Penultimate draft

[ipdf/sam.git] / chapters / Background / Floats / Visualisation.tex

diff --git a/chapters/Background/Floats/Visualisation.tex b/chapters/Background/Floats/Visualisation.tex
index 41bbda3..90b30b5 100644 (file)
--- a/chapters/Background/Floats/Visualisation.tex
+++ b/chapters/Background/Floats/Visualisation.tex
@@ -1,5 +1,7 @@
 
-Figure \ref{floats.pdf} shows the positive real numbers which can be represented exactly by an 8 bit floating point number encoded in the IEEE-754 format. We show two encodings using (1,2,5) and (1,3,4) bits to encode (sign, exponent, mantissa) respectively. For each distinct value of the exponent, the successive floating point representations lie on a straight line with constant slope. As the exponent increases, larger values are represented, but the distance between successive values increases; this can be seen in Figure\ref{}. The marked single point discontinuity at \verb/0x10/ and \verb/0x20/ occur when $e$ leaves the denormalised region and the encoding of $m$ changes. We have also plotted a fixed point representation for comparison; fixed point and integer representations appear on straight lines.
+To assist with understanding the limitations of floating point representations, we have produced a plot of the positive real numbers which can be represented exactly by an 8 bit floating point number encoded in the IEEE-754 format. We could not find any similar visualisations in the literature.
+
+In Figure \ref{floats.pdf} we show two encodings using (1,2,5) and (1,3,4) bits to encode (sign, exponent, mantissa) respectively. For each distinct value of the exponent, the successive floating point representations lie on a straight line with constant slope. As the exponent increases, larger values are represented, but the distance between successive values increases; this can be seen in Figure \ref{floats_diff.pdf}. The marked single point discontinuity at \verb/0x10/ and \verb/0x20/ occur when $e$ leaves the denormalised region and the encoding of $m$ changes. We have also plotted a fixed point representation for comparison; fixed point and integer representations appear on straight lines.
 
 \begin{comment}
 The earlier example $7.25$ would be converted to a (1,3,4) floating point representation as follows:
@@ -15,20 +17,21 @@ The earlier example $7.25$ would be converted to a (1,3,4) floating point repres
 This particular example can be encoded exactly; however as there are an infinite number of real values and only a finite number of floats, in general a value must be $7.26$ must be rounded or truncated at Step 3. 
 \end{comment}
 
+\input{chapters/Background/Floats/Operations}
 %\begin{figure}[H]
 %      \centering
 %\begin{minipage}[t]{0.45\textwidth}
        \begin{figure}[H]
                \centering
-               \includegraphics[width=0.8\textwidth]{figures/floats.pdf}\label{floats.pdf}
-               \caption{Positive 8-Bit Number Representations}
+               \includegraphics[width=0.8\textwidth]{figures/floats.pdf} 
+               \caption{Positive 8-Bit Number Representations} \label{floats.pdf}
        \end{figure}
 %\end{minipage}
 %\begin{minipage}[t]{0.45\textwidth}
        \begin{figure}[H]
                \centering
-               \includegraphics[width=0.8\textwidth]{figures/floats_diff.pdf}\label{floats_diff.pdf}
-               \caption{Difference between successive numbers}
+               \includegraphics[width=0.8\textwidth]{figures/floats_diff.pdf} 
+               \caption{Difference between successive numbers} \label{floats_diff.pdf}
        \end{figure}
 %\end{minipage}
 %      \caption{8 bit float and fixed point representations a) As mapped to real values b) The distance between each representation}\label{floats.pdf}