% Copyright 2008 by Christian Feuersaenger
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Free Documentation License.
%
% See the file doc/generic/pgf/licenses/LICENSE for more details.


\section{Floating Point Unit Library}
{\noindent {\emph{by Christian Feuers\"anger}}}
\label{pgfmath-floatunit}
\label{section-library-fpu}

\begingroup
\pgfqkeys{/pgf/number format}{sci}
\pgfkeys{/pgf/fpu}

\begin{pgflibrary}{fpu}
  The floating point unit (fpu) allows the full data range of
  scientific computing for use in \pgfname. Its core is the \pgfname\
  math routines for mantissa operations, leading to a reasonable
  trade--of between speed and accuracy. It does not require any
  third--party packages or external programs. 
\end{pgflibrary}

\subsection{Overview}
The fpu provides a replacement set of math commands which can be
installed in isolated placed to achieve large data ranges at
reasonable accuracy. It provides at least\footnote{To be more precise,
  the FPU's exponent is currently a 32 bit integer. That means it
  supports a significantly larger data range than an IEEE double
  precision number -- but if a future \TeX\ version may provide
  lowlevel access to doubles, this may change.} the IEEE double
precision data range, $\pgfmathprintnumber{-1e+324}, \dotsc,
\pgfmathprintnumber{+1e324}$. The absolute smallest number bigger than
zero is $\pgfmathprintnumber{1e-324}$. The FPU's relative precision is
at least $\pgfmathprintnumber{1e-4}$ although operations like addition
have a relative precision of $\pgfmathprintnumber{1e-6}$. 

Note that the library has not really been tested together with any
drawing operations. It should be used to work with arbitrary input
data which is then transformed somehow into \pgfname\ precision. This,
in turn, can be processed by \pgfname. 

\subsection{Usage}
\begin{key}{/pgf/fpu=\marg{boolean} (default true)}
  This key installs or uninstalls the FPU. The installation exchanges
  any routines of the standard math parser with those of the FPU:
  |\pgfmathadd| will be replaced with |\pgfmathfloatadd| and so
  on. Furthermore, any number will be parsed with
  |\pgfmathfloatparsenumber|. 

\begin{codeexample}[]
\pgfkeys{/pgf/fpu}
\pgfmathparse{1+1}\pgfmathresult
\end{codeexample}
  \noindent The FPU uses a lowlevel number representation consisting
  of flags, mantissa and exponent\footnote{Users should \emph{always}
    use high level routines to manipulate floating point numbers as
    the format may change in a future release.}. To avoid unnecessary
  format conversions, |\pgfmathresult| will usually contain such a
  cryptic number. Depending on the context, the result may need to be
  converted into something which is suitable for \pgfname\ processing
  (like coordinates) or may need to be typeset. The FPU provides such
  methods as well. 

%--------------------------------------------------
% \begin{codeexample}[]
% \begin{tikzpicture}
% 	\fill[red,fpu,/pgf/fpu/scale results=1e-10] (*1.234e10,*1e10) -- (*2e10,*2e10);
% \end{tikzpicture}
% \end{codeexample}
%--------------------------------------------------

  Use |fpu=false| to deactivate the FPU. This will restore any
  change. Please note that this is not necessary if the FPU is used
  inside of a \TeX\ group -- it will be deactivated afterwards
  anyway. 

  It does not hurt to call |fpu=true| or |fpu=false| multiple times. 

  Please note that if the |fixed point arithmetics| library of
  \pgfname\ will be activated after the FPU, the FPU will be
  deactivated automatically. 
\end{key}

\begin{key}{/pgf/fpu/output format=\mchoice{float,sci,fixed} (initially float)}
  This key allows to change the number format in which the FPU assigns
  |\pgfmathresult|. 
  
  The predefined choice |float| uses the low-level format used by the
  FPU. This is useful for further processing inside of any library. 
\begin{codeexample}[]
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=float}
\pgfmathparse{exp(50)*42}\pgfmathresult
\end{codeexample}

  The choice |sci| returns numbers in the format
  \meta{mantissa}|e|\meta{exponent}. It provides almost no
  computational overhead. 
\begin{codeexample}[]
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=sci}
\pgfmathparse{4.22e-8^-2}\pgfmathresult
\end{codeexample}

  The choice |fixed| returns normal fixed point numbers and provides
  the highest compatibility with the \pgfname\ engine. It is activated
  automatically in case the FPU scales results. 
\begin{codeexample}[]
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}
\pgfmathparse{sqrt(1e-12)}\pgfmathresult
\end{codeexample}
\end{key}

\begin{key}{/pgf/fpu/scale results=\marg{scale}}
  A feature which allows semi-automatic result scaling. Setting this
  key has two effects: first, the output format for \emph{any}
  computation will be set to |fixed| (assuming results will be
  processed by \pgfname's kernel). Second, any expression which starts
  with a star, |*|, will be multiplied with \meta{scale}. 
\end{key}

\begin{keylist}{
    /pgf/fpu/scale file plot x=\marg{scale},%
    /pgf/fpu/scale file plot y=\marg{scale},%
    /pgf/fpu/scale file plot z=\marg{scale}}%
  These keys will patch \pgfname's |plot file| command to
  automatically scale single coordinates by \meta{scale}. 
  
  The initial setting does not scale |plot file|.
\end{keylist}

\begin{command}{\pgflibraryfpuifactive\marg{true-code}\marg{false-code}}
  This command can be used to execute either \meta{true-code} or
  \meta{false-code}, depending on whether the FPU has been activated
  or not. 
\end{command}

\subsection{Comparison to the fixed point arithmetics library}
There are other ways to increase the data range and/or the precision
of \pgfname's math parser. One of them is the |fp| package, preferable
combined with \pgfname's |fixed point arithmetic| library. The
differences between the FPU and |fp| are: 
\begin{itemize}
\item The FPU supports at least the complete IEEE double precision
  number range, while |fp| covers only numbers of magnitude
  $\pm\pgfmathprintnumber{1e17}$. 
\item The FPU has a uniform relative precision of about 4--5 correct
  digits. The fixed point library has an absolute precision which may
  perform good in many cases -- but will fail at the ends of the data
  range (as every fixed point routines does). 
\item The FPU has potential to be faster than |fp| as it has access to
  fast mantissa operations using \pgfname's math capabilities (which
  use \TeX\ registers). 
\end{itemize}

\subsection{Command Reference and Programmer's Manual}

\subsubsection{Creating and Converting Floats}
\begin{command}{\pgfmathfloatparsenumber\marg{x}}
  Reads a number of arbitrary magnitude and precision and stores its
  result into |\pgfmathresult| as floating point number $m \cdot 10^e$
  with mantissa and exponent base~$10$. 

  The algorithm and the storage format is purely text-based. The
  number is stored as a triple of flags, a positive mantissa and an
  exponent, such as 
\begin{codeexample}[]
\pgfmathfloatparsenumber{2}
\pgfmathresult
\end{codeexample}
  Please do not rely on the low-level representation here, use
  |\pgfmathfloattomacro| (and its variants) and |\pgfmathfloatcreate|
  if you want to work with these components. 

  The flags encoded in |\pgfmathresult| are represented as a digit
  where `$0$' stands for the number $\pm 0\cdot 10^0$, `$1$' stands
  for a positive sign, `$2$' means a negative sign, `$3$' stands for
  `not a number', `$4$' means $+\infty$ and `$5$' stands for
  $-\infty$. 

  The mantissa is a normalized real number $m \in \mathbb{R}$, $1 \le
  m < 10$. It always contains a period and at least one digit after
  the period. The exponent is an integer. 

  Examples:
\begin{codeexample}[]
\pgfmathfloatparsenumber{0}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{0.2}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{42}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{20.5E+2}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{1e6}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{5.21513e-11}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E.
\end{codeexample}
  The argument \meta{x} may be given in fixed point format or the
  scientific ``e'' (or ``E'') notation. The scientific notation does not
  necessarily need to be normalized. The supported exponent range is
  (currently) only limited by the \TeX-integer range (which uses 31
  bit integer numbers). 
\end{command}

\begin{key}{/pgf/fpu/handlers/empty number=\marg{input}\marg{unreadable part}}
  This command key is invoked in case an empty string is parsed inside
  of |\pgfmathfloatparsenumber|. You can overwrite it to assign a
  replacement |\pgfmathresult| (in float!). 

  The initial setting is to invoke |invalid number|, see below.
\end{key}
\begin{key}{/pgf/fpu/handlers/invalid number=\marg{input}\marg{unreadable part}}
  This command key is invoked in case an invalid string is parsed
  inside of |\pgfmathfloatparsenumber|. You can overwrite it to assign
  a replacement |\pgfmathresult| (in float!). 

  The initial setting is to generate an error message.
\end{key}
\begin{key}{/pgf/fpu/handlers/wrong lowlevel format=\marg{input}\marg{unreadable part}}
  This command key is invoked whenever |\pgfmathfloattoregisters| or
  its variants encounter something which is not a properly formatted
  lowlevel floating point number. As for |invalid number|, this key
  may assign a new |\pgfmathresult| (in floating point) which will be
  used instead of the offending \meta{input}. 

  The initial setting is to generate an error message.
\end{key}

\begin{command}{\pgfmathfloatqparsenumber\marg{x}}
  The same as |\pgfmathfloatparsenumber|, but does not perform sanity checking.
\end{command}

\begin{command}{\pgfmathfloattofixed{\marg{x}}}
  Converts a number in floating point representation to a fixed point
  number. It is a counterpart to |\pgfmathfloatparsenumber|. The
  algorithm is purely text based and defines |\pgfmathresult| as a
  string sequence which represents the floating point number \meta{x}
  as a fixed point number (of arbitrary precision). 

\begin{codeexample}[]
\pgfmathfloatparsenumber{0.00052}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E
$\to$
\pgfmathfloattofixed{\pgfmathresult}
\pgfmathresult
\end{codeexample}

\begin{codeexample}[]
\pgfmathfloatparsenumber{123.456e4}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E
$\to$
\pgfmathfloattofixed{\pgfmathresult}
\pgfmathresult
\end{codeexample}
\end{command}

\begin{command}{\pgfmathfloattoint\marg{x}}
  Converts a number from low-level floating point representation to an
  integer (by truncating the fractional part). 
\begin{codeexample}[]
\pgfmathfloatparsenumber{123456}
\pgfmathfloattoint{\pgfmathresult}
\pgfmathresult
\end{codeexample}

See also |\pgfmathfloatint| which returns the result as float.
\end{command}

\begin{command}{\pgfmathfloattosci\marg{float}}
  Converts a number from low-level floating point representation to
  scientific format, $1.234e4$. The result will be assigned to the
  macro |\pgfmathresult|. 
\end{command}

\begin{command}{\pgfmathfloatvalueof\marg{float}}
  Expands a number from low-level floating point representation to
  scientific format, $1.234e4$. 

  Use |\pgfmathfloatvalueof| in contexts where only expandable macros are allowed.
\end{command}

\begin{command}{\pgfmathfloatcreate{\marg{flags}}{\marg{mantissa}}{\marg{exponent}}}
  Defines |\pgfmathresult| as the floating point number encoded by
  \meta{flags}, \meta{mantissa} and \meta{exponent}.
	
  All arguments are characters and will be expanded using |\edef|.
\begin{codeexample}[]
\pgfmathfloatcreate{1}{1.0}{327}
\pgfmathfloattomacro{\pgfmathresult}{\F}{\M}{\E}
Flags: \F; Mantissa \M; Exponent \E
\end{codeexample}
\end{command}

\begin{command}{\pgfmathfloatifflags\marg{floating point number}\marg{flag}\marg{true-code}\marg{false-code}}
  Invokes \meta{true-code} if the flag of \meta{floating point number}
  equals \meta{flag} and \meta{false-code} otherwise. 

  The argument \meta{flag} can be one of
  \begin{description}
  \item[0] to test for zero,
  \item[1] to test for positive numbers,
  \item[+] to test for positive numbers,
  \item[2] to test for negative numbers,
  \item[-] to test for negative numbers,
  \item[3] for ``not-a-number'',
  \item[4] for $+\infty$,
  \item[5] for $-\infty$.
  \end{description}
	
\begin{codeexample}[]
\pgfmathfloatparsenumber{42}
\pgfmathfloatifflags{\pgfmathresult}{0}{It's zero!}{It's not zero!}
\pgfmathfloatifflags{\pgfmathresult}{1}{It's positive!}{It's not positive!}
\pgfmathfloatifflags{\pgfmathresult}{2}{It's negative!}{It's not negative!}

% or, equivalently
\pgfmathfloatifflags{\pgfmathresult}{+}{It's positive!}{It's not positive!}
\pgfmathfloatifflags{\pgfmathresult}{-}{It's negative!}{It's not negative!}
\end{codeexample}
\end{command}


\begin{command}{\pgfmathfloattomacro{\marg{x}}{\marg{flagsmacro}}{\marg{mantissamacro}}{\marg{exponentmacro}}}
  Extracts the flags of a floating point number \meta{x} to
  \meta{flagsmacro}, the mantissa to \meta{mantissamacro} and the
  exponent to \meta{exponentmacro}. 
\end{command}

\begin{command}{\pgfmathfloattoregisters{\marg{x}}{\marg{flagscount}}{\marg{mantissadimen}}{\marg{exponentcount}}}
  Takes a floating point number \meta{x} as input and writes flags to
  count register \meta{flagscount}, mantissa to dimen register
  \meta{mantissadimen} and exponent to count register \meta{exponentcount}.

  Please note that this method rounds the mantissa to \TeX-precision.
\end{command}

\begin{command}{\pgfmathfloattoregisterstok{\marg{x}}{\marg{flagscount}}{\marg{mantissatoks}}{\marg{exponentcount}}}
  A variant of |\pgfmathfloattoregisters| which writes the
  mantissa into a token register. It maintains the full input
  precision. 
\end{command}

\begin{command}{\pgfmathfloatgetflags{\marg{x}}{\marg{flagscount}}}
  Extracts the flags of \meta{x} into the count register
  \meta{flagscount}. 
\end{command}

\begin{command}{\pgfmathfloatgetflagstomacro{\marg{x}}{\marg{macro}}}
  Extracts the flags of \meta{x} into the macro \meta{macro}. 
\end{command}

\begin{command}{\pgfmathfloatgetmantissa{\marg{x}}{\marg{mantissadimen}}}
  Extracts the mantissa of \meta{x} into the dimen register
  \meta{mantissadimen}. 
\end{command}

\begin{command}{\pgfmathfloatgetmantissatok{\marg{x}}{\marg{mantissatoks}}}
  Extracts the mantissa of \meta{x} into the token register
  \meta{mantissatoks}. 
\end{command}

\begin{command}{\pgfmathfloatgetexponent{\marg{x}}{\marg{exponentcount}}}
  Extracts the exponent of \meta{x} into the count register
  \meta{exponentcount}. 
\end{command}

\subsubsection{Symbolic Rounding Operations}

Commands in this section constitute the basic level implementations of
the rounding routines. They work symbolically, i.e.\ they operate on
text, not on numbers and allow arbitrarily large numbers. 

\begin{command}{\pgfmathroundto{\marg{x}}}
  Rounds a fixed point number to prescribed precision and writes the
  result to |\pgfmathresult|. 

  The desired precision can be configured with
  |/pgf/number format/precision|, see
  section~\ref{pgfmath-numberprinting}. This section does also contain
  application examples.  
	
  Any trailing zeros after the period are discarded. The algorithm is
  purely text based and allows to deal with precisions beyond \TeX's
  fixed point support. 

  As a side effect, the global boolean |\ifpgfmathfloatroundhasperiod|
  will be set to true if and only if the resulting mantissa has a
  period. Furthermore, |\ifpgfmathfloatroundmayneedrenormalize| will
  be set to true if and only if the rounding result's floating point
  representation would have a larger exponent than \meta{x}. 
\begin{codeexample}[]
\pgfmathroundto{1}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathroundto{4.685}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathroundto{19999.9996}
\pgfmathresult
\end{codeexample}
\end{command}

\begin{command}{\pgfmathroundtozerofill{\marg{x}}}
  A variant of |\pgfmathroundto| which always uses a fixed number of
  digits behind the period. It fills missing digits with zeros. 
\begin{codeexample}[]
\pgfmathroundtozerofill{1}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathroundto{4.685}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathroundtozerofill{19999.9996}
\pgfmathresult
\end{codeexample}
\end{command}

\begin{command}{\pgfmathfloatround{\marg{x}}}
  Rounds a normalized floating point number to a prescribed precision
  and writes the result to |\pgfmathresult|. 

  The desired precision can be configured with
  |/pgf/number format/precision|, see
  section~\ref{pgfmath-numberprinting}.  
	
  This method employs |\pgfmathroundto| to round the mantissa and
  applies renormalization if necessary. 

  As a side effect, the global boolean |\ifpgfmathfloatroundhasperiod|
  will be set to true if and only if the resulting mantissa has a
  period. 
\begin{codeexample}[]
\pgfmathfloatparsenumber{52.5864}
\pgfmathfloatround{\pgfmathresult}
\pgfmathfloattosci{\pgfmathresult}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathfloatparsenumber{9.995}
\pgfmathfloatround{\pgfmathresult}
\pgfmathfloattosci{\pgfmathresult}
\pgfmathresult
\end{codeexample}
\end{command}

\begin{command}{\pgfmathfloatroundzerofill{\marg{x}}}
  A variant of |\pgfmathfloatround| produces always the same number of
  digits after the period (it includes zeros if necessary). 
\begin{codeexample}[]
\pgfmathfloatparsenumber{52.5864}
\pgfmathfloatroundzerofill{\pgfmathresult}
\pgfmathfloattosci{\pgfmathresult}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathfloatparsenumber{9.995}
\pgfmathfloatroundzerofill{\pgfmathresult}
\pgfmathfloattosci{\pgfmathresult}
\pgfmathresult
\end{codeexample}
\end{command}



\subsubsection{Math Operations Commands}

This section describes some of the replacement commands in more
detail. 

Please note that these commands can be used even if the |fpu| as such
has not been activated -- it is sufficient to load the library. 

\begin{command}{\pgfmathfloat\meta{op}}
  Methods of this form constitute the replacement operations where
  \meta{op} can be any of the well-known math operations. 

  Thus, \declareandlabel{\pgfmathfloatadd} is the counterpart
  for |\pgfmathadd| and so on. The semantics and number of
  arguments is the same, but all input and output arguments are
  \emph{expected} to be floating point numbers. 
\end{command}

\begin{command}{\pgfmathfloattoextentedprecision{\marg{x}}}
  Renormalizes \meta{x} to extended precision mantissa, meaning
  $100 \le m < 1000$ instead of $1 \le m < 10$.
  
  The ``extended precision'' means we have higher accuracy when we apply
  pgfmath operations to mantissas. 
  
  The input argument is expected to be a normalized floating point
  number; the output argument is a non-normalized floating point number
  (well, normalized to extended precision). 
  
  The operation is supposed to be very fast.
\end{command}

\begin{command}{\pgfmathfloatsetextprecision\marg{shift}}
  Sets the precision used inside of |\pgfmathfloattoextentedprecision| to \meta{shift}.
  
  The different choices are
  
  \begin{tabular}{llrll}
    0 & normalization to &    $0$ & $\le m < 1$ 	& (disable extended precision)\\
    1 & normalization to &   $10$ & $\le m < 100$	\\
    2 & normalization to & 	$100$ & $\le m < 1000$	& (default of |\pgfmathfloattoextentedprecision|)\\
    3 & normalization to & $1000$ & $\le m < 10000$	\\
  \end{tabular}
\end{command}

\begin{command}{\pgfmathfloatlessthan{\marg{x}}{\marg{y}}}
  Defines |\pgfmathresult| as $1.0$ if $\meta{x} < \meta{y}$, but
  $0.0$ otherwise. It also sets the global \TeX-boolean
  |\pgfmathfloatcomparison| accordingly. The arguments \meta{x} and
  \meta{y} are expected to be numbers which have already been
  processed by |\pgfmathfloatparsenumber|. Arithmetic is carried out
  using \TeX-registers for exponent- and mantissa comparison. 
\end{command}

\begin{command}{\pgfmathfloatmultiplyfixed\marg{float}\marg{fixed}}
  Defines |\pgfmathresult| to be $\meta{float} \cdot \meta{fixed}$
  where \meta{float} is a floating point number and \meta{fixed} is a
  fixed point number. The computation is performed in floating point
  arithmetics, that means we compute $m \cdot \meta{fixed}$ and
  renormalize the result where $m$ is the mantissa of \meta{float}. 

  This operation renormalizes \meta{float} with
  |\pgfmathfloattoextentedprecision| before the operation, that means
  it is intended for relatively small arguments of \meta{fixed}. The
  result is a floating point number. 
\end{command}

\begin{command}{\pgfmathfloatifapproxequalrel\marg{a}\marg{b}\marg{true-code}\marg{false-code}}
  Computes the relative error between \meta{a} and \meta{b} (assuming
  \meta{b}$\neq 0$) and invokes \meta{true-code} if the relative error
  is below |/pgf/fpu/rel thresh| and \meta{false-code} if that is not
  the case. 

  The input arguments will be parsed with |\pgfmathfloatparsenumber|.

  \begin{key}{/pgf/fpu/rel thresh=\marg{number} (initially 1e-4)}
    A threshold used by |\pgfmathfloatifapproxequalrel| to decide
    whether numbers are approximately equal. 
  \end{key}
\end{command}


\begin{command}{\pgfmathfloatshift{\marg{x}}{\marg{num}}}
  Defines |\pgfmathresult| to be $\meta{x} \cdot 10^{\meta{num}}$. The
  operation is an arithmetic shift base ten and modifies only the
  exponent of \meta{x}. The argument \meta{num} is expected to be a
  (positive or negative) integer. 
\end{command}


\begin{command}{\pgfmathfloatabserror\marg{x}\marg{y}}
  Defines |\pgfmathresult| to be the absolute error between two
  floating point numbers $x$ and $y$, $\lvert x - y\rvert $ and
  returns the result as floating point number. 
\end{command}

\begin{command}{\pgfmathfloatrelerror\marg{x}\marg{y}}
  Defines |\pgfmathresult| to be the relative error between two
  floating point numbers $x$ and $y$, $\lvert x - y\rvert / \lvert y
  \rvert $ and returns the result as floating point number. 
\end{command}


\begin{command}{\pgfmathfloatint\marg{x}}
  Returns the integer part of the floating point number \meta{x}, by
  truncating any digits after the period. This methods truncates the
  absolute value $\rvert x \lvert$ to the next smaller integer and
  restores the original sign afterwards. 

  The result is returned as floating point number as well.

  See also |\pgfmathfloattoint| which returns the number in integer
  format. 
\end{command}

\begin{command}{\pgfmathlog{\marg{x}}}
  Defines |\pgfmathresult| to be the natural logarithm of \meta{x},
  $\ln(\meta{x})$. This method is logically the same as |\pgfmathln|,
  but it applies floating point arithmetics to read number \meta{x}
  and employs the logarithm identity
  \[ \ln(m \cdot 10^e) = \ln(m) + e \cdot \ln(10) \]
  to get the result. The factor $\ln(10)$ is a constant, so only
  $\ln(m)$ with $1 \le m < 10$ needs to be computed. This is done
  using standard pgf math operations. 

  Please note that \meta{x} needs to be a number, expression parsing
  is not possible here. 

  If \meta{x} is \emph{not} a bounded positive real number (for
  example $\meta{x} \le 0$), |\pgfmathresult| will be \emph{empty}, no
  error message will be generated. 
\begin{codeexample}[]
\pgfmathlog{1.452e-7}
\pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathlog{6.426e+8}
\pgfmathresult
\end{codeexample}
\end{command}

\subsubsection{Accessing the Original Math Routines for Programmers}

As soon as the library is loaded, every private math routine will be
copied to a new name. This allows library and package authors to
access the \TeX-register based math routines even if the FPU is
activated. And, of course, it allows the FPU as such to perform its
own mantissa computations.

The private implementations of \pgfname\ math commands, which are of
the form |\pgfmath|\meta{name}|@|, will be available
as|\pgfmath@basic@|\meta{name}|@| as soon as the library is loaded. 


\endgroup
