a program without changing the function it computes –Common-sub expression elimination –Copy propagation –Dead-code elimination –Constant folding CS308 Compiler Theory7

Common Subexpressions Common subexpression –Previously computed –The values of the variables not changed Local: CS308 Compiler Theory8

Common Subexpressions Global CS308 Compiler Theory9

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Copy Propagation Copy statements or Copies –u = v CS308 Compiler Theory11

Dead-Code Elimination Live variable –A variable is live at a point in a program if its value can be used subsequently; –otherwise, it is dead at that point. Constant folding –Deducing at compile time that the value of an expression is a constant and using the constant instead CS308 Compiler Theory12

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Code Motion An important modification that decreases the amount of code in a loop Loop-invariant computation –An expression that yields the same result independent of the number of times a loop is executed Code Motion takes loop-invariant computation before its loop CS308 Compiler Theory14 while (i <= limit-2) t = limit -2 while (i <= t)

Induction Variables and Reduction in Strength Induction variable –For an induction variable x, there is a positive or negative constant c such that each time x is assigned, its value increases by c Induction variables can be computed with a single increment (addition or subtraction) per loop iteration Strength reduction –The transformation of replacing an expensive operation, such as multiplication, by a cheaper one, such as addition Induction variables lead to –strength reduction –eliminate computation CS308 Compiler Theory15

Now We have: CS308 Compiler Theory16 Inside-out

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Test yourself E CS308 Compiler Theory18

Data-Flow Analysis Techniques that derive information about the flow of data along program execution paths Examples –One way to implement global common sub expression elimination requires us to determine whether two identical expressions evaluate to the same value along any possible execution path of the program. –If the result of an assignment is not used along any subsequent execution path, then we can eliminate the assignment as dead code. CS308 Compiler Theory19

The Data-Flow Abstraction Execution paths –Within one basic block, the program point after a statement is the same as the program point before the next statement. –If there is an edge from block B1 to block B2, then the program point after the last statement of B1 may be followed immediately by the program point before the first statement of B2. Define an execution path from point P1 to point Pn to be a sequence of points P1, P2,..., Pn such that for each i = 1, 2,..., n - 1, either 1. Pi is the point immediately preceding a statement and Pi+1 is the point immediately following that same statement, or 2. Pi is the end of some block and Pi+1 is the beginning of a successor block. Reaching definition –The definitions that may reach a program point along some path CS308 Compiler Theory20

The Data-Flow Analysis Schema data-flow value –represents an abstraction of the set of all possible program states that can be observed for a program point Domain –The set of possible data-flow values for the application. –Example: the domain of data-flow values for reaching definitions is the set of all subsets of definitions in the program. Denote the data-flow values before and after each statement s by IN[S] and OUT[S] Data-flow problem –to find a solution to a set of constraints on the IN [S] 'S and OUT[S] 'S, for all statements S. –Two sets of constraints: those based on the semantics of the statements ( "transfer functions" ) and those based on the flow of control. CS308 Compiler Theory21

Transfer Functions The data-flow values before and after a statement are constrained by the semantics of the statement. transfer function –Both a and b will have the same value after the b=a statement. –Transfer function of a statement s is denoted as f s Two flavors of transfer function –Information propagate forward along execution paths –Flow backwards up the execution paths CS308 Compiler Theory22

Control-Flow Constraints Simple for within a basic block –if a block B consists of statements Sl, S2,..., Sn in that order, then the control-flow value out of Si is the same as the control-flow value into Si+1. Complicated for between basic blocks CS308 Compiler Theory23

Data-Flow Schemas on Basic Blocks IN[B], OUT[B] –denote the data-flow values immediately before and immediately after basic block B IN[B] = IN[S1], OUT[B] = OUT[Sn] –Suppose block B consists of statements Sl,..., Sn, in that order. f B = f Sn ○ ○ f S2 ○ f S1 CS308 Compiler Theory24

Reaching Definitions A 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. A definition of a variable x is killed if there is any other definition of x anywhere along the path. Conservative –if we do not know whether a statement s is assigning a value to x, we must assume that it may assign to it. CS308 Compiler Theory25

Transfer Equations for Reaching Definitions Generates a definition d of variable u and Kills all other definitions in the program that define variable u Transfer function of definition d can be expressed as where gen d = {d}, the set of definitions generated by the statement, and kill d is the set of all other definitions of u in the program. CS308 Compiler Theory26

Transfer Equations for Reaching Definitions If Then CS308 Compiler Theory27

Transfer Equations for Reaching Definitions Suppose block B has n statements, with transfer functions for Then CS308 Compiler Theory28

Transfer Equations for Reaching Definitions The gen set contains all the definitions inside the block that are "visible" immediately after the block Downwards exposed –A definition is downwards exposed in a basic block only if it is not "killed" by a subsequent definition to the same variable inside the same basic block. –A basic block's kill set is simply the union of all the definitions killed by the individual statements. –kill=kill1 U kill2={d1,d2} –gen=gen2 U (gen1-kill2) = {d2} –f(x)={d2} U (x-{d1,d2}) //always includes d2 CS308 Compiler Theory29

Control-Flow Equations OUT[P] IN[B] whenever there is a control-flow edge from P to B. IN[B] needs to be no larger than the union of the reaching definitions of all the predecessor blocks CS308 Compiler Theory30

Iterative Algorithm for Reaching Definitions The reaching definitions problem is defined by the following equations: for all basic blocks B other than ENTRY CS308 Compiler Theory31

Iterative Algorithm for Reaching Definitions Algorithm : Reaching definitions. INPUT: A flow graph for which kill B and gen B have been computed for each block B. OUTPUT: IN[B ] and OUT[B] METHOD: CS308 Compiler Theory32

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Live-Variable Analysis In live-variable analysis we wish to know for variable x and point p whether the value of x at p could be used along some path in the flow graph starting at p. If so, we say x is live at p; otherwise, x is dead at p. Definitions: 1. def B : the set of variables defined in B prior to any use of that variable in B 2. use B : the set of variables whose values may be used in B prior to any definition of the variable. CS308 Compiler Theory34

Live-Variable Analysis Equations relating def and use: for all basic blocks B other than EXIT CS308 Compiler Theory35

Live-Variable Analysis Algorithm: Live-variable analysis. INPUT: A flow graph with def and use computed for each block. OUTPUT: IN[B] and OUT[B]. METHOD: CS308 Compiler Theory36

Available Expressions An expression x + y is available at a point p ： –if every path, from the entry node to p evaluates x + y, and after the last such evaluation prior to reaching p, there are no subsequent assignments to x or y. A block kills expression x + y : –if it assigns (or may assign) x or y and does not subsequently recompute x + y. A block generates expression x + y : –if it definitely evaluates x + y and does not subsequently define x or y. CS308 Compiler Theory37

Available Expressions The primary use of available-expression information is for detecting global common subexpressions. CS308 Compiler Theory38

Available Expressions Computation of the set of generated expressions –At point p set S of expressions is available, and q is the point after p, with statement x=y+z 1. Add to S the expression y + z. 2. Delete from S any expression involving variable x. Example ： CS308 Compiler Theory39

Available Expressions Let IN[B] be the set of expressions that are available before B OUT[B] be the same for the point following the end of B e_genB be the expressions generated by B e_killB be the set of expressions killed in B Then For all basic blocks B other than ENTRY CS308 Compiler Theory40

Available Expressions Algorithm: Available expressions. INPUT: A flow graph with e_killB and e_genB computed for each block B. The initial block is B1. OUTPUT: IN[B] and OUT[B]. METHOD: CS308 Compiler Theory41

Test yourself gen, kill, IN, OUT sets for each block e_gen, e_kill, IN, OUT sets for available expressions. def, use, IN, OUT sets for live variable analysis. CS308 Compiler Theory42