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<p class="caption"><span class="caption-text">Getting Started</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../installation.html">Installation</a></li>
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<li class="toctree-l2 current"><a class="current reference internal" href="#">Fused Softmax</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#motivations">Motivations</a></li>
<li class="toctree-l3"><a class="reference internal" href="#compute-kernel">Compute Kernel</a></li>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-getting-started-tutorials-02-fused-softmax-py"><span class="std std-ref">here</span></a>
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<div class="sphx-glr-example-title section" id="fused-softmax">
<span id="sphx-glr-getting-started-tutorials-02-fused-softmax-py"></span><h1>Fused Softmax<a class="headerlink" href="#fused-softmax" title="Permalink to this headline"></a></h1>
<p>In this tutorial, you will write a fused softmax layer that outperform’s PyTorch implementation and learn about:</p>
<ul class="simple">
<li><p>The benefits of kernel fusion for bandwidth-bound operations.</p></li>
<li><p>The syntax and usage of reduction operators in Triton.</p></li>
<li><p>The automatic vectorization capabilities of the Triton compiler.</p></li>
</ul>
<div class="section" id="motivations">
<h2>Motivations<a class="headerlink" href="#motivations" title="Permalink to this headline"></a></h2>
<p>Custom GPU kernels for elementwise additions are educationally valuable but won’t get you very far in practice.
Let us consider instead the case of a simple (numerically stabilized) softmax operation:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="c1"># Compute the row-wise softmax of x</span>
<span class="k">def</span> <span class="nf">naive_softmax</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="c1"># read MN elements ; write M elements</span>
<span class="n">x_max</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># read 2MN elements ; write MN elements</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">x_max</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="c1"># read MN elements ; write MN elements</span>
<span class="n">numerator</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1"># read MN elements ; write M elements</span>
<span class="n">denominator</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">numerator</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># read 2MN elements ; write MN elements</span>
<span class="n">ret</span> <span class="o">=</span> <span class="n">numerator</span> <span class="o">/</span> <span class="n">denominator</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="c1"># in total: read 7MN elements ; wrote 3MN + 2M elements</span>
<span class="k">return</span> <span class="n">ret</span>
</pre></div>
</div>
<p>When implemented naively in pytorch, computing <code class="code docutils literal notranslate"><span class="pre">y</span> <span class="pre">=</span> <span class="pre">naive_softmax(x)</span></code> for <span class="math notranslate nohighlight">\(x \in R^{M \times N}\)</span> requires reading <span class="math notranslate nohighlight">\(7MN\)</span> elements from DRAM and writing back <span class="math notranslate nohighlight">\(3MN + 2M\)</span> elements.
Instead, we want to write a custom “fused” pytorch operators that only reads X once and does all the necessary computations on-chip.
This would require reading and writing back only <span class="math notranslate nohighlight">\(MN\)</span> bytes, so we could expect a theoretical speed-up of 5x.
In practice, though, we expect less because our kernel will spend some time computing exponentials and moving data around in shared memory.</p>
</div>
<div class="section" id="compute-kernel">
<h2>Compute Kernel<a class="headerlink" href="#compute-kernel" title="Permalink to this headline"></a></h2>
<p>Our softmax kernel works as follows: each program loads a row of X and writes back a normalized row of Y. Note that one important limitation of Triton is that each block must have a power-of-two number of elements, which means that we need to guard the memory operations properly if we want to handle any possible input shapes:</p>
<blockquote>
<div><div class="highlight-C notranslate"><div class="highlight"><pre><span></span><span class="n">__global__</span> <span class="kt">void</span> <span class="n">softmax</span><span class="p">(</span><span class="kt">float</span><span class="o">*</span> <span class="n">Y</span><span class="p">,</span> <span class="kt">float</span><span class="o">*</span> <span class="n">X</span><span class="p">,</span> <span class="kt">int</span> <span class="n">stride_xm</span><span class="p">,</span> <span class="kt">int</span> <span class="n">stride_ym</span><span class="p">,</span> <span class="kt">int</span> <span class="n">M</span><span class="p">,</span> <span class="kt">int</span> <span class="n">N</span><span class="p">){</span>
<span class="c1">// row index</span>
<span class="kt">int</span> <span class="n">m</span> <span class="o">=</span> <span class="n">get_program_id</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
<span class="c1">// column indices</span>
<span class="kt">int</span> <span class="n">n</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span> <span class="p">...</span> <span class="n">BLOCK</span><span class="p">;</span>
<span class="c1">// the memory address of all the elements</span>
<span class="c1">// that we want to load can be computed as follows</span>
<span class="kt">float</span><span class="o">*</span> <span class="n">px</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">X</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">stride_xm</span> <span class="o">+</span> <span class="n">n</span><span class="p">;</span>
<span class="c1">// because BLOCK has to be a power of two</span>
<span class="c1">// (per Triton-C specs), it is important</span>
<span class="c1">// to guard each memory operation with predicates</span>
<span class="c1">// or we will read out of bounds</span>
<span class="kt">bool</span> <span class="n">check</span><span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">;</span>
<span class="kt">float</span> <span class="n">x</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">check</span> <span class="o">?</span> <span class="o">*</span><span class="nl">px</span> <span class="p">:</span> <span class="o">-</span><span class="n">F32_INFINITY</span><span class="p">;</span>
<span class="c1">// syntax for reduction in Triton is:</span>
<span class="c1">// x[..., OPERATOR, ...]</span>
<span class="c1">// ^</span>
<span class="c1">// index</span>
<span class="c1">// The operators currently supported are {min, max, +}</span>
<span class="kt">float</span> <span class="n">z</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="n">max</span><span class="p">];</span>
<span class="c1">// The exponential in Triton is fast but approximate</span>
<span class="c1">// (i.e., like __expf in CUDA)</span>
<span class="kt">float</span> <span class="n">num</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">);</span>
<span class="kt">float</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">num</span><span class="p">[</span><span class="o">+</span><span class="p">];</span>
<span class="c1">// The result of the reduction is now stored in y</span>
<span class="kt">float</span> <span class="n">y</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span> <span class="o">/</span> <span class="n">denom</span><span class="p">;</span>
<span class="c1">// We write it back</span>
<span class="kt">float</span><span class="o">*</span> <span class="n">py</span> <span class="p">[</span><span class="n">BLOCK</span><span class="p">]</span> <span class="o">=</span> <span class="n">Y</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">stride_ym</span> <span class="o">+</span> <span class="n">n</span><span class="p">;</span>
<span class="o">*?</span><span class="p">(</span><span class="n">check</span><span class="p">)</span><span class="n">py</span> <span class="o">=</span> <span class="n">y</span><span class="p">;</span>
<span class="p">}</span>
</pre></div>
</div>
</div></blockquote>
</div>
<div class="section" id="torch-bindings">
<h2>Torch Bindings<a class="headerlink" href="#torch-bindings" title="Permalink to this headline"></a></h2>
<p>We need to make sure that BLOCK is the smallest power of two
greater than the number of rows N of the input matrix.
Different values of BLOCK will result in different kernels</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">import</span> <span class="nn">triton</span>
<span class="c1"># Source code for the Triton kernel</span>
<span class="n">_src</span> <span class="o">=</span> <span class="s2">&quot;&quot;&quot;</span>
<span class="s2">__global__ void softmax(float* Y, float* X, int stride_ym, int stride_xm, int M, int N){</span>
<span class="s2"> int m = get_program_id(0);</span>
<span class="s2"> int n [BLOCK] = 0 ... BLOCK;</span>
<span class="s2"> float* px [BLOCK] = X + m*stride_xm + n;</span>
<span class="s2"> bool check[BLOCK] = n &lt; N;</span>
<span class="s2"> float x [BLOCK] = check ? *px : -F32_INFINITY;</span>
<span class="s2"> float z [BLOCK] = x - x[max];</span>
<span class="s2"> float num [BLOCK] = exp(z);</span>
<span class="s2"> float denom = num[+];</span>
<span class="s2"> float y [BLOCK] = num / denom;</span>
<span class="s2"> float* py [BLOCK] = Y + m*stride_ym + n;</span>
<span class="s2"> *?(check)py = y;</span>
<span class="s2">}</span>
<span class="s2">&quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">next_power_of_2</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="n">n</span> <span class="o">|=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span>
<span class="n">n</span> <span class="o">|=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="mi">2</span>
<span class="n">n</span> <span class="o">|=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="mi">4</span>
<span class="n">n</span> <span class="o">|=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="mi">8</span>
<span class="n">n</span> <span class="o">|=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="mi">16</span>
<span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">n</span>
<span class="n">_kernels</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">make_kernel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">device</span><span class="p">):</span>
<span class="n">BLOCK</span> <span class="o">=</span> <span class="n">next_power_of_2</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="n">BLOCK</span><span class="p">,</span> <span class="n">device</span><span class="p">)</span>
<span class="k">if</span> <span class="n">key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_kernels</span><span class="p">:</span>
<span class="n">defines</span> <span class="o">=</span> <span class="p">{</span><span class="s1">&#39;BLOCK&#39;</span><span class="p">:</span> <span class="n">BLOCK</span><span class="p">}</span>
<span class="n">_kernels</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">triton</span><span class="o">.</span><span class="n">kernel</span><span class="p">(</span><span class="n">_src</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">device</span><span class="p">,</span> <span class="n">defines</span><span class="o">=</span><span class="n">defines</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_kernels</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
<span class="k">class</span> <span class="nc">_softmax</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">autograd</span><span class="o">.</span><span class="n">Function</span><span class="p">):</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="c1"># constraints of the op</span>
<span class="k">assert</span> <span class="n">x</span><span class="o">.</span><span class="n">dtype</span> <span class="o">==</span> <span class="n">torch</span><span class="o">.</span><span class="n">float32</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">empty_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1"># *create launch grid*:</span>
<span class="c1"># here we just launch a grid of M programs</span>
<span class="n">M</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">shape</span>
<span class="n">grid</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">opt</span><span class="p">:</span> <span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">)</span>
<span class="c1"># *launch kernel*:</span>
<span class="n">kernel</span> <span class="o">=</span> <span class="n">make_kernel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">y</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
<span class="n">kernel</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">data_ptr</span><span class="p">(),</span> <span class="n">x</span><span class="o">.</span><span class="n">data_ptr</span><span class="p">(),</span> <span class="n">y</span><span class="o">.</span><span class="n">stride</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">x</span><span class="o">.</span><span class="n">stride</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
<span class="k">return</span> <span class="n">y</span>
<span class="n">softmax</span> <span class="o">=</span> <span class="n">_softmax</span><span class="o">.</span><span class="n">apply</span>
</pre></div>
</div>
</div>
<div class="section" id="unit-test">
<h2>Unit Test<a class="headerlink" href="#unit-test" title="Permalink to this headline"></a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">1823</span><span class="p">,</span> <span class="mi">781</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">)</span>
<span class="n">y_tri</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">y_ref</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">y_tri</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">y_ref</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">y_tri</span><span class="p">,</span> <span class="n">y_ref</span><span class="p">))</span>
</pre></div>
</div>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor([[2.0935e-03, 6.4551e-04, 9.8605e-05, ..., 3.3981e-04, 2.7386e-03,
9.1986e-05],
[7.0923e-04, 6.7521e-04, 5.1366e-04, ..., 9.8392e-04, 2.6547e-04,
6.9062e-04],
[1.4032e-04, 5.8826e-04, 1.1694e-03, ..., 6.6423e-04, 1.8178e-04,
6.7049e-04],
...,
[1.1767e-03, 4.2703e-03, 6.0596e-04, ..., 9.5274e-04, 1.1681e-03,
6.4924e-04],
[1.0772e-04, 7.4854e-04, 3.1912e-03, ..., 2.4980e-04, 1.9012e-03,
5.2567e-04],
[2.8518e-03, 8.1899e-04, 7.7046e-04, ..., 1.3403e-03, 5.3167e-04,
4.3268e-04]], device=&#39;cuda:0&#39;)
tensor([[2.0935e-03, 6.4551e-04, 9.8605e-05, ..., 3.3981e-04, 2.7386e-03,
9.1986e-05],
[7.0923e-04, 6.7521e-04, 5.1366e-04, ..., 9.8392e-04, 2.6547e-04,
6.9062e-04],
[1.4032e-04, 5.8826e-04, 1.1694e-03, ..., 6.6423e-04, 1.8178e-04,
6.7049e-04],
...,
[1.1767e-03, 4.2703e-03, 6.0596e-04, ..., 9.5274e-04, 1.1681e-03,
6.4924e-04],
[1.0772e-04, 7.4854e-04, 3.1912e-03, ..., 2.4980e-04, 1.9012e-03,
5.2567e-04],
[2.8518e-03, 8.1899e-04, 7.7046e-04, ..., 1.3403e-03, 5.3167e-04,
4.3268e-04]], device=&#39;cuda:0&#39;)
True
</pre></div>
</div>
<p>Seems to work!</p>
</div>
<div class="section" id="benchmarking">
<h2>Benchmarking<a class="headerlink" href="#benchmarking" title="Permalink to this headline"></a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">M</span> <span class="o">=</span> <span class="mi">4096</span>
<span class="n">Ns</span> <span class="o">=</span> <span class="p">[</span><span class="mi">128</span> <span class="o">*</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">50</span><span class="p">)]</span>
<span class="n">tri_ms</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">ref_ms</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">def_ms</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">N</span> <span class="ow">in</span> <span class="n">Ns</span><span class="p">:</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">gbps</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">ms</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">nelement</span><span class="p">()</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">element_size</span><span class="p">()</span> <span class="o">*</span> <span class="mf">1e-9</span> <span class="o">/</span> <span class="p">(</span><span class="n">ms</span> <span class="o">*</span> <span class="mf">1e-3</span><span class="p">)</span>
<span class="n">tri_ms</span> <span class="o">+=</span> <span class="p">[</span><span class="n">gbps</span><span class="p">(</span><span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">do_bench</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">)))]</span>
<span class="n">ref_ms</span> <span class="o">+=</span> <span class="p">[</span><span class="n">gbps</span><span class="p">(</span><span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">do_bench</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)))]</span>
<span class="n">def_ms</span> <span class="o">+=</span> <span class="p">[</span><span class="n">gbps</span><span class="p">(</span><span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">do_bench</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">naive_softmax</span><span class="p">(</span><span class="n">x</span><span class="p">)))]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;N&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Bandwidth (GB/s)&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Ns</span><span class="p">,</span> <span class="n">tri_ms</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Triton&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Ns</span><span class="p">,</span> <span class="n">ref_ms</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Torch&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Ns</span><span class="p">,</span> <span class="n">def_ms</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Naive&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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