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<p>Implementation of <a href="https://medium.com/hard-mode/the-legendary-fast-inverse-square-root-e51fee3b49d9" target="_blank">the inverse square root Root</a>.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdint&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;limits&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for inv_sqrt.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:ad219034bf5fba657f5035ec5a1d50f52" id="r_ad219034bf5fba657f5035ec5a1d50f52"><td class="memTemplParams" colspan="2">template&lt;typename T = double, char iterations = 2&gt; </td></tr>
<tr class="memitem:ad219034bf5fba657f5035ec5a1d50f52"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt</a> (T x)</td></tr>
<tr class="memdesc:ad219034bf5fba657f5035ec5a1d50f52"><td class="mdescLeft">&#160;</td><td class="mdescRight">for <code><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></code> <br /></td></tr>
<tr class="separator:ad219034bf5fba657f5035ec5a1d50f52"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa2703e5cf3fecde8becd9066b9666b97" id="r_aa2703e5cf3fecde8becd9066b9666b97"><td class="memTemplParams" colspan="2">template&lt;typename T = double&gt; </td></tr>
<tr class="memitem:aa2703e5cf3fecde8becd9066b9666b97"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt</a> (T number)</td></tr>
<tr class="memdesc:aa2703e5cf3fecde8becd9066b9666b97"><td class="mdescLeft">&#160;</td><td class="mdescRight">This is the function that calculates the fast inverse square root. The following code is the fast inverse square root with standard lib (cmath) More information can be found at <a href="https://www.linkedin.com/pulse/fast-inverse-square-root-still-armin-kassemi-langroodi" target="_blank">LinkedIn</a> <br /></td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d" id="r_aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
<tr class="memdesc:aa8dca7b867074164d5f45b0f3851269d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Self-test implementations. <br /></td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <br /></td></tr>
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Implementation of <a href="https://medium.com/hard-mode/the-legendary-fast-inverse-square-root-e51fee3b49d9" target="_blank">the inverse square root Root</a>. </p>
<p>Two implementation to calculate inverse inverse root, from Quake III Arena (C++ version) and with a standard library (<code>cmath</code>). This algorithm is used to calculate shadows in Quake III Arena. </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="ad219034bf5fba657f5035ec5a1d50f52" name="ad219034bf5fba657f5035ec5a1d50f52"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad219034bf5fba657f5035ec5a1d50f52">&#9670;&#160;</a></span>Fast_InvSqrt()</h2>
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template&lt;typename T = double, char iterations = 2&gt; </div>
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<tr>
<td class="memname">T Fast_InvSqrt </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>x</em></span></td><td>)</td>
<td></td>
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<p>for <code><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></code> </p>
<p>for assert for IO operations for numeric_limits</p>
<p>This is the function that calculates the fast inverse square root. The following code is the fast inverse square root implementation from Quake III Arena (Adapted for C++). More information can be found at <a href="https://en.wikipedia.org/wiki/Fast_inverse_square_root" target="_blank">Wikipedia</a> </p><dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">T</td><td>floating type </td></tr>
<tr><td class="paramname">iterations</td><td>inverse square root, the greater the number of iterations, the more exact the result will be (1 or 2). </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">x</td><td>value to calculate </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>the inverse square root </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 28</span> {</div>
<div class="line"><span class="lineno"> 29</span> <span class="keyword">using </span>Tint = <span class="keyword">typename</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/conditional.html">std::conditional</a>&lt;<span class="keyword">sizeof</span>(T) == 8, <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a>,</div>
<div class="line"><span class="lineno"> 30</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int32_t</a>&gt;::type;</div>
<div class="line"><span class="lineno"> 31</span> T y = x;</div>
<div class="line"><span class="lineno"> 32</span> T x2 = y * 0.5;</div>
<div class="line"><span class="lineno"> 33</span> </div>
<div class="line"><span class="lineno"> 34</span> Tint i =</div>
<div class="line"><span class="lineno"> 35</span> *<span class="keyword">reinterpret_cast&lt;</span>Tint *<span class="keyword">&gt;</span>(&amp;y); <span class="comment">// Store floating-point bits in integer</span></div>
<div class="line"><span class="lineno"> 36</span> </div>
<div class="line"><span class="lineno"> 37</span> i = (<span class="keyword">sizeof</span>(T) == 8 ? 0x5fe6eb50c7b537a9 : 0x5f3759df) -</div>
<div class="line"><span class="lineno"> 38</span> (i &gt;&gt; 1); <span class="comment">// Initial guess for Newton&#39;s method</span></div>
<div class="line"><span class="lineno"> 39</span> </div>
<div class="line"><span class="lineno"> 40</span> y = *<span class="keyword">reinterpret_cast&lt;</span>T *<span class="keyword">&gt;</span>(&amp;i); <span class="comment">// Convert new bits into float</span></div>
<div class="line"><span class="lineno"> 41</span> </div>
<div class="line"><span class="lineno"> 42</span> y = y * (1.5 - (x2 * y * y)); <span class="comment">// 1st iteration Newton&#39;s method</span></div>
<div class="line"><span class="lineno"> 43</span> <span class="keywordflow">if</span> (iterations == 2) {</div>
<div class="line"><span class="lineno"> 44</span> y = y * (1.5 - (x2 * y * y)); <span class="comment">// 2nd iteration, the more exact result</span></div>
<div class="line"><span class="lineno"> 45</span> }</div>
<div class="line"><span class="lineno"> 46</span> <span class="keywordflow">return</span> y;</div>
<div class="line"><span class="lineno"> 47</span>}</div>
<div class="ttc" id="aconditional_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/conditional.html">std::conditional</a></div></div>
<div class="ttc" id="ainteger_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></div></div>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Main function. </p>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 87</span> {</div>
<div class="line"><span class="lineno"> 88</span> <a class="code hl_function" href="#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementations</span></div>
<div class="line"><span class="lineno"> 89</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Fast inverse square root of 36 is: &quot;</span></div>
<div class="line"><span class="lineno"> 90</span> &lt;&lt; <a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt&lt;float, 1&gt;</a>(36.0f) &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 91</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Fast inverse square root of 36 is: &quot;</span></div>
<div class="line"><span class="lineno"> 92</span> &lt;&lt; <a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt&lt;double, 2&gt;</a>(36.0f) &lt;&lt; <span class="stringliteral">&quot; (2 iterations)&quot;</span></div>
<div class="line"><span class="lineno"> 93</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 94</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Fast inverse square root of 100 is: &quot;</span></div>
<div class="line"><span class="lineno"> 95</span> &lt;&lt; <a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt</a>(100.0f)</div>
<div class="line"><span class="lineno"> 96</span> &lt;&lt; <span class="stringliteral">&quot; (With default template type and iterations: double, 2)&quot;</span></div>
<div class="line"><span class="lineno"> 97</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 98</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Standard inverse square root of 36 is: &quot;</span></div>
<div class="line"><span class="lineno"> 99</span> &lt;&lt; <a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt&lt;float&gt;</a>(36.0f) &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 100</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Standard inverse square root of 100 is: &quot;</span></div>
<div class="line"><span class="lineno"> 101</span> &lt;&lt; <a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt</a>(100.0f)</div>
<div class="line"><span class="lineno"> 102</span> &lt;&lt; <span class="stringliteral">&quot; (With default template type: double)&quot;</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 103</span>}</div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
<div class="ttc" id="ainv__sqrt_8cpp_html_aa2703e5cf3fecde8becd9066b9666b97"><div class="ttname"><a href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt</a></div><div class="ttdeci">T Standard_InvSqrt(T number)</div><div class="ttdoc">This is the function that calculates the fast inverse square root. The following code is the fast inv...</div><div class="ttdef"><b>Definition</b> inv_sqrt.cpp:59</div></div>
<div class="ttc" id="ainv__sqrt_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations.</div><div class="ttdef"><b>Definition</b> inv_sqrt.cpp:68</div></div>
<div class="ttc" id="ainv__sqrt_8cpp_html_ad219034bf5fba657f5035ec5a1d50f52"><div class="ttname"><a href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt</a></div><div class="ttdeci">T Fast_InvSqrt(T x)</div><div class="ttdoc">for std::sqrt</div><div class="ttdef"><b>Definition</b> inv_sqrt.cpp:28</div></div>
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Here is the call graph for this function:</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa2703e5cf3fecde8becd9066b9666b97">&#9670;&#160;</a></span>Standard_InvSqrt()</h2>
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template&lt;typename T = double&gt; </div>
<table class="memname">
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<td class="memname">T Standard_InvSqrt </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>number</em></span></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>This is the function that calculates the fast inverse square root. The following code is the fast inverse square root with standard lib (cmath) More information can be found at <a href="https://www.linkedin.com/pulse/fast-inverse-square-root-still-armin-kassemi-langroodi" target="_blank">LinkedIn</a> </p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">T</td><td>floating type </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">number</td><td>value to calculate </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>the inverse square root </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 59</span> {</div>
<div class="line"><span class="lineno"> 60</span> T squareRoot = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">sqrt</a>(number);</div>
<div class="line"><span class="lineno"> 61</span> <span class="keywordflow">return</span> 1.0f / squareRoot;</div>
<div class="line"><span class="lineno"> 62</span>}</div>
<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">&#9670;&#160;</a></span>test()</h2>
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<td class="memname">static void test </td>
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<p>Self-test implementations. </p>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 68</span> {</div>
<div class="line"><span class="lineno"> 69</span> <span class="keyword">const</span> <span class="keywordtype">float</span> epsilon = 1e-3f;</div>
<div class="line"><span class="lineno"> 70</span> </div>
<div class="line"><span class="lineno"> 71</span> <span class="comment">/* Tests with multiple values */</span></div>
<div class="line"><span class="lineno"> 72</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt&lt;float&gt;</a>(100.0f) - 0.0998449f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 73</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt&lt;double&gt;</a>(36.0f) - 0.166667f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 74</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt</a>(12.0f) - 0.288423f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 75</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#aa2703e5cf3fecde8becd9066b9666b97">Standard_InvSqrt&lt;double&gt;</a>(5.0f) - 0.447141f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 76</span> </div>
<div class="line"><span class="lineno"> 77</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt&lt;float, 1&gt;</a>(100.0f) - 0.0998449f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 78</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt&lt;double, 1&gt;</a>(36.0f) - 0.166667f) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 79</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt</a>(12.0f) - 0.288423) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 80</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a>(<a class="code hl_function" href="#ad219034bf5fba657f5035ec5a1d50f52">Fast_InvSqrt&lt;double&gt;</a>(5.0f) - 0.447141) &lt; epsilon);</div>
<div class="line"><span class="lineno"> 81</span>}</div>
<div class="ttc" id="afabs_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a></div><div class="ttdeci">T fabs(T... args)</div></div>
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