: Compiler Design. 9.6 Available expressions 9.7 Busy expressions. Thomas R. Gross. Computer Science Department ETH Zurich, Switzerland

Transcription

1 : Compiler Design 9.6 Available expressions 9.7 Busy expressions Thomas R. Gross Computer Science Department ETH Zurich, Switzerland

2 Available expressions An expression a+b is available at a point P if a+b is evaluated on every path from ENTRY to P and there is no defininon of a or b on this path aoer a+b was evaluated Interested in set of expressions available at the start of a basic block B Set depends on paths that lead to P before_b Assume that a+b has no side effects Reading a memory locacon may have side effects too! Not a problem for JavaLi or Java 2

3 Available expressions e1: x = a + b if (Tcond) B1 B2?? Is a+b computed in e1 available for e2? Available at P Depends on B2 P e2: y = a + b B3 3

4 Available expressions e1: x = a + b if (Tcond) B1 B2 a = Is a+b computed in e1 available for e2? No a may have at P a value that differs from the value at e1. P e2: y = a + b B3 4

5 Available expressions e1: x = a + b if (Tcond) B1 B2 a = e3: z = a + b Is a+b available for e2? Yes either computed by e1 or by e3 P e2: y = a + b B3 5

6 Transfer funcnon e: = a + b B a+b is available at the end of basic block B if there is no stmt in B that follows e and that defines a or b Must be conservacve If a statement in the shaded region may modify a (or b) then a+b is not available at the end of B 6

7 Transfer funcnon P before_b d: a = B P aoer_b (Maybe) a+b is available at P before_b. a+b is not available at P aoer_b Unless there is a statement in B that follows d and re-computes a+b, see last slide DefiniNon d has killed the expression a+b 7

8 Transfer funcnon d: x = e: = a + b Define for basic block B gen B = { expr expr a+b is evaluated in B, neither a nor b subsequently defined in B } kill B = { expr a or b of expr a+b defined in B and a+b is not subsequently evaluated in B } Must be conservanve In gen only if we are sure the expression is evaluated In kill if there is a chance that one of the operands of a+b is defined 8

9 Transfer funcnon IN(B) d: x = e: = a + b B OUT(B) Want to find sets IN(B): set of expressions available at start of B OUT(B): set of expressions available at end of B Transfer funcnon captures how statements in B determine OUT(B) 9

10 Transfer funcnon... B1 B2 B3 Given OUT(B1). What should be IN(B2) and IN(B3)? IN(B2) = OUT(B1), IN(B3) = OUT(B1) An expression that is available a_er B1 is available at the start of B2 and the start of B3 10

11 Transfer funcnon B2 B3 P... B1 Given OUT(B2) and OUT(B3) What should be IN(B1)? Expression E is available at P if E is computed on every path from ENTRY to P (and none of the operands of E is killed subsequently) Must consider all paths leading to P E must be available on all paths 11

12 Transfer funcnon B2 B3... B1 Given OUT(Bi) IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) 12

13 Finding IN(B) and OUT(B) gen and kill capture what happens inside a basic block Some texts use e_gen and e_kill to discnguish between sets needed for reaching definicons These sets are different! We need IN and OUT for each basic block IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) OUT(B) = gen B (IN(B) kill B ) N basic blocks, 2 N sets IN / OUT 13

14 Finding IN(B) and OUT(B) N basic blocks, 2 N sets IN / OUT System with 2 N unknowns Solve by iteracng uncl a fixed point is found How to start iteranon? Safe assumpcon OUT[ENTRY] = 14

15 Finding IN(B) and OUT(B) Safe assumpnon OUT[ENTRY] = What about OUT[Bi] for Bi ENTRY? For reaching definicons, we wanted smallest set of definicons that reach OK if we say d reaches but it does not For available expressions, we want largest set of expressions that reach OK if expr is available but not included in set So start with a large approximanon and remove expressions that are clearly not available OUT[Bi] = U! U is the set of all expressions that appear in the program! 15

16 Finding available expressions OUT[ENTRY] = IniCalize OUT[B] = U for B ENTRY while (changes to any OUT(B)) { for (each basic block B ENTRY) { } } IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) OUT(B) = gen B (IN(B) kill B ) 16

17 Comments The algorithm (previous slide) idennfies an expression expr as available only if expr is truly available. InducCon on length of path If expr is not available (along some path) then expr is not in OUT(B) for a block B expr is not available for some predecessor B of B as we use to form IN[B] Again, the order of visinng nodes makers For speed of convergence, not correctness 17

18 Available expressions Is this a forward or a backward problem? 18

19 Available expressions Is this a forward or a backward problem? Forward How similar is it to reaching defininons? Reaching definicons: IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) Available expressions: IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) 19

20 Summary for data flow problems Compute two sets for each basic block B SoluCon = OUT(B) / IN(B) Context = IN(B) / OUT(B) Two sets to capture local effects Define a direccon Reaching definicons, available expressions: Forward Ancestor: predecessor Live variables: Backward Ancestor: successor Init SoluCon(ENTRY) / SoluCon(EXIT) Init remaining SoluCon sets Define a meet operator Reaching definicons, live variables: Available expressions: Iterate over basic blocks B Context(B) = Bi, Bi is ancestor of B in CFG SoluCon(Bi) SoluCon(B) = local_new B (Context(B) local_cut B ) 20

21 Dataflow problems Forward Backward 21

22 Dataflow problems Forward Backward Reaching definicons Live variables Available expressions 22

23 9.7 Busy expressions if ( ) { A[j] = 0; } else { A[j] = 1; } 23

24 IR view if ( ) { tmp1 = j * 4 tmp2 = &A + tmp1 *tmp2 = 0 } else { tmp3 = j * 4 tmp4 = &A + tmp3 *tmp4 = 1 } 24

25 Expression j*4 is evaluated in both branches of the ifstmt HoisNng j*4 to block above the if-stmt reduces the size of the object code No (immediate) speed benefit Expression j*4 is a (very) busy expression. An expression a+b is very busy at a point P if a+b is evaluated on all paths from P to EXIT and there is no defininon of a or b on a path between P and an evaluanon of a+b Interested in set of expressions available at the start of a basic block B Set depends on paths that start at P before_b 26

26 Transfer funcnon S: = a b B a+b is very busy at the start of basic block B if there is no stmt in B prior to S that defines a or b. 27

27 Transfer funcnon S : a = S: = a b a+b is not (very) busy at the start of basic block B as there is a stmt in B prior to S that defines a (stmt S ) 28

28 Transfer funcnon S: = a b Define for basic block B kill B = { expr an operand of expr is defined in B by stmt S and expr is not evaluated before S } gen B = { expr expr is evaluated in B prior to a definicon of any of its operands} Sets kill and gen: must be conservanve In gen only if we are sure the expression is evaluated In kill if there is a chance that the expression is killed 29

29 Transfer funcnon NoNce that we may be interested in the set of busy expressions at the start of a basic block Determined by statements in basic block and statements in subsequent basic blocks Set of busy expressions at the end of a basic block Determined by statements in subsequent basic blocks Another term: expression is downward exposed 30

30 Transfer funcnon IN(B) = {b 1, b 2,, b n } S: = a b For a basic block B we define IN(B) = { expr expr very busy at P before_b } OUT(B) = { expr expr very busy at P a_er_b } OUT(B) = {b 1, b 2,, v m } Expression b IN(B) b is evaluated in B prior to definicon of an operand, i.e. b gen B b OUT(B) and operands of b not defined in B, i.e. b kill B IN(B) = gen B (OUT(B) kill B ) 31

31 Transfer funcnon B2 B3... B1 Given IN(B1) What should be OUT(B2) and OUT(B3)? OUT(B2) = IN(B1), OUT(B3) = IN(B1) 32

32 Transfer funcnon... B1 B2 B3 Given IN(B2) and IN(B3) What should be OUT(B1)? An expression E is very busy at a point P if E is evaluated on all paths from P to EXIT (and there is no definicon of an operand of E between P and such an evaluacon) Must consider all paths starcng at P 33

33 Transfer funcnon... B1 B2 B3 Given IN(Bi) OUT(B) = Bi, Bi is successor of B in CFG IN(Bi) 34

34 Finding IN(B) and OUT(B) N basic blocks, 2 N sets IN / OUT System with 2 N unknowns Solve by iteracng uncl a fixed point is found How to start iteranon? Safe assumpcon IN[EXIT] = Nothing is very busy at the end IN(B) = U U set of all expressions in program For all B EXIT 35

35 CompuNng very busy expressions IN[EXIT] = IniCalize IN[B] = U for B EXIT while (changes to any IN(B)) { for (each basic block B EXIT) { OUT(B) = Bi, Bi is successor of B in CFG IN(Bi) IN(B) = gen B (OUT(B) kill B ) } } 36

36 Comments Compiler must move code to benefit from very busy expression informanon: code hoisnng if ( ) { tmp1 = j * 4 tmp2 = &A + tmp1 *tmp2 = 0 } else { } tmp3 = j * 4 tmp4 = &A + tmp3 *tmp4 = 1 tmp5 = j * 4 tmp6 = &A + tmp5 if ( ) { *tmp6 = 0 } else { } *tmp6 = 1 38

37 Comments There exists an approach to code monon that deals with all kinds of code monon Loop invariant removal Common sub-expression eliminacon Very busy expression hoiscng together (and also deals with other opnmizanons) and therefore code hoisnng (of busy expressions in isolanon) is not employed in modern compilers. 39

38 9.8 Dataflow problems Forward Backward Reaching definicons Live variables Available expressions Very busy expressions 40

39 Summary for data flow problems Compute two sets for each basic block B SoluCon = OUT(B) / IN(B) Context = IN(B) / OUT(B) Two sets to capture local effects (local_new, local_cut) Define a direccon Reaching definicons, available expressions: Forward Ancestor: predecessor Live variables, very busy expressions: Backward Ancestor: successor Init SoluCon(ENTRY) / SoluCon(EXIT) Init remaining SoluCon sets Define a meet operator Reaching definicons, live variables: Available expressions, very busy expressions: Iterate over basic blocks B Context(B) = Bi, Bi is ancestor of B in CFG SoluCon(Bi) SoluCon(B) = local_new B (Context(B) local_cut B ) 41

40 Compiler implementanon Framework with abstract funcnons to allow specificanon of IniCalizaCon of local_new, local_cut DirecCon Meet (confluence) operator IniCal IN and OUT sets Transfer funccon (depends on direccon) OUT = (IN local_cut) local_gen IN = (OUT local_cut) local_gen Use the framework to instannate specific data flow analyses 42

41 Class hierarchy 43

42 9.9 Code transformanons Compiler uses global dataflow informanon to implement code transformanons Covered only briefly in this class 44

43 9.9.1 Constant folding Need: constant propaganon informanon as discussed in 9.1 Or reaching definicons as discussed in 9.2 TransformaNon: if both operands are constant, compute value in compiler AwenCon: careful if the compiler does not run on target system ( cross-compiler ) Must make sure compiler produces result that would have been obtained at runcme TransformaNon may create new opportunines 45

44 9.9.2 Dead assignment eliminanon Need: live variable analysis Dead assignment: target of store is not live TransformaNon: remove useless assignment statement AssumpCon: Reuse of available expressions Ced to (designated temporary) variables TransformaNon may render other statements dead 46

45 9.9.3 Dead code eliminanon Need: constant propaganon informanon, reaching defininons Dead code: unreachable code For well-structured programs: if-then or if-then-else statements with condicon expressions that can be evaluated at compile Cme TransformaNon: remove unreachable code May render variables dead 47

46 9.9.4 Copy propaganon Various transformanons introduce copies x = a + b x = a + b if (..) { t = x z = a + b if (..) { } z = t y = a + b } y = t 48

47 Copy propaganon May want to eliminate the copies Replace t by x x = a + b x = a + b if (..) { t = x z = a + b if (..) { } z = x y = a + b } y = x Dead assignment eliminanon takes care of t=x 49

48 Copy propaganon May also have beneficial effects on register allocator Given copy stmt S: t = x Compiler can subsntute x for t at point P if 1. DefiniCon S is the only definicon of t that reaches P 2. On every path from S to P there are no assignments to x A path may go through P mulcple Cmes but not go through S a second Cme and scll must sacsfy this condicon CondiNon (1) is easily checked given reaching defininons 50

49 New dataflow problem: reaching copy statements For each basic block B gen B = {copy-stmt copy-stmt a=b appears in B and there is no subsequent definicon of b in B} kill B = {copy-stmt for the copy-stmt a=b there is a definicon of a or b in B (and copy-stmt does not appear a_er these definicons in B)} Note that different assignments (copy statements) a=b kill each other Transfer funcnon IN(B): Set of copies S: a=b such that every path to P before_b contains S and there is no definicon of b subsequent to the last occurrence of S OUT(B): to P a_er_b 51

50 Transfer funcnon OUT (ENTRY) = OUT(B) = U U set of all copy statements in program For all B ENTRY IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) OUT(B) = gen B (IN(B) kill B ) 52

51 Finding reaching copy statements OUT[ENTRY] = IniCalize OUT[B] = U for B ENTRY while (changes to any OUT(B)) { for (each basic block B ENTRY) { } } IN(B) = Bi, Bi is predecessor of B in CFG OUT(Bi) OUT(B) = gen B (IN(B) kill B ) 53

52 Example S1: x = y B1 B2 S2: y = B3 S3: x = z B4 S4: = x B5 S4: = x 54

53 Example gen = {S1: x=y} kill= {S3: x=z} S1: x = y B1 B2 S2: y = gen = {S3: x=z} kill= {S1: x=y} B3 S3: x = z gen = {} kill= {S1: x=y} B4 S4: = x B5 S4: = x 55

54 Example IN(B2) = IN(B3) = OUT(B1) = { S1: x=y } OUT(B2) = OUT(B3) = IN(B4) = OUT (B4) = { S3: x=z } IN(B5) = OUT(B2) OUT(B4) = Please note that for reaching defininons, the defininon of x in S1 reaches B5 (via B2) 56

55 9.10 must be conservanve Pointers not an issue for JavaLi must be careful if language allows addresses to be taken Risk of aliases Arrays don t include them in data flow computanon No effort to understand modificacon of individual array elements 57

56 Method invocanon Java (and JavaLi) allow only value parameters *and* there is not way to get a reference to a local variable Treat actual parameter as uses of a variable resp. evaluacon of an expression Other languages support pointers (and may have global variables) ret = foo (p1, p2,, pn) Variable ret is defined, p1, p2,, pn are used What is modified by foo? Any parameter pi? LocaCons reachable via pi? Globals? Must make worst-case assumpcon in the absence of bewer informacon 58

57 ConservaNsm int a, *b; d1: a = 3; d2: *b =. P:. For reaching defininons, d1 reaches P Can t be sure that *b writes a For constant propaganon, (a=3) is false at P *b might destroy value of a 59