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-% -*- mode: latex; TeX-master: "Vorbis_I_spec"; -*-
-%!TEX root = Vorbis_I_spec.tex
-\section{Helper equations} \label{vorbis:spec:helper}
-
-\subsection{Overview}
-
-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 referenced where appropriate.
-
-
-\subsection{Functions}
-
-\subsubsection{ilog} \label{vorbis:spec:ilog}
-
-The "ilog(x)" function returns the position number (1 through n) of the highest set bit in the two's complement integer value
-\varname{[x]}.  Values of \varname{[x]} less than zero are defined to return zero.
-
-\begin{programlisting}
-  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
-\end{programlisting}
-
-Examples:
-
-\begin{itemize}
- \item ilog(0) = 0;
- \item ilog(1) = 1;
- \item ilog(2) = 2;
- \item ilog(3) = 2;
- \item ilog(4) = 3;
- \item ilog(7) = 3;
- \item ilog(negative number) = 0;
-\end{itemize}
-
-
-
-
-\subsubsection{float32\_unpack} \label{vorbis:spec:float32:unpack}
-
-"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.
-
-\begin{programlisting}
-  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 ) )
-\end{programlisting}
-
-
-
-\subsubsection{lookup1\_values} \label{vorbis:spec:lookup1:values}
-
-"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
-\varname{[codebook\_entries]}.
-
-The return value for this function is defined to be 'the greatest
-integer value for which \varname{[return\_value]} to the power of
-\varname{[codebook\_dimensions]} is less than or equal to
-\varname{[codebook\_entries]}'.
-
-
-
-\subsubsection{low\_neighbor} \label{vorbis:spec:low:neighbor}
-
-"low\_neighbor(v,x)" finds the position \varname{n} in vector \varname{[v]} of
-the greatest value scalar element for which \varname{n} is less than
-\varname{[x]} and vector \varname{[v]} element \varname{n} is less
-than vector \varname{[v]} element \varname{[x]}.
-
-\subsubsection{high\_neighbor} \label{vorbis:spec:high:neighbor}
-
-"high\_neighbor(v,x)" finds the position \varname{n} in vector [v] of
-the lowest value scalar element for which \varname{n} is less than
-\varname{[x]} and vector \varname{[v]} element \varname{n} is greater
-than vector \varname{[v]} element \varname{[x]}.
-
-
-
-\subsubsection{render\_point} \label{vorbis:spec:render:point}
-
-"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.
-
-\begin{programlisting}
-  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
-\end{programlisting}
-
-
-
-\subsubsection{render\_line} \label{vorbis:spec:render:line}
-
-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.
-
-
-\begin{programlisting}
-  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] ) {
-
-             16) [err] = [err] - [adx]
-             17)   [y] = [y] + [sy]
-
-           } else {
-
-             18) [y] = [y] + [base]
-
-           }
-
-       19) vector [v] element [x] = [y]
-
-     }
-\end{programlisting}
-
-
-
-
-
-
-
-