1. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT Optimizing Compilers CISC 673 Spring 2009 Static Single Assignment John Cavazos University of Delaware

2. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 2 What is SSA? Many data-flow problems have been formulated To limit number of analyses, use single analysis to perform multiple transformations  SSA Compiler optimization algorithms enhanced or enabled by SSA:  Constant propagation  Dead code elimination  Global value numbering  Partial redundancy elimination  Register allocation

3. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 3 SSA Construction Algorithm ( High-level sketch ) 1. Insert  -functions 2. Rename values

4. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 4 SSA Construction Algorithm ( Less high-level sketch ) Insert  -functions at every join for every name Solve reaching definitions Rename each use to def that reaches it ( will be unique ) What’s wrong with this approach Too many  -functions! To do better, we need a more complex approach Builds maximal SSA

5. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 5 1.Insert  -functions a.) calculate dominance frontiers b.) find global names for each name, build list of blocks that define it c.) insert  -functions SSA Construction Algorithm ( Less high-level sketch )

6. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 6 Insert  -functions  global name n worklist ← Block(n) // blocks in which n is assigned  block b ∈ worklist  block d in b’s dominance frontier insert a  -function for n in d add d to worklist

7. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 7 Computing Dominance Frontiers Only join points are in DF(n) for some n Simple algorithm for computing dominance frontiers For each join point x ( i.e., |preds(x)| > 1 ) For each CFG predecessor of x Run up to ID OM (x ) in dominator tree, add x to DF(n) for each n between x and ID OM (x )

8. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 8 Dominance Frontiers nB0B1B2B3B4B5B6B7 DOM(n)00,10,1,20,1,30,1,3,40,1,3,50,1,3,60,1,7 IDOM(n)--0113331 DF(n)-- 776671 For each join point x For each CFG pred of x Run to ID OM (x ) in dom tree, add x to DF(n) for each n between x and ID OM (x ) B1B1 B2B2 B3B3 B4B4 B5B5 B6B6 B7B7 B0B0 B1B1 B2B2 B3B3 B4B4 B5B5 B6B6 B7B7 B0B0 Flow Graph Dominance Tree

9. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT Dominance Frontiers &  -Function Insertion Dominance Frontiers A definition at n forces a  -function at m iff n  D OM (m) but n  D OM (p) for some p  preds(m) DF(n ) is fringe just beyond region n dominates B1B1 B2B2 B3B3 B4B4 B5B5 B6B6 B7B7 B0B0  in 1 forces  -function in DF(1) = Ø (halt ) x ... x   (...) DF(4) is {6}, so  in 4 forces  -function in 6 x   (...)  in 6 forces  -function in DF(6) = {7} x   (...)  in 7 forces  -function in DF(7) = {1} For each assignment, we insert the  -functions

10. a   (a,a) b   (b,b) c   (c,c) d   (d,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i ... B0B0 b ... c ... d ... B2B2 a   (a,a) b   (b,b) c   ( c,c) d   (d,d) i   (i,i) a ... c ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 With all the  -functions Lots of new ops Renaming is next Assume a, b, c, & d defined before B 0 Excluding local names avoids  ’s for y & z

11. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 11 2.Rename variables in a pre-order walk over dominator tree (uses counter and a stack per global name) Staring with the root block, b a.) generate unique names for each  -function and push them on the appropriate stacks SSA Construction Algorithm ( Less high-level sketch )

12. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 12 2. Rename variables (cont’d) b.) rewrite each operation in the block i. Rewrite uses of global names with the current version (from the stack) ii. Rewrite definition by creating & pushing new name c.) fill in  -function parameters of successor blocks d.) recurse on b’s children in the dominator tree e.) pop names generated in b from stacks SSA Construction Algorithm ( Less high-level sketch )

13. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 13 SSA Construction Algorithm for each global name i counter[i]  0 stack[i]   call Rename(n 0 ) NewName(n) i  counter[n] counter[n]  counter[n] + 1 push n i onto stack[n] return n i Rename(b) for each  -function in b, x   (…) rename x as NewName(x) for each operation “x  y op z” in b rewrite y as top(stack[y]) rewrite z as top(stack[z]) rewrite x as NewName(x) for each successor of b in the CFG rewrite appropriate  parameters for each successor s of b in dom. tree Rename(s) for each operation “x  y op z” in b pop(stack[x]) Adding all the details...

14. a   (a,a) b   (b,b) c   (c,c) d   (d,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i ... B0B0 b ... c ... d ... B2B2 a   (a,a) b   (b,b) c   (c,c) d   (d,d) i   (i,i) a ... c ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 Counters Stacks 11110 a a0a0 b0b0 c0c0 d0d0 Before processing B 0 bcdi Assume a, b, c, & d defined before B 0 i has not been defined 20

15. a   (a,a) b   (b,b) c   (c,c) d   (d,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b ... c ... d ... B2B2 a   (a 0,a) b   (b 0,b) c   (c 0,c) d   (d 0,d) i   (i 0,i) a ... c ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 Counters Stacks 11111 abcdi a0a0 b0b0 c0c0 d0d0 End of B 0 i0i0 21

16. a   (a,a) b   (b,b) c   (c,c) d   (d,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b ... c ... d ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 Counters Stacks 32322 abcdi a0a0 b0b0 c0c0 d0d0 End of B 1 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 22

17. a   (a 2,a) b   (b 2,b) c   (c 3,c) d   (d 2,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 Counters Stacks 33432 abcdi a0a0 b0b0 c0c0 d0d0 End of B 2 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 b2b2 d2d2 c3c3 23

18. a   (a 2,a) b   (b 2,b) c   (c 3,c) d   (d 2,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a ... d ... B3B3 B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 i ≤ 100 Counters Stacks 33432 abcdi a0a0 b0b0 c0c0 d0d0 Before starting B 3 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 24

19. a   (a 2,a) b   (b 2,b) c   (c 3,c) d   (d 2,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d ... B4B4 c ... B5B5 d   (d,d) c   (c,c) b ... B6B6 i > 100 Counters Stacks 43442 abcdi a0a0 b0b0 c0c0 d0d0 End of B 3 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 a3a3 d3d3 25

20. a   (a 2,a) b   (b 2,b) c   (c 3,c) d   (d 2,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c ... B5B5 d   (d 4,d) c   (c 2,c) b ... B6B6 i > 100 Counters Stacks 43452 abcdi a0a0 b0b0 c0c0 d0d0 End of B 4 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 a3a3 d3d3 d4d4 26

21. a   (a 2,a) b   (b 2,b) c   (c 3,c) d   (d 2,d) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c 4 ... B5B5 d   (d 4,d 3 ) c   (c 2,c 4 ) b ... B6B6 i > 100 Counters Stacks 43552 abcdi a0a0 b0b0 c0c0 d0d0 End of B 5 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 a3a3 d3d3 c4c4 27

22. a   (a 2,a 3 ) b   (b 2,b 3 ) c   (c 3,c 5 ) d   (d 2,d 5 ) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c 4 ... B5B5 d 5   (d 4,d 3 ) c 5   (c 2,c 4 ) b 3 ... B6B6 i > 100 Counters Stacks 44662 abcdi a0a0 b0b0 c0c0 d0d0 End of B 6 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 a3a3 d3d3 c5c5 d5d5 b3b3 28

23. a   (a 2,a 3 ) b   (b 2,b 3 ) c   (c 3,c 5 ) d   (d 2,d 5 ) y  a+b z  c+d i  i+1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a) b 1   (b 0,b) c 1   (c 0,c) d 1   (d 0,d) i 1   (i 0,i) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c 4 ... B5B5 d 5   (d 4,d 3 ) c 5   (c 2,c 4 ) b 3 ... B6B6 i > 100 Counters Stacks 44662 abcdi a0a0 b0b0 c0c0 d0d0 Before B 7 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 29

24. a 4   (a 2,a 3 ) b 4   (b 2,b 3 ) c 6   (c 3,c 5 ) d 6   (d 2,d 5 ) y  a 4 +b 4 z  c 6 +d 6 i 2  i 1 +1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a 4 ) b 1   (b 0,b 4 ) c 1   (c 0,c 6 ) d 1   (d 0,d 6 ) i 1   (i 0,i 2 ) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c 4 ... B5B5 d 5   (d 4,d 3 ) c 5   (c 2,c 4 ) b 3 ... B6B6 i > 100 Counters Stacks 55773 abcdi a0a0 b0b0 c0c0 d0d0 End of B 7 i0i0 a1a1 b1b1 c1c1 d1d1 i1i1 a2a2 c2c2 a4a4 b4b4 c6c6 d6d6 i2i2 30

25. a 4   (a 2,a 3 ) b 4   (b 2,b 3 ) c 6   (c 3,c 5 ) d 6   (d 2,d 5 ) y  a 4 +b 4 z  c 6 +d 6 i 2  i 1 +1 B7B7 i > 100 i 0 ... B0B0 b 2 ... c 3 ... d 2 ... B2B2 a 1   (a 0,a 4 ) b 1   (b 0,b 4 ) c 1   (c 0,c 6 ) d 1   (d 0,d 6 ) i 1   (i 0,i 2 ) a 2 ... c 2 ... B1B1 a 3 ... d 3 ... B3B3 d 4 ... B4B4 c 4 ... B5B5 d 5   (d 4,d 3 ) c 5   (c 2,c 4 ) b 3 ... B6B6 i > 100 After renaming Semi-pruned SSA form We’re done … Semi-pruned  only names live in 2 or more blocks are “global names”. 31

26. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 26 SSA Construction Algorithm ( Pruned SSA ) What’s this “pruned SSA” stuff? Minimal SSA still contains extraneous  -functions Inserts some  -functions where they are dead Would like to avoid inserting them

27. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 27 SSA Construction Algorithm ( Two Ideas ) Semi-pruned SSA: discard names used in only one block  Significant reduction in total number of  -functions  Needs only local Live information ( cheap to compute ) Pruned SSA: only insert  -functions where their value is live  Inserts even fewer  -functions, but costs more to do  Requires global Live variable analysis ( more expensive ) In practice, both are simple modifications to step 1.

28. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 28 SSA Deconstruction At some point, we need executable code Few machines implement  operations Need to fix up the flow of values Basic idea Insert copies  -function pred’s Simple algorithm  Works in most cases Adds lots of copies  Many of them coalesce away X 17  (x 10,x 11 )...  x 17......  x 17 X 17  x 10 X 17  x 11