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<div class="headertitle"><div class="title">erf.h</div></div>
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<a href="erf_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">// Copyright © 2023 Apple Inc.</span></div>
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<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span> </div>
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<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="preprocessor">#pragma once</span></div>
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<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="preprocessor">#include <metal_math></span></div>
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<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span> </div>
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<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment">/*</span></div>
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<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment"> * Approximation to the error function.</span></div>
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<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment"> * Based on code from:</span></div>
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<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment"> * https://stackoverflow.com/questions/35148198/efficient-faithfully-rounded-implementation-of-error-function-erff#answer-35148199</span></div>
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<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment"> */</span></div>
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<div class="foldopen" id="foldopen00011" data-start="{" data-end="}">
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<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"><a class="line" href="erf_8h.html#a6ce199ee56105c67adbf8c48c019a8b2"> 11</a></span><span class="keywordtype">float</span> <a class="code hl_function" href="erf_8h.html#a6ce199ee56105c67adbf8c48c019a8b2">erf</a>(<span class="keywordtype">float</span> a) {</div>
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<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span> <span class="keywordtype">float</span> r, s, t, <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a>;</div>
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<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span> t = <a class="code hl_function" href="namespacemetal.html#a87c5122c60f9a12afceb9925a5b78ffb">metal::abs</a>(a);</div>
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<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span> s = a * a;</div>
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<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span> <span class="keywordflow">if</span> (t > 0.927734375f) {</div>
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<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span> <span class="comment">// maximum error 0.99527 ulp</span></div>
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<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(</div>
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<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span> -1.72853470e-5f, t, 3.83197126e-4f); <span class="comment">// -0x1.220000p-16,0x1.91cfb2p-12</span></div>
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<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span> <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a> = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(</div>
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<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span> -3.88396438e-3f, t, 2.42546219e-2f); <span class="comment">// -0x1.fd1438p-9, 0x1.8d6342p-6</span></div>
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<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a>);</div>
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<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, t, -1.06777877e-1f); <span class="comment">// -0x1.b55cb8p-4</span></div>
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<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, t, -6.34846687e-1f); <span class="comment">// -0x1.450aa0p-1</span></div>
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<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, t, -1.28717512e-1f); <span class="comment">// -0x1.079d0cp-3</span></div>
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<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, t, -t);</div>
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<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span> <span class="comment">// TODO, replace with expm1 when implemented</span></div>
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<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span> r = 1.0f - <a class="code hl_function" href="namespacemetal.html#ac2a0b3618d922ac014baac8189d44650">metal::exp</a>(r);</div>
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<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span> r = metal::copysign(r, a);</div>
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<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span> <span class="comment">// maximum error 0.98929 ulp</span></div>
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<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span> r = -5.96761703e-4f; <span class="comment">// -0x1.38e000p-11</span></div>
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<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, 4.99119423e-3f); <span class="comment">// 0x1.471a58p-8</span></div>
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<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, -2.67681349e-2f); <span class="comment">// -0x1.b691b2p-6</span></div>
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<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, 1.12819925e-1f); <span class="comment">// 0x1.ce1c44p-4</span></div>
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<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, -3.76125336e-1f); <span class="comment">// -0x1.812700p-2</span></div>
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<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, s, 1.28379166e-1f); <span class="comment">// 0x1.06eba8p-3</span></div>
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<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span> r = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(r, a, a);</div>
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<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span> }</div>
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<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span> <span class="keywordflow">return</span> r;</div>
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<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span>}</div>
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</div>
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<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span> </div>
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<div class="foldopen" id="foldopen00042" data-start="{" data-end="}">
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<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"><a class="line" href="erf_8h.html#a1846e0d683c7aff826bb32addcc3b885"> 42</a></span><span class="keywordtype">float</span> <a class="code hl_function" href="erf_8h.html#a1846e0d683c7aff826bb32addcc3b885">erfinv</a>(<span class="keywordtype">float</span> a) {</div>
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<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span> <span class="keyword">auto</span> t = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(a, 0.0f - a, 1.0f);</div>
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<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> t = <a class="code hl_function" href="namespacemetal.html#a423a9f4f2fc7ef5ec7eda061277b51b6">metal::log</a>(t);</div>
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<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> <span class="keywordtype">float</span> p;</div>
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<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> <span class="keywordflow">if</span> (<a class="code hl_function" href="namespacemetal.html#a87c5122c60f9a12afceb9925a5b78ffb">metal::abs</a>(t) > 6.125f) { <span class="comment">// maximum ulp error = 2.35793</span></div>
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<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span> p = 3.03697567e-10f; <span class="comment">// 0x1.4deb44p-32</span></div>
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<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 2.93243101e-8f); <span class="comment">// 0x1.f7c9aep-26</span></div>
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<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 1.22150334e-6f); <span class="comment">// 0x1.47e512p-20</span></div>
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<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 2.84108955e-5f); <span class="comment">// 0x1.dca7dep-16</span></div>
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<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 3.93552968e-4f); <span class="comment">// 0x1.9cab92p-12</span></div>
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<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 3.02698812e-3f); <span class="comment">// 0x1.8cc0dep-9</span></div>
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 4.83185798e-3f); <span class="comment">// 0x1.3ca920p-8</span></div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, -2.64646143e-1f); <span class="comment">// -0x1.0eff66p-2</span></div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 8.40016484e-1f); <span class="comment">// 0x1.ae16a4p-1</span></div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> } <span class="keywordflow">else</span> { <span class="comment">// maximum ulp error = 2.35002</span></div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> p = 5.43877832e-9f; <span class="comment">// 0x1.75c000p-28</span></div>
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<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 1.43285448e-7f); <span class="comment">// 0x1.33b402p-23</span></div>
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<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 1.22774793e-6f); <span class="comment">// 0x1.499232p-20</span></div>
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<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 1.12963626e-7f); <span class="comment">// 0x1.e52cd2p-24</span></div>
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<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, -5.61530760e-5f); <span class="comment">// -0x1.d70bd0p-15</span></div>
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<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, -1.47697632e-4f); <span class="comment">// -0x1.35be90p-13</span></div>
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<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 2.31468678e-3f); <span class="comment">// 0x1.2f6400p-9</span></div>
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<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 1.15392581e-2f); <span class="comment">// 0x1.7a1e50p-7</span></div>
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||
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<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, -2.32015476e-1f); <span class="comment">// -0x1.db2aeep-3</span></div>
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<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> p = <a class="code hl_function" href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a>(p, t, 8.86226892e-1f); <span class="comment">// 0x1.c5bf88p-1</span></div>
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<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> }</div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> <span class="keywordflow">return</span> a * p;</div>
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<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span>}</div>
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</div>
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<div class="ttc" id="aerf_8h_html_a1846e0d683c7aff826bb32addcc3b885"><div class="ttname"><a href="erf_8h.html#a1846e0d683c7aff826bb32addcc3b885">erfinv</a></div><div class="ttdeci">float erfinv(float a)</div><div class="ttdef"><b>Definition</b> erf.h:42</div></div>
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<div class="ttc" id="aerf_8h_html_a6ce199ee56105c67adbf8c48c019a8b2"><div class="ttname"><a href="erf_8h.html#a6ce199ee56105c67adbf8c48c019a8b2">erf</a></div><div class="ttdeci">float erf(float a)</div><div class="ttdef"><b>Definition</b> erf.h:11</div></div>
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<div class="ttc" id="anamespacemetal_html_a423a9f4f2fc7ef5ec7eda061277b51b6"><div class="ttname"><a href="namespacemetal.html#a423a9f4f2fc7ef5ec7eda061277b51b6">metal::log</a></div><div class="ttdeci">METAL_FUNC bfloat16_t log(bfloat16_t x)</div><div class="ttdef"><b>Definition</b> bf16_math.h:234</div></div>
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<div class="ttc" id="anamespacemetal_html_a6301a78d69ff14a06194ca85a0c7d326"><div class="ttname"><a href="namespacemetal.html#a6301a78d69ff14a06194ca85a0c7d326">metal::fma</a></div><div class="ttdeci">METAL_FUNC bfloat16_t fma(bfloat16_t x, bfloat16_t y, bfloat16_t z)</div><div class="ttdef"><b>Definition</b> bf16_math.h:234</div></div>
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<div class="ttc" id="anamespacemetal_html_a87c5122c60f9a12afceb9925a5b78ffb"><div class="ttname"><a href="namespacemetal.html#a87c5122c60f9a12afceb9925a5b78ffb">metal::abs</a></div><div class="ttdeci">METAL_FUNC bfloat16_t abs(bfloat16_t x)</div><div class="ttdef"><b>Definition</b> bf16_math.h:234</div></div>
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<div class="ttc" id="anamespacemetal_html_ac2a0b3618d922ac014baac8189d44650"><div class="ttname"><a href="namespacemetal.html#ac2a0b3618d922ac014baac8189d44650">metal::exp</a></div><div class="ttdeci">METAL_FUNC bfloat16_t exp(bfloat16_t x)</div><div class="ttdef"><b>Definition</b> bf16_math.h:234</div></div>
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<div class="ttc" id="atypes_2bf16_8h_html_aa21e554721eddcf127b7fcfa7fdc56bd"><div class="ttname"><a href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a></div><div class="ttdeci">uint32_t u</div><div class="ttdef"><b>Definition</b> bf16.h:17</div></div>
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