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<a href="expm1f_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>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span> </div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="preprocessor">#pragma once</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span> </div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="preprocessor">#include &lt;metal_math&gt;</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span> </div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment">// Original license copied below:</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="comment">// Copyright (c) 2015-2023 Norbert Juffa</span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="comment">// All rights reserved.</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="comment">//</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="comment">// Redistribution and use in source and binary forms, with or without</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="comment">// modification, are permitted provided that the following conditions</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span><span class="comment">// are met:</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span><span class="comment">//</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="comment">// 1. Redistributions of source code must retain the above copyright</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span><span class="comment">// notice, this list of conditions and the following disclaimer.</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="comment">//</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="comment">// 2. Redistributions in binary form must reproduce the above copyright</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span><span class="comment">// notice, this list of conditions and the following disclaimer in the</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="comment">// documentation and/or other materials provided with the distribution.</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="comment">//</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="comment">// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="comment">// &quot;AS IS&quot; AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="comment">// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span><span class="comment">// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="comment">// HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span><span class="comment">// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span><span class="comment">// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span><span class="comment">// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span><span class="comment">// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="comment">// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="comment">// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span> </div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span><span class="comment">/* Compute exponential base e minus 1. Maximum ulp error = 0.997458</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span><span class="comment"></span> </div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="comment"> i = rint(a/log(2)), f = a-i*log(2). Then expm1(a) = 2**i * (expm1(f)+1) - 1.</span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span><span class="comment"> Compute r = expm1(f). Then expm1(a)= 2 * (0.5 * 2**i * r + 0.5 * 2**i - 0.5).</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span><span class="comment"> With t = 0.5*2**i, expm1(a) = 2*(r * t + t-0.5). However, for best accuracy,</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="comment"> when i == 1, expm1(a)= 2*(r + 0.5), and when i == 0, expm1(a) = r.</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span><span class="comment"></span> </div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="comment"> NOTE: Scale factor b is only applied if i &lt; 0 or i &gt; 1 (should be power of 2)</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="comment">*/</span></div>
<div class="foldopen" id="foldopen00043" data-start="{" data-end="}">
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"><a class="line" href="expm1f_8h.html#adf20e03405fba634ca8d01acac24592e"> 43</a></span><span class="keywordtype">float</span> <a class="code hl_function" href="expm1f_8h.html#adf20e03405fba634ca8d01acac24592e">expm1f_scaled_unchecked</a>(<span class="keywordtype">float</span> a, <span class="keywordtype">float</span> b) {</div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> <span class="keywordtype">float</span> <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, j, r, s, t, <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a>, v, x, y;</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> <span class="keywordtype">int</span> i;</div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> </div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span> <span class="comment">// exp(a) = 2**i * exp(f); i = rintf (a / log(2))</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> j = fma(1.442695f, a, 12582912.f); <span class="comment">// 0x1.715476p0, 0x1.8p23</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> j = j - 12582912.0f; <span class="comment">// 0x1.8p23</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> i = (int)j;</div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a> = fma(j, -6.93145752e-1f, a);</div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> </div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> <span class="comment">// approximate r = exp(f)-1 on interval [-log(2)/2, +log(2)/2]</span></div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> s = <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a> * <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>;</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keywordflow">if</span> (a == 0.0f)</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> s = a; <span class="comment">// ensure -0 is passed through</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> <span class="comment">// err = 0.997458 ulp1 = 11081805</span></div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> r = 1.97350979e-4f; <span class="comment">// 0x1.9de000p-13</span></div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> r = fma(r, <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, 1.39309070e-3f); <span class="comment">// 0x1.6d30bcp-10</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> r = fma(r, <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, 8.33343994e-3f); <span class="comment">// 0x1.1111f6p-7</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> r = fma(r, <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, 4.16668020e-2f); <span class="comment">// 0x1.55559ep-5</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> r = fma(r, <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, 1.66666716e-1f); <span class="comment">// 0x1.55555cp-3</span></div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> r = fma(r, <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>, 4.99999970e-1f); <span class="comment">// 0x1.fffffep-2</span></div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a> = (j == 1) ? (<a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a> + 0.5f) : <a class="code hl_variable" href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a>;</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> v = fma(r, s, <a class="code hl_variable" href="types_2bf16_8h.html#aa21e554721eddcf127b7fcfa7fdc56bd">u</a>);</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> s = 0.5f * b;</div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> t = ldexp(s, i);</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> y = t - s;</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> x = (t - y) - s; <span class="comment">// double-float canonicalization of difference</span></div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> r = fma(v, t, x) + y;</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> r = r + r;</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> <span class="keywordflow">if</span> (j == 0)</div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> r = v;</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <span class="keywordflow">if</span> (j == 1)</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> r = v + v;</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> <span class="keywordflow">return</span> r;</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span>}</div>
</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> </div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span><span class="comment">/* Compute exponential base e minus 1. max ulp err = 0.99746 */</span></div>
<div class="foldopen" id="foldopen00080" data-start="{" data-end="}">
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"><a class="line" href="expm1f_8h.html#a87f66d30e185950f42ce3641783cdc40"> 80</a></span><span class="keywordtype">float</span> <a class="code hl_function" href="expm1f_8h.html#a87f66d30e185950f42ce3641783cdc40">expm1f</a>(<span class="keywordtype">float</span> a) {</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> <span class="keywordtype">float</span> r;</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> </div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> r = <a class="code hl_function" href="expm1f_8h.html#adf20e03405fba634ca8d01acac24592e">expm1f_scaled_unchecked</a>(a, 1.0f);</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> <span class="comment">/* handle severe overflow and underflow */</span></div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> <span class="keywordflow">if</span> (abs(a - 1.0f) &gt; 88.0f) {</div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> r = pow(2, a);</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> r = fma(r, r, -1.0f);</div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> }</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> <span class="keywordflow">return</span> r;</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span>}</div>
</div>
<div class="ttc" id="aexpm1f_8h_html_a87f66d30e185950f42ce3641783cdc40"><div class="ttname"><a href="expm1f_8h.html#a87f66d30e185950f42ce3641783cdc40">expm1f</a></div><div class="ttdeci">float expm1f(float a)</div><div class="ttdef"><b>Definition</b> expm1f.h:80</div></div>
<div class="ttc" id="aexpm1f_8h_html_adf20e03405fba634ca8d01acac24592e"><div class="ttname"><a href="expm1f_8h.html#adf20e03405fba634ca8d01acac24592e">expm1f_scaled_unchecked</a></div><div class="ttdeci">float expm1f_scaled_unchecked(float a, float b)</div><div class="ttdef"><b>Definition</b> expm1f.h:43</div></div>
<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>
<div class="ttc" id="atypes_2bf16_8h_html_af900396d7b72ff2a7002e8befe8cf8f1"><div class="ttname"><a href="types_2bf16_8h.html#af900396d7b72ff2a7002e8befe8cf8f1">f</a></div><div class="ttdeci">float f</div><div class="ttdef"><b>Definition</b> bf16.h:16</div></div>
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