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 <div class="subTitle">org.apache.commons.math3.util</div>
 <h2 title="Class Precision" class="title">Class Precision</h2>
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 <pre>public class <span class="strong">Precision</span>
 extends <a href="http://docs.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
 <div class="block">Utilities for comparing numbers.</div>
 <dl><dt><span class="strong">Since:</span></dt>
   <dd>3.0</dd>
 <dt><span class="strong">Version:</span></dt>
   <dd>$Id: Precision.java 1591835 2014-05-02 09:04:01Z tn $</dd></dl>
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 <table class="overviewSummary" border="0" cellpadding="3" cellspacing="0" summary="Field Summary table, listing fields, and an explanation">
 <caption><span>Fields</span><span class="tabEnd">&nbsp;</span></caption>
 <tr>
 <th class="colFirst" scope="col">Modifier and Type</th>
 <th class="colLast" scope="col">Field and Description</th>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static double</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#EPSILON">EPSILON</a></strong></code>
 <div class="block">
  Largest double-precision floating-point number such that
  <code>1 + EPSILON</code> is numerically equal to 1.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static double</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#SAFE_MIN">SAFE_MIN</a></strong></code>
 <div class="block">Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.</div>
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 </tr>
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 <h3>Method Summary</h3>
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 <caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
 <tr>
 <th class="colFirst" scope="col">Modifier and Type</th>
 <th class="colLast" scope="col">Method and Description</th>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static int</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double, double, double)">compareTo</a></strong>(double&nbsp;x,
          double&nbsp;y,
          double&nbsp;eps)</code>
 <div class="block">Compares two numbers given some amount of allowed error.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static int</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double, double, int)">compareTo</a></strong>(double&nbsp;x,
          double&nbsp;y,
          int&nbsp;maxUlps)</code>
 <div class="block">Compares two numbers given some amount of allowed error.</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double)">equals</a></strong>(double&nbsp;x,
       double&nbsp;y)</code>
 <div class="block">Returns true iff they are equal as defined by
  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><code>equals(x, y, 1)</code></a>.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)">equals</a></strong>(double&nbsp;x,
       double&nbsp;y,
       double&nbsp;eps)</code>
 <div class="block">Returns <code>true</code> if there is no double value strictly between the
  arguments or the difference between them is within the range of allowed
  error (inclusive).</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)">equals</a></strong>(double&nbsp;x,
       double&nbsp;y,
       int&nbsp;maxUlps)</code>
 <div class="block">Returns true if both arguments are equal or within the range of allowed
  error (inclusive).</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)">equals</a></strong>(float&nbsp;x,
       float&nbsp;y)</code>
 <div class="block">Returns true iff they are equal as defined by
  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><code>equals(x, y, 1)</code></a>.</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, float)">equals</a></strong>(float&nbsp;x,
       float&nbsp;y,
       float&nbsp;eps)</code>
 <div class="block">Returns true if both arguments are equal or within the range of allowed
  error (inclusive).</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)">equals</a></strong>(float&nbsp;x,
       float&nbsp;y,
       int&nbsp;maxUlps)</code>
 <div class="block">Returns true if both arguments are equal or within the range of allowed
  error (inclusive).</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                   double&nbsp;y)</code>
 <div class="block">Returns true if both arguments are NaN or neither is NaN and they are
  equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double)"><code>equals(x, y, 1)</code></a>.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double, double)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                   double&nbsp;y,
                   double&nbsp;eps)</code>
 <div class="block">Returns true if both arguments are NaN or are equal or within the range
  of allowed error (inclusive).</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double, double, int)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                   double&nbsp;y,
                   int&nbsp;maxUlps)</code>
 <div class="block">Returns true if both arguments are NaN or if they are equal as defined
  by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><code>equals(x, y, maxUlps)</code></a>.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                   float&nbsp;y)</code>
 <div class="block">Returns true if both arguments are NaN or neither is NaN and they are
  equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)"><code>equals(x, y, 1)</code></a>.</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float, float)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                   float&nbsp;y,
                   float&nbsp;eps)</code>
 <div class="block">Returns true if both arguments are NaN or are equal or within the range
  of allowed error (inclusive).</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float, float, int)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                   float&nbsp;y,
                   int&nbsp;maxUlps)</code>
 <div class="block">Returns true if both arguments are NaN or if they are equal as defined
  by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><code>equals(x, y, maxUlps)</code></a>.</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static boolean</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsWithRelativeTolerance(double, double, double)">equalsWithRelativeTolerance</a></strong>(double&nbsp;x,
                            double&nbsp;y,
                            double&nbsp;eps)</code>
 <div class="block">Returns <code>true</code> if there is no double value strictly between the
  arguments or the relative difference between them is smaller or equal
  to the given tolerance.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static double</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#representableDelta(double, double)">representableDelta</a></strong>(double&nbsp;x,
                   double&nbsp;originalDelta)</code>
 <div class="block">Computes a number <code>delta</code> close to <code>originalDelta</code> with
  the property that</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static double</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(double, int)">round</a></strong>(double&nbsp;x,
      int&nbsp;scale)</code>
 <div class="block">Rounds the given value to the specified number of decimal places.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static double</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(double, int, int)">round</a></strong>(double&nbsp;x,
      int&nbsp;scale,
      int&nbsp;roundingMethod)</code>
 <div class="block">Rounds the given value to the specified number of decimal places.</div>
 </td>
 </tr>
 <tr class="altColor">
 <td class="colFirst"><code>static float</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(float, int)">round</a></strong>(float&nbsp;x,
      int&nbsp;scale)</code>
 <div class="block">Rounds the given value to the specified number of decimal places.</div>
 </td>
 </tr>
 <tr class="rowColor">
 <td class="colFirst"><code>static float</code></td>
 <td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(float, int, int)">round</a></strong>(float&nbsp;x,
      int&nbsp;scale,
      int&nbsp;roundingMethod)</code>
 <div class="block">Rounds the given value to the specified number of decimal places.</div>
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 <h4>EPSILON</h4>
 <pre>public static final&nbsp;double EPSILON</pre>
 <div class="block"><p>
  Largest double-precision floating-point number such that
  <code>1 + EPSILON</code> is numerically equal to 1. This value is an upper
  bound on the relative error due to rounding real numbers to double
  precision floating-point numbers.
  </p>
  <p>
  In IEEE 754 arithmetic, this is 2<sup>-53</sup>.
  </p></div>
 <dl><dt><span class="strong">See Also:</span></dt><dd><a href="http://en.wikipedia.org/wiki/Machine_epsilon">Machine epsilon</a></dd></dl>
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 <pre>public static final&nbsp;double SAFE_MIN</pre>
 <div class="block">Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.
  <br/>
  In IEEE 754 arithmetic, this is also the smallest normalized
  number 2<sup>-1022</sup>.</div>
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 <pre>public static&nbsp;int&nbsp;compareTo(double&nbsp;x,
             double&nbsp;y,
             double&nbsp;eps)</pre>
 <div class="block">Compares two numbers given some amount of allowed error.</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - the first number</dd><dd><code>y</code> - the second number</dd><dd><code>eps</code> - the amount of error to allow when checking for equality</dd>
 <dt><span class="strong">Returns:</span></dt><dd><ul><li>0 if  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><code>equals(x, y, eps)</code></a></li>
        <li>&lt; 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><code>equals(x, y, eps)</code></a> &amp;&amp; x &lt; y</li>
        <li>> 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, double)"><code>equals(x, y, eps)</code></a> &amp;&amp; x > y</li></ul></dd></dl>
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 <pre>public static&nbsp;int&nbsp;compareTo(double&nbsp;x,
             double&nbsp;y,
             int&nbsp;maxUlps)</pre>
 <div class="block">Compares two numbers given some amount of allowed error.
  Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
  (or fewer) floating point numbers between them, i.e. two adjacent floating
  point numbers are considered equal.
  Adapted from <a
  href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
  Bruce Dawson</a></div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
  values between <code>x</code> and <code>y</code>.</dd>
 <dt><span class="strong">Returns:</span></dt><dd><ul><li>0 if  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><code>equals(x, y, maxUlps)</code></a></li>
        <li>&lt; 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><code>equals(x, y, maxUlps)</code></a> &amp;&amp; x &lt; y</li>
        <li>> 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double, double, int)"><code>equals(x, y, maxUlps)</code></a> &amp;&amp; x > y</li></ul></dd></dl>
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 <h4>equals</h4>
 <pre>public static&nbsp;boolean&nbsp;equals(float&nbsp;x,
              float&nbsp;y)</pre>
 <div class="block">Returns true iff they are equal as defined by
  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float, int)"><code>equals(x, y, 1)</code></a>.</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
 <dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal.</dd></dl>
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 <pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(float&nbsp;x,
                          float&nbsp;y)</pre>
 <div class="block">Returns true if both arguments are NaN or neither is NaN and they are
  equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float, float)"><code>equals(x, y, 1)</code></a>.</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
 <dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or both are NaN.</dd><dt><span class="strong">Since:</span></dt>
   <dd>2.2</dd></dl>
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 <h4>equals</h4>
 <pre>public static&nbsp;boolean&nbsp;equals(float&nbsp;x,
              float&nbsp;y,
              float&nbsp;eps)</pre>
 <div class="block">Returns true if both arguments are equal or within the range of allowed
  error (inclusive).</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>eps</code> - the amount of absolute error to allow.</dd>
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  Adapted from <a
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 <dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or within range of each other,
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  Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
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  point numbers are considered equal.
  Adapted from <a
  href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">
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 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
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 <dt><span class="strong">Returns:</span></dt><dd><code>true</code> if there are fewer than <code>maxUlps</code> floating
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 <dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd><dt><span class="strong">Since:</span></dt>
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  The value is rounded using the given method which is any method defined
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 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd><dd><code>roundingMethod</code> - Rounding method as defined in <a href="http://docs.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</dd>
 <dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd>
 <dt><span class="strong">Throws:</span></dt>
 <dd><code><a href="http://docs.oracle.com/javase/6/docs/api/java/lang/ArithmeticException.html?is-external=true" title="class or interface in java.lang">ArithmeticException</a></code> - if <code>roundingMethod == ROUND_UNNECESSARY</code>
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 <dd><code><a href="http://docs.oracle.com/javase/6/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</a></code> - if <code>roundingMethod</code> does not
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   <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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  The value is rounded using the <a href="http://docs.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true#ROUND_HALF_UP" title="class or interface in java.math"><code>BigDecimal.ROUND_HALF_UP</code></a> method.</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd>
 <dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd><dt><span class="strong">Since:</span></dt>
   <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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 <div class="block">Rounds the given value to the specified number of decimal places.
  The value is rounded using the given method which is any method defined
  in <a href="http://docs.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</div>
 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd><dd><code>roundingMethod</code> - Rounding method as defined in <a href="http://docs.oracle.com/javase/6/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</dd>
 <dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd>
 <dt><span class="strong">Throws:</span></dt>
 <dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if an exact operation is required but result is not exact</dd>
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 <dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value.</dd><dd><code>originalDelta</code> - Offset value.</dd>
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  differ by a representable floating number.</dd></dl>
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