+\section{Charles Babbage \cite{dodge_babbage, nature1871babbage}}
+
+Tributes to Charles Babbage. Might be interesting for historical background. Don't mention anything about floating point numbers though.
+
+\section{GPU Floating-Point Paranoia \cite{hillesland2004paranoia}}
+
+This paper discusses floating point representations on GPUs. They have reproduced the program \emph{Paranoia} by William Kahan for characterising floating point behaviour of computers (pre IEEE) for GPUs.
+
+
+There are a few remarks about GPU vendors not being very open about what they do or do not do with
+
+
+Unfortunately we only have the extended abstract, but a pretty good summary of the paper (written by the authors) is at: \url{www.cs.unc.edu/~ibr/projects/paranoia/}
+
+From the abstract:
+
+``... [GPUs are often similar to IEEE] However, we have found
+that GPUs do not adhere to IEEE standards for floating-point op-
+erations, nor do they give the information necessary to establish
+bounds on error for these operations ... ''
+
+and ``...Our goal is to determine the error bounds on floating-point op-
+eration results for quickly evolving graphics systems. We have cre-
+ated a tool to measure the error for four basic floating-point opera-
+tions: addition, subtraction, multiplication and division.''
+
+The implementation is only for windows and uses glut and glew and things.
+Implement our own version?
+
+\section{A floating-point technique for extending the available precision \cite{dekker1971afloating}}
+
+This is Dekker's formalisation of the Fast2Sum algorithm originally implemented by Kahn.
+
+\begin{align*}
+ z &= \text{RN}(x + y) \\
+ w &= \text{RN}(z - x) \\
+ zz &= \text{RN}(y - w) \\
+ \implies z + zz &= x + y
+\end{align*}
+
+There is a version for multiplication.
+
+I'm still not quite sure when this is useful. I haven't been able to find an example for $x$ and $y$ where $x + y \neq \text{Fast2Sum}(x, y)$.
+
+
+
+\chapter{General Notes}
+
+\section{Rounding Errors}
+
+They happen. There is ULP and I don't mean a political party.
+
+TODO: Probably say something more insightful. Other than "here is a graph that shows errors and we blame rounding".
+
+Results of \verb/calculatepi/
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/calculatepi.pdf}
+ \caption{Example of accumulated rounding errors in a numerical calculation}
+\end{figure}
+
+Tests with \verb/calculatepi/ show it's not quite as simple as just blindly replacing all your additions with Fast2Sum from Dekker\cite{dekker1971afloating}.
+ie: The graph looks exactly the same for single precision. \verb/calculatepi/ obviously also has multiplication ops in it which I didn't change. Will look at after sleep maybe.
+
+