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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl"></a><a class="link" href="floating_points_comparison_impl.html" title="Tolerance-based comparisons">Tolerance-based
  28           comparisons</a>
  29 </h5></div></div></div>
  30 <p>
  31             Assertions in the <span class="emphasis"><em>Unit Test Framework</em></span> use two kinds
  32             of comparison. For <code class="computeroutput"><span class="identifier">u</span></code>
  33             being close to zero with absolute tolerance <code class="computeroutput"><span class="identifier">eps</span></code>:
  34           </p>
  35 <pre class="programlisting"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span><span class="special">;</span> <span class="comment">// (abs)</span>
  36 </pre>
  37 <p>
  38             For <code class="computeroutput"><span class="identifier">u</span></code> and <code class="computeroutput"><span class="identifier">v</span></code> being close with relative tolerance
  39             <code class="computeroutput"><span class="identifier">eps</span></code>:
  40           </p>
  41 <pre class="programlisting">   <span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span> <span class="special">-</span> <span class="identifier">v</span><span class="special">)/</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span>
  42 <span class="special">&amp;&amp;</span> <span class="identifier">abs</span><span class="special">(</span><span class="identifier">u</span> <span class="special">-</span> <span class="identifier">v</span><span class="special">)/</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">v</span><span class="special">)</span> <span class="special">&lt;=</span> <span class="identifier">eps</span><span class="special">;</span> <span class="comment">// (rel)</span>
  43 </pre>
  44 <p>
  45             For rationale for choosing these formulae, see section <a class="link" href="floating_points_comparison_theory.html" title="Theory behind floating point comparisons">Floating
  46             point comparison algorithms</a>.
  47           </p>
  48 <p>
  49             Assertion <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_test_universal_macro.html" title="BOOST_TEST"><code class="computeroutput"><span class="identifier">BOOST_TEST</span></code></a> (when comparing floating-point
  50             numbers) uses the following algorithm:
  51           </p>
  52 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  53 <li class="listitem">
  54                 When either value <code class="computeroutput"><span class="identifier">u</span></code>
  55                 or <code class="computeroutput"><span class="identifier">v</span></code> is zero, evaluates
  56                 formula (abs) on the other value.
  57               </li>
  58 <li class="listitem">
  59                 When the specified tolerance is zero, performs direct (native) comparison
  60                 between <code class="computeroutput"><span class="identifier">u</span></code> and <code class="computeroutput"><span class="identifier">v</span></code>.
  61               </li>
  62 <li class="listitem">
  63                 Otherwise, performs formula (rel) on <code class="computeroutput"><span class="identifier">u</span></code>
  64                 and <code class="computeroutput"><span class="identifier">v</span></code>.
  65               </li>
  66 </ul></div>
  67 <div class="note"><table border="0" summary="Note">
  68 <tr>
  69 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
  70 <th align="left">Note</th>
  71 </tr>
  72 <tr><td align="left" valign="top">
  73 <p>
  74               Therefore in order to check if a number is close to zero with tolerance,
  75               you need to type:
  76             </p>
  77 <pre class="programlisting"><span class="identifier">BOOST_TEST</span><span class="special">(</span><span class="identifier">v</span> <span class="special">==</span> <span class="identifier">T</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">tt</span><span class="special">::</span><span class="identifier">tolerance</span><span class="special">(</span><span class="identifier">eps</span><span class="special">));</span>
  78 </pre>
  79 </td></tr>
  80 </table></div>
  81 <p>
  82             The compatibility assertions <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_close.html" title="BOOST_&lt;level&gt;_CLOSE"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_CLOSE</span></code></a> and <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_close_fraction.html" title="BOOST_&lt;level&gt;_CLOSE_FRACTION"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_CLOSE_FRACTION</span></code></a> perform formula
  83             (rel).
  84           </p>
  85 <p>
  86             The compatibility assertion <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_level_small.html" title="BOOST_&lt;level&gt;_SMALL"><code class="computeroutput"><span class="identifier">BOOST_</span><span class="special">&lt;</span><span class="identifier">level</span><span class="special">&gt;</span><span class="identifier">_SMALL</span></code></a> performs formula (abs).
  87           </p>
  88 <p>
  89             The <span class="emphasis"><em>Unit Test Framework</em></span> also provides unary predicate
  90             <code class="computeroutput"><a class="link" href="../../../../boost/math/fpc/small_with_tolerance.html" title="Class template small_with_tolerance">small_with_tolerance</a></code> and
  91             binary predicate predicate <code class="computeroutput"><a class="link" href="../../../../boost/math/fpc/close_at_tolerance.html" title="Class template close_at_tolerance">close_at_tolerance</a></code> that
  92             implement formula (abs) and (rel) respectively.
  93           </p>
  94 <h4>
  95 <a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.h0"></a>
  96             <span class="phrase"><a name="boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.tolerance_in_operator"></a></span><a class="link" href="floating_points_comparison_impl.html#boost_test.testing_tools.extended_comparison.floating_point.floating_points_comparison_impl.tolerance_in_operator">Tolerance
  97             in <code class="computeroutput"><span class="keyword">operator</span><span class="special">&lt;</span></code></a>
  98           </h4>
  99 <p>
 100             Tolerance-based computations also apply to <code class="computeroutput"><span class="keyword">operator</span><span class="special">&lt;</span></code> and other relational operators. The
 101             semantics are defined as follows:
 102           </p>
 103 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 104 <li class="listitem">
 105                 <span class="emphasis"><em>less-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-less</em></span>
 106                 and not <span class="emphasis"><em>close-at-tolerance</em></span>
 107               </li>
 108 <li class="listitem">
 109                 <span class="emphasis"><em>greater-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-greater</em></span>
 110                 and not <span class="emphasis"><em>close-at-tolerance</em></span>
 111               </li>
 112 <li class="listitem">
 113                 <span class="emphasis"><em>less-or-equal-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-less</em></span>
 114                 or <span class="emphasis"><em>close-at-tolerance</em></span>
 115               </li>
 116 <li class="listitem">
 117                 <span class="emphasis"><em>greater-or-equal-at-tolerance</em></span> &lt;==&gt; <span class="emphasis"><em>strictly-greater</em></span>
 118                 or <span class="emphasis"><em>close-at-tolerance</em></span>
 119               </li>
 120 </ul></div>
 121 <div class="note"><table border="0" summary="Note">
 122 <tr>
 123 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
 124 <th align="left">Note</th>
 125 </tr>
 126 <tr><td align="left" valign="top"><p>
 127               This implies that the exactly one of these: <code class="computeroutput"><span class="identifier">u</span>
 128               <span class="special">&lt;</span> <span class="identifier">v</span></code>,
 129               <code class="computeroutput"><span class="identifier">u</span> <span class="special">==</span>
 130               <span class="identifier">v</span></code>, <code class="computeroutput"><span class="identifier">u</span>
 131               <span class="special">&gt;</span> <span class="identifier">v</span></code>,
 132               passes with <a class="link" href="../../../utf_reference/testing_tool_ref/assertion_boost_test_universal_macro.html" title="BOOST_TEST"><code class="computeroutput"><span class="identifier">BOOST_TEST</span></code></a> at any given tolerance.
 133             </p></td></tr>
 134 </table></div>
 135 <div class="caution"><table border="0" summary="Caution">
 136 <tr>
 137 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td>
 138 <th align="left">Caution</th>
 139 </tr>
 140 <tr><td align="left" valign="top"><p>
 141               Relation <span class="emphasis"><em>less-at-tolerance</em></span> is not a <span class="emphasis"><em>Strict
 142               Weak Ordering</em></span> as it lacks the <span class="emphasis"><em>transitivity of
 143               the equivalence</em></span>; using it as predicate in <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">map</span></code> or any order-based STL algorithm
 144               would result in undefined behavior.
 145             </p></td></tr>
 146 </table></div>
 147 </div>
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