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| 22 | <title>Exact Floating Constants (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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| 55 | </head>
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| 56 |
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| 57 | <body lang="en">
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| 58 | <span id="Exact-Floating-Constants"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Handling-Infinity.html" accesskey="n" rel="next">Handling Infinity</a>, Previous: <a href="Error-Recovery.html" accesskey="p" rel="prev">Error Recovery</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-Constants"></span><h3 class="section">28.12 Exact Floating-Point Constants</h3>
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| 64 | <span id="index-exact-specification-of-floating_002dpoint-constants"></span>
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| 65 | <span id="index-floating_002dpoint-constants_002c-exact-specification-of"></span>
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| 66 |
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| 67 | <p>One of the frustrations that numerical programmers have suffered
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| 68 | with since the dawn of digital computers is the inability to
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| 69 | precisely specify numbers in their programs. On the early
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| 70 | decimal machines, that was not an issue: you could write a
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| 71 | constant <code>1e-30</code> and be confident of that exact value
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| 72 | being used in floating-point operations. However, when the
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| 73 | hardware works in a base other than 10, then human-specified
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| 74 | numbers have to be converted to that base, and then converted
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| 75 | back again at output time. The two base conversions are rarely
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| 76 | exact, and unwanted rounding errors are introduced.
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| 77 | </p>
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| 78 | <span id="index-hexademical-floating_002dpoint-constants"></span>
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| 79 | <p>As computers usually represent numbers in a base other than 10,
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| 80 | numbers often must be converted to and from different bases, and
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| 81 | rounding errors can occur during conversion. This problem is solved
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| 82 | in C using hexademical floating-point constants. For example,
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| 83 | <code>+0x1.fffffcp-1</code> is the number that is the IEEE 754 32-bit value
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| 84 | closest to, but below, <code>1.0</code>. The significand is represented as a
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| 85 | hexadecimal fraction, and the <em>power of two</em> is written in
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| 86 | decimal following the exponent letter <code>p</code> (the traditional
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| 87 | exponent letter <code>e</code> is not possible, because it is a hexadecimal
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| 88 | digit).
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| 89 | </p>
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| 90 | <p>In <code>printf</code> and <code>scanf</code> and related functions, you can use
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| 91 | the ‘<samp>%a</samp>’ and ‘<samp>%A</samp>’ format specifiers for writing and reading
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| 92 | hexadecimal floating-point values. ‘<samp>%a</samp>’ writes them with lower
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| 93 | case letters and ‘<samp>%A</samp>’ writes them with upper case letters. For
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| 94 | instance, this code reproduces our sample number:
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| 95 | </p>
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| 96 | <div class="example">
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| 97 | <pre class="example">printf ("%a\n", 1.0 - pow (2.0, -23));
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| 98 | -| 0x1.fffffcp-1
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| 99 | </pre></div>
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| 100 |
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| 101 | <p>The <code>strtod</code> family was similarly extended to recognize
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| 102 | numbers in that new format.
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| 103 | </p>
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| 104 | <p>If you want to ensure exact data representation for transfer of
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| 105 | floating-point numbers between C programs on different
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| 106 | computers, then hexadecimal constants are an optimum choice.
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| 107 | </p>
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| 108 |
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| 109 | <hr>
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| 110 | <div class="header">
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| 111 | <p>
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| 112 | Next: <a href="Handling-Infinity.html" accesskey="n" rel="next">Handling Infinity</a>, Previous: <a href="Error-Recovery.html" accesskey="p" rel="prev">Error Recovery</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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