-\section{Number Representations Trialed}
-\begin{itemize}
- \item IEEE-754 single, double, extended
- \item Custom implementation of Rationals with \verb/int64_t/
- \begin{itemize}
- \item Very limited since the integers grow exponentially and overflow
- \end{itemize}
- \item Custom implementation of Rationals with custom Arbitrary precision integers
- \begin{itemize}
- \item Actually works
- \item Implementation of division is too slow to be feasible
- \end{itemize}
- \item Custom rationals but with GMP arbitrary precision integers
- \begin{itemize}
- \item Our implementation of GCD is not feasible
- \end{itemize}
- \item Paranoid Numbers; store a operation tree of IEEE-754 floats and simplify the tree wherever \verb/FE_INEXACT/ is \emph{not} raised
- \begin{itemize}
- \item This was a really, really, really, bad idea
- \end{itemize}
- \item Just use GMP rationals already
- \begin{itemize}
- \item Works
- \end{itemize}
-
- \item MPFR floats
- \begin{itemize}
- \item They work, but they don't truly give arbitrary precision
- \item Because you have to specify the maximum precision
- \item However, this can be changed at runtime
- \item Future work: Trial MPFR floats changing the precision as needed
- \end{itemize}
-
-\end{itemize}
+\subsection{Na\"{i}ve Approach}
+
+A na\"{i}ve approach would be to replace all floating point operations with arbitrary precision operations, and this was in fact tried in early experiments. This approach requires use of the CPU renderer, as GLSL is restricted to floating point representations. A type definition \texttt{Real} on the CPU can be selected at compile time.
+
+Unfortunately truly arbitrary precision number representations were found to be far too inefficient for practical purposes --- for example, rendering a frame with GMP Rationals could take up to 60 seconds at the default view.
+
+
+\begin{comment}
+\subsubsection{Number Representations Trialed}
+
+\begin{enumerate}
+ \item IEEE-754 single, double, extended (control)
+ \item Custom implementation of Rationals with 64 bit integers
+ \item Custom implementation of Rationals with custom Arbitrary Precision Integers
+ \item Custom implementation of Rationals but with GMP integers
+ \item GMP Rationals
+ \item MPFR Arbitrary Precision Floats
+ \item iRRAM ``exact'' real arithmetic\cite{}
+\end{enumerate}
+\end{comment}
+
+\subsection{Intermediate Coordinate Systems}
+
+When an object is visible on the screen it is only necessary to render it accurately to within the nearest pixel.
+As shown in the Results Section \ref{}, introducing an intermediate coordinate system for a large number of objects and applying transformations to this coordinate system instead of individual objects produces the best results both in terms of reduced rounding errors using floating point arithmetic, and reduced number of required arbitrary precision operations.
+
+\subsection{Quadtree Document Division}
+
+An approach identified by Gow\cite{} is to construct intermediate coordinate systems as the user manipulates the view.
+
+