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   4 <title>Floating-Point Representation Distance (ULP), and Finding Adjacent Floating-Point Values</title>
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  27 <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
  28     Distance (ULP), and Finding Adjacent Floating-Point Values</a>
  29 </h2></div></div></div>
  30 <div class="toc"><dl class="toc">
  31 <dt><span class="section"><a href="next_float/nextafter.html">Finding the Next Representable
  32       Value in a Specific Direction (nextafter)</a></span></dt>
  33 <dt><span class="section"><a href="next_float/float_next.html">Finding the Next
  34       Greater Representable Value (float_next)</a></span></dt>
  35 <dt><span class="section"><a href="next_float/float_prior.html">Finding the Next
  36       Smaller Representable Value (float_prior)</a></span></dt>
  37 <dt><span class="section"><a href="next_float/float_distance.html">Calculating the
  38       Representation Distance Between Two floating-point Values (ULP) float_distance</a></span></dt>
  39 <dt><span class="section"><a href="next_float/float_advance.html">Advancing a floating-point
  40       Value by a Specific Representation Distance (ULP) float_advance</a></span></dt>
  41 <dt><span class="section"><a href="next_float/ulp.html">Obtaining the Size of a
  42       Unit In the Last Place - ULP</a></span></dt>
  43 </dl></div>
  44 <p>
  45       <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit of Least
  46       Precision or Unit in the Last Place</a> is the gap between two different,
  47       but as close as possible, floating-point numbers.
  48     </p>
  49 <p>
  50       Most decimal values, for example 0.1, cannot be exactly represented as floating-point
  51       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
  52       representable floating-point</a>.
  53     </p>
  54 <p>
  55       Functions are provided for finding adjacent greater and lesser floating-point
  56       values, and estimating the number of gaps between any two floating-point values.
  57     </p>
  58 <p>
  59       The floating-point type (FPT) must have has a fixed number of bits in the representation.
  60       The number of bits may set at runtime, but must be the same for all numbers.
  61       For example, <a href="http://shoup.net/ntl/doc/quad_float.txt" target="_top">NTL::quad_float</a>
  62       type (fixed 128-bit representation), <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
  63       type (arbitrary but fixed decimal digits, default 150) or <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
  64       <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>
  65       and<a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>
  66       are fixed at runtime, but <span class="bold"><strong>not</strong></span> a type that
  67       extends the representation to provide an exact representation for any number,
  68       for example <a href="http://keithbriggs.info/xrc.html" target="_top">XRC eXact Real in
  69       C</a>.
  70     </p>
  71 <p>
  72       The accuracy of mathematical functions can be assessed and displayed in terms
  73       of <a href="https://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">Unit in
  74       the Last Place</a>, often as a ulps plot or by binning the differences
  75       as a histogram. Samples are evaluated using the implementation under test and
  76       compared with 'known good' representation obtained using a more accurate method.
  77       Other implementations, often using arbitrary precision arithmetic, for example
  78       <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> are one source
  79       of references values. The other method, used widely in Boost.Math special functions,
  80       it to carry out the same algorithm, but using a higher precision type, typically
  81       using Boost.Multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>
  82       for 128-bit (about 35 decimal digit precision), or <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>
  83       (for 50 decimal digit precision).
  84     </p>
  85 <p>
  86       When converted to a particular machine representation, say <code class="computeroutput"><span class="keyword">double</span></code>,
  87       say using a <code class="computeroutput"><span class="keyword">static_cast</span></code>, the value
  88       is the nearest representation possible for the <code class="computeroutput"><span class="keyword">double</span></code>
  89       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
  90       in the last place (ULP)</a>, and can be obtained using the Boost.Math function
  91       <code class="computeroutput"><span class="identifier">ulp</span></code>. (Unless the algorithm
  92       is fundamentally flawed, something that should be revealed by 'sanity' checks
  93       using some independent sources).
  94     </p>
  95 <p>
  96       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
  97       plots reveal math-function accuracy</a>.
  98     </p>
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