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<a name="Representation-of-floating-point-numbers"></a>
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<p>
Next: <a href="Setting-up-your-IEEE-environment.html#Setting-up-your-IEEE-environment" accesskey="n" rel="next">Setting up your IEEE environment</a>, Up: <a href="IEEE-floating_002dpoint-arithmetic.html#IEEE-floating_002dpoint-arithmetic" accesskey="u" rel="up">IEEE floating-point arithmetic</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Representation-of-floating-point-numbers-1"></a>
<h3 class="section">45.1 Representation of floating point numbers</h3>
<a name="index-IEEE-format-for-floating-point-numbers"></a>
<a name="index-bias_002c-IEEE-format"></a>
<a name="index-exponent_002c-IEEE-format"></a>
<a name="index-sign-bit_002c-IEEE-format"></a>
<a name="index-mantissa_002c-IEEE-format"></a>
<p>The IEEE Standard for Binary Floating-Point Arithmetic defines binary
formats for single and double precision numbers.  Each number is composed
of three parts: a <em>sign bit</em> (<em>s</em>), an <em>exponent</em>
(<em>E</em>) and a <em>fraction</em> (<em>f</em>).  The numerical value of the
combination <em>(s,E,f)</em> is given by the following formula,
</p>
<div class="example">
<pre class="example">(-1)^s (1.fffff...) 2^E
</pre></div>

<a name="index-normalized-form_002c-IEEE-format"></a>
<a name="index-denormalized-form_002c-IEEE-format"></a>
<p>The sign bit is either zero or one.  The exponent ranges from a minimum value
<em>E_min</em> 
to a maximum value
<em>E_max</em> depending on the precision.  The exponent is converted to an 
unsigned number
<em>e</em>, known as the <em>biased exponent</em>, for storage by adding a
<em>bias</em> parameter,
<em>e = E + bias</em>.
The sequence <em>fffff...</em> represents the digits of the binary
fraction <em>f</em>.  The binary digits are stored in <em>normalized
form</em>, by adjusting the exponent to give a leading digit of <em>1</em>. 
Since the leading digit is always 1 for normalized numbers it is
assumed implicitly and does not have to be stored.
Numbers smaller than 
<em>2^(E_min)</em>
are be stored in <em>denormalized form</em> with a leading zero,
</p>
<div class="example">
<pre class="example">(-1)^s (0.fffff...) 2^(E_min)
</pre></div>

<a name="index-zero_002c-IEEE-format"></a>
<a name="index-infinity_002c-IEEE-format"></a>
<p>This allows gradual underflow down to 
<em>2^(E_min - p)</em> for <em>p</em> bits of precision. 
A zero is encoded with the special exponent of 
<em>2^(E_min - 1)</em> and infinities with the exponent of 
<em>2^(E_max + 1)</em>.
</p>
<a name="index-single-precision_002c-IEEE-format"></a>
<p>The format for single precision numbers uses 32 bits divided in the
following way,
</p>
<div class="smallexample">
<pre class="smallexample">seeeeeeeefffffffffffffffffffffff
    
s = sign bit, 1 bit
e = exponent, 8 bits  (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits
</pre></div>

<a name="index-double-precision_002c-IEEE-format"></a>
<p>The format for double precision numbers uses 64 bits divided in the
following way,
</p>
<div class="smallexample">
<pre class="smallexample">seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff

s = sign bit, 1 bit
e = exponent, 11 bits  (E_min=-1022, E_max=1023, bias=1023)
f = fraction, 52 bits
</pre></div>

<p>It is often useful to be able to investigate the behavior of a
calculation at the bit-level and the library provides functions for
printing the IEEE representations in a human-readable form.
</p>

<dl>
<dt><a name="index-gsl_005fieee_005ffprintf_005ffloat"></a>Function: <em>void</em> <strong>gsl_ieee_fprintf_float</strong> <em>(FILE * <var>stream</var>, const float * <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fieee_005ffprintf_005fdouble"></a>Function: <em>void</em> <strong>gsl_ieee_fprintf_double</strong> <em>(FILE * <var>stream</var>, const double * <var>x</var>)</em></dt>
<dd><p>These functions output a formatted version of the IEEE floating-point
number pointed to by <var>x</var> to the stream <var>stream</var>. A pointer is
used to pass the number indirectly, to avoid any undesired promotion
from <code>float</code> to <code>double</code>.  The output takes one of the
following forms,
</p>
<dl compact="compact">
<dt><code>NaN</code></dt>
<dd><p>the Not-a-Number symbol
</p>
</dd>
<dt><code>Inf, -Inf</code></dt>
<dd><p>positive or negative infinity
</p>
</dd>
<dt><code>1.fffff...*2^E, -1.fffff...*2^E</code></dt>
<dd><p>a normalized floating point number
</p>
</dd>
<dt><code>0.fffff...*2^E, -0.fffff...*2^E</code></dt>
<dd><p>a denormalized floating point number
</p>
</dd>
<dt><code>0, -0</code></dt>
<dd><p>positive or negative zero
</p>
</dd>
</dl>

<p>The output can be used directly in GNU Emacs Calc mode by preceding it
with <code>2#</code> to indicate binary.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fieee_005fprintf_005ffloat"></a>Function: <em>void</em> <strong>gsl_ieee_printf_float</strong> <em>(const float * <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fieee_005fprintf_005fdouble"></a>Function: <em>void</em> <strong>gsl_ieee_printf_double</strong> <em>(const double * <var>x</var>)</em></dt>
<dd><p>These functions output a formatted version of the IEEE floating-point
number pointed to by <var>x</var> to the stream <code>stdout</code>.
</p></dd></dl>

<p>The following program demonstrates the use of the functions by printing
the single and double precision representations of the fraction
<em>1/3</em>.  For comparison the representation of the value promoted from
single to double precision is also printed.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_ieee_utils.h&gt;

int
main (void) 
{
  float f = 1.0/3.0;
  double d = 1.0/3.0;

  double fd = f; /* promote from float to double */
  
  printf (&quot; f=&quot;); gsl_ieee_printf_float(&amp;f); 
  printf (&quot;\n&quot;);

  printf (&quot;fd=&quot;); gsl_ieee_printf_double(&amp;fd); 
  printf (&quot;\n&quot;);

  printf (&quot; d=&quot;); gsl_ieee_printf_double(&amp;d); 
  printf (&quot;\n&quot;);

  return 0;
}
</pre></div>

<p>The binary representation of <em>1/3</em> is <em>0.01010101... </em>.  The
output below shows that the IEEE format normalizes this fraction to give
a leading digit of 1,
</p>
<div class="smallexample">
<pre class="smallexample"> f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
 d= 1.0101010101010101010101010101010101010101010101010101*2^-2
</pre></div>

<p>The output also shows that a single-precision number is promoted to
double-precision by adding zeros in the binary representation.
</p>

  





<hr>
<div class="header">
<p>
Next: <a href="Setting-up-your-IEEE-environment.html#Setting-up-your-IEEE-environment" accesskey="n" rel="next">Setting up your IEEE environment</a>, Up: <a href="IEEE-floating_002dpoint-arithmetic.html#IEEE-floating_002dpoint-arithmetic" accesskey="u" rel="up">IEEE floating-point arithmetic</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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