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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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<a href="#typedef-members">Typedefs</a> &#124;
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<p>for IO operations
<a href="#details">More...</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="typedef-members" name="typedef-members"></a>
Typedefs</h2></td></tr>
<tr class="memitem:a006c8ebd11bb1e035815dc835a1a7a85" id="r_a006c8ebd11bb1e035815dc835a1a7a85"><td class="memItemLeft" align="right" valign="top">using&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a006c8ebd11bb1e035815dc835a1a7a85">Point</a></td></tr>
<tr class="memdesc:a006c8ebd11bb1e035815dc835a1a7a85"><td class="mdescLeft">&#160;</td><td class="mdescRight">structure of points containing two numbers, x and y, such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. <br /></td></tr>
<tr class="separator:a006c8ebd11bb1e035815dc835a1a7a85"><td class="memSeparator" colspan="2">&#160;</td></tr>
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Functions</h2></td></tr>
<tr class="memitem:ab37f3a7302a84179aae682c79d8390bf" id="r_ab37f3a7302a84179aae682c79d8390bf"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ab37f3a7302a84179aae682c79d8390bf">aliquot_sum</a> (const uint64_t num)</td></tr>
<tr class="memdesc:ab37f3a7302a84179aae682c79d8390bf"><td class="mdescLeft">&#160;</td><td class="mdescRight">to return the aliquot sum of a number <br /></td></tr>
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<tr class="memitem:abf7f2a6d91f1ca6c89698792aea3f188" id="r_abf7f2a6d91f1ca6c89698792aea3f188"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#abf7f2a6d91f1ca6c89698792aea3f188">approximate_pi</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; <a class="el" href="#a006c8ebd11bb1e035815dc835a1a7a85">Point</a> &gt; &amp;pts)</td></tr>
<tr class="memdesc:abf7f2a6d91f1ca6c89698792aea3f188"><td class="mdescLeft">&#160;</td><td class="mdescRight">This function uses the points in a given vector 'pts' (drawn at random) to return an approximation of the number π. <br /></td></tr>
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<tr class="memdesc:a971ce57e368f2f631cf1f4ff3f864049"><td class="mdescLeft">&#160;</td><td class="mdescRight">area of a <a href="https://en.wikipedia.org/wiki/Square" target="_blank">square</a> (l * l) <br /></td></tr>
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<tr class="memitem:ab31d141f7c5b551746b1eee0eb4dedca" id="r_ab31d141f7c5b551746b1eee0eb4dedca"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
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<tr class="memdesc:ab31d141f7c5b551746b1eee0eb4dedca"><td class="mdescLeft">&#160;</td><td class="mdescRight">area of a <a href="https://en.wikipedia.org/wiki/Rectangle" target="_blank">rectangle</a> (l * w) <br /></td></tr>
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<tr class="memitem:ab3b920cc56442abd92279ba23b50f4dc" id="r_ab3b920cc56442abd92279ba23b50f4dc"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:ab3b920cc56442abd92279ba23b50f4dc"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ab3b920cc56442abd92279ba23b50f4dc">triangle_area</a> (T base, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:ab3b920cc56442abd92279ba23b50f4dc"><td class="mdescLeft">&#160;</td><td class="mdescRight">area of a <a href="https://en.wikipedia.org/wiki/Triangle" target="_blank">triangle</a> (b * h / 2) <br /></td></tr>
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<tr class="memitem:a40e36c67da78d2131408c57ee091ad75" id="r_a40e36c67da78d2131408c57ee091ad75"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a40e36c67da78d2131408c57ee091ad75"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a40e36c67da78d2131408c57ee091ad75">circle_area</a> (T radius)</td></tr>
<tr class="memdesc:a40e36c67da78d2131408c57ee091ad75"><td class="mdescLeft">&#160;</td><td class="mdescRight">area of a <a href="https://en.wikipedia.org/wiki/Area_of_a_circle" target="_blank">circle</a> (pi <br /></td></tr>
<tr class="separator:a40e36c67da78d2131408c57ee091ad75"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a5de184925e68658f15415dd53954df4f" id="r_a5de184925e68658f15415dd53954df4f"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a5de184925e68658f15415dd53954df4f"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a5de184925e68658f15415dd53954df4f">parallelogram_area</a> (T base, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:a5de184925e68658f15415dd53954df4f"><td class="mdescLeft">&#160;</td><td class="mdescRight">area of a <a href="https://en.wikipedia.org/wiki/Parallelogram" target="_blank">parallelogram</a> (b * h) <br /></td></tr>
<tr class="separator:a5de184925e68658f15415dd53954df4f"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:abc46c784a297fc1d2eb8b33a327fba4c" id="r_abc46c784a297fc1d2eb8b33a327fba4c"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:abc46c784a297fc1d2eb8b33a327fba4c"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#abc46c784a297fc1d2eb8b33a327fba4c">cube_surface_area</a> (T length)</td></tr>
<tr class="memdesc:abc46c784a297fc1d2eb8b33a327fba4c"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface area of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> ( 6 * (l <br /></td></tr>
<tr class="separator:abc46c784a297fc1d2eb8b33a327fba4c"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ab7f29862d30df351c317eedd60a0c656" id="r_ab7f29862d30df351c317eedd60a0c656"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:ab7f29862d30df351c317eedd60a0c656"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ab7f29862d30df351c317eedd60a0c656">sphere_surface_area</a> (T radius)</td></tr>
<tr class="memdesc:ab7f29862d30df351c317eedd60a0c656"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface area of a <a href="https://en.wikipedia.org/wiki/Sphere" target="_blank">sphere</a> ( 4 * pi * r^2) <br /></td></tr>
<tr class="separator:ab7f29862d30df351c317eedd60a0c656"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ac5803413618fcfb922cb32c6db0fc864" id="r_ac5803413618fcfb922cb32c6db0fc864"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:ac5803413618fcfb922cb32c6db0fc864"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ac5803413618fcfb922cb32c6db0fc864">cylinder_surface_area</a> (T radius, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:ac5803413618fcfb922cb32c6db0fc864"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface area of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> (2 * pi * r * h + 2 * pi * r^2) <br /></td></tr>
<tr class="separator:ac5803413618fcfb922cb32c6db0fc864"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a3277e65a8f380e7632791975bfba0efb" id="r_a3277e65a8f380e7632791975bfba0efb"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a3277e65a8f380e7632791975bfba0efb"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a3277e65a8f380e7632791975bfba0efb">hemi_sphere_surface_area</a> (T radius)</td></tr>
<tr class="memdesc:a3277e65a8f380e7632791975bfba0efb"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface area of a <a href="https://en.wikipedia.org/wiki/Surface_area" target="_blank">hemi-sphere</a> ( 3 * pi * r^2) <br /></td></tr>
<tr class="separator:a3277e65a8f380e7632791975bfba0efb"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:af05567415a9ea36c254b54e3d5a2152a" id="r_af05567415a9ea36c254b54e3d5a2152a"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af05567415a9ea36c254b54e3d5a2152a">sum_of_divisor</a> (int num)</td></tr>
<tr class="memdesc:af05567415a9ea36c254b54e3d5a2152a"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to calculate the sum of all the proper divisor of an integer. <br /></td></tr>
<tr class="separator:af05567415a9ea36c254b54e3d5a2152a"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a8e6eede206201db0d1dbb618fa969bec" id="r_a8e6eede206201db0d1dbb618fa969bec"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a8e6eede206201db0d1dbb618fa969bec">are_amicable</a> (int x, int y)</td></tr>
<tr class="memdesc:a8e6eede206201db0d1dbb618fa969bec"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to check whether the pair is amicable or not. <br /></td></tr>
<tr class="separator:a8e6eede206201db0d1dbb618fa969bec"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a6c72f756a7bf1b9043c357e3fe7814ca" id="r_a6c72f756a7bf1b9043c357e3fe7814ca"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a6c72f756a7bf1b9043c357e3fe7814ca">is_factorial</a> (uint64_t n)</td></tr>
<tr class="memdesc:a6c72f756a7bf1b9043c357e3fe7814ca"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to check if the given number is factorial of some number or not. <br /></td></tr>
<tr class="separator:a6c72f756a7bf1b9043c357e3fe7814ca"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:abd8f794b2229b42876169ff841b6e444" id="r_abd8f794b2229b42876169ff841b6e444"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#abd8f794b2229b42876169ff841b6e444">is_prime</a> (int64_t num)</td></tr>
<tr class="memdesc:abd8f794b2229b42876169ff841b6e444"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to check if the given number is prime or not. <br /></td></tr>
<tr class="separator:abd8f794b2229b42876169ff841b6e444"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a91366864111e1fac29722ca45e02ea8f" id="r_a91366864111e1fac29722ca45e02ea8f"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a91366864111e1fac29722ca45e02ea8f">sieve</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; bool &gt; *vec)</td></tr>
<tr class="memdesc:a91366864111e1fac29722ca45e02ea8f"><td class="mdescLeft">&#160;</td><td class="mdescRight">Performs the sieve. <br /></td></tr>
<tr class="separator:a91366864111e1fac29722ca45e02ea8f"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ad09d59850865012a6fd95d89954c82e4" id="r_ad09d59850865012a6fd95d89954c82e4"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ad09d59850865012a6fd95d89954c82e4">print_primes</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; bool &gt; const &amp;<a class="el" href="../../de/d9b/prime__numbers_8cpp.html#a9575f3a51eeb8a57d657b3db6a4b441a">primes</a>)</td></tr>
<tr class="memdesc:ad09d59850865012a6fd95d89954c82e4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Prints all the indexes of true values in the passed <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>. <br /></td></tr>
<tr class="separator:ad09d59850865012a6fd95d89954c82e4"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ac37d3ba52eb296597d7a024ba8c4a5a5" id="r_ac37d3ba52eb296597d7a024ba8c4a5a5"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ac37d3ba52eb296597d7a024ba8c4a5a5">phiFunction</a> (uint64_t n)</td></tr>
<tr class="memdesc:ac37d3ba52eb296597d7a024ba8c4a5a5"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to calculate Euler's Totient. <br /></td></tr>
<tr class="separator:ac37d3ba52eb296597d7a024ba8c4a5a5"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a7e78996673df791014cfe540b183456a" id="r_a7e78996673df791014cfe540b183456a"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a7e78996673df791014cfe540b183456a">factorial</a> (uint8_t n)</td></tr>
<tr class="memdesc:a7e78996673df791014cfe540b183456a"><td class="mdescLeft">&#160;</td><td class="mdescRight">function to find factorial of given number <br /></td></tr>
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<tr class="memitem:aec65db4e5c7317323227f026fe50ef11" id="r_aec65db4e5c7317323227f026fe50ef11"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aec65db4e5c7317323227f026fe50ef11">integral_approx</a> (double lb, double ub, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a>&lt; double(double)&gt; &amp;func, double delta=.0001)</td></tr>
<tr class="memdesc:aec65db4e5c7317323227f026fe50ef11"><td class="mdescLeft">&#160;</td><td class="mdescRight">Computes integral approximation. <br /></td></tr>
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<tr class="memitem:a50936ee98f4d40f17823befc65a32aec" id="r_a50936ee98f4d40f17823befc65a32aec"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a50936ee98f4d40f17823befc65a32aec">test_eval</a> (double approx, double expected, double threshold)</td></tr>
<tr class="memdesc:a50936ee98f4d40f17823befc65a32aec"><td class="mdescLeft">&#160;</td><td class="mdescRight">Wrapper to evaluate if the approximated value is within <code>.XX%</code> threshold of the exact value. <br /></td></tr>
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<tr class="memitem:a2565c745aac2f9561cc6fd9e56d9b77a" id="r_a2565c745aac2f9561cc6fd9e56d9b77a"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a2565c745aac2f9561cc6fd9e56d9b77a">iterativeFactorial</a> (uint8_t n)</td></tr>
<tr class="memdesc:a2565c745aac2f9561cc6fd9e56d9b77a"><td class="mdescLeft">&#160;</td><td class="mdescRight">Calculates the factorial iteratively. <br /></td></tr>
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<tr class="memitem:afa39ec943a4836c878e1614fd89b146f" id="r_afa39ec943a4836c878e1614fd89b146f"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#afa39ec943a4836c878e1614fd89b146f">largestPower</a> (uint32_t n, const uint16_t &amp;p)</td></tr>
<tr class="memdesc:afa39ec943a4836c878e1614fd89b146f"><td class="mdescLeft">&#160;</td><td class="mdescRight">Function to calculate largest power. <br /></td></tr>
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<tr class="memitem:a04065193d190d605e1f0d0d93a87e244" id="r_a04065193d190d605e1f0d0d93a87e244"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a04065193d190d605e1f0d0d93a87e244">lcmSum</a> (const uint16_t &amp;num)</td></tr>
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<tr class="memitem:a8d8e81a7cd59644b311ef9adb268f5f0" id="r_a8d8e81a7cd59644b311ef9adb268f5f0"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a8d8e81a7cd59644b311ef9adb268f5f0">magic_number</a> (const uint64_t &amp;n)</td></tr>
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<tr class="memitem:afcd07701d73ed65cd616bcba02737f3d" id="r_afcd07701d73ed65cd616bcba02737f3d"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#afcd07701d73ed65cd616bcba02737f3d">power</a> (uint64_t a, uint64_t b, uint64_t c)</td></tr>
<tr class="memdesc:afcd07701d73ed65cd616bcba02737f3d"><td class="mdescLeft">&#160;</td><td class="mdescRight">This function calculates a raised to exponent b under modulo c using modular exponentiation. <br /></td></tr>
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<tr class="memitem:a6e2dff75c5de70455b90c799d6ad6967" id="r_a6e2dff75c5de70455b90c799d6ad6967"><td class="memTemplParams" colspan="2">template&lt;class T &gt; </td></tr>
<tr class="memitem:a6e2dff75c5de70455b90c799d6ad6967"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a6e2dff75c5de70455b90c799d6ad6967">n_choose_r</a> (T n, T r)</td></tr>
<tr class="memdesc:a6e2dff75c5de70455b90c799d6ad6967"><td class="mdescLeft">&#160;</td><td class="mdescRight">This is the function implementation of \( \binom{n}{r} \). <br /></td></tr>
<tr class="separator:a6e2dff75c5de70455b90c799d6ad6967"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a9236348755183644f1225e162d01ab14" id="r_a9236348755183644f1225e162d01ab14"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a9236348755183644f1225e162d01ab14"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a9236348755183644f1225e162d01ab14">square_perimeter</a> (T length)</td></tr>
<tr class="memdesc:a9236348755183644f1225e162d01ab14"><td class="mdescLeft">&#160;</td><td class="mdescRight">perimeter of a <a href="https://en.wikipedia.org/wiki/Square" target="_blank">square</a> (4 * l) <br /></td></tr>
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<tr class="memitem:a428769a16e9525e56588d7c7709d25a6" id="r_a428769a16e9525e56588d7c7709d25a6"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a428769a16e9525e56588d7c7709d25a6"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a428769a16e9525e56588d7c7709d25a6">rect_perimeter</a> (T length, T width)</td></tr>
<tr class="memdesc:a428769a16e9525e56588d7c7709d25a6"><td class="mdescLeft">&#160;</td><td class="mdescRight">perimeter of a <a href="https://en.wikipedia.org/wiki/Rectangle" target="_blank">rectangle</a> ( 2(l + w) ) <br /></td></tr>
<tr class="separator:a428769a16e9525e56588d7c7709d25a6"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a3d1e4db743b189f309327572663415f3" id="r_a3d1e4db743b189f309327572663415f3"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a3d1e4db743b189f309327572663415f3"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a3d1e4db743b189f309327572663415f3">triangle_perimeter</a> (T base, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>, T hypotenuse)</td></tr>
<tr class="memdesc:a3d1e4db743b189f309327572663415f3"><td class="mdescLeft">&#160;</td><td class="mdescRight">perimeter of a <a href="https://en.wikipedia.org/wiki/Triangle" target="_blank">triangle</a> (a + b + c) <br /></td></tr>
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<tr class="memitem:ad0acf82b7bc920182bf8322d1e103953" id="r_ad0acf82b7bc920182bf8322d1e103953"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:ad0acf82b7bc920182bf8322d1e103953"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ad0acf82b7bc920182bf8322d1e103953">circle_perimeter</a> (T radius)</td></tr>
<tr class="memdesc:ad0acf82b7bc920182bf8322d1e103953"><td class="mdescLeft">&#160;</td><td class="mdescRight">perimeter of a <a href="https://en.wikipedia.org/wiki/perimeter_of_a_circle" target="_blank">circle</a> (2 * pi * r) <br /></td></tr>
<tr class="separator:ad0acf82b7bc920182bf8322d1e103953"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a0efb235330ff48e14fd31faaccbcebb3" id="r_a0efb235330ff48e14fd31faaccbcebb3"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a0efb235330ff48e14fd31faaccbcebb3"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a0efb235330ff48e14fd31faaccbcebb3">parallelogram_perimeter</a> (T base, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:a0efb235330ff48e14fd31faaccbcebb3"><td class="mdescLeft">&#160;</td><td class="mdescRight">perimeter of a <a href="https://en.wikipedia.org/wiki/Parallelogram" target="_blank">parallelogram</a> 2(b + h) <br /></td></tr>
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<tr class="memitem:a8998ca7b1886d1d7d00aef3b457a9b1b" id="r_a8998ca7b1886d1d7d00aef3b457a9b1b"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a8998ca7b1886d1d7d00aef3b457a9b1b"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a8998ca7b1886d1d7d00aef3b457a9b1b">cube_surface_perimeter</a> (T length)</td></tr>
<tr class="memdesc:a8998ca7b1886d1d7d00aef3b457a9b1b"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface perimeter of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> ( 12 <br /></td></tr>
<tr class="separator:a8998ca7b1886d1d7d00aef3b457a9b1b"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa8592c3279c41a2c6d4d64eeb488f63f" id="r_aa8592c3279c41a2c6d4d64eeb488f63f"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:aa8592c3279c41a2c6d4d64eeb488f63f"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#aa8592c3279c41a2c6d4d64eeb488f63f">n_polygon_surface_perimeter</a> (T sides, T length)</td></tr>
<tr class="memdesc:aa8592c3279c41a2c6d4d64eeb488f63f"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface perimeter of a <a href="https://www.cuemath.com/measurement/perimeter-of-polygon/" target="_blank">n-polygon</a> ( n * l) <br /></td></tr>
<tr class="separator:aa8592c3279c41a2c6d4d64eeb488f63f"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a1d4df7a4e43a2eac1acc0ac610487c73" id="r_a1d4df7a4e43a2eac1acc0ac610487c73"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a1d4df7a4e43a2eac1acc0ac610487c73"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a1d4df7a4e43a2eac1acc0ac610487c73">cylinder_surface_perimeter</a> (T radius, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:a1d4df7a4e43a2eac1acc0ac610487c73"><td class="mdescLeft">&#160;</td><td class="mdescRight">surface perimeter of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> (2 * radius + 2 * height) <br /></td></tr>
<tr class="separator:a1d4df7a4e43a2eac1acc0ac610487c73"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a8a48be4d7f14e34c5c92925bc1cbf3bb" id="r_a8a48be4d7f14e34c5c92925bc1cbf3bb"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a8a48be4d7f14e34c5c92925bc1cbf3bb">power_of_two</a> (int n)</td></tr>
<tr class="memdesc:a8a48be4d7f14e34c5c92925bc1cbf3bb"><td class="mdescLeft">&#160;</td><td class="mdescRight">This function finds whether a number is power of 2 or not. <br /></td></tr>
<tr class="separator:a8a48be4d7f14e34c5c92925bc1cbf3bb"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aacb1411ef2029e81f249c21e17c96fdb" id="r_aacb1411ef2029e81f249c21e17c96fdb"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>&lt; long double &gt;, 2 &gt;&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aacb1411ef2029e81f249c21e17c96fdb">quadraticEquation</a> (long double a, long double b, long double c)</td></tr>
<tr class="memdesc:aacb1411ef2029e81f249c21e17c96fdb"><td class="mdescLeft">&#160;</td><td class="mdescRight">Quadratic equation calculator. <br /></td></tr>
<tr class="separator:aacb1411ef2029e81f249c21e17c96fdb"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae1ca505751f5a6d3977b86372cfe75ea" id="r_ae1ca505751f5a6d3977b86372cfe75ea"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae1ca505751f5a6d3977b86372cfe75ea">binomialCoeffSum</a> (uint64_t n)</td></tr>
<tr class="separator:ae1ca505751f5a6d3977b86372cfe75ea"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae413098478fa38acaac887b7654f0725" id="r_ae413098478fa38acaac887b7654f0725"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:ae413098478fa38acaac887b7654f0725"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#ae413098478fa38acaac887b7654f0725">cube_volume</a> (T length)</td></tr>
<tr class="memdesc:ae413098478fa38acaac887b7654f0725"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> <br /></td></tr>
<tr class="separator:ae413098478fa38acaac887b7654f0725"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a3fdc74c24697ec5bb5c3698c96117c12" id="r_a3fdc74c24697ec5bb5c3698c96117c12"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a3fdc74c24697ec5bb5c3698c96117c12"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a3fdc74c24697ec5bb5c3698c96117c12">rect_prism_volume</a> (T length, T width, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:a3fdc74c24697ec5bb5c3698c96117c12"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Cuboid" target="_blank">rectangular</a> prism. <br /></td></tr>
<tr class="separator:a3fdc74c24697ec5bb5c3698c96117c12"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a3fe35440c27758ecc2287e08217d63a7" id="r_a3fe35440c27758ecc2287e08217d63a7"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a3fe35440c27758ecc2287e08217d63a7"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a3fe35440c27758ecc2287e08217d63a7">cone_volume</a> (T radius, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>, double PI=3.14)</td></tr>
<tr class="memdesc:a3fe35440c27758ecc2287e08217d63a7"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Cone" target="_blank">cone</a> <br /></td></tr>
<tr class="separator:a3fe35440c27758ecc2287e08217d63a7"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a2d704a7b72a6b2db8b76c8581b577b2c" id="r_a2d704a7b72a6b2db8b76c8581b577b2c"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a2d704a7b72a6b2db8b76c8581b577b2c"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a2d704a7b72a6b2db8b76c8581b577b2c">triangle_prism_volume</a> (T base, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>, T depth)</td></tr>
<tr class="memdesc:a2d704a7b72a6b2db8b76c8581b577b2c"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Triangular_prism" target="_blank">triangular</a> prism. <br /></td></tr>
<tr class="separator:a2d704a7b72a6b2db8b76c8581b577b2c"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a94db02b3c9e55a69ac1696f30e2f761c" id="r_a94db02b3c9e55a69ac1696f30e2f761c"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a94db02b3c9e55a69ac1696f30e2f761c"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a94db02b3c9e55a69ac1696f30e2f761c">pyramid_volume</a> (T length, T width, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>)</td></tr>
<tr class="memdesc:a94db02b3c9e55a69ac1696f30e2f761c"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Pyramid_(geometry)" target="_blank">pyramid</a> <br /></td></tr>
<tr class="separator:a94db02b3c9e55a69ac1696f30e2f761c"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a34d66a77c19ce9b8b3a3d14352b34551" id="r_a34d66a77c19ce9b8b3a3d14352b34551"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a34d66a77c19ce9b8b3a3d14352b34551"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a34d66a77c19ce9b8b3a3d14352b34551">sphere_volume</a> (T radius, double PI=3.14)</td></tr>
<tr class="memdesc:a34d66a77c19ce9b8b3a3d14352b34551"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Sphere" target="_blank">sphere</a> <br /></td></tr>
<tr class="separator:a34d66a77c19ce9b8b3a3d14352b34551"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:abde24398be43538c62e4a496968e60ca" id="r_abde24398be43538c62e4a496968e60ca"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:abde24398be43538c62e4a496968e60ca"><td class="memTemplItemLeft" align="right" valign="top">T&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#abde24398be43538c62e4a496968e60ca">cylinder_volume</a> (T radius, T <a class="el" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>, double PI=3.14)</td></tr>
<tr class="memdesc:abde24398be43538c62e4a496968e60ca"><td class="mdescLeft">&#160;</td><td class="mdescRight">The volume of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> <br /></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>for IO operations </p>
<p>for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> and <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></p>
<p><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></p>
<p>for IO implementations</p>
<p>for io operations</p>
<p>Evaluate recurrence relation using <a href="https://www.hackerearth.com/practice/notes/matrix-exponentiation-1/" target="_blank">matrix exponentiation</a>.</p>
<p>Math algorithms.</p>
<p>Mathematical functions.</p>
<p>for I/O operations</p>
<p>Mathematical algorithms.</p>
<p>for cout</p>
<p>for M_PI definition and pow()</p>
<p>for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></p>
<p>for assert</p>
<p>Mathematical algorithms</p>
<p>for assert for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a> for IO operations</p>
<p>Mathematical algorithms</p>
<p>for assert for uint16_t datatype for IO operations</p>
<p>Mathematical algorithms</p>
<p>for assert for int32_t type for atoi</p>
<p>Mathematical algorithms</p>
<p>For assert For timing the sieve For IO operations For string handling For <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></p>
<p>for IO operations for assert</p>
<p>for assert</p>
<p>Math algorithms</p>
<p>for assert for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> and <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></p>
<p>Mathematical algorithms</p>
<p>for std::abs for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a> for assert</p>
<p>Maths algorithms</p>
<p>for assert for mathematical functions for passing in functions for IO operations</p>
<p>for math functions for fixed size data types for time to initialize rng for function pointers for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> for random number generation for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></p>
<p>for assert for integral types for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a> for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></p>
<p>for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> and <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> for assert</p>
<p>Given a recurrence relation; evaluate the value of nth term. For e.g., For fibonacci series, recurrence series is <code>f(n) = f(n-1) + f(n-2)</code> where <code>f(0) = 0</code> and <code>f(1) = 1</code>. Note that the method used only demonstrates recurrence relation with one variable (n), unlike <code>nCr</code> problem, since it has two (n, r)</p>
<h3><a class="anchor" id="autotoc_md84"></a>
Algorithm</h3>
<p>This problem can be solved using matrix exponentiation method. </p><dl class="section see"><dt>See also</dt><dd>here for simple <a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_exponentiation.cpp" target="_blank">number exponentiation algorithm</a> or <a href="https://en.wikipedia.org/wiki/Exponentiation_by_squaring" target="_blank">explaination here</a>. </dd></dl>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/AshishYUO" target="_blank">Ashish Daulatabad</a> for assert for IO operations for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a> STL</dd></dl>
<p>Mathematical algorithms</p>
<p>for assert for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></p>
<p>Maths algorithms.</p>
<p>for assert for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></p>
<p>Mathematical algorithms</p>
<p>for assert for M_PI definition and pow() for uint16_t datatype</p>
<p>Mathematical algorithms</p>
<p><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a> assert <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/trunc.html">std::trunc</a>, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a> <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a> <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a> <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setprecision.html">std::setprecision</a></p>
<p>Mathematical algorithms</p>
<p>for assert for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a> for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::uint32_t</a></p>
<p>Mathematical algorithms </p>
</div><h2 class="groupheader">Typedef Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a006c8ebd11bb1e035815dc835a1a7a85">&#9670;&#160;</a></span>Point</h2>
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<td class="memname">using <a class="el" href="#a006c8ebd11bb1e035815dc835a1a7a85">math::Point</a></td>
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<b>Initial value:</b><div class="fragment"><div class="line"> <span class="keyword">struct </span>{</div>
<div class="line"> <span class="keywordtype">double</span> x;</div>
<div class="line"> <span class="keywordtype">double</span> y;</div>
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<p>structure of points containing two numbers, x and y, such that 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. </p>
<div class="fragment"><div class="line"><span class="lineno"> 35</span> {</div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordtype">double</span> x;</div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordtype">double</span> y;</div>
<div class="line"><span class="lineno"> 38</span>};</div>
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<h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#ab37f3a7302a84179aae682c79d8390bf">&#9670;&#160;</a></span>aliquot_sum()</h2>
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<td class="memname">uint64_t math::aliquot_sum </td>
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<p>to return the aliquot sum of a number </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">num</td><td>The input number </td></tr>
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</dd>
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<div class="fragment"><div class="line"><span class="lineno"> 35</span> {</div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordflow">if</span> (num == 0 || num == 1) {</div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordflow">return</span> 0; <span class="comment">// The aliquot sum for 0 and 1 is 0</span></div>
<div class="line"><span class="lineno"> 38</span> }</div>
<div class="line"><span class="lineno"> 39</span> </div>
<div class="line"><span class="lineno"> 40</span> uint64_t sum = 0;</div>
<div class="line"><span class="lineno"> 41</span> </div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">for</span> (uint64_t i = 1; i &lt;= num / 2; i++) {</div>
<div class="line"><span class="lineno"> 43</span> <span class="keywordflow">if</span> (num % i == 0) {</div>
<div class="line"><span class="lineno"> 44</span> sum += i;</div>
<div class="line"><span class="lineno"> 45</span> }</div>
<div class="line"><span class="lineno"> 46</span> }</div>
<div class="line"><span class="lineno"> 47</span> </div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">sum</a>;</div>
<div class="line"><span class="lineno"> 49</span>}</div>
<div class="ttc" id="anamespacemachine__learning_html_a6f1c98c016ad34ff3d9f39372161bd35"><div class="ttname"><a href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">machine_learning::sum</a></div><div class="ttdeci">T sum(const std::vector&lt; std::valarray&lt; T &gt; &gt; &amp;A)</div><div class="ttdef"><b>Definition</b> vector_ops.hpp:232</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#abf7f2a6d91f1ca6c89698792aea3f188">&#9670;&#160;</a></span>approximate_pi()</h2>
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<td class="memname">double math::approximate_pi </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; <a class="el" href="#a006c8ebd11bb1e035815dc835a1a7a85">Point</a> &gt; &amp;</td> <td class="paramname"><span class="paramname"><em>pts</em></span></td><td>)</td>
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<p>This function uses the points in a given vector 'pts' (drawn at random) to return an approximation of the number π. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">pts</td><td>Each item of pts contains a point. A point is represented by the point structure (coded above). </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>an estimate of the number π. </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 47</span> {</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordtype">double</span> count = 0; <span class="comment">// Points in circle</span></div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">for</span> (<a class="code hl_struct" href="../../d8/dc8/struct_point.html">Point</a> p : pts) {</div>
<div class="line"><span class="lineno"> 50</span> <span class="keywordflow">if</span> ((p.x * p.x) + (p.y * p.y) &lt;= 1) {</div>
<div class="line"><span class="lineno"> 51</span> count++;</div>
<div class="line"><span class="lineno"> 52</span> }</div>
<div class="line"><span class="lineno"> 53</span> }</div>
<div class="line"><span class="lineno"> 54</span> <span class="keywordflow">return</span> 4.0 * count / <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(pts.size());</div>
<div class="line"><span class="lineno"> 55</span>}</div>
<div class="ttc" id="astruct_point_html"><div class="ttname"><a href="../../d8/dc8/struct_point.html">Point</a></div><div class="ttdef"><b>Definition</b> line_segment_intersection.cpp:12</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8e6eede206201db0d1dbb618fa969bec">&#9670;&#160;</a></span>are_amicable()</h2>
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<td class="memname">bool math::are_amicable </td>
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<p>Function to check whether the pair is amicable or not. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">x</td><td>First number. </td></tr>
<tr><td class="paramname">y</td><td>Second number. </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd><code>true</code> if the pair is amicable </dd>
<dd>
<code>false</code> if the pair is not amicable </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 58</span> {</div>
<div class="line"><span class="lineno"> 59</span> <span class="keywordflow">return</span> (<a class="code hl_function" href="#af05567415a9ea36c254b54e3d5a2152a">sum_of_divisor</a>(x) == y) &amp;&amp; (<a class="code hl_function" href="#af05567415a9ea36c254b54e3d5a2152a">sum_of_divisor</a>(y) == x);</div>
<div class="line"><span class="lineno"> 60</span>}</div>
<div class="ttc" id="anamespacemath_html_af05567415a9ea36c254b54e3d5a2152a"><div class="ttname"><a href="#af05567415a9ea36c254b54e3d5a2152a">math::sum_of_divisor</a></div><div class="ttdeci">int sum_of_divisor(int num)</div><div class="ttdoc">Function to calculate the sum of all the proper divisor of an integer.</div><div class="ttdef"><b>Definition</b> check_amicable_pair.cpp:31</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae1ca505751f5a6d3977b86372cfe75ea">&#9670;&#160;</a></span>binomialCoeffSum()</h2>
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<td class="memname">uint64_t math::binomialCoeffSum </td>
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<p>Function to calculate sum of binomial coefficients </p><dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">n</td><td>number </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>Sum of binomial coefficients of number </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 26</span> {</div>
<div class="line"><span class="lineno"> 27</span> <span class="comment">// Calculating 2^n</span></div>
<div class="line"><span class="lineno"> 28</span> <span class="keywordflow">return</span> (1 &lt;&lt; n);</div>
<div class="line"><span class="lineno"> 29</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a40e36c67da78d2131408c57ee091ad75">&#9670;&#160;</a></span>circle_area()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">T math::circle_area </td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span></td><td>)</td>
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<p>area of a <a href="https://en.wikipedia.org/wiki/Area_of_a_circle" target="_blank">circle</a> (pi </p>
<ul>
<li>r^2) <dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the circle </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>area of the circle </dd></dl>
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<div class="fragment"><div class="line"><span class="lineno"> 63</span> {</div>
<div class="line"><span class="lineno"> 64</span> <span class="keywordflow">return</span> M_PI * pow(radius, 2);</div>
<div class="line"><span class="lineno"> 65</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ad0acf82b7bc920182bf8322d1e103953">&#9670;&#160;</a></span>circle_perimeter()</h2>
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<td class="memname">T math::circle_perimeter </td>
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<p>perimeter of a <a href="https://en.wikipedia.org/wiki/perimeter_of_a_circle" target="_blank">circle</a> (2 * pi * r) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the circle </td></tr>
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</dd>
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<dl class="section return"><dt>Returns</dt><dd>perimeter of the circle </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 63</span> {</div>
<div class="line"><span class="lineno"> 64</span> <span class="keywordflow">return</span> 2 * M_PI * radius;</div>
<div class="line"><span class="lineno"> 65</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3fe35440c27758ecc2287e08217d63a7">&#9670;&#160;</a></span>cone_volume()</h2>
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<td class="memname">T math::cone_volume </td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span>, </td>
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<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>PI</em></span><span class="paramdefsep"> = </span><span class="paramdefval">3.14</span>&#160;)</td>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Cone" target="_blank">cone</a> </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>The radius of the base circle </td></tr>
<tr><td class="paramname">height</td><td>The height of the cone </td></tr>
<tr><td class="paramname">PI</td><td>The definition of the constant PI </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>The volume of the cone </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 53</span> {</div>
<div class="line"><span class="lineno"> 54</span> <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(radius, 2) * PI * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> / 3;</div>
<div class="line"><span class="lineno"> 55</span>}</div>
<div class="ttc" id="aavltree_8cpp_html_ae4a66d8b0c2b0d626aea45977e358c83"><div class="ttname"><a href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a></div><div class="ttdeci">int height(node *root)</div><div class="ttdef"><b>Definition</b> avltree.cpp:38</div></div>
<div class="ttc" id="apow_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a></div><div class="ttdeci">T pow(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#abc46c784a297fc1d2eb8b33a327fba4c">&#9670;&#160;</a></span>cube_surface_area()</h2>
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<td class="memname">T math::cube_surface_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span></td><td>)</td>
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<p>surface area of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> ( 6 * (l </p>
<ul>
<li>l)) <dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the cube </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface area of the cube </dd></dl>
</li>
</ul>
<div class="fragment"><div class="line"><span class="lineno"> 86</span> {</div>
<div class="line"><span class="lineno"> 87</span> <span class="keywordflow">return</span> 6 * length * length;</div>
<div class="line"><span class="lineno"> 88</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8998ca7b1886d1d7d00aef3b457a9b1b">&#9670;&#160;</a></span>cube_surface_perimeter()</h2>
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<td class="memname">T math::cube_surface_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span></td><td>)</td>
<td></td>
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<p>surface perimeter of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> ( 12 </p>
<ul>
<li>l) <dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the cube </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface perimeter of the cube </dd></dl>
</li>
</ul>
<div class="fragment"><div class="line"><span class="lineno"> 86</span> {</div>
<div class="line"><span class="lineno"> 87</span> <span class="keywordflow">return</span> 12 * length;</div>
<div class="line"><span class="lineno"> 88</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae413098478fa38acaac887b7654f0725">&#9670;&#160;</a></span>cube_volume()</h2>
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<td class="memname">T math::cube_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span></td><td>)</td>
<td></td>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Cube" target="_blank">cube</a> </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>The length of the cube </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the cube </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 28</span> {</div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(length, 3);</div>
<div class="line"><span class="lineno"> 30</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ac5803413618fcfb922cb32c6db0fc864">&#9670;&#160;</a></span>cylinder_surface_area()</h2>
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<td class="memname">T math::cylinder_surface_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
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<p>surface area of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> (2 * pi * r * h + 2 * pi * r^2) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the cylinder </td></tr>
<tr><td class="paramname">height</td><td>is the height of the cylinder </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface area of the cylinder </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 109</span> {</div>
<div class="line"><span class="lineno"> 110</span> <span class="keywordflow">return</span> 2 * M_PI * radius * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> + 2 * M_PI * pow(radius, 2);</div>
<div class="line"><span class="lineno"> 111</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1d4df7a4e43a2eac1acc0ac610487c73">&#9670;&#160;</a></span>cylinder_surface_perimeter()</h2>
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<td class="memname">T math::cylinder_surface_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span>, </td>
</tr>
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<p>surface perimeter of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> (2 * radius + 2 * height) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the cylinder </td></tr>
<tr><td class="paramname">height</td><td>is the height of the cylinder </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface perimeter of the cylinder </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 111</span> {</div>
<div class="line"><span class="lineno"> 112</span> <span class="keywordflow">return</span> (2 * radius) + (2 * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>);</div>
<div class="line"><span class="lineno"> 113</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#abde24398be43538c62e4a496968e60ca">&#9670;&#160;</a></span>cylinder_volume()</h2>
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<td class="memname">T math::cylinder_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>PI</em></span><span class="paramdefsep"> = </span><span class="paramdefval">3.14</span>&#160;)</td>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Cylinder" target="_blank">cylinder</a> </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>The radius of the base circle </td></tr>
<tr><td class="paramname">height</td><td>The height of the cylinder </td></tr>
<tr><td class="paramname">PI</td><td>The definition of the constant PI </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the cylinder </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 103</span> {</div>
<div class="line"><span class="lineno"> 104</span> <span class="keywordflow">return</span> PI * <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(radius, 2) * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>;</div>
<div class="line"><span class="lineno"> 105</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a7e78996673df791014cfe540b183456a">&#9670;&#160;</a></span>factorial()</h2>
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<td class="memname">uint64_t math::factorial </td>
<td>(</td>
<td class="paramtype">uint8_t</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>function to find factorial of given number </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>is the number which is to be factorialized </td></tr>
</table>
</dd>
</dl>
<dl class="section warning"><dt>Warning</dt><dd>Maximum value for the parameter is 20 as 21! cannot be represented in 64 bit unsigned int </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 29</span> {</div>
<div class="line"><span class="lineno"> 30</span> <span class="keywordflow">if</span> (n &lt; 20) {</div>
<div class="line"><span class="lineno"> 31</span> <span class="keywordflow">throw</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a>(<span class="stringliteral">&quot;maximum value is 20\n&quot;</span>);</div>
<div class="line"><span class="lineno"> 32</span> }</div>
<div class="line"><span class="lineno"> 33</span> <span class="keywordflow">if</span> (n == 0) {</div>
<div class="line"><span class="lineno"> 34</span> <span class="keywordflow">return</span> 1;</div>
<div class="line"><span class="lineno"> 35</span> }</div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordflow">return</span> n * <a class="code hl_function" href="#a7e78996673df791014cfe540b183456a">factorial</a>(n - 1);</div>
<div class="line"><span class="lineno"> 37</span>}</div>
<div class="ttc" id="ainvalid_argument_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a></div></div>
<div class="ttc" id="anamespacemath_html_a7e78996673df791014cfe540b183456a"><div class="ttname"><a href="#a7e78996673df791014cfe540b183456a">math::factorial</a></div><div class="ttdeci">uint64_t factorial(uint8_t n)</div><div class="ttdoc">function to find factorial of given number</div><div class="ttdef"><b>Definition</b> factorial.cpp:29</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3277e65a8f380e7632791975bfba0efb">&#9670;&#160;</a></span>hemi_sphere_surface_area()</h2>
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<td class="memname">T math::hemi_sphere_surface_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span></td><td>)</td>
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<p>surface area of a <a href="https://en.wikipedia.org/wiki/Surface_area" target="_blank">hemi-sphere</a> ( 3 * pi * r^2) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the hemi-sphere </td></tr>
</table>
</dd>
</dl>
<dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">T</td><td>datatype of radius </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface area of the hemi-sphere </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 121</span> {</div>
<div class="line"><span class="lineno"> 122</span> <span class="keywordflow">return</span> 3 * M_PI * pow(radius, 2);</div>
<div class="line"><span class="lineno"> 123</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aec65db4e5c7317323227f026fe50ef11">&#9670;&#160;</a></span>integral_approx()</h2>
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<td class="memname">double math::integral_approx </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>lb</em></span>, </td>
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<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>ub</em></span>, </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a>&lt; double(double)&gt; &amp;</td> <td class="paramname"><span class="paramname"><em>func</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>delta</em></span><span class="paramdefsep"> = </span><span class="paramdefval">.0001</span>&#160;)</td>
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<p>Computes integral approximation. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">lb</td><td>lower bound </td></tr>
<tr><td class="paramname">ub</td><td>upper bound </td></tr>
<tr><td class="paramname">func</td><td>function passed in </td></tr>
<tr><td class="paramname">delta</td><td></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>integral approximation of function from [lb, ub] </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 33</span> {</div>
<div class="line"><span class="lineno"> 34</span> <span class="keywordtype">double</span> result = 0;</div>
<div class="line"><span class="lineno"> 35</span> uint64_t numDeltas = <span class="keyword">static_cast&lt;</span>uint64_t<span class="keyword">&gt;</span>((ub - lb) / delta);</div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; numDeltas; i++) {</div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordtype">double</span> begin = lb + i * delta;</div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordtype">double</span> end = lb + (i + 1) * delta;</div>
<div class="line"><span class="lineno"> 39</span> result += delta * (func(begin) + func(end)) / 2;</div>
<div class="line"><span class="lineno"> 40</span> }</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">return</span> result;</div>
<div class="line"><span class="lineno"> 42</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a6c72f756a7bf1b9043c357e3fe7814ca">&#9670;&#160;</a></span>is_factorial()</h2>
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<td class="memname">bool math::is_factorial </td>
<td>(</td>
<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>Function to check if the given number is factorial of some number or not. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>number to be checked. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>true if number is a factorial returns true </dd>
<dd>
false if number is not a factorial </dd></dl>
<p>this loop is basically a reverse factorial calculation, where instead of multiplying we are dividing. We start at i = 2 since i = 1 has no impact division wise</p>
<p>if n was the sum of a factorial then it should be divided until it becomes 1</p>
<div class="fragment"><div class="line"><span class="lineno"> 27</span> {</div>
<div class="line"><span class="lineno"> 28</span> <span class="keywordflow">if</span> (n &lt;= 0) { <span class="comment">// factorial numbers are only ever positive Integers</span></div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 30</span> }</div>
<div class="line"><span class="lineno"> 31</span><span class="comment"></span> </div>
<div class="line"><span class="lineno"> 32</span><span class="comment"> /*!</span></div>
<div class="line"><span class="lineno"> 33</span><span class="comment"> * this loop is basically a reverse factorial calculation, where instead</span></div>
<div class="line"><span class="lineno"> 34</span><span class="comment"> * of multiplying we are dividing. We start at i = 2 since i = 1 has</span></div>
<div class="line"><span class="lineno"> 35</span><span class="comment"> * no impact division wise</span></div>
<div class="line"><span class="lineno"> 36</span><span class="comment"> */</span></div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordtype">int</span> i = 2;</div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">while</span> (n % i == 0) {</div>
<div class="line"><span class="lineno"> 39</span> n = n / i;</div>
<div class="line"><span class="lineno"> 40</span> i++;</div>
<div class="line"><span class="lineno"> 41</span> }</div>
<div class="line"><span class="lineno"> 42</span><span class="comment"></span> </div>
<div class="line"><span class="lineno"> 43</span><span class="comment"> /*!</span></div>
<div class="line"><span class="lineno"> 44</span><span class="comment"> * if n was the sum of a factorial then it should be divided until it</span></div>
<div class="line"><span class="lineno"> 45</span><span class="comment"> * becomes 1</span></div>
<div class="line"><span class="lineno"> 46</span><span class="comment"> */</span></div>
<div class="line"><span class="lineno"> 47</span> <span class="keywordflow">return</span> (n == 1);</div>
<div class="line"><span class="lineno"> 48</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#abd8f794b2229b42876169ff841b6e444">&#9670;&#160;</a></span>is_prime()</h2>
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<td class="memname">bool math::is_prime </td>
<td>(</td>
<td class="paramtype">int64_t</td> <td class="paramname"><span class="paramname"><em>num</em></span></td><td>)</td>
<td></td>
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<p>Function to check if the given number is prime or not. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">num</td><td>number to be checked. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>true if number is a prime </dd>
<dd>
false if number is not a prime. </dd></dl>
<p>Reduce all possibilities of a number which cannot be prime with the first 3 if, else if conditionals. Example: Since no even number, except 2 can be a prime number and the next prime we find after our checks is 5, we will start the for loop with i = 5. then for each loop we increment i by +6 and check if i or i+2 is a factor of the number; if it's a factor then we will return false. otherwise, true will be returned after the loop terminates at the terminating condition which is i*i &lt;= num</p>
<div class="fragment"><div class="line"><span class="lineno"> 31</span> {<span class="comment"></span></div>
<div class="line"><span class="lineno"> 32</span><span class="comment"> /*!</span></div>
<div class="line"><span class="lineno"> 33</span><span class="comment"> * Reduce all possibilities of a number which cannot be prime with the first</span></div>
<div class="line"><span class="lineno"> 34</span><span class="comment"> * 3 if, else if conditionals. Example: Since no even number, except 2 can</span></div>
<div class="line"><span class="lineno"> 35</span><span class="comment"> * be a prime number and the next prime we find after our checks is 5,</span></div>
<div class="line"><span class="lineno"> 36</span><span class="comment"> * we will start the for loop with i = 5. then for each loop we increment</span></div>
<div class="line"><span class="lineno"> 37</span><span class="comment"> * i by +6 and check if i or i+2 is a factor of the number; if it&#39;s a factor</span></div>
<div class="line"><span class="lineno"> 38</span><span class="comment"> * then we will return false. otherwise, true will be returned after the</span></div>
<div class="line"><span class="lineno"> 39</span><span class="comment"> * loop terminates at the terminating condition which is i*i &lt;= num</span></div>
<div class="line"><span class="lineno"> 40</span><span class="comment"> */</span></div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">if</span> (num &lt;= 1) {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 43</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (num == 2 || num == 3) {</div>
<div class="line"><span class="lineno"> 44</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 45</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (num % 2 == 0 || num % 3 == 0) {</div>
<div class="line"><span class="lineno"> 46</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 47</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordflow">for</span> (int64_t i = 5; i * i &lt;= num; i = i + 6) {</div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">if</span> (num % i == 0 || num % (i + 2) == 0) {</div>
<div class="line"><span class="lineno"> 50</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 51</span> }</div>
<div class="line"><span class="lineno"> 52</span> }</div>
<div class="line"><span class="lineno"> 53</span> }</div>
<div class="line"><span class="lineno"> 54</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 55</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a2565c745aac2f9561cc6fd9e56d9b77a">&#9670;&#160;</a></span>iterativeFactorial()</h2>
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<td class="memname">uint64_t math::iterativeFactorial </td>
<td>(</td>
<td class="paramtype">uint8_t</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>Calculates the factorial iteratively. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>Nth factorial. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>Factorial. </dd></dl>
<dl class="section note"><dt>Note</dt><dd>0! = 1. </dd></dl>
<dl class="section warning"><dt>Warning</dt><dd>Maximum=20 because there are no 128-bit integers in C++. 21! returns 1.419e+19, which is not 21! but (21! % UINT64_MAX). </dd></dl>
<dl class="section examples"><dt>Examples</dt><dd><a class="el" href="../../d6/d50/_2_users_2runner_2work_2_c-_plus-_plus_2_c-_plus-_plus_2math_2iterative_factorial_8cpp-example.html#a0">/Users/runner/work/C-Plus-Plus/C-Plus-Plus/math/iterative_factorial.cpp</a>.</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 47</span> {</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordflow">if</span> (n &gt; 20) {</div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">throw</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a>(<span class="stringliteral">&quot;Maximum n value is 20&quot;</span>);</div>
<div class="line"><span class="lineno"> 50</span> }</div>
<div class="line"><span class="lineno"> 51</span> </div>
<div class="line"><span class="lineno"> 52</span> <span class="comment">// 1 because it is the identity number of multiplication.</span></div>
<div class="line"><span class="lineno"> 53</span> uint64_t accumulator = 1;</div>
<div class="line"><span class="lineno"> 54</span> </div>
<div class="line"><span class="lineno"> 55</span> <span class="keywordflow">while</span> (n &gt; 1) {</div>
<div class="line"><span class="lineno"> 56</span> accumulator *= n;</div>
<div class="line"><span class="lineno"> 57</span> --n;</div>
<div class="line"><span class="lineno"> 58</span> }</div>
<div class="line"><span class="lineno"> 59</span> </div>
<div class="line"><span class="lineno"> 60</span> <span class="keywordflow">return</span> accumulator;</div>
<div class="line"><span class="lineno"> 61</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#afa39ec943a4836c878e1614fd89b146f">&#9670;&#160;</a></span>largestPower()</h2>
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<td class="memname">uint64_t math::largestPower </td>
<td>(</td>
<td class="paramtype">uint32_t</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
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<td class="paramtype">const uint16_t &amp;</td> <td class="paramname"><span class="paramname"><em>p</em></span>&#160;)</td>
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<p>Function to calculate largest power. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>number </td></tr>
<tr><td class="paramname">p</td><td>prime number </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>largest power </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 27</span> { </div>
<div class="line"><span class="lineno"> 28</span> <span class="comment">// Initialize result </span></div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordtype">int</span> x = 0; </div>
<div class="line"><span class="lineno"> 30</span> </div>
<div class="line"><span class="lineno"> 31</span> <span class="comment">// Calculate result </span></div>
<div class="line"><span class="lineno"> 32</span> <span class="keywordflow">while</span> (n) </div>
<div class="line"><span class="lineno"> 33</span> { </div>
<div class="line"><span class="lineno"> 34</span> n /= p; </div>
<div class="line"><span class="lineno"> 35</span> x += n; </div>
<div class="line"><span class="lineno"> 36</span> } </div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordflow">return</span> x; </div>
<div class="line"><span class="lineno"> 38</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a04065193d190d605e1f0d0d93a87e244">&#9670;&#160;</a></span>lcmSum()</h2>
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<td class="memname">uint64_t math::lcmSum </td>
<td>(</td>
<td class="paramtype">const uint16_t &amp;</td> <td class="paramname"><span class="paramname"><em>num</em></span></td><td>)</td>
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<p>Function to compute sum of euler totients in sumOfEulerTotient vector </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">num</td><td>input number </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>int Sum of LCMs, i.e. ∑LCM(i, num) from i = 1 to num </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 29</span> {</div>
<div class="line"><span class="lineno"> 30</span> uint64_t i = 0, j = 0;</div>
<div class="line"><span class="lineno"> 31</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;uint64_t&gt;</a> eulerTotient(num + 1);</div>
<div class="line"><span class="lineno"> 32</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;uint64_t&gt;</a> sumOfEulerTotient(num + 1);</div>
<div class="line"><span class="lineno"> 33</span> </div>
<div class="line"><span class="lineno"> 34</span> <span class="comment">// storing initial values in eulerTotient vector</span></div>
<div class="line"><span class="lineno"> 35</span> <span class="keywordflow">for</span> (i = 1; i &lt;= num; i++) {</div>
<div class="line"><span class="lineno"> 36</span> eulerTotient[i] = i;</div>
<div class="line"><span class="lineno"> 37</span> }</div>
<div class="line"><span class="lineno"> 38</span> </div>
<div class="line"><span class="lineno"> 39</span> <span class="comment">// applying totient sieve</span></div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">for</span> (i = 2; i &lt;= num; i++) {</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">if</span> (eulerTotient[i] == i) {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">for</span> (j = i; j &lt;= num; j += i) {</div>
<div class="line"><span class="lineno"> 43</span> eulerTotient[j] = eulerTotient[j] / i;</div>
<div class="line"><span class="lineno"> 44</span> eulerTotient[j] = eulerTotient[j] * (i - 1);</div>
<div class="line"><span class="lineno"> 45</span> }</div>
<div class="line"><span class="lineno"> 46</span> }</div>
<div class="line"><span class="lineno"> 47</span> }</div>
<div class="line"><span class="lineno"> 48</span> </div>
<div class="line"><span class="lineno"> 49</span> <span class="comment">// computing sum of euler totients</span></div>
<div class="line"><span class="lineno"> 50</span> <span class="keywordflow">for</span> (i = 1; i &lt;= num; i++) {</div>
<div class="line"><span class="lineno"> 51</span> <span class="keywordflow">for</span> (j = i; j &lt;= num; j += i) {</div>
<div class="line"><span class="lineno"> 52</span> sumOfEulerTotient[j] += eulerTotient[i] * i;</div>
<div class="line"><span class="lineno"> 53</span> }</div>
<div class="line"><span class="lineno"> 54</span> }</div>
<div class="line"><span class="lineno"> 55</span> </div>
<div class="line"><span class="lineno"> 56</span> <span class="keywordflow">return</span> ((sumOfEulerTotient[num] + 1) * num) / 2;</div>
<div class="line"><span class="lineno"> 57</span>}</div>
<div class="ttc" id="avector_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8d8e81a7cd59644b311ef9adb268f5f0">&#9670;&#160;</a></span>magic_number()</h2>
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<td class="memname">bool math::magic_number </td>
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<td class="paramtype">const uint64_t &amp;</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>Function to check if the given number is magic number or not. </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>number to be checked. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>if number is a magic number, returns true, else false. </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 32</span> {</div>
<div class="line"><span class="lineno"> 33</span> <span class="keywordflow">if</span> (n &lt;= 0) {</div>
<div class="line"><span class="lineno"> 34</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 35</span> }</div>
<div class="line"><span class="lineno"> 36</span> <span class="comment">// result stores the modulus of @param n with 9</span></div>
<div class="line"><span class="lineno"> 37</span> uint64_t result = n % 9;</div>
<div class="line"><span class="lineno"> 38</span> <span class="comment">// if result is 1 then the number is a magic number else not</span></div>
<div class="line"><span class="lineno"> 39</span> <span class="keywordflow">if</span> (result == 1) {</div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 41</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 43</span> }</div>
<div class="line"><span class="lineno"> 44</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a6e2dff75c5de70455b90c799d6ad6967">&#9670;&#160;</a></span>n_choose_r()</h2>
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template&lt;class T &gt; </div>
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<td class="memname">T math::n_choose_r </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>n</em></span>, </td>
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<p>This is the function implementation of \( \binom{n}{r} \). </p>
<p>We are calculating the ans with iterations instead of calculating three different factorials. Also, we are using the fact that \( \frac{n!}{r! (n-r)!} = \frac{(n - r + 1) \times \cdots \times n}{1 \times
\cdots \times r} \) </p><dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">T</td><td>Only for integer types such as long, int_64 etc </td></tr>
</table>
</dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>\( n \) in \( \binom{n}{r} \) </td></tr>
<tr><td class="paramname">r</td><td>\( r \) in \( \binom{n}{r} \) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>ans \( \binom{n}{r} \) </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 35</span> {</div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordflow">if</span> (r &gt; n / 2) {</div>
<div class="line"><span class="lineno"> 37</span> r = n - r; <span class="comment">// Because of the fact that nCr(n, r) == nCr(n, n - r)</span></div>
<div class="line"><span class="lineno"> 38</span> }</div>
<div class="line"><span class="lineno"> 39</span> T ans = 1;</div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt;= r; i++) {</div>
<div class="line"><span class="lineno"> 41</span> ans *= n - r + i;</div>
<div class="line"><span class="lineno"> 42</span> ans /= i;</div>
<div class="line"><span class="lineno"> 43</span> }</div>
<div class="line"><span class="lineno"> 44</span> <span class="keywordflow">return</span> ans;</div>
<div class="line"><span class="lineno"> 45</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8592c3279c41a2c6d4d64eeb488f63f">&#9670;&#160;</a></span>n_polygon_surface_perimeter()</h2>
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<td class="memname">T math::n_polygon_surface_perimeter </td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>sides</em></span>, </td>
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<p>surface perimeter of a <a href="https://www.cuemath.com/measurement/perimeter-of-polygon/" target="_blank">n-polygon</a> ( n * l) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the polygon </td></tr>
<tr><td class="paramname">sides</td><td>is the number of sides of the polygon </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface perimeter of the polygon </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 99</span> {</div>
<div class="line"><span class="lineno"> 100</span> <span class="keywordflow">return</span> sides * length;</div>
<div class="line"><span class="lineno"> 101</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a5de184925e68658f15415dd53954df4f">&#9670;&#160;</a></span>parallelogram_area()</h2>
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<td class="memname">T math::parallelogram_area </td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
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<p>area of a <a href="https://en.wikipedia.org/wiki/Parallelogram" target="_blank">parallelogram</a> (b * h) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">base</td><td>is the length of the bottom side of the parallelogram </td></tr>
<tr><td class="paramname">height</td><td>is the length of the tallest point in the parallelogram </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>area of the parallelogram </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 75</span> {</div>
<div class="line"><span class="lineno"> 76</span> <span class="keywordflow">return</span> base * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>;</div>
<div class="line"><span class="lineno"> 77</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0efb235330ff48e14fd31faaccbcebb3">&#9670;&#160;</a></span>parallelogram_perimeter()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">T math::parallelogram_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>base</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
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<p>perimeter of a <a href="https://en.wikipedia.org/wiki/Parallelogram" target="_blank">parallelogram</a> 2(b + h) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">base</td><td>is the length of the bottom side of the parallelogram </td></tr>
<tr><td class="paramname">height</td><td>is the length of the tallest point in the parallelogram </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>perimeter of the parallelogram </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 75</span> {</div>
<div class="line"><span class="lineno"> 76</span> <span class="keywordflow">return</span> 2 * (base + <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>);</div>
<div class="line"><span class="lineno"> 77</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ac37d3ba52eb296597d7a024ba8c4a5a5">&#9670;&#160;</a></span>phiFunction()</h2>
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<td class="memname">uint64_t math::phiFunction </td>
<td>(</td>
<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>Function to calculate Euler's Totient. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>the number to find the Euler's Totient of </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 39</span> {</div>
<div class="line"><span class="lineno"> 40</span> uint64_t result = n;</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">for</span> (uint64_t i = 2; i * i &lt;= n; i++) {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">if</span> (n % i != 0) <span class="keywordflow">continue</span>;</div>
<div class="line"><span class="lineno"> 43</span> <span class="keywordflow">while</span> (n % i == 0) n /= i;</div>
<div class="line"><span class="lineno"> 44</span> </div>
<div class="line"><span class="lineno"> 45</span> result -= result / i;</div>
<div class="line"><span class="lineno"> 46</span> }</div>
<div class="line"><span class="lineno"> 47</span> <span class="keywordflow">if</span> (n &gt; 1) result -= result / n;</div>
<div class="line"><span class="lineno"> 48</span> </div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">return</span> result;</div>
<div class="line"><span class="lineno"> 50</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#afcd07701d73ed65cd616bcba02737f3d">&#9670;&#160;</a></span>power()</h2>
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<td class="memname">uint64_t math::power </td>
<td>(</td>
<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>b</em></span>, </td>
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<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>c</em></span>&#160;)</td>
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<p>This function calculates a raised to exponent b under modulo c using modular exponentiation. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>integer base </td></tr>
<tr><td class="paramname">b</td><td>unsigned integer exponent </td></tr>
<tr><td class="paramname">c</td><td>integer modulo </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>a raised to power b modulo c </dd></dl>
<p>Initialize the answer to be returned</p>
<p>Update a if it is more than or equal to c</p>
<p>In case a is divisible by c;</p>
<p>If b is odd, multiply a with answer</p>
<p>b must be even now</p>
<p>b = b/2</p>
<div class="fragment"><div class="line"><span class="lineno"> 35</span> {</div>
<div class="line"><span class="lineno"> 36</span> uint64_t ans = 1; <span class="comment">/// Initialize the answer to be returned</span></div>
<div class="line"><span class="lineno"> 37</span> a = a % c; <span class="comment">/// Update a if it is more than or equal to c</span></div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">if</span> (a == 0) {</div>
<div class="line"><span class="lineno"> 39</span> <span class="keywordflow">return</span> 0; <span class="comment">/// In case a is divisible by c;</span></div>
<div class="line"><span class="lineno"> 40</span> }</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">while</span> (b &gt; 0) {<span class="comment"></span></div>
<div class="line"><span class="lineno"> 42</span><span class="comment"> /// If b is odd, multiply a with answer</span></div>
<div class="line"><span class="lineno"> 43</span><span class="comment"></span> <span class="keywordflow">if</span> (b &amp; 1) {</div>
<div class="line"><span class="lineno"> 44</span> ans = ((ans % c) * (a % c)) % c;</div>
<div class="line"><span class="lineno"> 45</span> }<span class="comment"></span></div>
<div class="line"><span class="lineno"> 46</span><span class="comment"> /// b must be even now</span></div>
<div class="line"><span class="lineno"> 47</span><span class="comment"></span> b = b &gt;&gt; 1; <span class="comment">/// b = b/2</span></div>
<div class="line"><span class="lineno"> 48</span> a = ((a % c) * (a % c)) % c;</div>
<div class="line"><span class="lineno"> 49</span> }</div>
<div class="line"><span class="lineno"> 50</span> <span class="keywordflow">return</span> ans;</div>
<div class="line"><span class="lineno"> 51</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8a48be4d7f14e34c5c92925bc1cbf3bb">&#9670;&#160;</a></span>power_of_two()</h2>
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<td class="memname">int math::power_of_two </td>
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<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>This function finds whether a number is power of 2 or not. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>value for which we want to check prints the result, as "Yes, the number n is a power of 2" or "No, the number is not a power of 2" without quotes </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>1 if <code>n</code> IS the power of 2 </dd>
<dd>
0 if n is NOT a power of 2 </dd></dl>
<p>result stores the bitwise and of n and n-1</p>
<div class="fragment"><div class="line"><span class="lineno"> 42</span> {<span class="comment"></span></div>
<div class="line"><span class="lineno"> 43</span><span class="comment"> /// result stores the</span></div>
<div class="line"><span class="lineno"> 44</span><span class="comment"> /// bitwise and of n and n-1</span></div>
<div class="line"><span class="lineno"> 45</span><span class="comment"></span> <span class="keywordtype">int</span> result = n &amp; (n - 1);</div>
<div class="line"><span class="lineno"> 46</span> </div>
<div class="line"><span class="lineno"> 47</span> <span class="keywordflow">if</span> (result == 0) {</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordflow">return</span> 1;</div>
<div class="line"><span class="lineno"> 49</span> }</div>
<div class="line"><span class="lineno"> 50</span> </div>
<div class="line"><span class="lineno"> 51</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 52</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ad09d59850865012a6fd95d89954c82e4">&#9670;&#160;</a></span>print_primes()</h2>
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<td class="memname">void math::print_primes </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; bool &gt; const &amp;</td> <td class="paramname"><span class="paramname"><em>primes</em></span></td><td>)</td>
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<p>Prints all the indexes of true values in the passed <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">primes</td><td>The vector that has been passed through <code>sieve(...)</code> </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 51</span> {</div>
<div class="line"><span class="lineno"> 52</span> <span class="keywordflow">for</span> (uint64_t i = 0; i &lt; <a class="code hl_function" href="../../de/d9b/prime__numbers_8cpp.html#a9575f3a51eeb8a57d657b3db6a4b441a">primes</a>.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/size.html">size</a>(); i++) {</div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">if</span> (<a class="code hl_function" href="../../de/d9b/prime__numbers_8cpp.html#a9575f3a51eeb8a57d657b3db6a4b441a">primes</a>[i]) {</div>
<div class="line"><span class="lineno"> 54</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; i &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"> 55</span> }</div>
<div class="line"><span class="lineno"> 56</span> }</div>
<div class="line"><span class="lineno"> 57</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="aprime__numbers_8cpp_html_a9575f3a51eeb8a57d657b3db6a4b441a"><div class="ttname"><a href="../../de/d9b/prime__numbers_8cpp.html#a9575f3a51eeb8a57d657b3db6a4b441a">primes</a></div><div class="ttdeci">std::vector&lt; int &gt; primes(size_t max)</div><div class="ttdef"><b>Definition</b> prime_numbers.cpp:12</div></div>
<div class="ttc" id="asize_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector/size.html">std::vector::size</a></div><div class="ttdeci">T size(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a94db02b3c9e55a69ac1696f30e2f761c">&#9670;&#160;</a></span>pyramid_volume()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">T math::pyramid_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>width</em></span>, </td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Pyramid_(geometry)" target="_blank">pyramid</a> </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>The length of the base shape (or base for triangles) </td></tr>
<tr><td class="paramname">width</td><td>The width of the base shape (or height for triangles) </td></tr>
<tr><td class="paramname">height</td><td>The height of the pyramid </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the pyramid </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 80</span> {</div>
<div class="line"><span class="lineno"> 81</span> <span class="keywordflow">return</span> length * width * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> / 3;</div>
<div class="line"><span class="lineno"> 82</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aacb1411ef2029e81f249c21e17c96fdb">&#9670;&#160;</a></span>quadraticEquation()</h2>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>&lt; long double &gt;, 2 &gt; math::quadraticEquation </td>
<td>(</td>
<td class="paramtype">long double</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
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<td class="paramtype">long double</td> <td class="paramname"><span class="paramname"><em>b</em></span>, </td>
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<td class="paramtype">long double</td> <td class="paramname"><span class="paramname"><em>c</em></span>&#160;)</td>
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<p>Quadratic equation calculator. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>quadratic coefficient. </td></tr>
<tr><td class="paramname">b</td><td>linear coefficient. </td></tr>
<tr><td class="paramname">c</td><td>constant </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>Array containing the roots of quadratic equation, incl. complex root. </dd></dl>
<dl class="section examples"><dt>Examples</dt><dd><a class="el" href="../../df/dcd/_2_users_2runner_2work_2_c-_plus-_plus_2_c-_plus-_plus_2math_2quadratic_equations_complex_numbers_8cpp-example.html#a1">/Users/runner/work/C-Plus-Plus/C-Plus-Plus/math/quadratic_equations_complex_numbers.cpp</a>.</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 55</span> {</div>
<div class="line"><span class="lineno"> 56</span> <span class="keywordflow">if</span> (a == 0) {</div>
<div class="line"><span class="lineno"> 57</span> <span class="keywordflow">throw</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/error/invalid_argument.html">std::invalid_argument</a>(<span class="stringliteral">&quot;quadratic coefficient cannot be 0&quot;</span>);</div>
<div class="line"><span class="lineno"> 58</span> }</div>
<div class="line"><span class="lineno"> 59</span> </div>
<div class="line"><span class="lineno"> 60</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> discriminant = b * b - 4 * a * c;</div>
<div class="line"><span class="lineno"> 61</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;std::complex&lt;long double&gt;</a>, 2&gt; solutions{0, 0};</div>
<div class="line"><span class="lineno"> 62</span> </div>
<div class="line"><span class="lineno"> 63</span> <span class="keywordflow">if</span> (discriminant == 0) {</div>
<div class="line"><span class="lineno"> 64</span> solutions[0] = -b * 0.5 / a;</div>
<div class="line"><span class="lineno"> 65</span> solutions[1] = -b * 0.5 / a;</div>
<div class="line"><span class="lineno"> 66</span> <span class="keywordflow">return</span> solutions;</div>
<div class="line"><span class="lineno"> 67</span> }</div>
<div class="line"><span class="lineno"> 68</span> </div>
<div class="line"><span class="lineno"> 69</span> <span class="comment">// Complex root (discriminant &lt; 0)</span></div>
<div class="line"><span class="lineno"> 70</span> <span class="comment">// Note that the left term (-b / 2a) is always real. The imaginary part</span></div>
<div class="line"><span class="lineno"> 71</span> <span class="comment">// appears when b^2 - 4ac &lt; 0, so sqrt(b^2 - 4ac) has no real roots. So,</span></div>
<div class="line"><span class="lineno"> 72</span> <span class="comment">// the imaginary component is i * (+/-)sqrt(abs(b^2 - 4ac)) / 2a.</span></div>
<div class="line"><span class="lineno"> 73</span> <span class="keywordflow">if</span> (discriminant &gt; 0) {</div>
<div class="line"><span class="lineno"> 74</span> <span class="comment">// Since discriminant &gt; 0, there are only real roots. Therefore,</span></div>
<div class="line"><span class="lineno"> 75</span> <span class="comment">// imaginary component = 0.</span></div>
<div class="line"><span class="lineno"> 76</span> solutions[0] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex&lt;long double&gt;</a>{</div>
<div class="line"><span class="lineno"> 77</span> (-b - <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(discriminant)) * 0.5 / a, 0};</div>
<div class="line"><span class="lineno"> 78</span> solutions[1] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex&lt;long double&gt;</a>{</div>
<div class="line"><span class="lineno"> 79</span> (-b + <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(discriminant)) * 0.5 / a, 0};</div>
<div class="line"><span class="lineno"> 80</span> <span class="keywordflow">return</span> solutions;</div>
<div class="line"><span class="lineno"> 81</span> }</div>
<div class="line"><span class="lineno"> 82</span> <span class="comment">// Since b^2 - 4ac is &lt; 0, for faster computation, -discriminant is</span></div>
<div class="line"><span class="lineno"> 83</span> <span class="comment">// enough to make it positive.</span></div>
<div class="line"><span class="lineno"> 84</span> solutions[0] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex&lt;long double&gt;</a>{</div>
<div class="line"><span class="lineno"> 85</span> -b * 0.5 / a, -<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(-discriminant) * 0.5 / a};</div>
<div class="line"><span class="lineno"> 86</span> solutions[1] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex&lt;long double&gt;</a>{</div>
<div class="line"><span class="lineno"> 87</span> -b * 0.5 / a, <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(-discriminant) * 0.5 / a};</div>
<div class="line"><span class="lineno"> 88</span> </div>
<div class="line"><span class="lineno"> 89</span> <span class="keywordflow">return</span> solutions;</div>
<div class="line"><span class="lineno"> 90</span>}</div>
<div class="ttc" id="aarray_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a></div></div>
<div class="ttc" id="acomplex_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a></div></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="#ab31d141f7c5b551746b1eee0eb4dedca">&#9670;&#160;</a></span>rect_area()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">T math::rect_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>width</em></span>&#160;)</td>
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<p>area of a <a href="https://en.wikipedia.org/wiki/Rectangle" target="_blank">rectangle</a> (l * w) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the rectangle </td></tr>
<tr><td class="paramname">width</td><td>is the width of the rectangle </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>area of the rectangle </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 40</span> {</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">return</span> length * width;</div>
<div class="line"><span class="lineno"> 42</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a428769a16e9525e56588d7c7709d25a6">&#9670;&#160;</a></span>rect_perimeter()</h2>
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<td class="memname">T math::rect_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>width</em></span>&#160;)</td>
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<p>perimeter of a <a href="https://en.wikipedia.org/wiki/Rectangle" target="_blank">rectangle</a> ( 2(l + w) ) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the rectangle </td></tr>
<tr><td class="paramname">width</td><td>is the width of the rectangle </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>perimeter of the rectangle </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 40</span> {</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">return</span> 2 * (length + width);</div>
<div class="line"><span class="lineno"> 42</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3fdc74c24697ec5bb5c3698c96117c12">&#9670;&#160;</a></span>rect_prism_volume()</h2>
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<td class="memname">T math::rect_prism_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>width</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
</tr>
</table>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Cuboid" target="_blank">rectangular</a> prism. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>The length of the base rectangle </td></tr>
<tr><td class="paramname">width</td><td>The width of the base rectangle </td></tr>
<tr><td class="paramname">height</td><td>The height of the rectangular prism </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the rectangular prism </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 41</span> {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">return</span> length * width * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a>;</div>
<div class="line"><span class="lineno"> 43</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a91366864111e1fac29722ca45e02ea8f">&#9670;&#160;</a></span>sieve()</h2>
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<td class="memname">void math::sieve </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; bool &gt; *</td> <td class="paramname"><span class="paramname"><em>vec</em></span></td><td>)</td>
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<p>Performs the sieve. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">vec</td><td>Array of bools, all initialised to true, where the number of elements is the highest number we wish to check for primeness </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 33</span> {</div>
<div class="line"><span class="lineno"> 34</span> (*vec)[0] = <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 35</span> (*vec)[1] = <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 36</span> </div>
<div class="line"><span class="lineno"> 37</span> <span class="comment">// The sieve sets values to false as they are found not prime</span></div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">for</span> (uint64_t n = 2; n &lt; vec-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/size.html">size</a>(); n++) {</div>
<div class="line"><span class="lineno"> 39</span> <span class="keywordflow">for</span> (uint64_t multiple = n &lt;&lt; 1; multiple &lt; vec-&gt;<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/size.html">size</a>();</div>
<div class="line"><span class="lineno"> 40</span> multiple += n) {</div>
<div class="line"><span class="lineno"> 41</span> (*vec)[multiple] = <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 42</span> }</div>
<div class="line"><span class="lineno"> 43</span> }</div>
<div class="line"><span class="lineno"> 44</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ab7f29862d30df351c317eedd60a0c656">&#9670;&#160;</a></span>sphere_surface_area()</h2>
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<td class="memname">T math::sphere_surface_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span></td><td>)</td>
<td></td>
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<p>surface area of a <a href="https://en.wikipedia.org/wiki/Sphere" target="_blank">sphere</a> ( 4 * pi * r^2) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>is the radius of the sphere </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>surface area of the sphere </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 97</span> {</div>
<div class="line"><span class="lineno"> 98</span> <span class="keywordflow">return</span> 4 * M_PI * pow(radius, 2);</div>
<div class="line"><span class="lineno"> 99</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a34d66a77c19ce9b8b3a3d14352b34551">&#9670;&#160;</a></span>sphere_volume()</h2>
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<td class="memname">T math::sphere_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>radius</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>PI</em></span><span class="paramdefsep"> = </span><span class="paramdefval">3.14</span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>The volume of a <a href="https://en.wikipedia.org/wiki/Sphere" target="_blank">sphere</a> </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">radius</td><td>The radius of the sphere </td></tr>
<tr><td class="paramname">PI</td><td>The definition of the constant PI </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the sphere </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 91</span> {</div>
<div class="line"><span class="lineno"> 92</span> <span class="keywordflow">return</span> PI * <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(radius, 3) * 4 / 3;</div>
<div class="line"><span class="lineno"> 93</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a971ce57e368f2f631cf1f4ff3f864049">&#9670;&#160;</a></span>square_area()</h2>
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<td class="memname">T math::square_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span></td><td>)</td>
<td></td>
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<p>area of a <a href="https://en.wikipedia.org/wiki/Square" target="_blank">square</a> (l * l) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the square </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>area of square </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 29</span> {</div>
<div class="line"><span class="lineno"> 30</span> <span class="keywordflow">return</span> length * length;</div>
<div class="line"><span class="lineno"> 31</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a9236348755183644f1225e162d01ab14">&#9670;&#160;</a></span>square_perimeter()</h2>
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<td class="memname">T math::square_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>length</em></span></td><td>)</td>
<td></td>
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<p>perimeter of a <a href="https://en.wikipedia.org/wiki/Square" target="_blank">square</a> (4 * l) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">length</td><td>is the length of the square </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>perimeter of square </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 28</span> {</div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordflow">return</span> 4 * length;</div>
<div class="line"><span class="lineno"> 30</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#af05567415a9ea36c254b54e3d5a2152a">&#9670;&#160;</a></span>sum_of_divisor()</h2>
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<td class="memname">int math::sum_of_divisor </td>
<td>(</td>
<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>num</em></span></td><td>)</td>
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<p>Function to calculate the sum of all the proper divisor of an integer. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">num</td><td>selected number. </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>Sum of the proper divisor of the number. </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 31</span> {</div>
<div class="line"><span class="lineno"> 32</span> <span class="comment">// Variable to store the sum of all proper divisors.</span></div>
<div class="line"><span class="lineno"> 33</span> <span class="keywordtype">int</span> sum = 1;</div>
<div class="line"><span class="lineno"> 34</span> <span class="comment">// Below loop condition helps to reduce Time complexity by a factor of</span></div>
<div class="line"><span class="lineno"> 35</span> <span class="comment">// square root of the number.</span></div>
<div class="line"><span class="lineno"> 36</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> div = 2; div * div &lt;= num; ++div) {</div>
<div class="line"><span class="lineno"> 37</span> <span class="comment">// Check &#39;div&#39; is divisor of &#39;num&#39;.</span></div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">if</span> (num % div == 0) {</div>
<div class="line"><span class="lineno"> 39</span> <span class="comment">// If both divisor are same, add once to &#39;sum&#39;</span></div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">if</span> (div == (num / div)) {</div>
<div class="line"><span class="lineno"> 41</span> sum += div;</div>
<div class="line"><span class="lineno"> 42</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 43</span> <span class="comment">// If both divisor are not the same, add both to &#39;sum&#39;.</span></div>
<div class="line"><span class="lineno"> 44</span> sum += (div + (num / div));</div>
<div class="line"><span class="lineno"> 45</span> }</div>
<div class="line"><span class="lineno"> 46</span> }</div>
<div class="line"><span class="lineno"> 47</span> }</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">sum</a>;</div>
<div class="line"><span class="lineno"> 49</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a50936ee98f4d40f17823befc65a32aec">&#9670;&#160;</a></span>test_eval()</h2>
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<td class="memname">void math::test_eval </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>approx</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>expected</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>threshold</em></span>&#160;)</td>
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</table>
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<p>Wrapper to evaluate if the approximated value is within <code>.XX%</code> threshold of the exact value. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">approx</td><td>aprroximate value </td></tr>
<tr><td class="paramname">exact</td><td>expected value </td></tr>
<tr><td class="paramname">threshold</td><td>values from [0, 1) </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 51</span> {</div>
<div class="line"><span class="lineno"> 52</span> assert(approx &gt;= expected * (1 - threshold));</div>
<div class="line"><span class="lineno"> 53</span> assert(approx &lt;= expected * (1 + threshold));</div>
<div class="line"><span class="lineno"> 54</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ab3b920cc56442abd92279ba23b50f4dc">&#9670;&#160;</a></span>triangle_area()</h2>
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<td class="memname">T math::triangle_area </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>base</em></span>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>&#160;)</td>
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<p>area of a <a href="https://en.wikipedia.org/wiki/Triangle" target="_blank">triangle</a> (b * h / 2) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">base</td><td>is the length of the bottom side of the triangle </td></tr>
<tr><td class="paramname">height</td><td>is the length of the tallest point in the triangle </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>area of the triangle </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 52</span> {</div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">return</span> base * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> / 2;</div>
<div class="line"><span class="lineno"> 54</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3d1e4db743b189f309327572663415f3">&#9670;&#160;</a></span>triangle_perimeter()</h2>
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<td class="memname">T math::triangle_perimeter </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>base</em></span>, </td>
</tr>
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<td class="paramkey"></td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>hypotenuse</em></span>&#160;)</td>
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<p>perimeter of a <a href="https://en.wikipedia.org/wiki/Triangle" target="_blank">triangle</a> (a + b + c) </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">base</td><td>is the length of the bottom side of the triangle </td></tr>
<tr><td class="paramname">height</td><td>is the length of the tallest point in the triangle </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>perimeter of the triangle </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 52</span> {</div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">return</span> base + <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> + hypotenuse;</div>
<div class="line"><span class="lineno"> 54</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a2d704a7b72a6b2db8b76c8581b577b2c">&#9670;&#160;</a></span>triangle_prism_volume()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">T math::triangle_prism_volume </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>base</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>height</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>depth</em></span>&#160;)</td>
</tr>
</table>
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<p>The volume of a <a href="https://en.wikipedia.org/wiki/Triangular_prism" target="_blank">triangular</a> prism. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">base</td><td>The length of the base triangle </td></tr>
<tr><td class="paramname">height</td><td>The height of the base triangles </td></tr>
<tr><td class="paramname">depth</td><td>The depth of the triangular prism (the height of the whole prism) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The volume of the triangular prism </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 67</span> {</div>
<div class="line"><span class="lineno"> 68</span> <span class="keywordflow">return</span> base * <a class="code hl_function" href="../../d8/dee/avltree_8cpp.html#ae4a66d8b0c2b0d626aea45977e358c83">height</a> * depth / 2;</div>
<div class="line"><span class="lineno"> 69</span>}</div>
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