\chapter{Literature Review}\label{Background}
-\rephrase{0. Here is a brilliant summary of the sections below}
+The first half of this chapter will be devoted to documents themselves, including: the representation and displaying of graphics primitives\cite{computergraphics2}, and how collections of these primitives are represented in document formats, focusing on widely used standards\cite{plrm, pdfref17, svg2011-1.1}.
-This chapter provides an overview of relevant literature. The areas of interest can be broadly grouped into two largely seperate categories; Documents and Number Representations.
+We will find that although there has been a great deal of research into the rendering, storing, editing, manipulation, and extension of document formats, modern standards are content to specify at best single precision IEEE-754 floating point arithmetic.
-The first half of this chapter will be devoted to documents themselves, including: the representation and rendering of low level graphics primitives, how collections of these primitives are represented in document formats, and the various standards for documents in use today.
+The research on arbitrary precision arithmetic applied to documents is very sparse; however arbitrary precision arithmetic itself is a very active field of research. Therefore, the second half of this chapter will be devoted to considering fixed precision floating point numbers as specified by the IEEE-754 standard, possible limitations in precision, and alternative number representations for increased or arbitrary precision arithmetic.
-We will find that although there has been a great deal of research into the rendering, storing, editing, manipulation, and extension of document formats, all popular document standards are content to specify at best a single precision IEEE-754 floating point number representations.
+In Chapter \ref{Progress}, we will discuss our findings so far with regards to arbitrary precision arithmetic applied to document formats, and expand upon the goals outlined in Chapture \ref{Proposal}.
-The research on arbitrary precision arithmetic applied to documents is very sparse; however arbitrary precision arithmetic itself is a very active field of research. Therefore, the second half of this chapter will be devoted to considering the IEEE-754 standard, its advantages and limitations, and possible alternative number representations to allow for arbitrary precision arithmetic.
+\section{Raster and Vector Images}\label{Raster and Vector Images}
+\input{chapters/Background_Raster-vs-Vector}
-In Chapter \ref{Progress}, we will discuss our findings so far with regards to arbitrary precision arithmetic applied to document formats.
+\section{Rasterising Vector Images}\label{Rasterising Vector Images}
+Throughout Section \ref{vector-vs-raster-graphics} we were careful to refer to ``modern'' display devices, which are raster based. It is of some historical significance that vector display devices were popular during the 70s and 80s, and papers oriented towards drawing on these devices can be found\cite{brassel1979analgorithm}. Whilst curves can be drawn at high resolution on vector displays, a major disadvantage was shading; by the early 90s the vast majority of computer displays were raster based\cite{computergraphics2}.
-\pagebreak
-
-\section{Raster and Vector Graphics}\label{vector-vs-raster-graphics}
-
-\rephrase{1. Here are the fundamentals of graphics (raster and vector, rendering)}
-
-At a fundamental level everything that is seen on a display device is represented as either a vector or raster image. These images can be stored as stand alone documents or embedded in a much more complex document format capable of containing many other types of information.
-
-A raster image's structure closely matches it's representation as shown on modern display hardware; the image is represented as a grid of filled square ``pixels''. Each pixel is the same size and contains information describing its colour. This representation is simple and also well suited to storing images as produced by cameras and scanners\cite{citationneeded}.
-
-The drawback of raster images is that by their very nature there can only be one level of detail. Figures \ref{vector-vs-raster} and \ref{vector-vs-raster-scaled} attempt to illustrate this by comparing raster images to vector images in a similar way to Worth and Packard\cite{worth2003xr}.
-
-Consider the right side of Figure \ref{vector-vs-raster}. This is a raster image which should be recognisable as an animal defined by fairly sharp edges. Figure \ref{vector-vs-raster-scaled} shows that zooming on the animal's face causes these edges to appear jagged. There is no information in the original image as to what should be displayed at a larger size, so each square shaped pixel is simply increased in size. A blurring effect will probably be visible in most PDF viewers; the software has attempted to make the ``edge'' appear more realistic using a technique called ``antialiasing'' which averages neighbouring pixels in the original image in order to generate extra pixels in the scaled image\cite{citationneeded}.\footnote{The exact appearance of the images at different zoom levels will depend greatly on the PDF viewer or printer used to display this report. On the author's display using the Atril (1.6.0) viewer, the top images appear to be pixel perfect mirror images at a 100\% scale. In the bottom raster image, antialiasing is not applied at zoom levels above $125\%$ and the effect of scaling is quite noticable.}
-
-%\footnote{\noindent This behaviour may be configured in some PDF viewers (Adobe Reader) whilst others (Evince, Atril, Okular) will choose whether or not to bother with antialiasing based on the zoom level. For best results experiment with changing the zoom level in your PDF viewer.\footnotemark}\footnotetext{On the author's hardware, the animals in the vector and raster images should appear mirrored pixel for pixel; but they may vary slightly on other PDF viewers or display devices.}
-
-In contrast, the left sides of Figures \ref{vector-vs-raster} and \ref{vector-vs-raster-scaled} are a vector image. A vector image contains information about a number of geometric shapes. To display this image on modern display hardware, the coordinates are transformed according to the view and \emph{then} the image is converted into a raster like representation. Whilst the raster image merely appears to contain edges, the vector image actually contains information about these edges, meaning they can be displayed ``infinitely sharply'' at any level of detail\cite{citationneeded} --- or they could be if the coordinates are stored with enough precision (see Section \ref{}). Thus, vector images are well suited to high quality digital art\footnote{Figure \ref{vector-vs-raster} is not to be taken as an example of this.} and text\cite{citationneeded}.
-
-
-\rephrase{Woah, an entire page with only one citation ham fisted in after I had written the rest... and the ``actually writing it'' phase of the Lit Review is off to a great start.}
-
-\newlength\imageheight
-\newlength\imagewidth
-\settoheight\imageheight{\includegraphics{figures/fox-raster.png}}
-\settowidth\imagewidth{\includegraphics{figures/fox-raster.png}}
-
-%Height: \the\imageheight
-%Width: \the\imagewidth
-
-
-\begin{figure}[H]
- \centering
- \includegraphics[scale=0.7528125]{figures/fox-vector.pdf}
- \includegraphics[scale=0.7528125]{figures/fox-raster.png}
- \caption{Original Vector and Raster Images}\label{vector-vs-raster}
-\end{figure} % As much as I hate to break up the party, these fit best over the page (at the moment)
-\begin{figure}[H]
- \centering
- \includegraphics[scale=0.7528125, viewport=210 85 280 150,clip, width=0.45\textwidth]{figures/fox-vector.pdf}
- \includegraphics[scale=0.7528125, viewport=0 85 70 150,clip, width=0.45\textwidth]{figures/fox-raster.png}
- \caption{Scaled Vector and Raster Images}\label{vector-vs-raster-scaled}
-\end{figure}
+Hearn and Baker's textbook ``Computer Graphics''\cite{computergraphics2} gives a comprehensive overview of graphics from physical display technologies through fundamental drawing algorithms to popular graphics APIs. This section will examine algorithms for drawing two dimensional geometric primitives on raster displays as discussed in ``Computer Graphics'' and the relevant literature. Informal tutorials are abundant on the internet\cite{elias2000graphics}. This section is by no means a comprehensive survey of the literature but intends to provide some idea of the computations which are required to render a document.
-\section{Rendering Vector Images}
+\subsection{Straight Lines}\label{Straight Lines}
+\input{chapters/Background_Lines}
-Throughout Section \ref{vector-vs-raster-graphics} we were careful to refer to ``modern'' display devices, which are raster based. It is of some historical significance that vector display devices were popular during the 70s and 80s, and so algorithms for drawing a vector image directly without rasterisation exist. An example is the shading of polygons which is somewhat more complicated on a vector display than a raster display\cite{brassel1979analgorithm, lane1983analgorithm}.
+\subsection{Spline Curves}\label{Spline Curves}
-All modern displays of practical interest are raster based. In this section we explore the structure of vector graphics in more detail, and how different primitives are rendered.
+Splines are continuous curves formed from piecewise polynomial segments. A polynomial of $n$th degree is defined by $n$ constants $\{a_0, a_1, ... a_n\}$ and:
+\begin{align}
+ y(x) &= \displaystyle\sum_{k=0}^n a_k x^k
+\end{align}
-\rephrase{After the wall of citationless text in Section \ref{vector-vs-raster-graphics} we should probably redeem ourselves a bit here}
-\subsection{Bezier Curves}
-
-The bezier curve is of vital importance in vector graphics.
-
-\rephrase{Things this section lacks}
-\begin{itemize}
- \item Who came up with them (presumably it was a guy named Bezier)
- \item Flesh out how they evolved or came into use?
- \item Naive algorithm
- \item De Casteljau Algorithm
-\end{itemize}
+A straight line is simply a polynomial of $0$th degree. Splines may be rasterised by sampling of $y(x)$ at a number of points $x_i$ and rendering straight lines between $(x_i, y_i)$ and $(x_{i+1}, y_{i+1})$ as discussed in Section \ref{Straight Lines}. More direct algorithms for drawing splines based upon Brasenham and Wu's algorithms also exist\cite{citationneeded}.
-Recently, Goldman presented an argument that Bezier's could be considered as fractal in nature, a fractal being the fixed point of an iterated function system\cite{goldman_thefractal}. Goldman's proof depends upon a modification to the De Casteljau Subdivision algorithm. Whilst we will not go the details of the proof, or attempt comment on its theoretical value, it is interesting to note that Goldman's algorithm is not only worse than the De Casteljau algorithm upon which it was based, but it also performs worse than a naive Bezier rendering algorithm. Figure \ref{bezier-goldman} shows our results using implementations of the various algorithms in python.
+There are many different ways to define a spline. One approach is to specify ``knots'' on the spline and solve for the cooefficients to generate a cubic spline ($n = 3$) passing through the points. Alternatively, special polynomials may be defined using ``control'' points which themselves are not part of the curve; these are convenient for graphical based editors. Bezier splines are the most straight forward way to define a curve in the standards considered in Section \ref{Document Representations}
+\subsubsection{Bezier Curves}
+\input{chapters/Background_Bezier}
-\begin{figure}[H]
- \centering
- \includegraphics[width=0.7\textwidth]{figures/bezier-goldman.png}
- \caption{Performance of Bezier Subdivision Algorithms}\label{bezier-goldman}
-\end{figure}
+\subsection{Font Rendering}
-\rephrase{Does the Goldman bit need to be here? Probably NOT. Do I need to check very very carefully that I haven't made a mistake before saying this? YES. Will I have made a mistake? Probably.}
+Donald Knuth's 1986 textbook ``Metafont'' blargh
-\subsection{Shapes}
-Shapes are just bezier curves joined together.
-\subsubsection{Approximating a Circle Using Cubic Beziers}
+\subsection{Shading}
-An example of a shape is a circle. We used some algorithm on wikipedia that I'm sure is in Literature somewhere
-\cite{citationneeded} and made a circle. It's in my magical ipython notebook with the De Casteljau stuff.
+Algorithms for shading on vector displays involved drawing equally spaced lines in the region with endpoints defined by the boundaries of the region\cite{brassel1979analgorithm}. Apart from being unrealistic, these techniques required a computationally expensive sorting of vertices\cite{lane1983analgorithm}.
-\subsection{Text}
-Text is just Bezier Curves, I think we proved out point with the circle, but maybe find some papers to cite\cite{citationneeded}
+On raster displays, shading is typically based upon Lane's algorithm of 1983\cite{lane1983analgorithm}. Lane's algorithm relies on the ability to ``subtract'' fill from a region. This algorithm is now implemented in the GPU \rephrase{stencil buffer-y and... stuff} \cite{kilgard2012gpu}
+\subsection{Compositing and the Painter's Model}\label{Compositing and the Painter's Model}
-\subsection{Shading}
+So far we have discussed techniques for rendering vector graphics primitives in isolation, with no regard to the overall structure of a document which may contain many thousands of primitives. A straight forward approach would be to render all elements sequentially to the display, with the most recently drawn pixels overwriting lower elements. Such an approach is particularly inconvenient for anti-aliased images where colours must appear to smoothly blur between the edge of a primitive and any drawn underneath it.
-Shading is actually extremely complicated! \cite{brassel1979analgorithm, lane1983analgorithm}
-\rephrase{Sure, but do we care enough to talk about it? We will run out of space at this rate}
+Colour raster displays are based on an additive red-green-blue $(r,g,b)$ colour representation which matches the human eye's response to light\cite{computergraphics2}. In 1984, Porter and Duff introduced a fourth colour channel for rasterised images called the ``alpha'' channel, analogous to the transparency of a pixel\cite{porter1984compositing}. In compositing models, elements can be rendered seperately, with the four colour channels of successively drawn elements being combined according to one of several possible operations.
-\subsection{Other Things}
-We don't really care about other things in this report.
+In the ``painter's model'' as described by the SVG standard, Porter and Duff's ``over'' operation is used when rendering one primitive over another\cite{svg2011-1.1}.
+Given an existing pixel $P_1$ with colour values $(r_1, g_1, b_1, a_1)$ and a pixel $P_2$ with colours $(r_2, g_2, b_2, a_2)$ to be painted over $P_1$, the resultant pixel $P_T$ has colours given by:
+\begin{align}
+ a_T &= 1 - (1-a_1)(1-a_2) \\
+ r_T &= (1 - a_2)r_1 + r_2 \quad \text{(similar for $g_T$ and $b_T$)}
+\end{align}
+It should be apparent that alpha values of $1$ correspond to an opaque pixel; that is, when $a_2 = 1$ the resultant pixel $P_T$ is the same as $P_2$.
+When the final pixel is actually drawn on an rgb display, the $(r, g, b)$ components are $(r_T/a_T, g_T/a_T, b_T/a_T)$.
-\rephrase{6. Sort of starts here... or at least background does}
+The PostScript and PDF standards, as well as the OpenGL API also use a painter's model for compositing. However, PostScript does not include an alpha channel, so $P_T = P_2$ always\cite{plrm}. Figure \ref{SVG} illustrates the painter's model for partially transparent shapes as they would appear in both the SVG and PDF models.
-\subsection{Rendering Vector Graphics on the GPU}
+\subsection{Rasterisation on the CPU and GPU}
-Traditionally, vector graphics have been rasterized by the CPU before being sent to the GPU for drawing\cite{kilgard2012gpu}. Lots of people would like to change this \cite{worth2003xr, loop2007rendering, rice2008openvg, kilgard2012gpu, green2007improved} ... \rephrase{All of these are things David found except kilgard which I thought I found and then realised David already had it :S}
+Traditionally, vector graphics have been rasterized by the CPU before being sent to the GPU for drawing\cite{kilgard2012gpu}. Lots of people would like to change this \cite{worth2003xr, loop2007rendering, rice2008openvg, kilgard2012gpu, green2007improved}.
\rephrase{2. Here are the ways documents are structured ... we got here eventually}
-\section{Document Representations}
+\section{Document Representations}\label{Document Representations}
-\rephrase{The file format can be either human readable\footnote{For some definition of human and some definition of readable} or binary\footnote{So, our viewer is basically a DOM style but stored in a binary format}. Can also be compressed or not. Here we are interested in how the document is interpreted or traversed in order to produce graphics output.}
+The representation of information, particularly for scientific purposes, has changed dramatically over the last few decades. For example, Brassel's 1979 paper referenced earlier has been produced on a mechanical type writer. Although the paper discusses an algorithm for shading on computer displays, the figures illustrating this algorithm have not been generated by a computer, but drawn by Brassel's assistant\cite{brassel1979analgorithm}. In contrast, modern papers such as Barnes et. al's recent paper on embedding 3d images in PDF documents\cite{barnes2013embeddding} can themselves be an interactive proof of concept.
-\subsection{Interpreted Model}
+In this section we will consider various approaches and motivations to specifying the structure and appearance of a document, including: early interpreted formats (PostScript, \TeX, DVI), the Document Object Model popular in standards for web based documents (HTML, SVG), and Adobe's ubiquitous Portable Document Format (PDF). Some of these formats were discussed in a recent paper ``Pixels Or Perish'' by Hayes\cite{hayes2012pixelsor} who argues for greater interactivity in the PDF standard.
+
+\subsection{Interpreted Document Formats}
+\input{chapters/Background_Interpreted}
-\rephrase{Did I just invent that terminology or did I read it in a paper? Is there actually existing terminology for this that sounds similar enough to ``Document Object Model'' for me to compare them side by side?}
\begin{itemize}
\item This model treats a document as the source code program which produces graphics
\item Arose from the desire to produce printed documents using computers (which were still limited to text only displays).
\item Typed by hand or (later) generated by a GUI program
\item PostScript --- largely supersceded by PDF on the desktop but still used by printers\footnote{Desktop pdf viewers can still cope with PS, but I wonder if a smartphone pdf viewer would implement it?}
- \item \TeX --- Predates PostScript! {\LaTeX } is being used to create this very document and until now I didn't even have it here!
+ \item \TeX --- Predates PostScript, similar idea
\begin{itemize}
- \item I don't really want to go down the path of investigating the billion steps involved in getting \LaTeX into an actually viewable format
- \item There are interpreters (usually WYSIWYG editors) for \LaTeX though
\item Maybe if \LaTeX were more popular there would be desktop viewers that converted \LaTeX directly into graphics
\end{itemize}
\item Potential for dynamic content, interactivity; dynamic PostScript, enhanced Postscript
- \item Scientific Computing --- Mathematica, Matlab, IPython Notebook --- The document and the code that produces it are stored together
\item Problems with security --- Turing complete, can be exploited easily
\end{itemize}
-\subsection{Crippled Interpreted Model}
+\pagebreak
+\subsection{Document Object Model}\label{Document Object Model}
+\input{chapters/Background_DOM}
-\rephrase{I'm pretty sure I made that one up}
+\subsection{The Portable Document Format}
-\begin{itemize}
- \item PDF is PostScript but without the Turing Completeness
- \item Solves security issues, more efficient
-\end{itemize}
-\subsection{Document Object Model}
+\subsection{Scientific Computation Packages}
-\begin{itemize}
- \item DOM = Tree of nodes; node may have attributes, children, data
- \item XML (SGML) is the standard language used to represent documents in the DOM
- \item XML is plain text
- \item SVG is a standard for a vector graphics language conforming to XML (ie: a DOM format)
- \item CSS style sheets represent more complicated styling on objects in the DOM
-\end{itemize}
-
-\subsection{Blurring the Line --- Javascript}
+The document and the code that produces it are one and the same.
\begin{itemize}
- \item The document is expressed in DOM format using XML/HTML/SVG
- \item A Javascript program is run which can modify the DOM
- \item At a high level this may be simply changing attributes of elements dynamically
- \item For low level control there is canvas2D and even WebGL which gives direct access to OpenGL functions
- \item Javascript can be used to make a HTML/SVG interactive
- \begin{itemize}
- \item Overlooking the fact that the SVG standard already allows for interactive elements...
- \end{itemize}
- \item Javascript is now becoming used even in desktop environments and programs (Windows 8, GNOME 3, Cinnamon, Game Maker Studio) ({\bf shudder})
- \item There are also a range of papers about including Javascript in PDF ``Pixels or Perish'' being the only one we have actually read\cite{hayes2012pixels}
+ \item Numerical computation packages such as Mathematica and Maple use arbitrary precision floats
\begin{itemize}
- \item I have no idea how this works; PDF is based on PostScript... it seems very circular to be using a programming language to modify a document that is modelled on being a (non turing complete) program
- \item This is yet more proof that people will converge towards solutions that ``work'' rather than those that are optimal or elegant
- \item I guess it's too much effort to make HTML look like PDF (or vice versa) so we could phase one out
+ \item Mathematica is not open source which is an issue when publishing scientific research (because people who do not fork out money for Mathematica cannot verify results)
+ \item What about Maple? \cite{HFP} and \cite{fousse2007mpfr} both mention it being buggy.
+ \item Octave and Matlab use fixed precision doubles
\end{itemize}
+ \item IPython is pretty cool guys
\end{itemize}
-\subsection{Why do we still use static PDFs}
+\section{Precision in Modern Document Formats}
-Despite their limitations, we still use static, boring old PDFs. Particularly in scientific communication.
+We briefly summarise the requirements of the standards discussed so far in regards to the precision of mathematical operations:
\begin{itemize}
- \item They are portable; you can write an amazing document in Mathematica/Matlab but it
- \item Scientific journals would need to adapt to other formats and this is not worth the effort
- \item No network connection is required to view a PDF (although DRM might change this?)
- \item All rescources are stored in a single file; a website is stored accross many seperate files (call this a ``distributed'' document format?)
- \item You can create PDFs easily using desktop processing WYSIWYG editors; WYSIWYG editors for web based documents are worthless due to the more complex content
- \item Until Javascript becomes part of the PDF standard, including Javascript in PDF documents will not become widespread
- \item Once you complicate a PDF by adding Javascript, it becomes more complicated to create; it is simply easier to use a series of static figures than to embed a shader in your document. Even for people that know WebGL.
+ \item {\bf PostScript} predates the IEEE-754 standard and originally specified a floating point representation with ? bits of exponent and ? bits of mantissa. Version ? of the PostScript standard changed to specify IEEE-754 binary32 ``single precision'' floats.
+ \item {\bf PDF} has also specified IEEE-754 binary32 since version ?. Importantly, the standard states that this is a \emph{maximum} precision; documents created with higher precision would not be viewable in Adobe Reader.
+ \item {\bf SVG} specifies a minimum of IEEE-754 binary32 but recommends more bits be used internally
+ \item {\bf Javascript} uses binary32 floats for all operations, and does not distinguish between integers and floats.
+ \item {\bf Python} uses binary64 floats
+ \item {\bf Matlab} uses binary64 floats
+ \item {\bf Mathematica} uses some kind of terrifying symbolic / arbitrary float combination
+ \item {\bf Maple} is similar but by many accounts horribly broken
+
\end{itemize}
-\rephrase{3. Here are the ways document standards specify precision (or don't)}
-\section{Precision in Modern Document Formats}
+\rephrase{4. Here is IEEE-754 which is what these standards use}
-\rephrase{All the above is very interesting and provides important context, but it is not actually directly related to the problem of infinite precision which we are going to try and solve. Sorry to make you read it all.}
+\section{Real Number Representations}
+We have found that PostScript, PDF, and SVG document standards all restrict themselves to IEEE floating point number representations of coordinates. This is unsurprising as the IEEE standard has been successfully adopted almost universally by hardware manufactures and programming language standards since the early 1990s. In the traditional view of a document as a static, finite sheet of paper, there is little motivation for enhanced precision.
+In this section we will begin by investigating floating point numbers as defined in the IEEE standard and their limitations. We will then consider alternative number representations including fixed point numbers, arbitrary precision floats, rational numbers, p-adic numbers and symbolic representations. \rephrase{Oh god I am still writing about IEEE floats let alone all those other things}
-\begin{itemize}
- \item Implementations of PostScript and PDF must by definition restrict themselves to IEEE binary32 ``single precision''\footnote{The original IEEE-754 defined single, double and extended precisions; in the revision these were renamed to binary32, binary64 and binary128 to explicitly state the base and number of bits}
- floating point number representations in order to conform to the standards\cite{plrm, pdfref17}.
- \item Implementations of SVG are by definition required to use IEEE binary32 as a {\bf minimum}. ``High Quality'' SVG viewers are required to use at least IEEE binary64.\cite{svg2011-1.1}
- \item Numerical computation packages such as Mathematica and Maple use arbitrary precision floats
- \begin{itemize}
- \item Mathematica is not open source which is an issue when publishing scientific research (because people who do not fork out money for Mathematica cannot verify results)
- \item What about Maple? \cite{HFP} and \cite{fousse2007mpfr} both mention it being buggy.
- \item Octave and Matlab use fixed precision doubles
- \end{itemize}
-\end{itemize}
-
-The use of IEEE binary32 floats in the PostScript and PDF standards is not surprising if we consider that these documents are oriented towards representing static pages. They don't actually need higher precision to do this; 32 bits is more than sufficient for A4 paper.
-
-\rephrase{4. Here is IEEE-754 which is what these standards use}
-
-\section{Representation of Numbers}
+\rephrase{Reorder to start with Integers, General Floats, then go to IEEE, then other things}
-Although this project has been motivated by a desire for more flexible document formats, the fundamental source of limited precision in vector document formats is the restriction to IEEE floating point numbers for representation of coordinates.
+\subsection{IEEE Floating Points}
-Whilst David Gow will be focusing on structures \rephrase{and the use of multiple coordinate systems} to represent a document so as to avoid or reduce these limitations\cite{proposalGow}, the focus of our own research will be \rephrase{increased precision in the representation of real numbers so as to get away with using a single global coordinate system}.
+Although the concept of a floating point representation has been attributed to various early computer scientists including Charles Babbage\cite{citationneeded}, it is widely accepted that William Kahan and his colleagues working on the IEEE-754 standard in the 1980s are the ``fathers of modern floating point computation''\cite{citationneeded}. The original IEEE-754 standard specified the encoding, number of bits, rounding methods, and maximum acceptable errors for the basic floating point operations for base $B = 2$ floats. It also specifies ``exceptions'' --- mechanisms by which a program can detect an error such as division by zero\footnote{Kahan has argued that exceptions in IEEE-754 are conceptually different to Exceptions as defined in several programming languages including C++ and Java. An IEEE exception is intended to prevent an error by its detection, whilst an exception in those languages is used to indicate an error has already occurred\cite{}}. We will restrict ourselves to considering $B = 2$, since it was found that this base in general gives the smallest rounding errors\cite{HFP}, although it is worth noting that different choices of base had been used historically\cite{goldman1991whatevery}, and the IEEE-854 and later the revised IEEE-754 standard specify a decimal representation $B = 10$ intended for use in financial applications.
-\subsection{The IEEE Standard}
+\subsection{Floating Point Definition}
-\subsection{Floating Point Number Representations}
+A floating point number $x$ is commonly represented by a tuple of integers $(s, e, m)$ in base $B$ as\cite{HFP, ieee2008-754}:
\begin{align*}
x &= (-1)^{s} \times m \times B^{e}
\end{align*}
-$B = 2$, although IEEE also defines decimal representations for $B = 10$ --- these are useful in financial software\cite{ieee2008-754}.
+Where $s$ is the sign and may be zero or one, $m$ is commonly called the ``mantissa'' and $e$ is the exponent.
+The name ``floating point'' refers to the equivelance of the $\times B^e$ operation to a shifting of a decimal point along the mantissa. This contrasts with a ``fixed point'' representation where $x$ is the sum of two fixed size numbers representing the integer and fractional part.
-\rephrase{Aside: Are decimal representations for a document format eg: CAD also useful because you can then use metric coordinate systems?}
+In the IEEE-754 standard, for a base of $B = 2$, numbers are encoded in continuous memory by a fixed number of bits, with $s$ occupying 1 bit, followed by $e$ and $m$ occupying a number of bits specified by the precision; 5 and 10 for a binary16 or ``half precision'' float, 8 and 23 for a binary32 or ``single precision'' and 15 and 52 for a binary64 or ``double precision'' float\cite{HFP, ieee2008-754}.
-\subsubsection{Precision}
+\subsection{Precision and Rounding}
-The floats map an infinite set of real numbers onto a discrete set of representations.
+Real values which cannot be represented exactly in a floating point representation must be rounded. The results of a floating point operation will in general be such values and thus there is a rounding error possible in any floating point operation. Goldberg's assertively titled 1991 paper ``What Every Computer Scientist Needs to Know about Floating Point Arithmetic'' provides a comprehensive overview of issues in floating point arithmetic and relates these to the 1984 version of the IEEE-754 standard\cite{goldberg1991whatevery}. More recently, after the release of the revised IEEE-754 standard in 2008, a textbook ``Handbook Of Floating Point Arithmetic'' has been published which provides a thourough review of literature relating to floating point arithmetic in both software and hardware\cite{HFP}.
-\rephrase{Figure: 8 bit ``minifloats'' (all 255 of them) clearly showing the ``precision vs range'' issue}
-The most a result can be rounded in conversion to a floating point number is the units in last place; $m_{N} \times B^{e}$.
+Figure \ref{minifloat.pdf} shows the positive real numbers which can be represented exactly by an 8 bit base $B = 2$ floating point number; and illustrates that a set of fixed precision floating point numbers forms a discrete approximation of the reals. There are only $2^7 = 256$ numbers in this set, which means it is easier to see some of the properties of floats that would be unclear using one of the IEEE-754 encodings. The first set of points corresponds to using 2 and 5 bits to encode $e$ and $m$ whilst the second set of points corresponds to a 3 and 4 bit encoding. This allows us to see the trade off between the precision and range of real values represented.
-\rephrase{Even though that paper that claims double is the best you will ever need because the error can be as much as the size of a bacterium relative to the distance to the moon}\cite{} \rephrase{there are many cases where increased number of bits will not save you}.\cite{HFP}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/minifloat.pdf} \\
+ \includegraphics[width=0.8\textwidth]{figures/minifloat_diff.pdf}
+ \caption{The mapping of 8 bit floats to reals}
+\end{figure}
+
+\subsection{Floating Point Operations}
+Floating point operations can in principle be performed using integer operations, but specialised Floating Point Units (FPUs) are an almost universal component of modern processors\cite{citationneeded}. The improvement of FPUs remains highly active in several areas including: efficiency\cite{seidel2001onthe}; accuracy of operations\cite{dieter2007lowcost}; and even the adaptation of algorithms originally used in software for reducing the overal error of a sequence of operations\cite{kadric2013accurate}. In this section we will consider the algorithms for floating point operations without focusing on the hardware implementation of these algorithms.
-\rephrase{5. Here are limitations of IEEE-754 floating point numbers on compatible hardware}
-\subsection{Limitations Imposed By CPU}
+\subsection{Some sort of Example(s) or Floating Point Mayhem}
-CPU's are restricted in their representation of floating point numbers by the IEEE standard.
+\rephrase{Eg: $f(x) = |x|$ calculated from sqrt and squaring}
+\rephrase{Eg: Massive rounding errors from calculatepi}
+
+\rephrase{Eg: Actual graphics things :S}
\subsection{Limitations Imposed By Graphics APIs and/or GPUs}
\item OpenGL standards specify: binary16, binary32, binary64
\item OpenVG aims to become a standard API for SVG viewers but the API only uses binary32 and hardware implementations may use less than this internally\cite{rice2008openvg}
\item It seems that IEEE has not been entirely successful; although all modern CPUs and GPUs are able to read and write IEEE floating point types, many do not conform to the IEEE standard in how they represent floating point numbers internally.
+ \item \rephrase{Blog post alert} \url{https://dolphin-emu.org/blog/2014/03/15/pixel-processing-problems/}
\end{itemize}
\rephrase{7. Sod all that, let's just use an arbitrary precision library (AND THUS WE FINALLY GET TO THE POINT)}
-\subsection{Alternate Number Representations}
-
-\rephrase{They exist\cite{HFP}}.
+\subsection{Arbitrary Precision Floating Point Numbers}
-Do it all using MFPR\cite{}, she'll be right.
+An arbitrary precision floating point number simply uses extra bits to store extra precision. Do it all using MFPR\cite{fousse2007mpfr}, she'll be right.
\rephrase{8. Here is a brilliant summary of sections 7- above}
Hopefully we have brought together the radically different areas of interest together in some sort of coherant fashion.
In the next chapter we will talk about how we have succeeded in rendering a rectangle. It will be fun. I am looking forward to it.
+\rephrase{Oh dear this is not going well}
+
git.ucc.asn.au / ipdf / sam.git / blobdiff
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