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<title>Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values</title>
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.next_float"></a><a class="link" href="next_float.html" title="Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values">Floating-Point Representation
    Distance (ULP), and Finding Adjacent Floating-Point Values</a>
</h2></div></div></div>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="next_float/nextafter.html">Finding the Next Representable
      Value in a Specific Direction (nextafter)</a></span></dt>
<dt><span class="section"><a href="next_float/float_next.html">Finding the Next
      Greater Representable Value (float_next)</a></span></dt>
<dt><span class="section"><a href="next_float/float_prior.html">Finding the Next
      Smaller Representable Value (float_prior)</a></span></dt>
<dt><span class="section"><a href="next_float/float_distance.html">Calculating the
      Representation Distance Between Two floating-point Values (ULP) float_distance</a></span></dt>
<dt><span class="section"><a href="next_float/float_advance.html">Advancing a floating-point
      Value by a Specific Representation Distance (ULP) float_advance</a></span></dt>
<dt><span class="section"><a href="next_float/ulp.html">Obtaining the Size of a
      Unit In the Last Place - ULP</a></span></dt>
</dl></div>
<p>
      <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit of Least
      Precision or Unit in the Last Place</a> is the gap between two different,
      but as close as possible, floating-point numbers.
    </p>
<p>
      Most decimal values, for example 0.1, cannot be exactly represented as floating-point
      values, but will be stored as the <a href="http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding" target="_top">closest
      representable floating-point</a>.
    </p>
<p>
      Functions are provided for finding adjacent greater and lesser floating-point
      values, and estimating the number of gaps between any two floating-point values.
    </p>
<p>
      The floating-point type (FPT) must have has a fixed number of bits in the representation.
      The number of bits may set at runtime, but must be the same for all numbers.
      For example, <a href="http://shoup.net/ntl/doc/quad_float.txt" target="_top">NTL::quad_float</a>
      type (fixed 128-bit representation), <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
      type (arbitrary but fixed decimal digits, default 150) or <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
      <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>
      and<a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>
      are fixed at runtime, but <span class="bold"><strong>not</strong></span> a type that
      extends the representation to provide an exact representation for any number,
      for example <a href="http://keithbriggs.info/xrc.html" target="_top">XRC eXact Real in
      C</a>.
    </p>
<p>
      The accuracy of mathematical functions can be assessed and displayed in terms
      of <a href="https://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit in
      the Last Place</a>, often as a ulps plot or by binning the differences
      as a histogram. Samples are evaluated using the implementation under test and
      compared with 'known good' representation obtained using a more accurate method.
      Other implementations, often using arbitrary precision arithmetic, for example
      <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> are one source
      of references values. The other method, used widely in Boost.Math special functions,
      it to carry out the same algorithm, but using a higher precision type, typically
      using Boost.Multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>
      for 128-bit (about 35 decimal digit precision), or <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
      (for 50 decimal digit precision).
    </p>
<p>
      When converted to a particular machine representation, say <code class="computeroutput"><span class="keyword">double</span></code>,
      say using a <code class="computeroutput"><span class="keyword">static_cast</span></code>, the value
      is the nearest representation possible for the <code class="computeroutput"><span class="keyword">double</span></code>
      type. This value cannot be 'wrong' by more than half a <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit
      in the last place (ULP)</a>, and can be obtained using the Boost.Math function
      <code class="computeroutput"><span class="identifier">ulp</span></code>. (Unless the algorithm
      is fundamentally flawed, something that should be revealed by 'sanity' checks
      using some independent sources).
    </p>
<p>
      See some discussion and example plots by Cleve Moler of Mathworks <a href="https://blogs.mathworks.com/cleve/2017/01/23/ulps-plots-reveal-math-function-accurary/" target="_top">ulps
      plots reveal math-function accuracy</a>.
    </p>
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