X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground%2FStandards%2FPrecision.tex;h=0a9511cff9d5daf8fa92ac9202b0bb4fc8b4d9b4;hp=afb5f24684dba48a0deb5abc5a44733c8de3d4b9;hb=689e433b348588e05f47d23385dbf05d239c95d2;hpb=67c33e185edd820e2ea6fef334455645bacd432b

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-We briefly summarise the requirements of the standards discussed so far in regards to the precision of mathematical operations.
-
 \subsection{PostScript}
 The PostScript reference describes a ``Real'' object for representing coordinates and values as follows: ``Real objects approximate mathematical real numbers within a much larger interval, but with limited precision; they are implemented as floating-point numbers''\cite{plrm}. There is no reference to the precision of mathematical operations, but the implementation limits \emph{suggest} a range of $\pm10^{38}$ ``approximate'' and the smallest values not rounded to zero are $\pm10^{-38}$ ``approximate''.
 
 \subsection{PDF}
 PDF defines ``Real'' objects in a similar way to PostScript, but suggests a range of $\pm3.403\times10^{38}$ and smallest non-zero values of $\pm1.175\times10^{38}$\cite{pdfref17}. A note in the PDF 1.7 manual mentions that Acrobat 6 now uses IEEE-754 single precision floats, but ``previous versions used 32-bit fixed point numbers'' and ``... Acrobat 6 still converts floating-point numbers to fixed point for some components''.
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 \subsection{{\TeX} and METAFONT}
 
 In ``The METAFONT book'' Knuth appears to describe coordinates as fixed point numbers: ``The computer works internally with coordinates that are integer multiples of $\frac{1}{65536} \approx 0.00002$ of the width of a pixel''\cite{knuth1983metafont}. \footnote{This corresponds to using $16$ bits for the fractional component of a fixed point representation} There is no mention of precision in ``The {\TeX} book''. In 2007 Beebe claimed that {\TeX} uses a $14.16$ fixed point encoding, and that this was due to the lack of standardised floating point arithmetic on computers at the time; a problem that the IEEE-754 was designed to solve\cite{beebe2007extending}. Beebe also suggested that {\TeX} and METAFONT could now be modified to use IEEE-754 arithmetic.
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 \subsection{SVG}
 
 The SVG standard specifies a minimum precision equivelant to that of ``single precision floats'' (presumably referring to IEEE-754) with a range of \verb/-3.4e+38F/ to \verb/+3.4e+38F/, and states ``It is recommended that higher precision floating point storage and computation be performed on operations such as
 coordinate system transformations to provide the best possible precision and to prevent round-off errors.''\cite{svg2011-1.1} An SVG Viewer may refer to itself as ``High Quality'' if it uses a minimum of ``double precision'' floats.
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 \subsection{Javascript}
-%We include Javascript here due to its relation with the SVG, HTML5 and PDF standards.
+We include Javascript here due to its relation with the SVG, HTML5 and PDF standards.
 According to the EMCA-262 standard, ``The Number type has exactly 18437736874454810627 (that is, $2^64-^53+3$) values, 
 representing the double-precision 64-bit format IEEE 754 values as specified in the IEEE Standard for Binary Floating-Point Arithmetic''\cite{ecma-262}. 
 The Number type does differ slightly from IEEE-754 in that there is only a single valid representation of ``Not a Number'' (NaN). The EMCA-262 does not define an ``integer'' representation.
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