Reaching Definitions Global Opt: Reaching Definitions ♦ Concept of definition and use Using Program Analysis ♦ a=x+y for Optimization ♦ is a definition of a. ♦ is a use of x and y. ♦ A definition reaches a use if value written Advanced Compiler Techniques 2005 by definition may be read by use. Erik Stenman Virtutech Advanced Compiler Techniques 2 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Reaching Definitions and Computing Reaching Definitions Constant Propagation Global Opt: Reaching Definitions & Constant Prop Global Opt: Reaching Definitions ♦ Compute with sets of definitions: ♦ Is a use of a variable a constant? ♦ Represent sets using bit vectors. ♦ Check all reaching definitions. ♦ Each definition has a position in bit vector. ♦ If all assign variable to same constant. ♦ At each basic block, compute: ♦ Then use is in fact a constant. ♦ Definitions that reach start of block. ♦ Can replace variable with constant. ♦ Definitions that reach end of block. ♦ Do computation by simulating execution of program until the fixed point is reached. Advanced Compiler Techniques Advanced Compiler Techniques 3 4 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Formalizing Analysis Dataflow Equations Global Opt: Reaching Definitions Global Opt: Reaching Definitions ♦ Each basic block has ♦ IN[b i ] = OUT[b 1 ] ∪ ... ∪ OUT[b n ] ♦ IN - set of definitions that reach beginning of where b 1 , ..., b n are predecessors of b i block ♦ OUT[b i ] = (IN[b i ] - KILL[b i ]) ∪ GEN[b i ] ♦ OUT - set of definitions that reach end of block ♦ IN[entry] = 0000000 ♦ GEN - set of definitions generated in block ♦ KILL - set of definitions killed in the block ♦ Result: system of equations. ♦ Compiler scans each basic block to derive GEN and KILL sets. Advanced Compiler Techniques Advanced Compiler Techniques 5 6 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er /
IN[0] = 0000000 Solving Equations GEN[0] = 1110000 KILL[0] = 0000011 s 1 =0; OUT[0] =(IN[0] -KILL[0]) ∪ GEN[0]= ♦ Use fix point algorithm. a 2 =4; 0000000-0000011 ∪ 1110000= 1110000 ♦ Initialize with solution of Global Opt: Reaching Definitions Global Opt: Reaching Definitions i 3 =0; IN[2]=OUT[0] OUT[b i ] = 0000000 k==0 GEN[2] = 0000100 KILL[2] = 0001000 ♦ Repeatedly apply equations: OUT[2]=(IN[2]-0001000) ∪ 0000100 IN[1]=OUT[0] GEN[1] = 0001000 b 4 =1; b 5 =2; ♦ IN[b i ] = OUT[b 1 ] ∪ ... ∪ OUT[b n ] KILL[1] = 0000100 OUT[1]=(IN[1]-0000100) ∪ 0001000 IN[3]=OUT[1] ∪ OUT[2] ♦ OUT[b i ] = (IN[b i ] - KILL[b i ]) ∪ GEN[b i ] GEN[3] = 0000000 i<n KILL[3] = 0000000 ♦ Until reach fixed point, i.e., until equation OUT[3]=IN[3] application has no further effect. s 6 =s+a*b; return s i 7 =i+1; ♦ Use a worklist to track which equation IN[5]=OUT[3] applications may have further effect. IN[4]=OUT[3] GEN[5] = 0000000 GEN[4] = 0000011 KILL[5] = 0000000 KILL[4] = 1010000 OUT[5]=IN[5] OUT[4]=(IN[4]-1010000) ∪ 0000011 Advanced Compiler Techniques Advanced Compiler Techniques 7 8 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Reaching Definitions Algorithm Questions for all nodes n 2N Global Opt: Reaching Definitions, summary OUT[n] = ; ; // Or OUT[n] = GEN[n]; Global Opt: Reaching Definitions Changed = N ; // N = all nodes in graph ♦ Does the algorithm halt? while (Changed != ; ) // Until fixed point reached. ♦ yes, because transfer function is monotonic. choose n 2 Changed; // Node from worklist ♦ if increase IN, increase OUT. Changed=Changed-{n}; // Remove from worklist ♦ in limit, all bits are 1. OldOut = OUT[n] // Remember old result ♦ If bit is 1, is there always an execution in which IN[n] = ; ; // Calculate IN as join corresponding definition reaches basic block? for all nodes p 2 predecessors (n) // of predecessors. IN[n]=IN[n] ∪ OUT[p]; ♦ If bit is 0, does the corresponding definition ever OUT[n]=(IN[n]-KILL[n]) ∪ GEN[n]; // Recalculate OUT reach basic block? if (OUT[n] != OldOut) // If OUT[n] changed ♦ Concept of conservative analysis. for all nodes s 2 successors (n) Changed=Changed ∪ {s}; //Add succs to worklist Advanced Compiler Techniques Advanced Compiler Techniques 9 10 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Computing Available Available Expressions Expressions Global Opt: Available Expressions Global Opt: Available Expressions ♦ An expression x+y is available at a point p if ♦ Represent sets of expressions using bit vectors. ♦ every path from the initial node to p evaluates x+y ♦ Each expression corresponds to a bit. before reaching p, ♦ Run dataflow algorithm similar to reaching ♦ and there are no assignments to x or y after the definitions. evaluation but before p. ♦ Big difference: ♦ Available Expression information can be used to ♦ Definition reaches a basic block if it comes from ANY do global (across basic blocks) CSE. predecessor in CFG. ♦ If an expression is available at use, there is no ♦ Expression is available at a basic block only if it is need to re-evaluate it. available from ALL predecessors in CFG. Advanced Compiler Techniques Advanced Compiler Techniques 11 12 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er /
Global CSE Transform 0000 0000 a=x+y; a=x+y; t 1 =a; Expressions x==0 Expressions Global Opt: Available Expressions & CSE x==0 1001 1001 1: x+y 1: x+y Global Opt: Available Expressions x=z; 2: i<n 2: i<n x=z; b=x+y; 3: i+c 3: i+c b=x+y; t 1 =b; 4: x==0 4: x==0 1000 1000 i=x+y; i=t 1 ; Must use same temp 1000 1000 for CSE in all blocks i<n i<n 1100 1100 1100 1100 c=x+y; c=t 1 ; d=x+y d=t 1 i=i+c; i=i+c; Advanced Compiler Techniques Advanced Compiler Techniques 13 14 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Formalizing Analysis Dataflow Equations ♦ Each basic block has ♦ IN[b i ] = OUT[b 1 ] ∩ ... ∩ OUT[b n ] Global Opt: Available Expressions Global Opt: Available Expressions IN - set of expressions that reach beginning of block. ♦ where b 1 , ..., b n are predecessors of b i OUT - set of expressions that reach end of block. ♦ OUT[b i ] = (IN[b i ] - KILL[b i ]) ∪ GEN[b i ] GEN - set of expressions generated in block. ♦ IN[entry] = 0000 KILL - set of expressions killed in the block. ♦ GEN[ x=z; b=x+y ] = 1000 ♦ Result: system of equations. ♦ KILL[ x=z; b=x+y ] = 1001 ♦ Compiler scans each basic block to derive GEN and KILL sets. Advanced Compiler Techniques Advanced Compiler Techniques 15 16 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / Solving Equations Available Expressions Algorithm ♦ Use fix point algorithm. for all nodes n 2N // E is set of all expressions. Global Opt: Available Expressions Global Opt: Available Expressions OUT[n] = E ; // OUT[n] = E -KILL[n]; ♦ IN[entry]=0000 Changed = N ; // N = all nodes in graph ♦ Initialize with solution of while (Changed != ; ) OUT[b i ] = 1111 choose n 2 Changed; Changed=Changed-{n}; ♦ Repeatedly apply equations: IN[n] = E ; ♦ IN[b i ] = OUT[b 1 ] ∩ ... ∩ OUT[b n ] OldOut = OUT[n] ♦ OUT[b i ] = (IN[b i ] - KILL[b i ]) ∪ GEN[b i ] for all nodes p 2 predecessors (n) IN[n]=IN[n] ∩ OUT[p]; ♦ Use a worklist to track which equation OUT[n]=(IN[n]-KILL[n]) ∪ GEN[n]; applications may have further effect. if (OUT[n] != OldOut) for all nodes s 2 successors (n) Changed=Changed ∪ {s}; Advanced Compiler Techniques Advanced Compiler Techniques 17 18 ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er / ht t p: / / l am p. epf l . ch/ t eachi ng/ advancedCom pi l er /
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