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<title>Comparison of Cube Root Finding Algorithms</title>
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.roots.root_comparison.cbrt_comparison"></a><a class="link" href="cbrt_comparison.html" title="Comparison of Cube Root Finding Algorithms">Comparison
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of Cube Root Finding Algorithms</a>
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</h4></div></div></div>
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<p>
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In the table below, the cube root of 28 was computed for three <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
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types</a> floating-point types, and one <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
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type <a href="../../../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>
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using 50 decimal digit precision, using four algorithms.
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</p>
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<p>
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The 'exact' answer was computed using a 100 decimal digit type:
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</p>
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<pre class="programlisting"><span class="identifier">cpp_bin_float_100</span> <span class="identifier">full_answer</span> <span class="special">(</span><span class="string">"3.036588971875662519420809578505669635581453977248111123242141654169177268411884961770250390838097895"</span><span class="special">);</span>
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</pre>
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<p>
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Times were measured using <a href="../../../../../../../libs/timer/doc/index.html" target="_top">Boost.Timer</a>
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using <code class="computeroutput"><span class="keyword">class</span> <span class="identifier">cpu_timer</span></code>.
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<span class="emphasis"><em>Its</em></span> is the number of iterations taken to find
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the root.
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</li>
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<li class="listitem">
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<span class="emphasis"><em>Times</em></span> is the CPU time-taken in arbitrary units.
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</li>
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<li class="listitem">
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<span class="emphasis"><em>Norm</em></span> is a normalized time, in comparison to the
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quickest algorithm (with value 1.00).
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</li>
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<li class="listitem">
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<span class="emphasis"><em>Dis</em></span> is the distance from the nearest representation
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of the 'exact' root in bits. Distance from the 'exact' answer is measured
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by using function <a href="../../../../../../../libs/math/doc/html/math_toolkit/next_float/float_distance.html" target="_top">Boost.Math
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float_distance</a>. One or two bits distance means that all results
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are effectively 'correct'. Zero means 'exact' - the nearest <a href="http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding" target="_top">representable</a>
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value for the floating-point type.
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</li>
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</ul></div>
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<p>
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The cube-root function is a simple function, and is a contrived example
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for root-finding. It does allow us to investigate some of the factors controlling
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efficiency that may be extrapolated to more complex functions.
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</p>
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<p>
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The program used was <a href="../../../../../example/root_finding_algorithms.cpp" target="_top">root_finding_algorithms.cpp</a>.
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100000 evaluations of each floating-point type and algorithm were used
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and the CPU times were judged from repeat runs to have an uncertainty of
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10 %. Comparing MSVC for <code class="computeroutput"><span class="keyword">double</span></code>
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and <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
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(which are identical on this patform) may give a guide to uncertainty of
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timing.
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</p>
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<p>
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The requested precision was set as follows:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Precision Requested
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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TOMS748
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</p>
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</td>
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<td>
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<p>
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numeric_limits<T>::digits - 2
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Newton
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.6)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Halley
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.4)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Schröder
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.4)
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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The C++ Standard cube root function <a href="http://en.cppreference.com/w/cpp/numeric/math/cbrt" target="_top">std::cbrt</a>
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is only defined for built-in or fundamental types, so cannot be used
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with any User-Defined floating-point types like <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
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This, and that the cube function is so impeccably-behaved, allows the
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implementer to use many tricks to achieve a fast computation. On some
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platforms,<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cbrt</span></code> appeared several times as quick
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as the more general <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cbrt</span></code>,
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on other platforms / compiler options <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cbrt</span></code>
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is noticeably faster. In general, the results are highly dependent
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on the code-generation / processor architecture selection compiler
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options used. One can assume that the standard library will have been
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compiled with options <span class="emphasis"><em>nearly</em></span> optimal for the platform
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it was installed on, where as the user has more choice over the options
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used for Boost.Math. Pick something too general/conservative and performance
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suffers, while selecting options that make use of the latest instruction
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set opcodes speed's things up noticeably.
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</li>
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<li class="listitem">
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Two compilers in optimise mode were compared: GCC 4.9.1 using Netbeans
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IDS and Microsoft Visual Studio 2013 (Update 1) on the same hardware.
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The number of iterations seemed consistent, but the relative run-times
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surprisingly different.
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</li>
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<li class="listitem">
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<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cbrt</span></code> allows use with <span class="emphasis"><em>any
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user-defined floating-point type</em></span>, conveniently <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
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It too can take some advantage of the good-behaviour of the cube function,
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compared to the more general implementation in the nth root-finding
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examples. For example, it uses a polynomial approximation to generate
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a better guess than dividing the exponent by three, and can avoid the
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complex checks in <a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson
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iteration</a> required to prevent the search going wildly off-track.
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For a known precision, it may also be possible to fix the number of
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iterations, allowing inlining and loop unrolling. It also algebraically
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simplifies the Halley steps leading to a big reduction in the number
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of floating point operations required compared to a "black box"
|
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implementation that calculates the derivatives seperately and then
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combines them in the Halley code. Typically, it was found that computation
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using type <code class="computeroutput"><span class="keyword">double</span></code> took
|
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a few times longer when using the various root-finding algorithms directly
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rather than the hand coded/optimized <code class="computeroutput"><span class="identifier">cbrt</span></code>
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routine.
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</li>
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<li class="listitem">
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The importance of getting a good guess can be seen by the iteration
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count for the multiprecision case: here we "cheat" a little
|
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and use the cube-root calculated to double precision as the initial
|
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guess. The limitation of this tactic is that the range of possible
|
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(exponent) values may be less than the multiprecision type.
|
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</li>
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<li class="listitem">
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|
For <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
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types</a>, there was little to choose between the three derivative
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methods, but for <a href="../../../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>,
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<a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson
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iteration</a> was twice as fast. Note that the cube-root is an extreme
|
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test case as the cost of calling the functor is so cheap that the runtimes
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are largely dominated by the complexity of the iteration code.
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</li>
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<li class="listitem">
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Compiling with optimisation halved computation times, and any differences
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between algorithms became nearly negligible. The optimisation speed-up
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of the <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
|
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Algorithm 748: enclosing zeros of continuous functions</a> was
|
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especially noticable.
|
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</li>
|
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<li class="listitem">
|
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Using a multiprecision type like <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
|
|
for a precision of 50 decimal digits took a lot longer, as expected
|
|
because most computation uses software rather than 64-bit floating-point
|
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hardware. Speeds are often more than 50 times slower.
|
|
</li>
|
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<li class="listitem">
|
|
Using <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>,
|
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<a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
|
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748: enclosing zeros of continuous functions</a> was much slower
|
|
showing the benefit of using derivatives. <a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson
|
|
iteration</a> was found to be twice as quick as either of the second-derivative
|
|
methods: this is an extreme case though, the function and its derivatives
|
|
are so cheap to compute that we're really measuring the complexity
|
|
of the boilerplate root-finding code.
|
|
</li>
|
|
<li class="listitem">
|
|
For multiprecision types only one or two extra <span class="emphasis"><em>iterations</em></span>
|
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are needed to get the remaining 35 digits, whatever the algorithm used.
|
|
(The time taken was of course much greater for these types).
|
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</li>
|
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<li class="listitem">
|
|
Using a 100 decimal-digit type only doubled the time and required only
|
|
a very few more iterations, so the cost of extra precision is mainly
|
|
the underlying cost of computing more digits, not in the way the algorithm
|
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works. This confirms previous observations using <a href="http://www.shoup.net/ntl/" target="_top">NTL
|
|
A Library for doing Number Theory</a> high-precision types.
|
|
</li>
|
|
</ul></div>
|
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<h6>
|
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<a name="math_toolkit.roots.root_comparison.cbrt_comparison.h0"></a>
|
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<span class="phrase"><a name="math_toolkit.roots.root_comparison.cbrt_comparison.program_root_finding_algorithms_"></a></span><a class="link" href="cbrt_comparison.html#math_toolkit.roots.root_comparison.cbrt_comparison.program_root_finding_algorithms_">Program
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root_finding_algorithms.cpp, Microsoft Visual C++ version 12.0, Dinkumware
|
|
standard library version 610, Win32, x64<br> 1000000 evaluations of each
|
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of 5 root_finding algorithms.</a>
|
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</h6>
|
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<div class="table">
|
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<a name="math_toolkit.roots.root_comparison.cbrt_comparison.cbrt_4"></a><p class="title"><b>Table 12.1. Cube root(28) for float, double, long double and cpp_bin_float_50</b></p>
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<div class="table-contents"><table class="table" summary="Cube root(28) for float, double, long double and cpp_bin_float_50">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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float
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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double
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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long d
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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cpp50
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<td class="auto-generated"> </td>
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<td class="auto-generated"> </td>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Algorithm
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</p>
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</td>
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<td>
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<p>
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Its
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</p>
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</td>
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<td>
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<p>
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Times
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</p>
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</td>
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<td>
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<p>
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Norm
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</p>
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</td>
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<td>
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<p>
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Dis
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</p>
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</td>
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<td>
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</td>
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<td>
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<p>
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Its
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</p>
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</td>
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<td>
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<p>
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Times
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</p>
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</td>
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<td>
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<p>
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Norm
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</p>
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</td>
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<td>
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<p>
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Dis
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</p>
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</td>
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<td>
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</td>
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<td>
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<p>
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Its
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</p>
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</td>
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<td>
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Times
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</p>
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</td>
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<td>
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<p>
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Norm
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</p>
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<p>
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Dis
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</p>
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</td>
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<td>
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</td>
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<td>
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<p>
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Its
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</p>
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</td>
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<td>
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Times
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</p>
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</td>
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<td>
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<p>
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Norm
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</p>
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</td>
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<td>
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<p>
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Dis
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cbrt
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</p>
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</td>
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<td>
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<p>
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0
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</p>
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</td>
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<td>
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<p>
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46875
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</p>
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</td>
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<td>
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<p>
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<span class="blue">1.0</span>
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</p>
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</td>
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<td>
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<p>
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0
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</p>
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0
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46875
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</p>
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</td>
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<td>
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<p>
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<span class="blue">1.0</span>
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</p>
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</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
46875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4906250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
TOMS748
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
234375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">5.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
437500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">9.3</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
437500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">9.3</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
66218750
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">15.</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Newton
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
109375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
125000
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
140625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4531250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Halley
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
125000
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
156250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
156250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
10625000
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Schröder
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
140625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
187500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
203125
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">4.3</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
13109375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.9
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><h6>
|
|
<a name="math_toolkit.roots.root_comparison.cbrt_comparison.h1"></a>
|
|
<span class="phrase"><a name="math_toolkit.roots.root_comparison.cbrt_comparison.program_root_finding_algorithms0"></a></span><a class="link" href="cbrt_comparison.html#math_toolkit.roots.root_comparison.cbrt_comparison.program_root_finding_algorithms0">Program
|
|
root_finding_algorithms.cpp, GNU C++ version 4.9.2, GNU libstdc++ version
|
|
20141030, Win32, x64<br> 1000000 evaluations of each of 5 root_finding
|
|
algorithms.</a>
|
|
</h6>
|
|
<div class="table">
|
|
<a name="math_toolkit.roots.root_comparison.cbrt_comparison.cbrt_4_0"></a><p class="title"><b>Table 12.2. Cube root(28) for float, double, long double and cpp_bin_float_50</b></p>
|
|
<div class="table-contents"><table class="table" summary="Cube root(28) for float, double, long double and cpp_bin_float_50">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
float
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
long d
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
cpp50
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<td class="auto-generated"> </td>
|
|
<td class="auto-generated"> </td>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Algorithm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
cbrt
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
46875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
46875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
46875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3500000
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
TOMS748
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
187500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
406250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">8.7</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
10
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
609375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">13.</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
44531250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">14.</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Newton
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
93750
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
109375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
171875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3140625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Halley
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
93750
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
125000
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
218750
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">4.7</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
7171875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Schröder
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
109375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
171875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3.7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
281250
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="red">6.0</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
8703125
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break">
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
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<div class="spirit-nav">
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