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| 22 | <title>Scaling by the Base (GNU C Language Manual)</title>
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| 32 | <link href="Floating-Point-in-Depth.html" rel="up" title="Floating Point in Depth">
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| 57 | <body lang="en">
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| 58 | <span id="Scaling-by-the-Base"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Rounding-Control.html" accesskey="n" rel="next">Rounding Control</a>, Previous: <a href="Signed-Zeros.html" accesskey="p" rel="prev">Signed Zeros</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="Scaling-by-Powers-of-the-Base"></span><h3 class="section">28.16 Scaling by Powers of the Base</h3>
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| 64 | <span id="index-scaling-floating-point-by-powers-of-the-base"></span>
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| 65 | <span id="index-floating_002dpoint-scaling-by-powers-of-the-base"></span>
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| 66 |
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| 67 | <p>We have discussed rounding errors several times in this chapter,
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| 68 | but it is important to remember that when results require no more
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| 69 | bits than the exponent and significand bits can represent, those results
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| 70 | are <em>exact</em>.
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| 71 | </p>
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| 72 | <p>One particularly useful exact operation is scaling by a power of
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| 73 | the base. While one, in principle, could do that with code like
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| 74 | this:
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| 75 | </p>
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| 76 | <div class="example">
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| 77 | <pre class="example">y = x * pow (2.0, (double)k); /* <span class="roman">Undesirable scaling: avoid!</span> */
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| 78 | </pre></div>
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| 79 |
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| 80 | <p>that is not advisable, because it relies on the quality of the
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| 81 | math-library power function, and that happens to be one of the
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| 82 | most difficult functions in the C math library to make accurate.
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| 83 | What is likely to happen on many systems is that the returned
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| 84 | value from <code>pow</code> will be close to a power of two, but
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| 85 | slightly different, so the subsequent multiplication introduces
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| 86 | rounding error.
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| 87 | </p>
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| 88 | <p>The correct, and fastest, way to do the scaling is either via the
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| 89 | traditional C library function, or with its C99 equivalent:
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| 90 | </p>
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| 91 | <div class="example">
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| 92 | <pre class="example">y = ldexp (x, k); /* <span class="roman">Traditional pre-C99 style.</span> */
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| 93 | y = scalbn (x, k); /* <span class="roman">C99 style.</span> */
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| 94 | </pre></div>
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| 95 |
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| 96 | <p>Both functions return <code>x * 2**k</code>.
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| 97 | See <a href="https://www.gnu.org/software/libc/manual/html_node/Normalization-Functions.html#Normalization-Functions">Normalization Functions</a> in <cite>The GNU C Library Reference Manual</cite>.
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| 98 | </p>
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| 99 |
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| 102 |
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