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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
+<html>
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+<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-15"/>
+<title>Ogg Vorbis Documentation</title>
+
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+<div id="xiphlogo">
+ <a href="http://www.xiph.org/"><img src="fish_xiph_org.png" alt="Fish Logo and Xiph.Org"/></a>
+</div>
+
+<h1>Ogg Vorbis I format specification: helper equations</h1>
+
+<h1>Overview</h1>
+
+<p>The equations below are used in multiple places by the Vorbis codec
+specification. Rather than cluttering up the main specification
+documents, they are defined here and linked in the main documents
+where appropriate.</p>
+
+<h2><a name="log">ilog</a></h2>
+
+<p>The "ilog(x)" function returns the position number (1 through n) of the
+highest set bit in the two's complement integer value
+<tt>[x]</tt>. Values of <tt>[x]</tt> less than zero are defined to return zero.</p>
+
+<pre>
+ 1) [return_value] = 0;
+ 2) if ( [x] is greater than zero ){
+
+ 3) increment [return_value];
+ 4) logical shift [x] one bit to the right, padding the MSb with zero
+ 5) repeat at step 2)
+
+ }
+
+ 6) done
+</pre>
+
+<p>Examples:</p>
+
+<ul>
+<li>ilog(0) = 0;</li>
+<li>ilog(1) = 1;</li>
+<li>ilog(2) = 2;</li>
+<li>ilog(3) = 2;</li>
+<li>ilog(4) = 3;</li>
+<li>ilog(7) = 3;</li>
+<li>ilog(negative number) = 0;</li>
+</ul>
+
+<h2><a name="float32_unpack">float32_unpack</a></h2>
+
+<p>"float32_unpack(x)" is intended to translate the packed binary
+representation of a Vorbis codebook float value into the
+representation used by the decoder for floating point numbers. For
+purposes of this example, we will unpack a Vorbis float32 into a
+host-native floating point number.</p>
+
+<pre>
+ 1) [mantissa] = [x] bitwise AND 0x1fffff (unsigned result)
+ 2) [sign] = [x] bitwise AND 0x80000000 (unsigned result)
+ 3) [exponent] = ( [x] bitwise AND 0x7fe00000) shifted right 21 bits (unsigned result)
+ 4) if ( [sign] is nonzero ) then negate [mantissa]
+ 5) return [mantissa] * ( 2 ^ ( [exponent] - 788 ) )
+</pre>
+
+<h2><a name="lookup1_values">lookup1_values</a></h2>
+
+<p>"lookup1_values(codebook_entries,codebook_dimensions)" is used to
+compute the correct length of the value index for a codebook VQ lookup
+table of lookup type 1. The values on this list are permuted to
+construct the VQ vector lookup table of size
+<tt>[codebook_entries]</tt>.</p>
+
+<p>The return value for this function is defined to be 'the greatest
+integer value for which <tt>[return_value] to the power of
+[codebook_dimensions] is less than or equal to
+[codebook_entries]</tt>'.</p>
+
+<h2><a name="low_neighbor">low_neighbor</a></h2>
+
+<p>"low_neighbor(v,x)" finds the position <i>n</i> in vector [v] of
+the greatest value scalar element for which <i>n</i> is less than
+<tt>[x]</tt> and <tt>vector [v] element <i>n</i> is less
+than vector [v] element [x]</tt>.</p>
+
+<h2><a name="high_neighbor">high_neighbor</a></h2>
+
+<p>"high_neighbor(v,x)" finds the position <i>n</i> in vector [v] of
+the lowest value scalar element for which <i>n</i> is less than
+<tt>[x]</tt> and <tt>vector [v] element <i>n</i> is greater
+than vector [v] element [x]</tt>.</p>
+
+<h2><a name="render_point">render_point</a></h2>
+
+<p>"render_point(x0,y0,x1,y1,X)" is used to find the Y value at point X
+along the line specified by x0, x1, y0 and y1. This function uses an
+integer algorithm to solve for the point directly without calculating
+intervening values along the line.</p>
+
+<pre>
+ 1) [dy] = [y1] - [y0]
+ 2) [adx] = [x1] - [x0]
+ 3) [ady] = absolute value of [dy]
+ 4) [err] = [ady] * ([X] - [x0])
+ 5) [off] = [err] / [adx] using integer division
+ 6) if ( [dy] is less than zero ) {
+
+ 7) [Y] = [y0] - [off]
+
+ } else {
+
+ 8) [Y] = [y0] + [off]
+
+ }
+
+ 9) done
+</pre>
+
+<h2><a name="render_line">render_line</a></h2>
+
+<p>Floor decode type one uses the integer line drawing algorithm of
+"render_line(x0, y0, x1, y1, v)" to construct an integer floor
+curve for contiguous piecewise line segments. Note that it has not
+been relevant elsewhere, but here we must define integer division as
+rounding division of both positive and negative numbers toward zero.</p>
+
+<pre>
+ 1) [dy] = [y1] - [y0]
+ 2) [adx] = [x1] - [x0]
+ 3) [ady] = absolute value of [dy]
+ 4) [base] = [dy] / [adx] using integer division
+ 5) [x] = [x0]
+ 6) [y] = [y0]
+ 7) [err] = 0
+
+ 8) if ( [dy] is less than 0 ) {
+
+ 9) [sy] = [base] - 1
+
+ } else {
+
+ 10) [sy] = [base] + 1
+
+ }
+
+ 11) [ady] = [ady] - (absolute value of [base]) * [adx]
+ 12) vector [v] element [x] = [y]
+
+ 13) iterate [x] over the range [x0]+1 ... [x1]-1 {
+
+ 14) [err] = [err] + [ady];
+ 15) if ( [err] >= [adx] ) {
+
+ 15) [err] = [err] - [adx]
+ 16) [y] = [y] + [sy]
+
+ } else {
+
+ 17) [y] = [y] + [base]
+
+ }
+
+ 18) vector [v] element [x] = [y]
+
+ }
+</pre>
+
+<div id="copyright">
+ The Xiph Fish Logo is a
+ trademark (&trade;) of Xiph.Org.<br/>
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