<html lang="en">
<head>
<title>Floating Point Concepts - The GNU C Library</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="The GNU C Library">
<meta name="generator" content="makeinfo 4.13">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Floating-Type-Macros.html#Floating-Type-Macros" title="Floating Type Macros">
<link rel="next" href="Floating-Point-Parameters.html#Floating-Point-Parameters" title="Floating Point Parameters">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
This file documents the GNU C library.

This is Edition 0.12, last updated 2007-10-27,
of `The GNU C Library Reference Manual', for version
2.8 (Sourcery G++ Lite 2011.03-41).

Copyright (C) 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2001, 2002,
2003, 2007, 2008, 2010 Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being ``Free Software Needs Free Documentation''
and ``GNU Lesser General Public License'', the Front-Cover texts being
``A GNU Manual'', and with the Back-Cover Texts as in (a) below.  A
copy of the license is included in the section entitled "GNU Free
Documentation License".

(a) The FSF's Back-Cover Text is: ``You have the freedom to
copy and modify this GNU manual.  Buying copies from the FSF
supports it in developing GNU and promoting software freedom.''-->
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
  pre.display { font-family:inherit }
  pre.format  { font-family:inherit }
  pre.smalldisplay { font-family:inherit; font-size:smaller }
  pre.smallformat  { font-family:inherit; font-size:smaller }
  pre.smallexample { font-size:smaller }
  pre.smalllisp    { font-size:smaller }
  span.sc    { font-variant:small-caps }
  span.roman { font-family:serif; font-weight:normal; } 
  span.sansserif { font-family:sans-serif; font-weight:normal; } 
--></style>
<link rel="stylesheet" type="text/css" href="../cs.css">
</head>
<body>
<div class="node">
<a name="Floating-Point-Concepts"></a>
<p>
Next:&nbsp;<a rel="next" accesskey="n" href="Floating-Point-Parameters.html#Floating-Point-Parameters">Floating Point Parameters</a>,
Up:&nbsp;<a rel="up" accesskey="u" href="Floating-Type-Macros.html#Floating-Type-Macros">Floating Type Macros</a>
<hr>
</div>

<h5 class="subsubsection">A.5.3.1 Floating Point Representation Concepts</h5>

<p>This section introduces the terminology for describing floating point
representations.

   <p>You are probably already familiar with most of these concepts in terms
of scientific or exponential notation for floating point numbers.  For
example, the number <code>123456.0</code> could be expressed in exponential
notation as <code>1.23456e+05</code>, a shorthand notation indicating that the
mantissa <code>1.23456</code> is multiplied by the base <code>10</code> raised to
power <code>5</code>.

   <p>More formally, the internal representation of a floating point number
can be characterized in terms of the following parameters:

     <ul>
<li><a name="index-sign-_0028of-floating-point-number_0029-3770"></a>The <dfn>sign</dfn> is either <code>-1</code> or <code>1</code>.

     <li><a name="index-base-_0028of-floating-point-number_0029-3771"></a><a name="index-radix-_0028of-floating-point-number_0029-3772"></a>The <dfn>base</dfn> or <dfn>radix</dfn> for exponentiation, an integer greater
than <code>1</code>.  This is a constant for a particular representation.

     <li><a name="index-exponent-_0028of-floating-point-number_0029-3773"></a>The <dfn>exponent</dfn> to which the base is raised.  The upper and lower
bounds of the exponent value are constants for a particular
representation.

     <p><a name="index-bias-_0028of-floating-point-number-exponent_0029-3774"></a>Sometimes, in the actual bits representing the floating point number,
the exponent is <dfn>biased</dfn> by adding a constant to it, to make it
always be represented as an unsigned quantity.  This is only important
if you have some reason to pick apart the bit fields making up the
floating point number by hand, which is something for which the GNU
library provides no support.  So this is ignored in the discussion that
follows.

     <li><a name="index-mantissa-_0028of-floating-point-number_0029-3775"></a><a name="index-significand-_0028of-floating-point-number_0029-3776"></a>The <dfn>mantissa</dfn> or <dfn>significand</dfn> is an unsigned integer which is a
part of each floating point number.

     <li><a name="index-precision-_0028of-floating-point-number_0029-3777"></a>The <dfn>precision</dfn> of the mantissa.  If the base of the representation
is <var>b</var>, then the precision is the number of base-<var>b</var> digits in
the mantissa.  This is a constant for a particular representation.

     <p><a name="index-hidden-bit-_0028of-floating-point-number-mantissa_0029-3778"></a>Many floating point representations have an implicit <dfn>hidden bit</dfn> in
the mantissa.  This is a bit which is present virtually in the mantissa,
but not stored in memory because its value is always 1 in a normalized
number.  The precision figure (see above) includes any hidden bits.

     <p>Again, the GNU library provides no facilities for dealing with such
low-level aspects of the representation. 
</ul>

   <p>The mantissa of a floating point number represents an implicit fraction
whose denominator is the base raised to the power of the precision.  Since
the largest representable mantissa is one less than this denominator, the
value of the fraction is always strictly less than <code>1</code>.  The
mathematical value of a floating point number is then the product of this
fraction, the sign, and the base raised to the exponent.

   <p><a name="index-normalized-floating-point-number-3779"></a>We say that the floating point number is <dfn>normalized</dfn> if the
fraction is at least <code>1/</code><var>b</var>, where <var>b</var> is the base.  In
other words, the mantissa would be too large to fit if it were
multiplied by the base.  Non-normalized numbers are sometimes called
<dfn>denormal</dfn>; they contain less precision than the representation
normally can hold.

   <p>If the number is not normalized, then you can subtract <code>1</code> from the
exponent while multiplying the mantissa by the base, and get another
floating point number with the same value.  <dfn>Normalization</dfn> consists
of doing this repeatedly until the number is normalized.  Two distinct
normalized floating point numbers cannot be equal in value.

   <p>(There is an exception to this rule: if the mantissa is zero, it is
considered normalized.  Another exception happens on certain machines
where the exponent is as small as the representation can hold.  Then
it is impossible to subtract <code>1</code> from the exponent, so a number
may be normalized even if its fraction is less than <code>1/</code><var>b</var>.)

   </body></html>