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| 22 | <title>Round-Trip Base Conversion (GNU C Language Manual)</title>
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| 57 | <body lang="en">
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| 58 | <span id="Round_002dTrip-Base-Conversion"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Further-Reading.html" accesskey="n" rel="next">Further Reading</a>, Previous: <a href="Complex-Arithmetic.html" accesskey="p" rel="prev">Complex Arithmetic</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="Round_002dTrip-Base-Conversion-1"></span><h3 class="section">28.20 Round-Trip Base Conversion</h3>
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| 64 | <span id="index-round_002dtrip-base-conversion"></span>
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| 65 | <span id="index-base-conversion-_0028floating-point_0029"></span>
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| 66 | <span id="index-floating_002dpoint-round_002dtrip-base-conversion"></span>
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| 67 |
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| 68 | <p>Most numeric programs involve converting between base-2 floating-point
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| 69 | numbers, as represented by the computer, and base-10 floating-point
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| 70 | numbers, as entered and handled by the programmer. What might not be
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| 71 | obvious is the number of base-2 bits vs. base-10 digits required for
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| 72 | each representation. Consider the following tables showing the number of
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| 73 | decimal digits representable in a given number of bits, and vice versa:
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| 74 | </p>
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| 75 | <table>
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| 76 | <tr><td width="50%">binary in</td><td width="10%">24</td><td width="10%">53</td><td width="10%">64</td><td width="10%">113</td><td width="10%">237</td></tr>
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| 77 | <tr><td width="50%">decimal out</td><td width="10%">9</td><td width="10%">17</td><td width="10%">21</td><td width="10%">36</td><td width="10%">73</td></tr>
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| 78 | </table>
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| 79 |
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| 80 | <table>
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| 81 | <tr><td width="50%">decimal in</td><td width="10%">7</td><td width="10%">16</td><td width="10%">34</td><td width="10%">70</td></tr>
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| 82 | <tr><td width="50%">binary out</td><td width="10%">25</td><td width="10%">55</td><td width="10%">114</td><td width="10%">234</td></tr>
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| 83 | </table>
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| 84 |
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| 85 | <p>We can compute the table numbers with these two functions:
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| 86 | </p>
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| 87 | <div class="example">
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| 88 | <pre class="example">int
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| 89 | matula(int nbits)
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| 90 | { /* <span class="roman">Return output decimal digits needed for nbits-bits input.</span> */
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| 91 | return ((int)ceil((double)nbits / log2(10.0) + 1.0));
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| 92 | }
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| 93 |
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| 94 | int
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| 95 | goldberg(int ndec)
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| 96 | { /* <span class="roman">Return output bits needed for ndec-digits input.</span> */
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| 97 | return ((int)ceil((double)ndec / log10(2.0) + 1.0));
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| 98 | }
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| 99 | </pre></div>
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| 100 |
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| 101 | <p>One significant observation from those numbers is that we cannot
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| 102 | achieve correct round-trip conversion between the decimal and
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| 103 | binary formats in the same storage size! For example, we need 25
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| 104 | bits to represent a 7-digit value from the 32-bit decimal format,
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| 105 | but the binary format only has 24 available. Similar
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| 106 | observations hold for each of the other conversion pairs.
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| 107 | </p>
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| 108 | <p>The general input/output base-conversion problem is astonishingly
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| 109 | complicated, and solutions were not generally known until the
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| 110 | publication of two papers in 1990 that are listed later near the end
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| 111 | of this chapter. For the 128-bit formats, the worst case needs more
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| 112 | than 11,500 decimal digits of precision to guarantee correct rounding
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| 113 | in a binary-to-decimal conversion!
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| 114 | </p>
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| 115 | <p>For further details see the references for Bennett Goldberg and David
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| 116 | Matula.
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| 117 | </p>
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| 118 |
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| 119 | <hr>
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| 120 | <div class="header">
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| 121 | <p>
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| 122 | Next: <a href="Further-Reading.html" accesskey="n" rel="next">Further Reading</a>, Previous: <a href="Complex-Arithmetic.html" accesskey="p" rel="prev">Complex Arithmetic</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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