GRAPHITE: Polyhedral Analyses and Optimizations for GCC Sebastian Pop 1 , Albert Cohen 2 , edric Bastoul 2 , Sylvain Girbal 2 , C´ e Silber 1 , Nicolas Vasilache 2 Georges-Andr´ 1 CRI/ENSMP 2 Alchemy/INRIA, LRI/Paris Sud 11 University June, 2006 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Architecture of GCC and Loop Nest Optimizer C C++ Java F95 Ada GENERIC GIMPLE Analyses LNO GIMPLE + CFG + SSA + Loops − aliasing − data dependences − number of iterations Machine description RTL x86 ppc arm S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Problems with Classical LNO Transforms “source to source” modifies the compiled program difficult to undo order of transforms fixed once for all invalidated data deps: ad-hoc correction or rebuild difficult to compose S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Problems with Classical LNO Transforms “source to source” modifies the compiled program difficult to undo order of transforms fixed once for all invalidated data deps: ad-hoc correction or rebuild difficult to compose solved in WRaP-IT (from 2002 at INRIA on ORC/Open64) GRAPHITE = WRaP-IT for GCC S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE : Representation on Top of Gimple-SSA Statements + parametric affine inequalities 1 a domain = bounds of enclosing loops 2 a list of access functions 3 a schedule = execution time i j m n cst for (i=0; i<m; i++) 2 3 i ≥ 0 1 0 0 0 0 for (j=5; j<n; j++) 6 7 − 1 0 1 0 − 1 − i + m ≥ − 1 6 7 A[2*i][j+1] = ... 6 7 0 1 0 0 5 j ≥ 5 4 5 0 − 1 0 1 − 1 − j + n ≥ − 1 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE : Representation on Top of Gimple-SSA Statements + parametric affine inequalities 1 a domain = bounds of enclosing loops 2 a list of access functions 3 a schedule = execution time for (i=0; i<m; i++) 2 3 i j m n cst for (j=5; j<n; j++) 2 0 0 0 0 2 ∗ i 4 5 A[2*i][j+1] = ... j + 1 0 1 0 0 1 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE : Representation on Top of Gimple-SSA Statements + parametric affine inequalities 1 a domain = bounds of enclosing loops 2 a list of access functions 3 a schedule = execution time GRAPHITE(1 , 2 , 3) extends LAMBDA(1 , 2) GRAPHITE: Gimple Represented As Polyhedra (with interchangeable envelopes) S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE versus LAMBDA common part: unimodular transform data and iteration order transform regions: extended from loops to SCoP “static control parts”: sequences, affine conditions and loops GRAPHITE knows about the sequence! enables more loop transforms: fusion, fission, tiling, software pipelining, scheduling S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE versus LAMBDA common part: unimodular transform data and iteration order transform regions: extended from loops to SCoP “static control parts”: sequences, affine conditions and loops GRAPHITE knows about the sequence! enables more loop transforms: fusion, fission, tiling, software pipelining, scheduling S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
GRAPHITE versus LAMBDA common part: unimodular transform data and iteration order transform regions: extended from loops to SCoP “static control parts”: sequences, affine conditions and loops GRAPHITE knows about the sequence! enables more loop transforms: fusion, fission, tiling, software pipelining, scheduling S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Operational Semantics (How Program Works) build a scheduling function S [ [ stmt ] ] → time sequence [ [ s 1 ; s 2 ] ]: trivial S [ [ s 1 ] ] = t S [ [ s 2 ] ] = t + 1 loop [ [ loop 1 s end 1 ] ]: add new dimensions S [ [ loop 1 ] ] = t S [ [ s ] ] = ( t , i 1 , 0) i 1 indexes loop 1 iterations: dynamic time S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Operational Semantics (How Program Works) build a scheduling function S [ [ stmt ] ] → time sequence [ [ s 1 ; s 2 ] ]: trivial S [ [ s 1 ] ] = t S [ [ s 2 ] ] = t + 1 loop [ [ loop 1 s end 1 ] ]: add new dimensions S [ [ loop 1 ] ] = t S [ [ s ] ] = ( t , i 1 , 0) i 1 indexes loop 1 iterations: dynamic time S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Example S0; S1; for (i=0; i<m; i++) { S2; » – i j m n cst for (j=5; j<n; j++) S [ [ S 0] ] = 0 0 0 0 0 S3; } S4; S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Example S0; S1; » – i j m n cst for (i=0; i<m; i++) { S [ [ S 0] ] = 0 0 0 0 0 S2; for (j=5; j<n; j++) S3; » – i j m n cst S [ [ S 1] ] = } 0 0 0 0 1 S4; S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Example » i j m n cst – S [ [ S 0] ] = S0; 0 0 0 0 0 S1; for (i=0; i<m; i++) { » i j m n cst – S2; S [ [ S 1] ] = 0 0 0 0 1 for (j=5; j<n; j++) S3; } 2 3 i j m n cst S4; 0 0 0 0 2 6 7 S [ [ S 2] ] = 6 7 1 0 0 0 0 4 5 0 0 0 0 0 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Example » – i j m n cst S [ [ S 0] ] = 0 0 0 0 0 » – i j m n cst S [ [ S 1] ] = S0; 0 0 0 0 1 S1; for (i=0; i<m; i++) { 2 3 S2; i j m n cst 0 0 0 0 2 for (j=5; j<n; j++) 6 7 S [ [ S 2] ] = 6 7 1 0 0 0 0 4 5 S3; 0 0 0 0 0 } S4; i j m n cst 2 3 0 0 0 0 2 6 7 1 0 0 0 0 6 7 S [ [ S 3] ] = 6 7 0 0 0 0 1 6 7 6 7 0 1 0 0 0 4 5 0 0 0 0 0 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Example » – i j m n cst S [ [ S 0] ] = 0 0 0 0 0 S0; S1; » – i j m n cst S [ [ S 1] ] = 0 0 0 0 1 for (i=0; i<m; i++) { S2; for (j=5; j<n; j++) 2 3 i j m n cst S3; 0 0 0 0 2 6 7 S [ [ S 2] ] = } 6 7 1 0 0 0 0 4 5 S4; 0 0 0 0 0 » i – j m n cst i j m n cst S [ [ S 4] ] = 2 3 0 0 0 0 3 0 0 0 0 2 6 7 1 0 0 0 0 6 7 S [ [ S 3] ] = 6 7 0 0 0 0 1 6 7 6 7 0 1 0 0 0 4 5 0 0 0 0 0 S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Separation Example i j m n cst 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 scheduling matrix S [ [ S 3] ] S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Separation Example i j m n cst 0 0 0 0 2 i j m n cst 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 i j m n cst 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 separate static / dynamic schedules S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Separation Example i j m n cst cst 2 0 0 0 0 2 i j m n cst 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 i j m n cst 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 static scheduling vector fusion, fission, code motion S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Separation Example i j m n cst cst 2 0 0 0 0 2 i j m n cst 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 i j m n cst m n cst 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Parameter scheduling matrix shifting S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
Schedule: Separation Example i j m n cst cst 2 0 0 0 0 2 i j m n cst 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 i j m n cst m n cst 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 i j 1 0 0 1 Iteration scheduling matrix interchange, skewing, reversal S.Pop, A.Cohen, C.Bastoul, S.Girbal, G.A.Silber, N.Vasilache GRAPHITE: Polyhedral Analyses and Optimizations for GCC
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