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| 22 | <title>Exact Floating-Point (GNU C Language Manual)</title>
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| 23 |
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| 24 | <meta name="description" content="Exact Floating-Point (GNU C Language Manual)">
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| 29 | <link href="index.html" rel="start" title="Top">
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| 30 | <link href="Symbol-Index.html" rel="index" title="Symbol Index">
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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.html" rel="next" title="Rounding">
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| 55 | </head>
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| 56 |
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| 57 | <body lang="en">
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| 58 | <span id="Exact-Floating_002dPoint"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Rounding.html" accesskey="n" rel="next">Rounding</a>, Previous: <a href="Exception-Flags.html" accesskey="p" rel="prev">Exception Flags</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="Exact-Floating_002dPoint-Arithmetic"></span><h3 class="section">28.6 Exact Floating-Point Arithmetic</h3>
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| 64 | <span id="index-exact-floating_002dpoint-arithmetic"></span>
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| 65 | <span id="index-floating_002dpoint-arithmetic_002c-exact"></span>
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| 66 |
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| 67 | <p>As long as the numbers are exactly representable (fractions whose
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| 68 | denominator is a power of 2), and intermediate results do not require
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| 69 | rounding, then floating-point arithmetic is <em>exact</em>. It is easy
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| 70 | to predict how many digits are needed for the results of arithmetic
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| 71 | operations:
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| 72 | </p>
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| 73 | <ul>
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| 74 | <li> addition and subtraction of two <var>n</var>-digit values with the
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| 75 | <em>same</em> exponent require at most <code><var>n</var> + 1</code> digits, but
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| 76 | when the exponents differ, many more digits may be needed;
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| 77 |
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| 78 | </li><li> multiplication of two <var>n</var>-digit values requires exactly
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| 79 | 2 <var>n</var> digits;
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| 80 |
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| 81 | </li><li> although integer division produces a quotient and a remainder of
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| 82 | no more than <var>n</var>-digits, floating-point remainder and square
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| 83 | root may require an unbounded number of digits, and the quotient
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| 84 | can need many more digits than can be stored.
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| 85 |
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| 86 | </li></ul>
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| 87 |
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| 88 | <p>Whenever a result requires more than <var>n</var> digits, rounding
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| 89 | is needed.
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| 90 | </p>
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| 91 |
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| 92 |
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| 93 |
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| 94 |
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| 95 | </body>
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| 96 | </html>
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