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<title>Obtaining the Size of a Unit In the Last Place - ULP</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.next_float.ulp"></a><a class="link" href="ulp.html" title="Obtaining the Size of a Unit In the Last Place - ULP">Obtaining the Size of a
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Unit In the Last Place - ULP</a>
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</h3></div></div></div>
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<p>
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Function <code class="computeroutput"><span class="identifier">ulp</span></code> gives the size
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of a unit-in-the-last-place for a specified floating-point value.
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</p>
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<h5>
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<a name="math_toolkit.next_float.ulp.h0"></a>
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<span class="phrase"><a name="math_toolkit.next_float.ulp.synopsis"></a></span><a class="link" href="ulp.html#math_toolkit.next_float.ulp.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">ulp</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">FPT</span><span class="special">></span>
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<span class="identifier">FPT</span> <span class="identifier">ulp</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">FPT</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">FPT</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
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<span class="identifier">FPT</span> <span class="identifier">ulp</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">FPT</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.next_float.ulp.h1"></a>
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<span class="phrase"><a name="math_toolkit.next_float.ulp.description_ulp"></a></span><a class="link" href="ulp.html#math_toolkit.next_float.ulp.description_ulp">Description
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- ulp</a>
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</h5>
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<p>
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Returns one <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">unit
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in the last place</a> of <span class="emphasis"><em>x</em></span>.
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</p>
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<p>
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Corner cases are handled as follows:
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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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If the argument is a NaN, then raises a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</li>
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<li class="listitem">
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If the argument is an infinity, then raises an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>.
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</li>
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<li class="listitem">
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If the argument is zero then returns the smallest representable value:
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for example for type <code class="computeroutput"><span class="keyword">double</span></code>
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this would be either <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code> or <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">()</span></code> depending whether denormals are supported
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(which have the values <code class="computeroutput"><span class="number">2.2250738585072014e-308</span></code>
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and <code class="computeroutput"><span class="number">4.9406564584124654e-324</span></code>
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respectively).
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</li>
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<li class="listitem">
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If the result is too small to represent, then returns the smallest representable
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value.
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</li>
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<li class="listitem">
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Always returns a positive value such that <code class="computeroutput"><span class="identifier">ulp</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">ulp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)</span></code>.
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</li>
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</ul></div>
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<p>
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<span class="bold"><strong>Important:</strong></span> The behavior of this function
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is aligned to that of <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Math.html#ulp%28double%29" target="_top">Java's
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ulp function</a>, please note however that this function should only
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ever be used for rough and ready calculations as there are enough corner
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cases to trap even careful programmers. In particular:
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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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The function is asymmetrical, which is to say, given <code class="computeroutput"><span class="identifier">u</span>
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<span class="special">=</span> <span class="identifier">ulp</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code> if <code class="computeroutput"><span class="identifier">x</span>
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<span class="special">></span> <span class="number">0</span></code>
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then <code class="computeroutput"><span class="identifier">x</span> <span class="special">+</span>
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<span class="identifier">u</span></code> is the next floating-point
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value, but <code class="computeroutput"><span class="identifier">x</span> <span class="special">-</span>
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<span class="identifier">u</span></code> is not necessarily the previous
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value. Similarly, if <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> <span class="number">0</span></code>
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then <code class="computeroutput"><span class="identifier">x</span> <span class="special">-</span>
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<span class="identifier">u</span></code> is the previous floating-point
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value, but <code class="computeroutput"><span class="identifier">x</span> <span class="special">+</span>
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<span class="identifier">u</span></code> is not necessarily the next
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value. The corner cases occur at power of 2 boundaries.
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</li>
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<li class="listitem">
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When the argument becomes very small, it may be that there is no floating-point
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value that represents one ULP. Whether this is the case or not depends
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not only on whether the hardware may <span class="emphasis"><em>sometimes</em></span> support
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denormals (as signalled by <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">FPT</span><span class="special">>::</span><span class="identifier">has_denorm</span></code>),
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but also whether these are currently enabled at runtime (for example
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on SSE hardware, the DAZ or FTZ flags will disable denormal support).
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In this situation, the <code class="computeroutput"><span class="identifier">ulp</span></code>
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function may return a value that is many orders of magnitude too large.
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</li>
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</ul></div>
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<p>
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In light of the issues above, we recommend that:
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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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To move between adjacent floating-point values always use <a class="link" href="float_next.html" title="Finding the Next Greater Representable Value (float_next)">float_next</a>,
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<a class="link" href="float_prior.html" title="Finding the Next Smaller Representable Value (float_prior)">float_prior</a>
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or <a class="link" href="nextafter.html" title="Finding the Next Representable Value in a Specific Direction (nextafter)">nextafter</a>
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(<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">nextafter</span></code> is another candidate, but
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our experience is that this also often breaks depending which optimizations
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and hardware flags are in effect).
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</li>
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<li class="listitem">
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To move several floating-point values away use <a class="link" href="float_advance.html" title="Advancing a floating-point Value by a Specific Representation Distance (ULP) float_advance">float_advance</a>.
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</li>
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<li class="listitem">
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To calculate the edit distance between two floats use <a href="../../../../../../libs/math/doc/html/math_toolkit/next_float/float_distance.html" target="_top">Boost.Math
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float_distance</a>.
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</li>
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</ul></div>
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<p>
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There is none the less, one important use case for this function:
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</p>
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<p>
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If it is known that the true result of some function is x<sub>t</sub> and the calculated
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result is x<sub>c</sub>, then the error measured in ulp is simply <code class="literal">fabs(x<sub>t</sub> -
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x<sub>c</sub>) / ulp(x<sub>t</sub>)</code>.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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