polly/www/example_manual_matmul.html - llvm-project - Git at Google

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 <!--=====================================================================-->
 <h1>Polly: Execute the individual Polly passes manually</h1>
 <!--=====================================================================-->

 <p>
 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.</p>

 The files used and created in this example are available in the Polly checkout
 in the folder <em>www/experiments/matmul</em>. They can be created automatically
 by running the <em>www/experiments/matmul/runall.sh</em> script.

 <ol>
 <li><h4>Create LLVM-IR from the C code</h4>

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

 <pre class="code">clang -S -emit-llvm matmul.c -o matmul.s</pre>
 </li>


 <li><h4>Load Polly automatically when calling the 'opt' tool</h4>

 Polly is not built into opt or bugpoint, but it is a shared library that needs
 to be loaded into these tools explicitally. The Polly library is called
 LVMPolly.so. For a cmake build it is available in the build/lib/ directory,
 autoconf creates the same file in
 build/tools/polly/{Release+Asserts|Asserts|Debug}/lib. For convenience we create
 an alias that automatically loads Polly if 'opt' is called.
 <pre class="code">
 export PATH_TO_POLLY_LIB="~/polly/build/lib/"
 alias opt="opt -load ${PATH_TO_POLLY_LIB}/LLVMPolly.so"</pre>
 </li>

 <li><h4>Prepare the LLVM-IR for Polly</h4>

 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 canonicalication passes. For this
 example only three passes are necessary. To get good coverage on more
 complicated input files often more canonicalization passes are needed. pollycc
 contains a list of passes that have shown to be beneficial.
 <pre class="code">opt -S -mem2reg -loop-simplify -polly-indvars matmul.s &gt; matmul.preopt.ll</pre></li>

 <li><h4>Show the SCoPs detected by Polly (optional)</h4>

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

 <pre class="code">opt -basicaa -polly-cloog -analyze -q matmul.preopt.ll</pre>

 <pre>
 init_array():
 for (c2=0;c2&lt;=1023;c2++) {
   for (c4=0;c4&lt;=1023;c4++) {
     Stmt_5(c2,c4);
   }
 }

 main():
 for (c2=0;c2&lt;=1023;c2++) {
   for (c4=0;c4&lt;=1023;c4++) {
     Stmt_4(c2,c4);
     for (c6=0;c6&lt;=1023;c6++) {
       Stmt_6(c2,c4,c6);
     }
   }
 }
 </pre>
 </li>
 <li><h4>Highlight the detected SCoPs in the CFGs of the program (requires graphviz/dotty)</h4>

 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.

 <pre class="code">opt -basicaa -view-scops -disable-output matmul.preopt.ll
 opt -basicaa -view-scops-only -disable-output matmul.preopt.ll</pre>
 The output for the different functions<br />
 view-scops:
 <a href="experiments/matmul/scops.main.dot.png">main</a>,
 <a href="experiments/matmul/scops.init_array.dot.png">init_array</a>,
 <a href="experiments/matmul/scops.print_array.dot.png">print_array</a><br />
 view-scops-only:
 <a href="experiments/matmul/scopsonly.main.dot.png">main</a>,
 <a href="experiments/matmul/scopsonly.init_array.dot.png">init_array</a>,
 <a href="experiments/matmul/scopsonly.print_array.dot.png">print_array</a>
 </li>

 <li><h4>View the polyhedral representation of the SCoPs</h4>
 <pre class="code">opt -basicaa -polly-scops -analyze matmul.preopt.ll</pre>
 <pre>
 [...]
 Printing analysis 'Polly - Create polyhedral description of Scops' for region:
 '%1 =&gt;&nbsp;%17' in function 'init_array':
    Context:
    { [] }
    Statements {
    	Stmt_5
            Domain&nbsp;:=
                { Stmt_5[i0, i1]&nbsp;: i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 };
            Scattering&nbsp;:=
                { Stmt_5[i0, i1] -&gt; scattering[0, i0, 0, i1, 0] };
            WriteAccess&nbsp;:=
                { Stmt_5[i0, i1] -&gt; MemRef_A[1037i0 + i1] };
            WriteAccess&nbsp;:=
                { Stmt_5[i0, i1] -&gt; MemRef_B[1047i0 + i1] };
    	FinalRead
            Domain&nbsp;:=
                { FinalRead[0] };
            Scattering&nbsp;:=
                { FinalRead[i0] -&gt; scattering[200000000, o1, o2, o3, o4] };
            ReadAccess&nbsp;:=
                { FinalRead[i0] -&gt; MemRef_A[o0] };
            ReadAccess&nbsp;:=
                { FinalRead[i0] -&gt; MemRef_B[o0] };
    }
 [...]
 Printing analysis 'Polly - Create polyhedral description of Scops' for region:
 '%1 =&gt;&nbsp;%17' in function 'main':
    Context:
    { [] }
    Statements {
    	Stmt_4
            Domain&nbsp;:=
                { Stmt_4[i0, i1]&nbsp;: i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 };
            Scattering&nbsp;:=
                { Stmt_4[i0, i1] -&gt; scattering[0, i0, 0, i1, 0, 0, 0] };
            WriteAccess&nbsp;:=
                { Stmt_4[i0, i1] -&gt; MemRef_C[1067i0 + i1] };
    	Stmt_6
            Domain&nbsp;:=
                { Stmt_6[i0, i1, i2]&nbsp;: i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 and i2 &gt;= 0 and i2 &lt;= 1023 };
            Scattering&nbsp;:=
                { Stmt_6[i0, i1, i2] -&gt; scattering[0, i0, 0, i1, 1, i2, 0] };
            ReadAccess&nbsp;:=
                { Stmt_6[i0, i1, i2] -&gt; MemRef_C[1067i0 + i1] };
            ReadAccess&nbsp;:=
                { Stmt_6[i0, i1, i2] -&gt; MemRef_A[1037i0 + i2] };
            ReadAccess&nbsp;:=
                { Stmt_6[i0, i1, i2] -&gt; MemRef_B[i1 + 1047i2] };
            WriteAccess&nbsp;:=
                { Stmt_6[i0, i1, i2] -&gt; MemRef_C[1067i0 + i1] };
    	FinalRead
            Domain&nbsp;:=
                { FinalRead[0] };
            Scattering&nbsp;:=
                { FinalRead[i0] -&gt; scattering[200000000, o1, o2, o3, o4, o5, o6] };
            ReadAccess&nbsp;:=
                { FinalRead[i0] -&gt; MemRef_C[o0] };
            ReadAccess&nbsp;:=
                { FinalRead[i0] -&gt; MemRef_A[o0] };
            ReadAccess&nbsp;:=
                { FinalRead[i0] -&gt; MemRef_B[o0] };
    }
 [...]
 </pre>
 </li>

 <li><h4>Show the dependences for the SCoPs</h4>
 <pre class="code">opt -basicaa -polly-dependences -analyze matmul.preopt.ll</pre>
 <pre>Printing analysis 'Polly - Calculate dependences for SCoP' for region:
 'for.cond =&gt; for.end28' in function 'init_array':
    Must dependences:
        {  }
    May dependences:
        {  }
    Must no source:
        {  }
    May no source:
        {  }
 Printing analysis 'Polly - Calculate dependences for SCoP' for region:
 'for.cond =&gt; for.end48' in function 'main':
    Must dependences:
        {  Stmt_4[i0, i1] -&gt; Stmt_6[i0, i1, 0]&nbsp;:
               i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023;
           Stmt_6[i0, i1, i2] -&gt; Stmt_6[i0, i1, 1 + i2]&nbsp;:
               i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 and i2 &gt;= 0 and i2 &lt;= 1022;
           Stmt_6[i0, i1, 1023] -&gt; FinalRead[0]&nbsp;:
               i1 &lt;= 1091540 - 1067i0 and i1 &gt;= -1067i0 and i1 &gt;= 0 and i1 &lt;= 1023;
           Stmt_6[1023, i1, 1023] -&gt; FinalRead[0]&nbsp;:
               i1 &gt;= 0 and i1 &lt;= 1023
        }
    May dependences:
        {  }
    Must no source:
        {  Stmt_6[i0, i1, i2] -&gt; MemRef_A[1037i0 + i2]&nbsp;:
               i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 and i2 &gt;= 0 and i2 &lt;= 1023;
           Stmt_6[i0, i1, i2] -&gt; MemRef_B[i1 + 1047i2]&nbsp;:
               i0 &gt;= 0 and i0 &lt;= 1023 and i1 &gt;= 0 and i1 &lt;= 1023 and i2 &gt;= 0 and i2 &lt;= 1023;
           FinalRead[0] -&gt; MemRef_A[o0];
           FinalRead[0] -&gt; MemRef_B[o0]
           FinalRead[0] -&gt; MemRef_C[o0]&nbsp;:
               o0 &gt;= 1092565 or (exists (e0 = [(o0)/1067]: o0 &lt;= 1091540 and o0 &gt;= 0
               and 1067e0 &lt;= -1024 + o0 and 1067e0 &gt;= -1066 + o0)) or o0 &lt;= -1;
        }
    May no source:
        {  }
 </pre></li>

 <li><h4>Export jscop files</h4>

 Polly can export the polyhedral representation in so called jscop files. Jscop
 files contain the polyhedral representation stored in a JSON file.
 <pre class="code">opt -basicaa -polly-export-jscop matmul.preopt.ll</pre>
 <pre>Writing SCoP 'for.cond =&gt; for.end28' in function 'init_array' to './init_array___%for.cond---%for.end28.jscop'.
 Writing SCoP 'for.cond =&gt; for.end48' in function 'main' to './main___%for.cond---%for.end48.jscop'.
 </pre></li>

 <li><h4>Import the changed jscop files and print the updated SCoP structure
 (optional)</h4>
 <p>Polly can reimport jscop files, in which the schedules of the statements are
 changed. These changed schedules are used to descripe transformations.
 It is possible to import different jscop files by providing the postfix
 of the jscop file that is imported.</p>
 <p> We apply three different transformations on the SCoP in the main function.
 The jscop files describing these transformations are hand written (and available
 in <em>www/experiments/matmul</em>).

 <h5>No Polly</h5>

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

 <pre class="code">
 opt matmul.preopt.ll -basicaa \
     -polly-import-jscop \
     -polly-cloog -analyze
 </pre>
 <pre>
 [...]
 main():
 for (c2=0;c2&ltg;=1535;c2++) {
   for (c4=0;c4&ltg;=1535;c4++) {
     Stmt_4(c2,c4);
     for (c6=0;c6&ltg;=1535;c6++) {
       Stmt_6(c2,c4,c6);
     }
   }
 }
 [...]
 </pre>
 <h5>Interchange (and Fission to allow the interchange)</h5>
 <p>We split the loops and can now apply an interchange of the loop dimensions that
 enumerate Stmt_6.</p>
 <pre class="code">
 opt matmul.preopt.ll -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged \
     -polly-cloog -analyze
 </pre>
 <pre>
 [...]
 Reading JScop '%1 =&gt; %17' in function 'main' from './main___%1---%17.jscop.interchanged'.
 [...]
 main():
 for (c2=0;c2&lt;=1535;c2++) {
   for (c4=0;c4&lt;=1535;c4++) {
     Stmt_4(c2,c4);
   }
 }
 for (c2=0;c2&lt;=1535;c2++) {
   for (c4=0;c4&lt;=1535;c4++) {
     for (c6=0;c6&lt;=1535;c6++) {
       Stmt_6(c2,c6,c4);
     }
   }
 }
 [...]
 </pre>
 <h5>Interchange + Tiling</h5>
 <p>In addition to the interchange we tile now the second loop nest.</p>

 <pre class="code">
 opt matmul.preopt.ll -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled \
     -polly-cloog -analyze
 </pre>
 <pre>
 [...]
 Reading JScop '%1 =&gt; %17' in function 'main' from './main___%1---%17.jscop.interchanged+tiled'.
 [...]
 main():
 for (c2=0;c2&lt;=1535;c2++) {
   for (c4=0;c4&lt;=1535;c4++) {
     Stmt_4(c2,c4);
   }
 }
 for (c2=0;c2&lt;=1535;c2+=64) {
   for (c3=0;c3&lt;=1535;c3+=64) {
     for (c4=0;c4&lt;=1535;c4+=64) {
       for (c5=c2;c5&lt;=c2+63;c5++) {
         for (c6=c4;c6&lt;=c4+63;c6++) {
           for (c7=c3;c7&lt;=c3+63;c7++) {
             Stmt_6(c5,c7,c6);
           }
         }
       }
     }
   }
 }
 [...]
 </pre>
 <h5>Interchange + Tiling + Strip-mining to prepare vectorization</h5>
 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.
 <pre class="code">
 opt matmul.preopt.ll -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
     -polly-cloog -analyze </pre>

 <pre>
 [...]
 Reading JScop '%1 =&gt; %17' in function 'main' from './main___%1---%17.jscop.interchanged+tiled+vector'.
 [...]
 main():
 for (c2=0;c2&lt;=1535;c2++) {
   for (c4=0;c4&lt;=1535;c4++) {
     Stmt_4(c2,c4);
   }
 }
 for (c2=0;c2&lt;=1535;c2+=64) {
   for (c3=0;c3&lt;=1535;c3+=64) {
     for (c4=0;c4&lt;=1535;c4+=64) {
       for (c5=c2;c5&lt;=c2+63;c5++) {
         for (c6=c4;c6&lt;=c4+63;c6++) {
           for (c7=c3;c7&lt;=c3+63;c7+=4) {
             for (c8=c7;c8&lt;=c7+3;c8++) {
               Stmt_6(c5,c8,c6);
             }
           }
         }
       }
     }
   }
 }
 [...]
 </pre>

 </li>

 <li><h4>Codegenerate the SCoPs</h4>
 <p>
 This generates new code for the SCoPs detected by polly.
 If -polly-import is present, transformations specified in the imported openscop
 files will be applied.</p>
 <pre class="code">opt matmul.preopt.ll | opt -O3 &gt; matmul.normalopt.ll</pre>
 <pre class="code">
 opt -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged \
     -polly-codegen matmul.preopt.ll \
    | opt -O3 &gt; matmul.polly.interchanged.ll</pre>
 <pre>
 Reading JScop '%1 =&gt; %19' in function 'init_array' from
     './init_array___%1---%19.jscop.interchanged'.
 File could not be read: No such file or directory
 Reading JScop '%1 =&gt; %17' in function 'main' from
     './main___%1---%17.jscop.interchanged'.
 </pre>
 <pre class="code">
 opt -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled \
     -polly-codegen matmul.preopt.ll \
    | opt -O3 &gt; matmul.polly.interchanged+tiled.ll</pre>
 <pre>
 Reading JScop '%1 =&gt; %19' in function 'init_array' from
     './init_array___%1---%19.jscop.interchanged+tiled'.
 File could not be read: No such file or directory
 Reading JScop '%1 =&gt; %17' in function 'main' from
     './main___%1---%17.jscop.interchanged+tiled'.
 </pre>
 <pre class="code">
 opt -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
     -polly-codegen -enable-polly-vector matmul.preopt.ll \
    | opt -O3 &gt; matmul.polly.interchanged+tiled+vector.ll</pre>
 <pre>
 Reading JScop '%1 =&gt; %19' in function 'init_array' from
     './init_array___%1---%19.jscop.interchanged+tiled+vector'.
 File could not be read: No such file or directory
 Reading JScop '%1 =&gt; %17' in function 'main' from
     './main___%1---%17.jscop.interchanged+tiled+vector'.
 </pre>
 <pre class="code">
 opt -basicaa \
     -polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
     -polly-codegen -enable-polly-vector -enable-polly-openmp matmul.preopt.ll \
   | opt -O3 &gt; matmul.polly.interchanged+tiled+openmp.ll</pre>
 <pre>
 Reading JScop '%1 =&gt; %19' in function 'init_array' from
     './init_array___%1---%19.jscop.interchanged+tiled+vector'.
 File could not be read: No such file or directory
 Reading JScop '%1 =&gt; %17' in function 'main' from
     './main___%1---%17.jscop.interchanged+tiled+vector'.
 </pre>

 <li><h4>Create the executables</h4>

 Create one executable optimized with plain -O3 as well as a set of executables
 optimized in different ways with Polly. One changes only the loop structure, the
 other adds tiling, the next adds vectorization and finally we use OpenMP
 parallelism.
 <pre class="code">
 llc matmul.normalopt.ll -o matmul.normalopt.s &amp;&amp; \
     gcc matmul.normalopt.s -o matmul.normalopt.exe
 llc matmul.polly.interchanged.ll -o matmul.polly.interchanged.s &amp;&amp; \
     gcc matmul.polly.interchanged.s -o matmul.polly.interchanged.exe
 llc matmul.polly.interchanged+tiled.ll -o matmul.polly.interchanged+tiled.s &amp;&amp; \
     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 &amp;&amp; \
     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 &amp;&amp; \
     gcc -lgomp matmul.polly.interchanged+tiled+vector+openmp.s -o matmul.polly.interchanged+tiled+vector+openmp.exe </pre>

 <li><h4>Compare the runtime of the executables</h4>

 By comparing the runtimes of the different code snippets we see that a simple
 loop interchange gives here the largest performance boost. However by adding
 vectorization and by using OpenMP we can further improve the performance
 significantly.
 <pre class="code">time ./matmul.normalopt.exe</pre>
 <pre>42.68 real, 42.55 user, 0.00 sys</pre>
 <pre class="code">time ./matmul.polly.interchanged.exe</pre>
 <pre>04.33 real, 4.30 user, 0.01 sys</pre>
 <pre class="code">time ./matmul.polly.interchanged+tiled.exe</pre>
 <pre>04.11 real, 4.10 user, 0.00 sys</pre>
 <pre class="code">time ./matmul.polly.interchanged+tiled+vector.exe</pre>
 <pre>01.39 real, 1.36 user, 0.01 sys</pre>
 <pre class="code">time ./matmul.polly.interchanged+tiled+vector+openmp.exe</pre>
 <pre>00.66 real, 2.58 user, 0.02 sys</pre>
 </li>
 </ol>

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	<!--=====================================================================-->
	<h1>Polly: Execute the individual Polly passes manually</h1>
	<!--=====================================================================-->

	<p>
	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.</p>

	The files used and created in this example are available in the Polly checkout
	in the folder <em>www/experiments/matmul</em>. They can be created automatically
	by running the <em>www/experiments/matmul/runall.sh</em> script.

	<ol>
	<li><h4>Create LLVM-IR from the C code</h4>

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

	<pre class="code">clang -S -emit-llvm matmul.c -o matmul.s</pre>
	</li>


	<li><h4>Load Polly automatically when calling the 'opt' tool</h4>

	Polly is not built into opt or bugpoint, but it is a shared library that needs
	to be loaded into these tools explicitally. The Polly library is called
	LVMPolly.so. For a cmake build it is available in the build/lib/ directory,
	autoconf creates the same file in
	build/tools/polly/{Release+Asserts\|Asserts\|Debug}/lib. For convenience we create
	an alias that automatically loads Polly if 'opt' is called.
	<pre class="code">
	export PATH_TO_POLLY_LIB="~/polly/build/lib/"
	alias opt="opt -load ${PATH_TO_POLLY_LIB}/LLVMPolly.so"</pre>
	</li>

	<li><h4>Prepare the LLVM-IR for Polly</h4>

	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 canonicalication passes. For this
	example only three passes are necessary. To get good coverage on more
	complicated input files often more canonicalization passes are needed. pollycc
	contains a list of passes that have shown to be beneficial.
	<pre class="code">opt -S -mem2reg -loop-simplify -polly-indvars matmul.s > matmul.preopt.ll</pre></li>

	<li><h4>Show the SCoPs detected by Polly (optional)</h4>

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

	<pre class="code">opt -basicaa -polly-cloog -analyze -q matmul.preopt.ll</pre>

	<pre>
	init_array():
	for (c2=0;c2<=1023;c2++) {
	for (c4=0;c4<=1023;c4++) {
	Stmt_5(c2,c4);
	}
	}

	main():
	for (c2=0;c2<=1023;c2++) {
	for (c4=0;c4<=1023;c4++) {
	Stmt_4(c2,c4);
	for (c6=0;c6<=1023;c6++) {
	Stmt_6(c2,c4,c6);
	}
	}
	}
	</pre>
	</li>
	<li><h4>Highlight the detected SCoPs in the CFGs of the program (requires graphviz/dotty)</h4>

	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.

	<pre class="code">opt -basicaa -view-scops -disable-output matmul.preopt.ll
	opt -basicaa -view-scops-only -disable-output matmul.preopt.ll</pre>
	The output for the different functions<br />
	view-scops:
	<a href="experiments/matmul/scops.main.dot.png">main</a>,
	<a href="experiments/matmul/scops.init_array.dot.png">init_array</a>,
	<a href="experiments/matmul/scops.print_array.dot.png">print_array</a><br />
	view-scops-only:
	<a href="experiments/matmul/scopsonly.main.dot.png">main</a>,
	<a href="experiments/matmul/scopsonly.init_array.dot.png">init_array</a>,
	<a href="experiments/matmul/scopsonly.print_array.dot.png">print_array</a>
	</li>

	<li><h4>View the polyhedral representation of the SCoPs</h4>
	<pre class="code">opt -basicaa -polly-scops -analyze matmul.preopt.ll</pre>
	<pre>
	[...]
	Printing analysis 'Polly - Create polyhedral description of Scops' for region:
	'%1 => %17' in function 'init_array':
	Context:
	{ [] }
	Statements {
	Stmt_5
	Domain :=
	{ Stmt_5[i0, i1] : i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 };
	Scattering :=
	{ Stmt_5[i0, i1] -> scattering[0, i0, 0, i1, 0] };
	WriteAccess :=
	{ Stmt_5[i0, i1] -> MemRef_A[1037i0 + i1] };
	WriteAccess :=
	{ Stmt_5[i0, i1] -> MemRef_B[1047i0 + i1] };
	FinalRead
	Domain :=
	{ FinalRead[0] };
	Scattering :=
	{ FinalRead[i0] -> scattering[200000000, o1, o2, o3, o4] };
	ReadAccess :=
	{ FinalRead[i0] -> MemRef_A[o0] };
	ReadAccess :=
	{ FinalRead[i0] -> MemRef_B[o0] };
	}
	[...]
	Printing analysis 'Polly - Create polyhedral description of Scops' for region:
	'%1 => %17' in function 'main':
	Context:
	{ [] }
	Statements {
	Stmt_4
	Domain :=
	{ Stmt_4[i0, i1] : i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 };
	Scattering :=
	{ Stmt_4[i0, i1] -> scattering[0, i0, 0, i1, 0, 0, 0] };
	WriteAccess :=
	{ Stmt_4[i0, i1] -> MemRef_C[1067i0 + i1] };
	Stmt_6
	Domain :=
	{ Stmt_6[i0, i1, i2] : i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 and i2 >= 0 and i2 <= 1023 };
	Scattering :=
	{ Stmt_6[i0, i1, i2] -> scattering[0, i0, 0, i1, 1, i2, 0] };
	ReadAccess :=
	{ Stmt_6[i0, i1, i2] -> MemRef_C[1067i0 + i1] };
	ReadAccess :=
	{ Stmt_6[i0, i1, i2] -> MemRef_A[1037i0 + i2] };
	ReadAccess :=
	{ Stmt_6[i0, i1, i2] -> MemRef_B[i1 + 1047i2] };
	WriteAccess :=
	{ Stmt_6[i0, i1, i2] -> MemRef_C[1067i0 + i1] };
	FinalRead
	Domain :=
	{ FinalRead[0] };
	Scattering :=
	{ FinalRead[i0] -> scattering[200000000, o1, o2, o3, o4, o5, o6] };
	ReadAccess :=
	{ FinalRead[i0] -> MemRef_C[o0] };
	ReadAccess :=
	{ FinalRead[i0] -> MemRef_A[o0] };
	ReadAccess :=
	{ FinalRead[i0] -> MemRef_B[o0] };
	}
	[...]
	</pre>
	</li>

	<li><h4>Show the dependences for the SCoPs</h4>
	<pre class="code">opt -basicaa -polly-dependences -analyze matmul.preopt.ll</pre>
	<pre>Printing analysis 'Polly - Calculate dependences for SCoP' for region:
	'for.cond => for.end28' in function 'init_array':
	Must dependences:
	{ }
	May dependences:
	{ }
	Must no source:
	{ }
	May no source:
	{ }
	Printing analysis 'Polly - Calculate dependences for SCoP' for region:
	'for.cond => for.end48' in function 'main':
	Must dependences:
	{ Stmt_4[i0, i1] -> Stmt_6[i0, i1, 0] :
	i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023;
	Stmt_6[i0, i1, i2] -> Stmt_6[i0, i1, 1 + i2] :
	i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 and i2 >= 0 and i2 <= 1022;
	Stmt_6[i0, i1, 1023] -> FinalRead[0] :
	i1 <= 1091540 - 1067i0 and i1 >= -1067i0 and i1 >= 0 and i1 <= 1023;
	Stmt_6[1023, i1, 1023] -> FinalRead[0] :
	i1 >= 0 and i1 <= 1023
	}
	May dependences:
	{ }
	Must no source:
	{ Stmt_6[i0, i1, i2] -> MemRef_A[1037i0 + i2] :
	i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 and i2 >= 0 and i2 <= 1023;
	Stmt_6[i0, i1, i2] -> MemRef_B[i1 + 1047i2] :
	i0 >= 0 and i0 <= 1023 and i1 >= 0 and i1 <= 1023 and i2 >= 0 and i2 <= 1023;
	FinalRead[0] -> MemRef_A[o0];
	FinalRead[0] -> MemRef_B[o0]
	FinalRead[0] -> MemRef_C[o0] :
	o0 >= 1092565 or (exists (e0 = [(o0)/1067]: o0 <= 1091540 and o0 >= 0
	and 1067e0 <= -1024 + o0 and 1067e0 >= -1066 + o0)) or o0 <= -1;
	}
	May no source:
	{ }
	</pre></li>

	<li><h4>Export jscop files</h4>

	Polly can export the polyhedral representation in so called jscop files. Jscop
	files contain the polyhedral representation stored in a JSON file.
	<pre class="code">opt -basicaa -polly-export-jscop matmul.preopt.ll</pre>
	<pre>Writing SCoP 'for.cond => for.end28' in function 'init_array' to './init_array___%for.cond---%for.end28.jscop'.
	Writing SCoP 'for.cond => for.end48' in function 'main' to './main___%for.cond---%for.end48.jscop'.
	</pre></li>

	<li><h4>Import the changed jscop files and print the updated SCoP structure
	(optional)</h4>
	<p>Polly can reimport jscop files, in which the schedules of the statements are
	changed. These changed schedules are used to descripe transformations.
	It is possible to import different jscop files by providing the postfix
	of the jscop file that is imported.</p>
	<p> We apply three different transformations on the SCoP in the main function.
	The jscop files describing these transformations are hand written (and available
	in <em>www/experiments/matmul</em>).

	<h5>No Polly</h5>

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

	<pre class="code">
	opt matmul.preopt.ll -basicaa \
	-polly-import-jscop \
	-polly-cloog -analyze
	</pre>
	<pre>
	[...]
	main():
	for (c2=0;c2&ltg;=1535;c2++) {
	for (c4=0;c4&ltg;=1535;c4++) {
	Stmt_4(c2,c4);
	for (c6=0;c6&ltg;=1535;c6++) {
	Stmt_6(c2,c4,c6);
	}
	}
	}
	[...]
	</pre>
	<h5>Interchange (and Fission to allow the interchange)</h5>
	<p>We split the loops and can now apply an interchange of the loop dimensions that
	enumerate Stmt_6.</p>
	<pre class="code">
	opt matmul.preopt.ll -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged \
	-polly-cloog -analyze
	</pre>
	<pre>
	[...]
	Reading JScop '%1 => %17' in function 'main' from './main___%1---%17.jscop.interchanged'.
	[...]
	main():
	for (c2=0;c2<=1535;c2++) {
	for (c4=0;c4<=1535;c4++) {
	Stmt_4(c2,c4);
	}
	}
	for (c2=0;c2<=1535;c2++) {
	for (c4=0;c4<=1535;c4++) {
	for (c6=0;c6<=1535;c6++) {
	Stmt_6(c2,c6,c4);
	}
	}
	}
	[...]
	</pre>
	<h5>Interchange + Tiling</h5>
	<p>In addition to the interchange we tile now the second loop nest.</p>

	<pre class="code">
	opt matmul.preopt.ll -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled \
	-polly-cloog -analyze
	</pre>
	<pre>
	[...]
	Reading JScop '%1 => %17' in function 'main' from './main___%1---%17.jscop.interchanged+tiled'.
	[...]
	main():
	for (c2=0;c2<=1535;c2++) {
	for (c4=0;c4<=1535;c4++) {
	Stmt_4(c2,c4);
	}
	}
	for (c2=0;c2<=1535;c2+=64) {
	for (c3=0;c3<=1535;c3+=64) {
	for (c4=0;c4<=1535;c4+=64) {
	for (c5=c2;c5<=c2+63;c5++) {
	for (c6=c4;c6<=c4+63;c6++) {
	for (c7=c3;c7<=c3+63;c7++) {
	Stmt_6(c5,c7,c6);
	}
	}
	}
	}
	}
	}
	[...]
	</pre>
	<h5>Interchange + Tiling + Strip-mining to prepare vectorization</h5>
	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.
	<pre class="code">
	opt matmul.preopt.ll -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
	-polly-cloog -analyze </pre>

	<pre>
	[...]
	Reading JScop '%1 => %17' in function 'main' from './main___%1---%17.jscop.interchanged+tiled+vector'.
	[...]
	main():
	for (c2=0;c2<=1535;c2++) {
	for (c4=0;c4<=1535;c4++) {
	Stmt_4(c2,c4);
	}
	}
	for (c2=0;c2<=1535;c2+=64) {
	for (c3=0;c3<=1535;c3+=64) {
	for (c4=0;c4<=1535;c4+=64) {
	for (c5=c2;c5<=c2+63;c5++) {
	for (c6=c4;c6<=c4+63;c6++) {
	for (c7=c3;c7<=c3+63;c7+=4) {
	for (c8=c7;c8<=c7+3;c8++) {
	Stmt_6(c5,c8,c6);
	}
	}
	}
	}
	}
	}
	}
	[...]
	</pre>

	</li>

	<li><h4>Codegenerate the SCoPs</h4>
	<p>
	This generates new code for the SCoPs detected by polly.
	If -polly-import is present, transformations specified in the imported openscop
	files will be applied.</p>
	<pre class="code">opt matmul.preopt.ll \| opt -O3 > matmul.normalopt.ll</pre>
	<pre class="code">
	opt -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged \
	-polly-codegen matmul.preopt.ll \
	\| opt -O3 > matmul.polly.interchanged.ll</pre>
	<pre>
	Reading JScop '%1 => %19' in function 'init_array' from
	'./init_array___%1---%19.jscop.interchanged'.
	File could not be read: No such file or directory
	Reading JScop '%1 => %17' in function 'main' from
	'./main___%1---%17.jscop.interchanged'.
	</pre>
	<pre class="code">
	opt -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled \
	-polly-codegen matmul.preopt.ll \
	\| opt -O3 > matmul.polly.interchanged+tiled.ll</pre>
	<pre>
	Reading JScop '%1 => %19' in function 'init_array' from
	'./init_array___%1---%19.jscop.interchanged+tiled'.
	File could not be read: No such file or directory
	Reading JScop '%1 => %17' in function 'main' from
	'./main___%1---%17.jscop.interchanged+tiled'.
	</pre>
	<pre class="code">
	opt -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
	-polly-codegen -enable-polly-vector matmul.preopt.ll \
	\| opt -O3 > matmul.polly.interchanged+tiled+vector.ll</pre>
	<pre>
	Reading JScop '%1 => %19' in function 'init_array' from
	'./init_array___%1---%19.jscop.interchanged+tiled+vector'.
	File could not be read: No such file or directory
	Reading JScop '%1 => %17' in function 'main' from
	'./main___%1---%17.jscop.interchanged+tiled+vector'.
	</pre>
	<pre class="code">
	opt -basicaa \
	-polly-import-jscop -polly-import-jscop-postfix=interchanged+tiled+vector \
	-polly-codegen -enable-polly-vector -enable-polly-openmp matmul.preopt.ll \
	\| opt -O3 > matmul.polly.interchanged+tiled+openmp.ll</pre>
	<pre>
	Reading JScop '%1 => %19' in function 'init_array' from
	'./init_array___%1---%19.jscop.interchanged+tiled+vector'.
	File could not be read: No such file or directory
	Reading JScop '%1 => %17' in function 'main' from
	'./main___%1---%17.jscop.interchanged+tiled+vector'.
	</pre>

	<li><h4>Create the executables</h4>

	Create one executable optimized with plain -O3 as well as a set of executables
	optimized in different ways with Polly. One changes only the loop structure, the
	other adds tiling, the next adds vectorization and finally we use OpenMP
	parallelism.
	<pre class="code">
	llc matmul.normalopt.ll -o matmul.normalopt.s && \
	gcc matmul.normalopt.s -o matmul.normalopt.exe
	llc matmul.polly.interchanged.ll -o matmul.polly.interchanged.s && \
	gcc matmul.polly.interchanged.s -o matmul.polly.interchanged.exe
	llc matmul.polly.interchanged+tiled.ll -o matmul.polly.interchanged+tiled.s && \
	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 && \
	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 && \
	gcc -lgomp matmul.polly.interchanged+tiled+vector+openmp.s -o matmul.polly.interchanged+tiled+vector+openmp.exe </pre>

	<li><h4>Compare the runtime of the executables</h4>

	By comparing the runtimes of the different code snippets we see that a simple
	loop interchange gives here the largest performance boost. However by adding
	vectorization and by using OpenMP we can further improve the performance
	significantly.
	<pre class="code">time ./matmul.normalopt.exe</pre>
	<pre>42.68 real, 42.55 user, 0.00 sys</pre>
	<pre class="code">time ./matmul.polly.interchanged.exe</pre>
	<pre>04.33 real, 4.30 user, 0.01 sys</pre>
	<pre class="code">time ./matmul.polly.interchanged+tiled.exe</pre>
	<pre>04.11 real, 4.10 user, 0.00 sys</pre>
	<pre class="code">time ./matmul.polly.interchanged+tiled+vector.exe</pre>
	<pre>01.39 real, 1.36 user, 0.01 sys</pre>
	<pre class="code">time ./matmul.polly.interchanged+tiled+vector+openmp.exe</pre>
	<pre>00.66 real, 2.58 user, 0.02 sys</pre>
	</li>
	</ol>

	</div>
	</body>
	</html>