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| 22 | <title>Rounding (GNU C Language Manual)</title>
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| 31 | <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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| 32 | <link href="Floating-Point-in-Depth.html" rel="up" title="Floating Point in Depth">
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| 33 | <link href="Rounding-Issues.html" rel="next" title="Rounding Issues">
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| 57 | <body lang="en">
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| 58 | <span id="Rounding"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Rounding-Issues.html" accesskey="n" rel="next">Rounding Issues</a>, Previous: <a href="Exact-Floating_002dPoint.html" accesskey="p" rel="prev">Exact Floating-Point</a>, Up: <a href="Floating-Point-in-Depth.html" accesskey="u" rel="up">Floating Point in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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| 61 | </div>
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| 62 | <hr>
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| 63 | <span id="Rounding-1"></span><h3 class="section">28.7 Rounding</h3>
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| 64 | <span id="index-rounding"></span>
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| 65 |
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| 66 | <p>When floating-point arithmetic produces a result that can’t fit
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| 67 | exactly in the significand of the type that’s in use, it has to
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| 68 | <em>round</em> the value. The basic arithmetic operations—addition,
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| 69 | subtraction, multiplication, division, and square root—always produce
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| 70 | a result that is equivalent to the exact, possibly infinite-precision
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| 71 | result rounded to storage precision according to the current rounding
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| 72 | rule.
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| 73 | </p>
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| 74 | <p>Rounding sets the <code>FE_INEXACT</code> exception flag (see <a href="Exception-Flags.html">Exception Flags</a>). This enables programs to determine that rounding has
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| 75 | occurred.
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| 76 | </p>
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| 77 | <p>Rounding consists of adjusting the exponent to bring the significand
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| 78 | back to the required base-point alignment, then applying the current
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| 79 | <em>rounding rule</em> to squeeze the significand into the fixed
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| 80 | available size.
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| 81 | </p>
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| 82 | <p>The current rule is selected at run time from four options. Here they
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| 83 | are:
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| 84 | </p>
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| 85 | <ul class="no-bullet">
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| 86 | <li>* <em>round-to-nearest</em>, with ties rounded to an even integer;
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| 87 |
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| 88 | </li><li>* <em>round-up</em>, towards <code>+Infinity</code>;
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| 89 |
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| 90 | </li><li>* <em>round-down</em>, towards <code>-Infinity</code>;
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| 91 |
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| 92 | </li><li>* <em>round-towards-zero</em>.
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| 93 | </li></ul>
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| 94 |
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| 95 | <p>Under those four rounding rules, a decimal value
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| 96 | <code>-1.2345</code> that is to be rounded to a four-digit result would
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| 97 | become <code>-1.234</code>, <code>-1.234</code>, <code>-1.235</code>, and
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| 98 | <code>-1.234</code>, respectively.
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| 99 | </p>
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| 100 | <p>The default rounding rule is <em>round-to-nearest</em>, because that has
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| 101 | the least bias, and produces the lowest average error. When the true
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| 102 | result lies exactly halfway between two representable machine numbers,
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| 103 | the result is rounded to the one that ends with an even digit.
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| 104 | </p>
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| 105 | <p>The <em>round-towards-zero</em> rule was common on many early computer
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| 106 | designs, because it is the easiest to implement: it just requires
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| 107 | silent truncation of all extra bits.
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| 108 | </p>
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| 109 | <p>The two other rules, <em>round-up</em> and <em>round-down</em>, are
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| 110 | essential for implementing <em>interval arithmetic</em>, whereby
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| 111 | each arithmetic operation produces lower and upper bounds that
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| 112 | are guaranteed to enclose the exact result.
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| 113 | </p>
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| 114 | <p>See <a href="Rounding-Control.html">Rounding Control</a>, for details on getting and setting the
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| 115 | current rounding mode.
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| 116 | </p>
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| 117 | <hr>
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| 118 | <div class="header">
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| 119 | <p>
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| 120 | Next: <a href="Rounding-Issues.html" accesskey="n" rel="next">Rounding Issues</a>, Previous: <a href="Exact-Floating_002dPoint.html" accesskey="p" rel="prev">Exact Floating-Point</a>, Up: <a href="Floating-Point-in-Depth.html" accesskey="u" rel="up">Floating Point in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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