How to manually use the Individual pieces of Polly

Execute the individual Polly passes manually

This example presents the individual passes that are involved when optimizing code with Polly. We show how to execute them individually and explain for each which analysis is performed or what transformation is applied. In this example the polyhedral transformation is user-provided to show how much performance improvement can be expected by an optimal automatic optimizer.

1. Create LLVM-IR from the C code

Polly works on LLVM-IR. Hence it is necessary to translate the source files into LLVM-IR. If more than one file should be optimized the files can be combined into a single file with llvm-link.

clang -S -emit-llvm matmul.c -Xclang -disable-O0-optnone -o matmul.ll

2. Prepare the LLVM-IR for Polly

Polly is only able to work with code that matches a canonical form. To translate the LLVM-IR into this form we use a set of canonicalization passes. They are scheduled by using ‘-polly-canonicalize’.

opt -S -polly-canonicalize matmul.ll -o matmul.preopt.ll

3. Show the SCoPs detected by Polly (optional)

To understand if Polly was able to detect SCoPs, we print the structure of the detected SCoPs. In our example two SCoPs are detected. One in ‘init_array’ the other in ‘main’.

$ opt -basic-aa -polly-ast -analyze matmul.preopt.ll -polly-process-unprofitable -polly-use-llvm-names

:: isl ast :: init_array :: %for.cond1.preheader---%for.end19

if (1)

    for (int c0 = 0; c0 <= 1535; c0 += 1)
      for (int c1 = 0; c1 <= 1535; c1 += 1)
        Stmt_for_body3(c0, c1);

else
    {  /* original code */ }

:: isl ast :: main :: %for.cond1.preheader---%for.end30

if (1)

    for (int c0 = 0; c0 <= 1535; c0 += 1)
      for (int c1 = 0; c1 <= 1535; c1 += 1) {
        Stmt_for_body3(c0, c1);
        for (int c2 = 0; c2 <= 1535; c2 += 1)
          Stmt_for_body8(c0, c1, c2);
      }

else
    {  /* original code */ }

4. Highlight the detected SCoPs in the CFGs of the program (requires graphviz/dotty)

Polly can use graphviz to graphically show a CFG in which the detected SCoPs are highlighted. It can also create ‘.dot’ files that can be translated by the ‘dot’ utility into various graphic formats.

$ opt -polly-use-llvm-names -basic-aa -view-scops -disable-output matmul.preopt.ll
$ opt -polly-use-llvm-names -basic-aa -view-scops-only -disable-output matmul.preopt.ll

The output for the different functions:

5. View the polyhedral representation of the SCoPs

$ opt -polly-use-llvm-names -basic-aa -polly-scops -analyze matmul.preopt.ll -polly-process-unprofitable

[...]Printing analysis 'Polly - Create polyhedral description of Scops' for region: 'for.cond1.preheader => for.end19' in function 'init_array':
    Function: init_array
    Region: %for.cond1.preheader---%for.end19
    Max Loop Depth:  2
        Invariant Accesses: {
        }
        Context:
        {  :  }
        Assumed Context:
        {  :  }
        Invalid Context:
        {  : 1 = 0 }
        Arrays {
            float MemRef_A[*][1536]; // Element size 4
            float MemRef_B[*][1536]; // Element size 4
        }
        Arrays (Bounds as pw_affs) {
            float MemRef_A[*][ { [] -> [(1536)] } ]; // Element size 4
            float MemRef_B[*][ { [] -> [(1536)] } ]; // Element size 4
        }
        Alias Groups (0):
            n/a
        Statements {
            Stmt_for_body3
                Domain :=
                    { Stmt_for_body3[i0, i1] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535 };
                Schedule :=
                    { Stmt_for_body3[i0, i1] -> [i0, i1] };
                MustWriteAccess :=      [Reduction Type: NONE] [Scalar: 0]
                    { Stmt_for_body3[i0, i1] -> MemRef_A[i0, i1] };
                MustWriteAccess :=      [Reduction Type: NONE] [Scalar: 0]
                    { Stmt_for_body3[i0, i1] -> MemRef_B[i0, i1] };
        }
[...]Printing analysis 'Polly - Create polyhedral description of Scops' for region: 'for.cond1.preheader => for.end30' in function 'main':
    Function: main
    Region: %for.cond1.preheader---%for.end30
    Max Loop Depth:  3
    Invariant Accesses: {
    }
    Context:
    {  :  }
    Assumed Context:
    {  :  }
    Invalid Context:
    {  : 1 = 0 }
    Arrays {
        float MemRef_C[*][1536]; // Element size 4
        float MemRef_A[*][1536]; // Element size 4
        float MemRef_B[*][1536]; // Element size 4
    }
    Arrays (Bounds as pw_affs) {
        float MemRef_C[*][ { [] -> [(1536)] } ]; // Element size 4
        float MemRef_A[*][ { [] -> [(1536)] } ]; // Element size 4
        float MemRef_B[*][ { [] -> [(1536)] } ]; // Element size 4
    }
    Alias Groups (0):
        n/a
    Statements {
        Stmt_for_body3
            Domain :=
                { Stmt_for_body3[i0, i1] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535 };
            Schedule :=
                { Stmt_for_body3[i0, i1] -> [i0, i1, 0, 0] };
            MustWriteAccess :=  [Reduction Type: NONE] [Scalar: 0]
                { Stmt_for_body3[i0, i1] -> MemRef_C[i0, i1] };
        Stmt_for_body8
            Domain :=
                { Stmt_for_body8[i0, i1, i2] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535 and 0 <= i2 <= 1535 };
            Schedule :=
                { Stmt_for_body8[i0, i1, i2] -> [i0, i1, 1, i2] };
            ReadAccess :=       [Reduction Type: NONE] [Scalar: 0]
                { Stmt_for_body8[i0, i1, i2] -> MemRef_C[i0, i1] };
            ReadAccess :=       [Reduction Type: NONE] [Scalar: 0]
                { Stmt_for_body8[i0, i1, i2] -> MemRef_A[i0, i2] };
            ReadAccess :=       [Reduction Type: NONE] [Scalar: 0]
                { Stmt_for_body8[i0, i1, i2] -> MemRef_B[i2, i1] };
            MustWriteAccess :=  [Reduction Type: NONE] [Scalar: 0]
                { Stmt_for_body8[i0, i1, i2] -> MemRef_C[i0, i1] };
    }

6. Show the dependences for the SCoPs

$ opt -basic-aa -polly-use-llvm-names -polly-dependences -analyze matmul.preopt.ll -polly-process-unprofitable

[...]Printing analysis 'Polly - Calculate dependences' for region: 'for.cond1.preheader => for.end19' in function 'init_array':
        RAW dependences:
                {  }
        WAR dependences:
                {  }
        WAW dependences:
                {  }
        Reduction dependences:
                n/a
        Transitive closure of reduction dependences:
                {  }
[...]Printing analysis 'Polly - Calculate dependences' for region: 'for.cond1.preheader => for.end30' in function 'main':
        RAW dependences:
                { Stmt_for_body3[i0, i1] -> Stmt_for_body8[i0, i1, 0] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535; Stmt_for_body8[i0, i1, i2] -> Stmt_for_body8[i0, i1, 1 + i2] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535 and 0 <= i2 <= 1534 }
        WAR dependences:
                {  }
        WAW dependences:
                { Stmt_for_body3[i0, i1] -> Stmt_for_body8[i0, i1, 0] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535; Stmt_for_body8[i0, i1, i2] -> Stmt_for_body8[i0, i1, 1 + i2] : 0 <= i0 <= 1535 and 0 <= i1 <= 1535 and 0 <= i2 <= 1534 }
        Reduction dependences:
                n/a
        Transitive closure of reduction dependences:
                {  }

7. Export jscop files

$ opt -basic-aa -polly-use-llvm-names -polly-export-jscop matmul.preopt.ll -polly-process-unprofitable

[...]Writing JScop '%for.cond1.preheader---%for.end19' in function 'init_array' to './init_array___%for.cond1.preheader---%for.end19.jscop'.

Writing JScop '%for.cond1.preheader---%for.end30' in function 'main' to './main___%for.cond1.preheader---%for.end30.jscop'.

8. Import the changed jscop files and print the updated SCoP structure (optional)

Polly can reimport jscop files, in which the schedules of the statements are changed. These changed schedules are used to describe transformations. It is possible to import different jscop files by providing the postfix of the jscop file that is imported.

We apply three different transformations on the SCoP in the main function. The jscop files describing these transformations are hand written (and available in docs/experiments/matmul).

No Polly

As a baseline we do not call any Polly code generation, but only apply the normal -O3 optimizations.

$ opt -basic-aa -polly-use-llvm-names matmul.preopt.ll -polly-import-jscop -polly-ast -analyze -polly-process-unprofitable

[...]
:: isl ast :: main :: %for.cond1.preheader---%for.end30

if (1)

    for (int c0 = 0; c0 <= 1535; c0 += 1)
      for (int c1 = 0; c1 <= 1535; c1 += 1) {
        Stmt_for_body3(c0, c1);
        for (int c3 = 0; c3 <= 1535; c3 += 1)
          Stmt_for_body8(c0, c1, c3);
      }

else
    {  /* original code */ }
[...]

Loop Interchange (and Fission to allow the interchange)

We split the loops and can now apply an interchange of the loop dimensions that enumerate Stmt_for_body8.

$ opt -basic-aa -polly-use-llvm-names matmul.preopt.ll -polly-import-jscop -polly-import-jscop-postfix=interchanged -polly-ast -analyze -polly-process-unprofitable

[...]
:: isl ast :: main :: %for.cond1.preheader---%for.end30

if (1)

    {
      for (int c1 = 0; c1 <= 1535; c1 += 1)
        for (int c2 = 0; c2 <= 1535; c2 += 1)
          Stmt_for_body3(c1, c2);
      for (int c1 = 0; c1 <= 1535; c1 += 1)
        for (int c2 = 0; c2 <= 1535; c2 += 1)
          for (int c3 = 0; c3 <= 1535; c3 += 1)
            Stmt_for_body8(c1, c3, c2);
    }

else
    {  /* original code */ }
[...]

Interchange + Tiling

In addition to the interchange we now tile the second loop nest.

$ opt -basic-aa -polly-use-llvm-names matmul.preopt.ll -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled -polly-ast -analyze -polly-process-unprofitable

[...]
:: isl ast :: main :: %for.cond1.preheader---%for.end30

if (1)

    {
      for (int c1 = 0; c1 <= 1535; c1 += 1)
        for (int c2 = 0; c2 <= 1535; c2 += 1)
          Stmt_for_body3(c1, c2);
      for (int c1 = 0; c1 <= 1535; c1 += 64)
        for (int c2 = 0; c2 <= 1535; c2 += 64)
          for (int c3 = 0; c3 <= 1535; c3 += 64)
            for (int c4 = c1; c4 <= c1 + 63; c4 += 1)
              for (int c5 = c3; c5 <= c3 + 63; c5 += 1)
                for (int c6 = c2; c6 <= c2 + 63; c6 += 1)
                  Stmt_for_body8(c4, c6, c5);
    }

else
    {  /* original code */ }
[...]

Interchange + Tiling + Strip-mining to prepare vectorization

To later allow vectorization we create a so called trivially parallelizable loop. It is innermost, parallel and has only four iterations. It can be replaced by 4-element SIMD instructions.

$ opt -basic-aa -polly-use-llvm-names matmul.preopt.ll -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled -polly-ast -analyze -polly-process-unprofitable

[...]
:: isl ast :: main :: %for.cond1.preheader---%for.end30

if (1)

    {
      for (int c1 = 0; c1 <= 1535; c1 += 1)
        for (int c2 = 0; c2 <= 1535; c2 += 1)
          Stmt_for_body3(c1, c2);
      for (int c1 = 0; c1 <= 1535; c1 += 64)
        for (int c2 = 0; c2 <= 1535; c2 += 64)
          for (int c3 = 0; c3 <= 1535; c3 += 64)
            for (int c4 = c1; c4 <= c1 + 63; c4 += 1)
              for (int c5 = c3; c5 <= c3 + 63; c5 += 1)
                for (int c6 = c2; c6 <= c2 + 63; c6 += 4)
                  for (int c7 = c6; c7 <= c6 + 3; c7 += 1)
                    Stmt_for_body8(c4, c7, c5);
    }

else
    {  /* original code */ }
[...]

9. Codegenerate the SCoPs

This generates new code for the SCoPs detected by polly. If -polly-import-jscop is present, transformations specified in the imported jscop files will be applied.

$ opt -S matmul.preopt.ll | opt -S -O3 -o matmul.normalopt.ll

$ opt -S matmul.preopt.ll -basic-aa -polly-use-llvm-names -polly-import-jscop -polly-import-jscop-postfix=interchanged -polly-codegen -polly-process-unprofitable | opt -S -O3 -o matmul.polly.interchanged.ll

Reading JScop '%for.cond1.preheader---%for.end19' in function 'init_array' from './init_array___%for.cond1.preheader---%for.end19.jscop.interchanged'.
File could not be read: No such file or directory
Reading JScop '%for.cond1.preheader---%for.end30' in function 'main' from './main___%for.cond1.preheader---%for.end30.jscop.interchanged'.

$ opt -S matmul.preopt.ll -basic-aa -polly-use-llvm-names -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled -polly-codegen -polly-process-unprofitable | opt -S -O3 -o matmul.polly.interchanged+tiled.ll

Reading JScop '%for.cond1.preheader---%for.end19' in function 'init_array' from './init_array___%for.cond1.preheader---%for.end19.jscop.interchanged+tiled'.
File could not be read: No such file or directory
Reading JScop '%for.cond1.preheader---%for.end30' in function 'main' from './main___%for.cond1.preheader---%for.end30.jscop.interchanged+tiled'.

$ opt -S matmul.preopt.ll -basic-aa -polly-use-llvm-names -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector -polly-codegen -polly-vectorizer=polly -polly-process-unprofitable | opt -S -O3 -o matmul.polly.interchanged+tiled+vector.ll

Reading JScop '%for.cond1.preheader---%for.end19' in function 'init_array' from './init_array___%for.cond1.preheader---%for.end19.jscop.interchanged+tiled+vector'.
File could not be read: No such file or directory
Reading JScop '%for.cond1.preheader---%for.end30' in function 'main' from './main___%for.cond1.preheader---%for.end30.jscop.interchanged+tiled+vector'.

$ opt -S matmul.preopt.ll -basic-aa -polly-use-llvm-names -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector -polly-codegen -polly-vectorizer=polly -polly-parallel -polly-process-unprofitable | opt -S -O3 -o matmul.polly.interchanged+tiled+openmp.ll

Reading JScop '%for.cond1.preheader---%for.end19' in function 'init_array' from './init_array___%for.cond1.preheader---%for.end19.jscop.interchanged+tiled+vector'.
File could not be read: No such file or directory
Reading JScop '%for.cond1.preheader---%for.end30' in function 'main' from './main___%for.cond1.preheader---%for.end30.jscop.interchanged+tiled+vector'.

10. Create the executables

$ llc matmul.normalopt.ll -o matmul.normalopt.s -relocation-model=pic
$ gcc matmul.normalopt.s -o matmul.normalopt.exe
$ llc matmul.polly.interchanged.ll -o matmul.polly.interchanged.s -relocation-model=pic
$ gcc matmul.polly.interchanged.s -o matmul.polly.interchanged.exe
$ llc matmul.polly.interchanged+tiled.ll -o matmul.polly.interchanged+tiled.s -relocation-model=pic
$ gcc matmul.polly.interchanged+tiled.s -o matmul.polly.interchanged+tiled.exe
$ llc matmul.polly.interchanged+tiled+vector.ll -o matmul.polly.interchanged+tiled+vector.s -relocation-model=pic
$ gcc matmul.polly.interchanged+tiled+vector.s -o matmul.polly.interchanged+tiled+vector.exe
$ llc matmul.polly.interchanged+tiled+vector+openmp.ll -o matmul.polly.interchanged+tiled+vector+openmp.s -relocation-model=pic
$ gcc matmul.polly.interchanged+tiled+vector+openmp.s -lgomp -o matmul.polly.interchanged+tiled+vector+openmp.exe

11. Compare the runtime of the executables

By comparing the runtimes of the different code snippets we see that a simple loop interchange gives here the largest performance boost. However in this case, adding vectorization and using OpenMP degrades the performance.

$ time ./matmul.normalopt.exe

real    0m11.295s
user    0m11.288s
sys     0m0.004s
$ time ./matmul.polly.interchanged.exe

real    0m0.988s
user    0m0.980s
sys     0m0.008s
$ time ./matmul.polly.interchanged+tiled.exe

real    0m0.830s
user    0m0.816s
sys     0m0.012s
$ time ./matmul.polly.interchanged+tiled+vector.exe

real    0m5.430s
user    0m5.424s
sys     0m0.004s
$ time ./matmul.polly.interchanged+tiled+vector+openmp.exe

real    0m3.184s
user    0m11.972s
sys     0m0.036s