THE FINAL COUNTDOWN

[ipdf/sam.git] / chapters / Background / Standards / Precision.tex

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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''.
 
 \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''.

-\begin{comment}
+%\begin{comment}

 \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.
 \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.

-\end{comment}
+%\end{comment}

 
 \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
 
 \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.
-\begin{comment}
+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 for view transformations.
+%\begin{comment}

 \subsection{Javascript}
 \subsection{Javascript}

-%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, 
+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, $2^{64}-2^{53}+3$) 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.
 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.

-\end{comment}
+%\end{comment}