%--------------------------------------------
%
% Package pgfplots
%
% Provides a user-friendly interface to create function plots (normal
% plots, semi-logplots and double-logplots).
% 
% It is based on Till Tantau's PGF package.
%
% Copyright 2007/2008 by Christian Feuersänger.
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
% 
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
% 
% You should have received a copy of the GNU General Public License
% along with this program.  If not, see <http://www.gnu.org/licenses/>.
%
%--------------------------------------------

\newif\ifpgfplots@usefpu
\pgfkeys{%
	/pgfplots/use fpu/.is if=pgfplots@usefpu,
	/pgfplots/use fpu=true,
}

% only useful inside of \ifx (equals only itsself)
\def\pgfplots@EOI{\pgfplots@EOI}%

% Produces an \aftergroup statement for each single token in #1.
%
% ATTENTION: you *can't* use braces inside of '#1'!
\def\pgfplotsqaftergroupeach#1{%
	\pgfplotsqaftergroupeach@#1\pgfplots@EOI
}
\def\pgfplotsqaftergroupeach@#1{%
	\gdef\pgfplots@TMP{#1}%
	\ifx\pgfplots@TMP\pgfplots@EOI
	\else
		\aftergroup#1%
		\expandafter\pgfplotsqaftergroupeach@
	\fi
}

\def\pgfplotsaftergroupcollectinto#1#2\pgfplots@EOI{%
	\def#1{#2}%
}%

% Assigns list contents #2 to a list macro #1.
%
% The list contents may be provided in one of two formats:
% a) in the  list format 'first\\second\\thirst\\'
% or
% b) in the PGF foreach list format 'first,second,third'.
\def\pgfplots@assign@list#1#2{%
	\pgfplotslistnew#1{#2}%
}%

% Sets the boolean \ifpgfplots@is@old@list@format  to true if and only
% if the input is a list in the format '{first\\second\\}'.
%
%
% Usage:
% \pgfplots@check@backwards@compatible@list@format <argument>'\\'\pgfplots@EOI
% you NEED to append '\\\pgfplots@EOI' at the end.
\def\pgfplots@check@backwards@compatible@list@format#1\pgfplots@EOI{%
	\pgfplotslist@check@backslash@list#1\pgfplotslist@EOI
	\ifpgfplotslist@is@backslash@terminated
		\pgfplots@is@old@list@formattrue
	\else
		\pgfplots@is@old@list@formatfalse
	\fi
}%

% Issues an error message if the **LaTeX** package '#1' is not loaded
% (An error is also raised if the document is not a LaTeX document).
% #1: the required LaTeX package
% #2: the feature which requires this package
\def\pgfplots@assert@LaTeX@package@loaded#1#2{%
	\pgfutil@IfUndefined{@ifpackageloaded}{%
		\pgfplots@error{Sorry, #2 only works with LaTeX (it relies on the LaTeX package #1)}%
	}{%
		\pgfutil@IfUndefined{ver@#1.sty}% this here has been extracted from ltclass.dtx. \@ifpackageloaded is only usable in preamble.
			{\pgfplots@error{Sorry, use need \string\usepackage{#1} in your preamble for '#2'}}%
			{\relax}%
	}%
}%
\def\pgfplots@assert@tikzinternal@exists#1{%
	\pgfutil@IfUndefined{#1}{%
		\pgfplots@error{Sorry, pgfplots relies on the existance of the tikz internal macro '#1'.
		 Unfortunately, this does no longer exist ... you will need to get an updated version of pgfplots to fix this problem.}%
		\expandafter\let\csname #1\endcsname=\pgfutil@empty
	}{}%
}%


% A future-version compatibility method which checks whether the
% macro '#2' exists.
%
% We assume that '#2' usually contains the value of the tikz key '#1'.
% Example:
% /tikz/variable is usually stored into '\tikz@plot@var'.
%
% However, this may change in future versions.
%
% So, we do the following:
% 1. check whether '#2' exists, if yet: ok.
% 2. if not, check whether the value is stored directly into '/tikz/#1'.
% 3. if not, check whether the value is stored directly into '/pgf/#1'.
% 4. If everything fails, provide an error message.
%
% #1: a tikz key without key prefix (/tikz/ and /pgf/ will be prepended).
% #2: a macro name WITHOUT backslash.
% #3: the default value if everything fails.
\def\pgfplots@gettikzinternal@keyval#1#2#3{%
	\pgfutil@IfUndefined{#2}{%
		\pgfkeysifdefined{/tikz/#1}{%
			\pgfkeysgetvalue{/tikz/#1}\pgfplots@TMP
			\expandafter\let\csname #1\endcsname=\pgfplots@TMP
		}{%
			\pgfkeysifdefined{/pgf/#1}{%
				\pgfkeysgetvalue{/pgf/#1}\pgfplots@TMP
				\expandafter\let\csname #1\endcsname=\pgfplots@TMP
			}{%
				\expandafter\def\csname #1\endcsname{#3}%
				{\t@pgfplots@tokc={#3}%
				\pgfplots@warning{Sorry, could not find value of '/tikz/#1'. Assuming '\the\t@pgfplots@tokc'.}%
				}%
			}%
		}%
	}{}%
}%

\def\pgfplots@getcurrent@plothandler#1{%
	\pgfutil@IfUndefined{tikz@plot@handler}{%
		\pgfplots@error{Sorry, can't get the current plot handler. It appears that tikz and pgfplots is no longer compatible!? You will need to get a newer version of pgfplots.}%
		\let#1=\pgfplothandlerlineto
	}{%
		\let#1=\tikz@plot@handler
	}%
}



% Converts an arbitrary command (without arguments) to a string in which all characters
% have category 12.
%
% #1: a macro name (which takes no arguments)
% #2: a macro name which will be assigned to '#1' converted to string.
%
% This uses '\meaning#1' hackery.
\def\pgfplots@command@to@string#1#2{%
	\expandafter\pgfplots@command@to@string@@\meaning#1\pgfplots@EOI{#2}%
}%
\xdef\pgfplots@glob@TMPa{\meaning\pgfutil@empty}%
\expandafter\def\expandafter\pgfplots@command@to@string@@\pgfplots@glob@TMPa#1\pgfplots@EOI#2{%
	\def#2{#1}%
}%
\def\pgfplotscommandtostring#1#2{\pgfplots@command@to@string{#1}{#2}}%

% As \pgfplots@command@to@string, but it works for commands accepting
% arguments. More precisely, #2 will be a string (!) of the form
% <argument pattern>{ <body> }
% such that you can write
% \def\test #2
% into a file to restore the original macro. This IGNORES \long
% (sorry)
\def\pgfplots@command@with@args@to@string#1#2{%
	\expandafter\PGFPLOTS@CS@WITH@ARGS@TO@STRING\meaning#1\EOI{#2}%
}%
{
\let\CATCODE=\catcode
\let\LONG=\long
\CATCODE`\m=12
\CATCODE`\a=12
\CATCODE`\c=12
\CATCODE`\r=12
\CATCODE`\o=12
\CATCODE`\:=12
\CATCODE`\-=12
\CATCODE`\>=12
\LONG\gdef\PGFPLOTS@CS@WITH@ARGS@TO@STRING #1macro:#2->#3\EOI#4{\def#4{#2{#3}}}%
}%

% Defines \pgfplotsretval to be '#1' but without leading and trailing
% spaces.
\def\pgfplotsutil@trim#1{%
	\pgfkeys@spdef\pgfplotsretval{#1}%
}%

% Invokes '#2' if the token(s) '#1' are actually are defined control
% sequence and '#3' if not.
%
% This method accepts different values '#1' than \pgfutil@ifundefined.
% It is indended to autodetect values provided in the user interface
% (ui),
% that means '#1' can be anything. But I still like to know whether it
% is a control sequence.
%
% \pgfplotsutil@ifdefinedui{\table} -> yes if \table is defined.
% \pgfplotsutil@ifdefinedui{\csname abc\endcsname} -> no
% \pgfplotsutil@ifdefinedui{a_file_name} -> no
\def\pgfplotsutil@ifdefinedui#1#2#3{%
	{\toks0={#1}% this construction is necessary to handle '#' characters in '#1'
	\xdef\pgfplotsutil@@arg@ifdefined{\the\toks0}%
	}%
	\pgfplots@command@to@string\pgfplotsutil@@arg@ifdefined\pgfplotsutil@@arg@ifdefined
	\expandafter\pgfplotsutil@trim\expandafter{\pgfplotsutil@@arg@ifdefined}%
	\pgfutil@IfUndefined{\pgfplotsretval}{#3}{#2}%
}%
% The same as \pgfplotsutil@ifdefinedui but appends the suffix '#2' to
% '#1' after '#1' has been normalized.
% It invokes '#3' if '#1#2' is defined and '#4' if not.
\def\pgfplotsutil@ifdefinedui@withsuffix#1#2#3#4{%
	{\toks0={#1}%
	\xdef\pgfplotsutil@@arg@ifdefined{\the\toks0}%
	}%
	\pgfplots@command@to@string\pgfplotsutil@@arg@ifdefined\pgfplotsutil@@arg@ifdefined
	\expandafter\pgfplotsutil@trim\expandafter{\pgfplotsutil@@arg@ifdefined}%
	\pgfutil@IfUndefined{\pgfplotsretval #2}{#4}{#3}%
}%

% Writes the command name of '#1' without the leading backslash to
% macro #2.
\def\pgfplotsutil@getcommandname#1#2{%
	\begingroup
		\escapechar=-1
		\xdef\pgfplots@glob@TMPa{\string#1}%
	\endgroup
	\let#2=\pgfplots@glob@TMPa
}

{
\catcode`\%=12
\gdef\pgfplots@PERCENT@TEXT{%}
\catcode`\^^I=12
\gdef\pgfplots@TAB{^^I}
\catcode`\#=12 \gdef\pgfplots@ROUTE{#}}

% Usage:
% \pgfplots@if{pgfplots@scaled@ticks}{true-code}{false-code}
%
% it is to be used if the tex boolean is only known as string, not as
% macro.
\def\pgfplots@if#1#2#3{%
	\csname if#1\endcsname
		#2%
	\else
		#3%
	\fi
}%

% Executes '#2' if the number '#1' is zero and '#3' if not.
% Example:
% \pgfplotsmath@ifzero{0}{It's zero!}{It's not zero}
% \pgfplotsmath@ifzero{0.0}{It's zero!}{It's not zero}
%
% The argument must be assignable to a TeX dimension as 'pt', but
% without units.
\def\pgfplotsmath@ifzero#1#2#3{%
	\begingroup
	\pgf@xa=#1pt
	\ifdim\pgf@xa=0pt
		\gdef\pgfplots@glob@TMPa{#2}%
	\else
		\gdef\pgfplots@glob@TMPa{#3}%
	\fi
	\endgroup
	\pgfplots@glob@TMPa
}%

% Invokes code `#4' if |#1 - #2| <= #3 and `#5' if not.
%
% #1,#2 are dimension (registers or numbers) with unit.
% #3: absolute thresold (dimen)
\def\pgfplotsmath@ifapproxequal@dim#1#2#3#4#5{%
	\begingroup
	\pgf@xa=#1\relax
	\pgf@xb=#2\relax
	\advance\pgf@xa by-\pgf@xb
	\ifdim\pgf@xa<0pt
		\multiply\pgf@xa by-1
	\fi
	\ifdim\pgf@xa>#3\relax
		\gdef\pgfplots@glob@TMPa{#5}%
	\else
		\gdef\pgfplots@glob@TMPa{#4}%
	\fi
	\endgroup
	\pgfplots@glob@TMPa
}

\def\pgfplotsmathmin#1#2{%
	\ifdim#1 pt<#2 pt
		\edef\pgfmathresult{#1}%
	\else
		\edef\pgfmathresult{#2}%
	\fi
}%
\def\pgfplotsmathmax#1#2{%
	\ifdim#1 pt>#2 pt
		\edef\pgfmathresult{#1}%
	\else
		\edef\pgfmathresult{#2}%
	\fi
}%
\def\pgfplotsmathlessthan#1#2{\ifdim#1pt<#2pt \pgfmathfloatcomparisontrue\else\pgfmathfloatcomparisonfalse\fi}

% Re-define two-argument function for min and max.
% The pgfmath engine now uses a variable number of arguments.
\def\pgfplotsmathfloatmax#1#2{%
	\pgfmathfloatlessthan{#1}{#2}%
	\ifpgfmathfloatcomparison
		\edef\pgfmathresult{#2}%
	\else
		\edef\pgfmathresult{#1}%
	\fi
}
\def\pgfplotsmathfloatmin#1#2{%
	\pgfmathfloatlessthan{#1}{#2}%
	\ifpgfmathfloatcomparison
		\edef\pgfmathresult{#1}%
	\else
		\edef\pgfmathresult{#2}%
	\fi
}

% Defines \pgfmathresult to be #1 * 10^{#2}.
%
% #1 a macro without unit suffix.
% #2 an integer number (may be a register)
\def\pgfplotsmathmultiplypowten@#1#2{%
	\begingroup
	\pgf@xa=#1pt
	\ifnum#2<0
		\ifcase-#2
		\or\divide\pgf@xa by10
		\or\divide\pgf@xa by100
		\or\divide\pgf@xa by1000
		\or\divide\pgf@xa by10000
		\or\divide\pgf@xa by100000
		\fi
	\else
		\ifcase#2
		\or\multiply\pgf@xa by10
		\or\multiply\pgf@xa by100
		\or\multiply\pgf@xa by1000
		\or\multiply\pgf@xa by10000
		\or\multiply\pgf@xa by100000
		\fi
	\fi
	\edef\pgfmathresult{\pgf@sys@tonumber{\pgf@xa}}%
	\pgfmath@smuggleone\pgfmathresult
	\endgroup
}%

% defines \pgf@x to be the scalar product between points #1 and #2
% #1, #2 are pgfpoint commands
\def\pgfplotsscalarproductofvectors#1#2{%
	\begingroup
	\pgf@process{#1}%
	\edef\pgfplots@scalarprod@a{\pgf@sys@tonumber\pgf@x}%
	\edef\pgfplots@scalarprod@b{\pgf@sys@tonumber\pgf@y}%
	\pgf@process{#2}%
	\pgf@x=\pgfplots@scalarprod@a\pgf@x
	\advance\pgf@x by\pgfplots@scalarprod@b\pgf@y
	\global\pgf@x=\pgf@x
	\endgroup
}%

% converts a comma-separated list (PGF foreach)  to my internal list
% structure.
\long\def\pgfplots@foreach@to@list#1\to#2{%
	\global\pgfplotslistnewempty\pgfplots@glob@TMPa
	\begingroup
	\foreach \pgfplots@i in {#1} {%
		\expandafter\pgfplotslistpushbackglobal\pgfplots@i\to\pgfplots@glob@TMPa
	}%
	\endgroup
	\pgfplotslistcopy\pgfplots@glob@TMPa\to#2\relax
}


% Removes duplicates in a comma separated list and creates a new list
% into the macro \pgfplotsretval. The list doesn't need to be sorted,
% but it should not contain too much elements as the runtime for this
% simple method is quadratic.
%
% #1: a CSV list.
% Assigns \pgfplotsretval
\def\pgfplotsutil@unify@short@csv@list#1{%
	\begingroup
		\def\pgfplotsretval{}%
		\pgfplotsutilforeachcommasep{#1}\as\pgfplots@unify@cur{%
			\pgfutil@ifundefined{pgfp@unify@\pgfplots@unify@cur @@@}{%
				\expandafter\def\csname pgfp@unify@\pgfplots@unify@cur @@@\endcsname{1}%
				\ifx\pgfplotsretval\pgfutil@empty
					\let\pgfplotsretval=\pgfplots@unify@cur%
				\else
					\t@pgfplots@toka=\expandafter{\pgfplotsretval}%
					\t@pgfplots@tokb=\expandafter{\pgfplots@unify@cur}%
					\edef\pgfplotsretval{\the\t@pgfplots@toka,\the\t@pgfplots@tokb}%
				\fi
			}{}%
		}%
		\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Simply invokes the code of PGF key #1 with value #2, that means
% #1#2\pgfeov
\def\pgfplots@invoke@pgfkeyscode#1#2{%
	\pgfkeysvalueof{#1}#2\pgfeov
}

% Usage:
% \pgfplots@letcsname pgfplots@xtick={pgfplots@kram@x}
% -> invokes \csname ... \endcsname for both args.
\def\pgfplots@letcsname#1=#2{%
	\expandafter\let\expandafter\pgfplots@loc@TMPc\csname #2\endcsname
	\expandafter\let\csname #1\endcsname=\pgfplots@loc@TMPc
}%

% Inside of a pgfplots axis, this method (together with
% \pgfplotsqpointxyz for 3d axes) is THE
% point method. If you override the mapping to PGF coordinates here,
% all other \pgfplotspoint* commands should inherit the changes as well.
\def\pgfplotsqpointxy#1#2{%
  \global\pgf@x=#1\pgf@xx%
  \global\advance\pgf@x by #2\pgf@yx%
  \global\pgf@y=#1\pgf@xy%
  \global\advance\pgf@y by #2\pgf@yy}

% Inside of a pgfplots axis, this method (together with
% \pgfplotsqpointxy for 2d axes) is THE
% point method. If you override the mapping to PGF coordinates here,
% all other \pgfplotspoint* commands should inherit the changes as well.
\def\pgfplotsqpointxyz#1#2#3{%
  \global\pgf@x=#1\pgf@xx%
  \global\advance\pgf@x by #2\pgf@yx%
  \global\advance\pgf@x by #3\pgf@zx%
  \global\pgf@y=#1\pgf@xy%
  \global\advance\pgf@y by #2\pgf@yy%
  \global\advance\pgf@y by #3\pgf@zy}

% A "quick" quick variant of \pgfqpointxy which assumes that 
% the X unit vector is ( e_xx,0 )^T  and the Y unit vector is ( 0,
% e_yy)^T.
%
% In words, the unit vectors are orthogonal. This is the usual case
% for two-dimensional plots and shall be optimized.
\def\pgfplotsqpointxy@orthogonal#1#2{%
	\global\pgf@x=#1\pgf@xx%
	\global\pgf@y=#2\pgf@yy}


% Takes a domain as input and generates a foreach argument which
% samples from the domain.
% Writes the result to \pgfplotsretval
% INPUT:
% 	#1:#2 the domain
% 	#3: the number of samples
% OUTPUT:
% 	\pgfplotsretval a foreach specification.
\def\pgfplots@domain@to@foreach#1:#2\relax#3{%
	\pgfmathparse{#1}%
	\let\pgfplots@loc@TMPa=\pgfmathresult%
	\pgfmathparse{#2}%
	\let\pgfplots@loc@TMPb=\pgfmathresult%
	\pgfmathparse{\pgfplots@loc@TMPa+(\pgfplots@loc@TMPb-\pgfplots@loc@TMPa)/(#3-1)}%
	\edef\pgfplotsretval{\pgfplots@loc@TMPa,\pgfmathresult,...,\pgfplots@loc@TMPb}%
}

{
	\catcode`\;=\active
	\catcode`\:=\active
	\gdef\pgfplots@activesemicolon{;}%
	\gdef\pgfplots@activecolon{:}%
}

% checks whether ';' is active and replaces a sequence of commands
% accordingly.
%
% @see \pgfplots@appendto@activesemicolon@switcher
\def\pgfplots@checkandpreparefor@active@semicolon{%
	\ifnum\the\catcode`\;=\active\relax
		\pgfplots@checkandpreparefor@active@semicolon@
	\fi
}%
\def\pgfplots@checkandpreparefor@active@semicolon@{}%

% Adds all commands '#1' to the sequence of commands which will be
% issued in case ';' is active.
\def\pgfplots@appendto@activesemicolon@switcher#1{%
	\expandafter\def\expandafter\pgfplots@checkandpreparefor@active@semicolon@\expandafter{%
		\pgfplots@checkandpreparefor@active@semicolon@
		#1%
	}%
}%

% Invokes '#3' for every element in the comma separated list '#1'.
% during '#3', the macro '#2' will be set to the current list element.
%
% In contrast to \foreach of tikz, this processing is NOT scoped by
% TeX groups. However, it can be nested.
%
% Please note that no trimming of white spaces is performed.
%
% This is the (accidentally) the same as \pgfplotsforeachentryinCSV.
\long\def\pgfplotsutilforeachcommasep#1\as#2#3{%
	\pgfplotsforeachentryinCSVisterminated@loop{#2}{#3}#1,\pgfplots@EOI
}%

% Usage:
% \pgfplotsforeachentryinCSV{\value}{1,2,3,4.5,6.7,10}{The value is \value\par}
%
% The loop is ungrouped.
%
% This is the same as \pgfplotsutilforeachcommasep, I must have been
% sleeping somehow.
%
% @see \pgfplotsutilforeachcommasep
% @see \pgfplotsforeachentryinCSVisterminated
% @see \pgfplotsforeachungrouped
% This loop can be nested.
\long\def\pgfplotsforeachentryinCSV#1#2#3{%
	\pgfplotsforeachentryinCSVisterminated@loop{#1}{#3}#2,\pgfplots@EOI
}%
% A variant of \pgfplotsforeachentryinCSV where a trailing comma
% indicates the end of input.
% Example:
%
% \pgfplotsforeachentryinCSV{\value}{1,2,3,4.5,6.7,10,}{The value is \value\par}
%                                                    ^
%
% It is permissable to provide an empty list (without trailing comma)
%
% This loop can be nested.
\long\def\pgfplotsforeachentryinCSVisterminated#1#2#3{%
	\pgfplotsforeachentryinCSVisterminated@loop{#1}{#3}#2\pgfplots@EOI
}%
% #1 : the loop macro
% #2 : the loop BODY
% #3 : the loop LIST
% (note the different sequence)
\long\def\pgfplotsforeachentryinCSVisterminated@loop#1#2{%
	\pgfutil@ifnextchar\pgfplots@EOI{%
		\pgfutil@gobble
	}{%
		\pgfplotsforeachentryinCSV@next{#1}{#2}%
	}%
}%
\long\def\pgfplotsforeachentryinCSV@next#1#2#3,{%
	\def#1{#3}%
	#2\relax
	\pgfplotsforeachentryinCSVisterminated@loop{#1}{#2}%
}%


% Discards any token up to (and including) \relax.
\def\pgfplots@gobble@until@relax#1\relax{}%
\def\pgfplots@gobble@until@EOI#1\pgfplots@EOI{}%

% A (simple) replacement for \foreach which 
% 1. does NOT scope the argument
% 2. allows to use PGF Math routines for loops ( '...' notation)
% instead of TeX registers.
%
% Usage:
% \pgfplotsforeachungrouped \x in {1,2,3,4,5} {<code>}
% 
% Some details:
% 1. If #2 is a UNIFORM SAMPLING RANGE, the algorithm is supposed to be
%   most effective. This is considered to be the case for the syntax
%   \pgfplotsforeachungrouped \x in {1,2,...,8}.
% 2. If case (1.) is not used, \foreach \x in {#2} {} is invoked to
% generate a temporary list. Afterwards, this list is invoked without
% scopes.
%
% This loop *can* be nested.
\def\pgfplotsforeachungrouped#1{%
	\pgfutil@ifnextchar/{\pgfplotsforeachungrouped@{#1}}{\pgfplotsforeachungrouped@{#1}/{}}%
}%
\long\def\pgfplotsforeachungrouped@#1/#2in #3#4{%
	\def\pgfplots@foreach@loc@TMPa{#2}%
	\ifx\pgfplots@foreach@loc@TMPa\pgfutil@empty
		\pgfplotsforeachungrouped@isuniform{#3}%
		\ifpgfplots@loc@tmp
			% special handling:
			% \foreach #1 in {a,b,...,c} can be improved:
			\pgfplotsforeachungroupeduniform@{#1}#3\relax{#4}%
		\else
			% invoke  \foreach #1 in {#3} and transport results:
			\pgfplotsforeachungroupednonuniform@#1{#3}{#4}%
		\fi
	\else
		% invoke  \foreach #1/#2 in {#3} and transport results:
		\pgfplotsforeachungroupedslashed@#1/#2{#3}{#4}%
	\fi
}
\let\pgfplots@original@pgfmathadd@=\pgfmathadd@

\long\def\pgfplotsforeachungroupeduniform@#1#2,#3,...,#4\relax#5{%
	% Compute mesh width!
	\pgfmathparse{#2}%
	\let\pgfplots@foreach@loc@TMPa=\pgfmathresult
	\pgfmathparse{#3}%
	\let\pgfplots@foreach@loc@TMPb=\pgfmathresult
	\pgfmathsubtract@{\pgfplots@foreach@loc@TMPb}{\pgfplots@foreach@loc@TMPa}%
	% Use \pgfmath engine for the loop:
	% mesh width:
	\let\pgfplots@foreach@loc@meshwidth=\pgfmathresult
	\pgfmathparse{0}% invoke the parser - in case the fpu is active.
	\pgfmathlessthan@{\pgfplots@foreach@loc@meshwidth}{\pgfmathresult}%
	% the loop will run while ( NOT \pgfplots@foreach@loc@cmp{<cur>}{<last>} )
	\ifdim\pgfmathresult pt=1pt
		\def\pgfplots@foreach@loc@cmp{\pgfmathlessthan@}%
	\else
		\def\pgfplots@foreach@loc@cmp{\pgfmathgreaterthan@}%
	\fi
	%
	\pgfmathparse{#4 + 0.5*\pgfplots@foreach@loc@meshwidth}%
	\let\pgfplots@foreach@loc@TMPb=\pgfmathresult
	%
	\t@pgfplots@toka={#5}%
	% to allow nesting without additional TeX groups of
	% \pgfplotsforeachungroupeduniform, I introduce this loop
	% structure here which does not need ANY state macro:
	\edef\pgfplots@foreach@loc@TMPc{%
		\noexpand\pgfplotsforeachungroupeduniform@loop@mathengine
			{\expandafter\noexpand\pgfplots@foreach@loc@cmp}% #1= comparison fct
			{\pgfplots@foreach@loc@TMPa}% #2 =lower limit (ITERATES)
			{\pgfplots@foreach@loc@TMPb}% #3 = upper limit
			{\noexpand#1}% #4 = the loop macro name
			{\pgfplots@foreach@loc@meshwidth}% #5 =  h
			{\the\t@pgfplots@toka}% #6 = the code to invoke
	}%
	\pgfplots@foreach@loc@TMPc
}%
\long\def\pgfplotsforeachungroupeduniform@loop@mathengine#1#2#3#4#5#6{%
	#1{#2}{#3}%
	\ifdim\pgfmathresult pt=0pt
		\pgfutil@in@{.0\relax}{#2\relax}%
		\ifpgfutil@in@
			\def\pgfplotsforeach@loc@TMP##1.0\relax{%
				\def#4{##1}%
			}%
			\pgfplotsforeach@loc@TMP#2\relax
		\else
			\def#4{#2}%
		\fi
		#6\relax
		\pgfmathadd@{#2}{#5}%
		\t@pgfplots@toka={{#1}}%
		\t@pgfplots@tokb=\expandafter{\pgfmathresult}%
		\t@pgfplots@tokc={{#3}{#4}{#5}{#6}}%
		% loop!
		\edef\pgfplots@loc@TMPa{\noexpand\pgfplotsforeachungroupeduniform@loop@mathengine
			\the\t@pgfplots@toka
			{\the\t@pgfplots@tokb}%
			\the\t@pgfplots@tokc}%
		\expandafter\pgfplots@loc@TMPa
	\fi
}%

\long\def\pgfplotsforeachungroupednonuniform@#1#2#3{%
	\pgfplotsapplistXXglobalnewempty
	\foreach\pgfplots@foreach@loc@TMPa in {#2} {%
		\expandafter\pgfplotsapplistXXglobalpushback\expandafter{\pgfplots@foreach@loc@TMPa,}%
	}%
	\pgfplotsapplistXXgloballet\pgfplots@foreach@loc@TMPa
	\pgfplotsapplistXXglobalclear
	\expandafter\pgfplotsforeachentryinCSVisterminated\expandafter\pgfplots@foreach@loc@TMPa\expandafter{\pgfplots@foreach@loc@TMPa}{%
		\let#1=\pgfplots@foreach@loc@TMPa
		#3%
	}%
}
\long\def\pgfplotsforeachungroupedslashed@#1/#2#3#4{%
	\pgfplotsapplistXXglobalnewempty
	\foreach\pgfplots@foreach@loc@TMPa/\pgfplots@foreach@loc@TMPb in {#3} {%
		\edef\pgfplots@foreach@loc@TMPa{\pgfplots@foreach@loc@TMPa/\pgfplots@foreach@loc@TMPb}%
		\expandafter\pgfplotsapplistXXglobalpushback\expandafter{\pgfplots@foreach@loc@TMPa,}%
	}%
	\pgfplotsapplistXXgloballet\pgfplots@foreach@loc@TMPa
	\pgfplotsapplistXXglobalclear
	\expandafter\pgfplotsforeachentryinCSVisterminated\expandafter\pgfplots@foreach@loc@TMPa\expandafter{\pgfplots@foreach@loc@TMPa}{%
		\expandafter\pgfplotsforeachungroupedslashed@@\pgfplots@foreach@loc@TMPa\pgfplots@EOI
		\let#1=\pgfplots@foreach@loc@TMPa
		\let#2=\pgfplots@foreach@loc@TMPb
		#4%
	}%
}
\def\pgfplotsforeachungroupedslashed@@#1/#2\pgfplots@EOI{%
	\def\pgfplots@foreach@loc@TMPa{#1}%
	\def\pgfplots@foreach@loc@TMPb{#2}%
}%

% Sets \ifpgfplots@loc@tmp to true if and only if '#1' is of the form
% #1 = <number1>,<number2>,...,<number3>.
\def\pgfplotsforeachungrouped@isuniform#1{%
	\edef\pgfplots@foreach@loc@TMPa{#1}%
	\expandafter\pgfplotsforeachungrouped@isuniform@\pgfplots@foreach@loc@TMPa,,,,\relax
}%
\def\pgfplotsforeachungrouped@isuniform@#1,#2,#3,#4,#5\relax{%
	\def\pgfplots@foreach@loc@TMPa{#5}%
	\def\pgfplots@foreach@loc@TMPb{,,,}%
	\pgfplots@loc@tmpfalse
	\ifx\pgfplots@foreach@loc@TMPa\pgfplots@foreach@loc@TMPb
		\def\pgfplots@foreach@loc@TMPa{#3}%
		\def\pgfplots@foreach@loc@TMPb{...}%
		\ifx\pgfplots@foreach@loc@TMPa\pgfplots@foreach@loc@TMPb
			\pgfutil@in@,{#4}%
			\ifpgfutil@in@
			\else
				% ok, we REALLY have a uniform range!
				\pgfplots@loc@tmptrue
			\fi
		\fi
	\fi
}%

% A variant of \pgfplotsforeachungrouped ( or \foreach ) which
% *invokes* with #1 set to the current iterate.
%
% Example:
% 	\pgfplotsinvokeforeach{a,b,c,d} 
% 		{\pgfkeys{key #1/.style={otherstyle #1}}}
% -> will invoke
%  \pgfkeys{key a/.style={otherstyle a}}
%  \pgfkeys{key b/.style={otherstyle b}}
%  \pgfkeys{key c/.style={otherstyle c}}
%  \pgfkeys{key d/.style={otherstyle d}}
% Note that \pgfplotsforeachungrouped \d in {a,b,c,d} 
% 		{\pgfkeys{key \d/.style={}}}
% would not work:
%  \pgfkeys{key a/.style={otherstyle \d}}
%  \pgfkeys{key b/.style={otherstyle \d}}
%  \pgfkeys{key c/.style={otherstyle \d}}
%  \pgfkeys{key d/.style={otherstyle \d}}
% such an application would need expansion control.
%
% #1: the iterates. Can be any argument as you would supply it to
% \foreach, for example \foreach \x in {1,2,...,10}
% 	-> #1 = {1,2,...,10}
% #2: the loop body. It can contain the parameter string '#1' which
% will be set to each element in the iterates list in turn.
%
% This method is actually just a light-weight adapter around
% \pgfplotsforeachungrouped.
\long\def\pgfplotsinvokeforeach#1#2{%
	\long\def\pgfplotsinvokeforeach@@##1{#2}%
	\pgfplotsforeachungrouped \pgfplotsinvokeforeach@ in {#1} {%
		\expandafter\pgfplotsinvokeforeach@@\expandafter{\pgfplotsinvokeforeach@}%
	}%
}

% Allows to provide an output routine for
% \pgfplotsforeachlogarithmicungrouped which changes the number format
% of \pgfmathresult.
%
% Example:
% \pgfplotsforeachlogarithmicformatresultwith{\pgfmathfloattofixed{\pgfmathresult}}
%
% #1: is code which modifies \pgfmathresult.
\def\pgfplotsforeachlogarithmicformatresultwith#1{%
	\def\pgfplotsforeachlogarithmicungrouped@finalizeresult{#1}%
}

% \pgfplotsforeachlogarithmicungrouped[<samples>] \x/\logx in {a:b} {#3}
%
% samples \x between a and b using a logarithmic scale.
%
% During '#3', \x will contain the sample and \logx the logarithm of
% \x.
%
% Both, #1 and #2 will be provided as floating point numbers unless
% configured otherwise with \pgfplotsforeachlogarithmicsetoutput,
% see above.
%
% Arithmethics will be carried out with the math class
% '\pgfplotsforeachlogarithmicmathid'.
% This macro should expand to something which is usable inside of
% \pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{....}
% the default is 'float'
% 
% FIXME : THIS CAN'T BE NESTED YET!
\def\pgfplotsforeachlogarithmicungrouped[#1]#2/#3{\pgfplotsforeachlogarithmicungrouped@[#1]{#2}{#3}}%
\long\def\pgfplotsforeachlogarithmicungrouped@[#1]#2#3in #4#5{%
	% define
	% 	\pgfplots@foreach@loc@TMPd = N in fixed point
	% 	\pgfplots@foreach@loc@TMPc = N-1 in float :
	\edef\pgfplots@plot@samples@@{#1}%
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{one}%
	\let\pgfplots@foreach@loc@TMPa=\pgfmathresult
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{parsenumber}{\pgfplots@plot@samples@@}%
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{op}{subtract}{{\pgfmathresult}{\pgfplots@foreach@loc@TMPa}}%
	\let\pgfplots@foreach@loc@TMPc=\pgfmathresult
	%
	\edef\pgfplots@loc@TMPb{#4}%
	\expandafter\pgfplotsforeachlogarithmicungrouped@readdomain\pgfplots@loc@TMPb\relax
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{if is bounded}{\pgfplots@foreach@loc@TMPb}{%
	}{%
		\pgfplotsforeachlogarithmicmath@unbounded@domain{#4}%
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{parsenumber}{-10000}%
		\let\pgfplots@foreach@loc@TMPb=\pgfmathresult
	}%
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{if is bounded}{\pgfplots@foreach@loc@TMPa}{%
	}{%
		\pgfplotsforeachlogarithmicmath@unbounded@domain{#4}%
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{parsenumber}{-10000}%
		\let\pgfplots@foreach@loc@TMPa=\pgfmathresult
	}%
	% compute mesh width into \pgfplots@foreach@loc@TMPc:
	%  \pgfplots@foreach@loc@TMPc := h := ( log(xmax) - log(xmin) ) / (N-1)
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{op}{subtract}{{\pgfplots@foreach@loc@TMPb}{\pgfplots@foreach@loc@TMPa}}%
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{op}{divide}{{\pgfmathresult}{\pgfplots@foreach@loc@TMPc}}%
	\let\pgfplots@foreach@loc@TMPc=\pgfmathresult
	%
	% apply local scoping here:
	\edef\pgfplots@foreach@loc@TMPd{%
		\noexpand\c@pgf@counta=\the\c@pgf@counta\noexpand\relax
	}%
	\long\def\pgfplots@foreach@loc@TMPf{#5}%
	\c@pgf@counta=0
	\pgfutil@loop
	\ifnum\c@pgf@counta<\pgfplots@plot@samples@@
		% compute exp(log(min) + i * h)
		\edef\pgfplots@loc@TMPa{\the\c@pgf@counta}%
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{parsenumber}{\pgfplots@loc@TMPa}%
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{op}{multiply}{{\pgfmathresult}{\pgfplots@foreach@loc@TMPc}}%
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{op}{add}{{\pgfmathresult}{\pgfplots@foreach@loc@TMPa}}%
		\let\pgfplots@loc@TMPa=\pgfmathresult
		\pgfplotsforeachlogarithmicungrouped@finalizeresult%
		\let#3=\pgfmathresult
		\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{exp}{\pgfplots@loc@TMPa}%
		\pgfplotsforeachlogarithmicungrouped@finalizeresult%
		\let#2=\pgfmathresult
		%
		% ok, invoke it!
		% -> store \c@pgf@counta before doing so. We need to scope
		%  manually here.
		\edef\pgfplots@foreach@loc@TMPe{\the\c@pgf@counta}%
		\pgfplots@foreach@loc@TMPf\relax
		\c@pgf@counta=\pgfplots@foreach@loc@TMPe\relax
		\advance\c@pgf@counta by 1
	\pgfutil@repeat
	% restore scoped variables:
	\pgfplots@foreach@loc@TMPd
}

\def\pgfplotsforeachlogarithmicmath@unbounded@domain#1{%
	\pgfplots@error{Sorry, the provided domain #1 is unsuitable for logarithmic sampling (applying the log results in unbounded values)}%
}%

% can be used to convert the number format from float to something
% else.
\def\pgfplotsforeachlogarithmicungrouped@finalizeresult{}%


\def\pgfplotsforeachlogarithmicmathid{float}

% defines 
%  \pgfplots@foreach@loc@TMPa := log(firstintervalpt)
% and
%  \pgfplots@foreach@loc@TMPb := log(secondintervalpt)
\def\pgfplotsforeachlogarithmicungrouped@readdomain#1:#2\relax{%
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{log}{#1}%
	\let\pgfplots@foreach@loc@TMPa=\pgfmathresult
	\pgfplotscoordmath{\pgfplotsforeachlogarithmicmathid}{log}{#2}%
	\let\pgfplots@foreach@loc@TMPb=\pgfmathresult
}

% invokes '#2' if #1 expands (\edef) to the empty string or '#3' if not.
\def\pgfplots@ifempty#1#2#3{%
	\edef\pgfplots@loc@TMPa{#1}%
	\ifx\pgfplots@loc@TMPa\pgfutil@empty #2\else #3\fi
}%

{
	\catcode`\#=12
	\catcode`\1=12
	\gdef\pgfplotsutil@add@to@macro@@one{#1}%
	\gdef\pgfplotsutil@add@to@macro@@two{#1#2}%
}

\def\pgfplotsutil@ifmacrodependsononearg#1#2#3{%
	\edef\pgfplots@loc@TMPa{\noexpand\pgfutil@in@{\pgfplotsutil@add@to@macro@@one}{\meaning#1}}%
	\pgfplots@loc@TMPa
	\ifpgfutil@in@
		#2%
	\else
		#3%
	\fi
}%

\def\pgfplotsutil@ifmacrodependsontwoargs#1#2#3{%
	\edef\pgfplots@util@add@to@macro@@{\noexpand\pgfutil@in@{\pgfplotsutil@add@to@macro@@two}{\meaning#1}}%
	\pgfplots@util@add@to@macro@@
	\ifpgfutil@in@
		#2%
	\else
		#3%
	\fi
}%

% Appends '#2' to macro '#1'.
%
% #1: a macro which takes either no arguments or exactly one.
% #2: some token text. May depend on '#1' if #1 is a macro which
% depends on one arg.
\def\pgfplotsutil@add@to@macro#1#2{%
	\pgfplotsutil@ifmacrodependsononearg#1{%
		\pgfplotsutil@ifmacrodependsontwoargs#1{%
			\pgfplotsutil@add@to@macro@twoargs{#1}{#2}%
		}{%
			\pgfplotsutil@add@to@macro@onearg{#1}{#2}%
		}%
	}{%
		\pgfplotsutil@add@to@macro@noargs{#1}{#2}%
	}%
}%

\def\pgfplotsutil@add@to@macro@twoargs#1#2{%
	\t@pgfplots@tokc=\expandafter{#1{##1}{##2}}%
	\t@pgfplots@tokb={#2}%
	\edef\pgfplots@util@add@to@macro@@{\the\t@pgfplots@tokc\the\t@pgfplots@tokb}%
	\expandafter\def\expandafter#1\expandafter##\expandafter1\expandafter##\expandafter2\expandafter{\pgfplots@util@add@to@macro@@}%
}

\def\pgfplotsutil@add@to@macro@onearg#1#2{%
	\t@pgfplots@tokc=\expandafter{#1{##1}}%
	\t@pgfplots@tokb={#2}%
	\edef\pgfplots@util@add@to@macro@@{\the\t@pgfplots@tokc\the\t@pgfplots@tokb}%
	\expandafter\def\expandafter#1\expandafter##\expandafter1\expandafter{\pgfplots@util@add@to@macro@@}%
}

\def\pgfplotsutil@add@to@macro@noargs#1#2{%
	\t@pgfplots@tokc=\expandafter{#1}%
	\t@pgfplots@tokb=\expandafter{#2}%
	\edef#1{\the\t@pgfplots@tokc\the\t@pgfplots@tokb}%
}
\def\pgfplots@logfileopen#1{%
	\immediate\write-1{PGFPlots: reading {#1}}%
	\pgfutil@IfUndefined{tikzifexternalizingcurrent}{%
		% version not up-to-date or external lib not loaded?
	}{%
		% adjust the .dep file only if we are externalizing.
		% Otherwise, it may be overwritten by \pgfplotstableread
		% commands which are *before* the tikzpicture (and the picture
		% will be replaced by its graphics)
		\tikzifexternalizingcurrent{\tikzpicturedependsonfile{#1}}{}%
	}%
}%

% Expands #2 using \edef and invokes #1 with the resulting string.
%
% DEPRECATED
% Example:
%   \pgf@xa=7.9pt
%   \pgfplotsutil@edef@invoke\pgfmathexp@{{\pgf@sys@tonumber{\pgf@xa}}}%
% will invoke
%   \pgfmathexp@{7.9}
\def\pgfplotsutil@edef@invoke#1#2{%
	\edef\pgfutil@edef@invoke@{#2}%
	\expandafter#1\pgfutil@edef@invoke@
}

% defines \pgfmathresult to be the INTEGER result of (#1 mod #2).
%
% \pgfplotsmathmod{2}{4} -> 2
% \pgfplotsmathmod{9}{5} -> 4
\def\pgfplotsmathmodint#1#2{%
	\begingroup
	\c@pgf@counta=#2\relax
	\c@pgf@countb=#1\relax
	\ifnum\c@pgf@counta=2
		\ifodd\c@pgf@countb
			\def\pgfmathresult{1}%
		\else
			\def\pgfmathresult{0}%
		\fi
	\else
		\c@pgf@countc=\c@pgf@countb
		\ifnum\c@pgf@counta=0
			\pgfplots@error{Can't compute \the\c@pgf@countb\space mod \the\c@pgf@counta\space -- this yields division by zero (second argument is zero)!}%
			\edef\pgfmathresult{#1}%
		\else
			\divide\c@pgf@countc by\c@pgf@counta
			\multiply\c@pgf@countc by\c@pgf@counta
			\advance\c@pgf@countb by-\c@pgf@countc
			\edef\pgfmathresult{\the\c@pgf@countb}%
		\fi
	\fi
	\pgfmath@smuggleone\pgfmathresult
	\endgroup
}%

% Advances a number stored in a macro and writes the result back into
% the macro.
% #1 is a macro containing a number. 
\def\pgfplotsutil@advancestringcounter#1{%
	\begingroup
		\c@pgf@counta=#1\relax
		\advance\c@pgf@counta by1
		\edef#1{\the\c@pgf@counta}%
		\pgfmath@smuggleone#1%
	\endgroup
}%
\def\pgfplotsutil@advancestringcounter@global#1{%
	\begingroup
		\c@pgf@counta=#1\relax
		\advance\c@pgf@counta by1
		\xdef#1{\the\c@pgf@counta}%
	\endgroup
}%

\newif\ifpgfplotsloopcontinue

% A loop construct which invokes '#1' and continues the loop if the
% boolean \ifpgfplotsloopcontinue is true and stops if it is false.
%
% #1: a statement which is expected to set \ifpgfplotsloopcontinue
% #2: the loop body.
\def\pgfplotsloop#1#2{%
	#1\relax
	\ifpgfplotsloopcontinue
		#2\relax
		\def\pgfplotsloop@{\pgfplotsloop{#1}{#2}}%
		\expandafter\pgfplotsloop@
	\fi
}%

% String comparison of '#1' and '#2'.
% Defines 
% \pgfplotsretval=0  if #1 == #2 
% \pgfplotsretval=1  if #1 < #2
% \pgfplotsretval=2  if #1 > #2
\def\pgfplotsutilstrcmp#1#2{%
	\begingroup
	\def\pgfplots@strcmp@arga{#1}%
	\pgfplots@command@to@string\pgfplots@strcmp@arga\pgfplots@strcmp@arga
	\def\pgfplots@strcmp@argb{#2}%
	\pgfplots@command@to@string\pgfplots@strcmp@argb\pgfplots@strcmp@argb
	%
	%
	\def\pgfplotsretval{0}%
	%
	\pgfplotsloop{%
		\if0\pgfplotsretval
			% get next token of arga:
			\expandafter\pgfplotsutilstrcmp@popfront\pgfplots@strcmp@arga\relax\relax
			\let\pgfplots@strcmp@arga=\pgfplots@strcmp@rest
			\let\pgfplots@strcmp@arga@=\pgfplots@strcmp@cur
			%
			% get next token of argb:
			\expandafter\pgfplotsutilstrcmp@popfront\pgfplots@strcmp@argb\relax\relax
			\let\pgfplots@strcmp@argb=\pgfplots@strcmp@rest
			\let\pgfplots@strcmp@argb@=\pgfplots@strcmp@cur
			%
			% if one of them is empty: set retval and break loop.
			\ifx\pgfplots@strcmp@arga@\pgfplotsutilstrcmp@relaxtext
				\ifx\pgfplots@strcmp@argb@\pgfplotsutilstrcmp@relaxtext
				\else
					\def\pgfplotsretval{1}%
				\fi
				\pgfplotsloopcontinuefalse
			\else
				\ifx\pgfplots@strcmp@argb@\pgfplotsutilstrcmp@relaxtext
					\def\pgfplotsretval{2}%
					\pgfplotsloopcontinuefalse
				\else
					\pgfplotsloopcontinuetrue
				\fi
			\fi
		\else
			\pgfplotsloopcontinuefalse
		\fi
	}{%
		%
		\ifnum\expandafter`\pgfplots@strcmp@arga@<\expandafter`\pgfplots@strcmp@argb@
			\def\pgfplotsretval{1}%
			\let\pgfplots@strcmp@arga=\pgfutil@empty
		\else
			\ifnum\expandafter`\pgfplots@strcmp@arga@>\expandafter`\pgfplots@strcmp@argb@
				\def\pgfplotsretval{2}%
				\let\pgfplots@strcmp@arga=\pgfutil@empty
			\fi
		\fi
	}%
	%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsutilstrcmp@relaxtext{\relax}%
\def\pgfplotsutilstrcmp@popfront#1#2\relax{%
	\def\pgfplots@strcmp@cur{#1}%
	\def\pgfplots@strcmp@rest{#2}%
}%

% Takes two dimensional cartesian coordinates #1,#2 and 
% defines the macros #3 and #4 to contain the associated polar
% coordinates.
%
% #1: x
% #2: y
% #3: a macro which will be filled with the angle (in degrees)
% #4: a macro which will be filled with the radius
\def\pgfplotsmathcarttopol#1#2#3#4{%
	\pgfplotsmathcarttopol@{#1}{#2}{#3}{#4}{default}%
}%
\def\pgfplotsmathcarttopolbasic#1#2#3#4{%
	\pgfplotsmathcarttopol@{#1}{#2}{#3}{#4}{pgfbasic}%
}%
% #5: the math engine to use (\pgfplotscoordmath argument)
\def\pgfplotsmathcarttopol@#1#2#3#4#5{%
	\begingroup
	\pgfplotscoordmath{#5}{parsenumber}{#1}%
	\let\pgfplots@x=\pgfmathresult
	\pgfplotscoordmath{#5}{parsenumber}{#2}%
	\let\pgfplots@y=\pgfmathresult
	\pgfplotscoordmath{#5}{op}{veclen}{{\pgfplots@x}{\pgfplots@y}}%
	\let\pgfplots@r=\pgfmathresult
	\pgfplotscoordmath{#5}{if is}{\pgfplots@r}{0}{%
		\pgfplotscoordmath{#5}{parsenumber}{0}%
	}{%
		\pgfplotscoordmath{#5}{op}{divide}{{\pgfplots@x}{\pgfplots@r}}%
		\let\pgfplots@quot=\pgfmathresult
		\pgfplotscoordmath{#5}{op}{acos}{{\pgfplots@quot}}%
		%
		\pgfplotscoordmath{#5}{if is}{\pgfplots@y}{-}{%
			\let\pgfplots@acos=\pgfmathresult
			\pgfplotscoordmath{#5}{parsenumber}{-1}%
			\pgfplotscoordmath{#5}{op}{multiply}{{\pgfmathresult}{\pgfplots@acos}}%
		}{}%
	}%
	\let\pgfplots@deg=\pgfmathresult
	\xdef\pgfplots@glob@TMPa{\noexpand\def\noexpand#3{\pgfplots@deg}\noexpand\def\noexpand#4{\pgfplots@r}}%
	\endgroup
	\pgfplots@glob@TMPa
}%

% Takes two dimensional polar coordinates #1,#2 and 
% defines the macros #3 and #4 to contain the associated cartesian
% coordinates.
%
% #1: angle (in degrees)
% #2: radius
% #3: a macro which will be filled with the x value
% #4: a macro which will be filled with the y value
\def\pgfplotsmathpoltocart#1#2#3#4{%
	\pgfplotsmathpoltocart@{#1}{#2}{#3}{#4}{default}%
}%
\def\pgfplotsmathpoltocartbasic#1#2#3#4{%
	\pgfplotsmathpoltocart@{#1}{#2}{#3}{#4}{pgfbasic}%
}%
% #5: the math engine to use (\pgfplotscoordmath argument)
\def\pgfplotsmathpoltocart@#1#2#3#4#5{%
	\begingroup
	\pgfplotscoordmath{#5}{parsenumber}{#1}%
	\let\pgfplots@deg=\pgfmathresult
	\pgfplotscoordmath{#5}{parsenumber}{#2}%
	\let\pgfplots@r=\pgfmathresult
	\pgfplotscoordmath{#5}{op}{sin}{{\pgfplots@deg}}%
	\pgfplotscoordmath{#5}{op}{multiply}{{\pgfplots@r}{\pgfmathresult}}%
	\let\pgfplots@y=\pgfmathresult
	\pgfplotscoordmath{#5}{op}{cos}{{\pgfplots@deg}}%
	\pgfplotscoordmath{#5}{op}{multiply}{{\pgfplots@r}{\pgfmathresult}}%
	\let\pgfplots@x=\pgfmathresult
	\xdef\pgfplots@glob@TMPa{\noexpand\def\noexpand#3{\pgfplots@x}\noexpand\def\noexpand#4{\pgfplots@y}}%
	\endgroup
	\pgfplots@glob@TMPa
}%



\pgfutil@IfUndefined{pgfmathdeclarefunction}{%
	% BACKWARDS COMPATIBILITY: We have PGF 2.00 :
	\def\pgfplotsmathdeclarepseudoconstant#1#2{%
		\t@pgfplots@toka=\expandafter{\csname pgfmath#1@\endcsname}%
		\t@pgfplots@tokb={\pgfmath@postfunction}%
		\expandafter\edef\csname pgfmath@parsefunction@#1\endcsname{\the\t@pgfplots@toka\the\t@pgfplots@tokb}%
		\expandafter\edef\csname pgfmath#1\endcsname{\the\t@pgfplots@toka}%
		\expandafter\def\csname pgfmath#1@\endcsname{#2}%
	}%
	\let\pgfplotsmathredeclarepseudoconstant=\pgfplotsmathdeclarepseudoconstant
}{%
	\def\pgfplotsmathdeclarepseudoconstant#1#2{%
		\pgfmathdeclarefunction*{#1}{0}{#2}%
	}%
	\def\pgfplotsmathredeclarepseudoconstant#1#2{%
		\pgfmathredeclarefunction{#1}{#2}%
	}%
}%

\def\pgfplotsmathfloatln@#1{%
	\pgfmathfloatln@{#1}%
}

% Defines a macro which computes ln(x)/ln(#2) in float.
%
% #1: a macro name without backslash
% #2: the log base as fixed point number. Maybe the empty string in
% which case the natural log is used.
%
% It will define the macros:
% <macro>##1    -> calls \pgfmathfloatparsenumber and then the '@' variant:
% <macro>@##1   -> expects and returns result in float
% <macro>@tofixed##1 -> expects result in float, returns it in fixed (or an empty string)
% <macro>inv@##1   -> the inverse (exponential) function, expects float, returns float
% %
% <macro>logbase   -> the basis as number
% <macro>logofbasefixed -> log(#2)
% <macro>invlogofbasefixed -> 1/log(#2)
% <macro>logofbasefloat -> log(#2) in float
% <macro>invlogofbasefloat -> 1/log(#2) in float
\def\pgfplotsmathdefinemacrolnbase#1#2{%
	\expandafter\edef\csname #1logbase\endcsname{#2}%
	\expandafter\ifx\csname #1logbase\endcsname\pgfutil@empty
		\expandafter\edef\csname #1logbase\endcsname{2.718281828459}%
		\expandafter\def\csname #1logofbasefixed\endcsname{1}%
		\expandafter\def\csname #1invlogofbasefixed\endcsname{1}%
	\else
		\pgfmathlog@{\csname #1logbase\endcsname}%
		\expandafter\let\csname #1logofbasefixed\endcsname\pgfmathresult%
		\expandafter\pgfmath@basic@reciprocal@\expandafter{\pgfmathresult}%
		\expandafter\let\csname #1invlogofbasefixed\endcsname\pgfmathresult%
	\fi
	%
	\pgfmathfloatparsenumber{\csname #1logofbasefixed\endcsname}%
	\expandafter\let\csname #1logofbasefloat\endcsname\pgfmathresult%
	\pgfmathfloatparsenumber{\csname #1invlogofbasefixed\endcsname}%
	\expandafter\let\csname #1invlogofbasefloat\endcsname\pgfmathresult%
	%
	\expandafter\def\csname #1@tofixed\endcsname##1{%
		\pgfmathlog@float{##1}%
		\ifx\pgfmathresult\pgfutil@empty
		\else
			\expandafter\pgfmath@basic@multiply@\expandafter
				{\pgfmathresult}%
				{\csname #1invlogofbasefixed\endcsname}%
		\fi
	}%
	\expandafter\def\csname #1@\endcsname##1{%
		\expandafter\csname #1@tofixed\endcsname{##1}%
		\ifx\pgfmathresult\pgfutil@empty
			\pgfmathfloatcreate{3}{0.0}{0}%
		\else
			\pgfmathfloatparsenumber{\pgfmathresult}%
		\fi
	}%
	\expandafter\def\csname #1\endcsname##1{%
		\pgfmathfloatparsenumber{##1}%
		\csname #1@\endcsname{\pgfmathresult}%
	}%
	%
	\expandafter\def\csname #1inv@\endcsname##1{%
		\expandafter\pgfmathfloatmultiply@\expandafter{##1}{\csname #1logofbasefloat\endcsname}%
		\pgfmathfloatexp@{\pgfmathresult}%
	}%
}%

% Usage:
% \pgfplotsutilstrreplace{<token>}{<replacement>}{<string>}
%
% -> will assign the modified string into \pgfplotsretval.
%
% #1: the string to search (one or more tokens)
% #2: zero, one or more tokens which will be inserted instead of '#1'.
% #3: the string to search in
\long\def\pgfplotsutilstrreplace#1#2#3{%
	\def\pgfplotsretval{}%
	\long\def\pgfplotsutil@search@and@replace@@##1#1##2\pgfplots@EOI{%
		\expandafter\def\expandafter\pgfplotsretval\expandafter{\pgfplotsretval ##1#2}%
		\pgfplotsutil@search@and@replace@loop{#1}{##2}%
	}%
	\pgfplotsutil@search@and@replace@loop{#1}{#3}%
}
\long\def\pgfplotsutil@search@and@replace@loop#1#2{%
	\pgfutil@in@{#1}{#2}%
	\ifpgfutil@in@
		\def\pgfplots@loc@TMPa{\pgfplotsutil@search@and@replace@@ #2\pgfplots@EOI}%
	\else
		\expandafter\def\expandafter\pgfplotsretval\expandafter{\pgfplotsretval #2}%
		\let\pgfplots@loc@TMPa=\relax
	\fi
	\pgfplots@loc@TMPa
}%

% strcmp: invokes #3 if #1=#2 (top level expansion only) and it
% invokes #4 if #1!=#2.
\long\def\pgfplotsutilifstringequal#1#2#3#4{%
	\def\pgfplotsifstringequal@{#1}%
	\def\pgfplotsifstringequal@@{#2}%
	\ifx\pgfplotsifstringequal@\pgfplotsifstringequal@@ #3\else #4\fi
}%

% this command is in pgfplotscore.code.tex:
% \pgfplotsiffileexists

% EXPERIMENTAL OPTIMIZATION:
\let\pgfmathparsex@orig=\pgfmathparse
\def\pgfmathparsex#1{%
	\begingroup
	\message{WORKING ON #1...}%
	\tracingmacros=2 \tracingcommands=2
	\let\pgfmath@parse@@@operator\relax
	\let\pgfmath@parse@@operator\relax
	\let\pgfmath@parse@operator\relax
	\let\pgfmath@stack@push@operand\pgfutil@gobble
	\let\pgfmath@base=\pgfmath@empty
	\let\pgfmath@number=\pgfmath@empty
	\let\pgfmath@parse@number@failed=\relax
    \edef\pgfmath@expression{#1}%
	\expandafter\pgfmathparsex@first\pgfmath@expression\pgfmathparsex@EOI%
	\if1\pgfmathparsex@recover@
		\xdef\pgfmathparsex@@{\noexpand\pgfmathparsex@orig{\pgfmath@expression}}%
	\else
		\xdef\pgfmathparsex@@{\noexpand\def\noexpand\pgfmathresult{\pgfmathresult}}%
	\fi
	\endgroup
	\pgfmathparsex@@
}%
\def\pgfmathparsex@first#1#2\pgfmathparsex@EOI{%
	\def\pgfmath@token@next{#1}%
	\pgfmath@parse@number{#2}%
}%
\def\pgfmathparsex@recover{\def\pgfmathparsex@recover@{1}}%
\def\pgfmathparsex@recover@{0}%


% compares YYYY-MM-DD < YYYY-MM-DD
%
% \pgfplotsutilifdatelessthan 2010-01-01\cmp 2011-01-01\relax{<true>}{<false>}
\long\def\pgfplotsutilifdatelessthan#1-#2-#3\cmp #4-#5-#6\relax#7#8{%
	\long\def\pgfplotsutilifdatelessthan@true{#7}%
	\long\def\pgfplotsutilifdatelessthan@false{#8}%
	\ifnum#1<#4\relax
		\pgfplotsutilifdatelessthan@true
	\else
		\ifnum#1>#4\relax
			\pgfplotsutilifdatelessthan@false
		\else
			\ifnum#2<#5\relax
				\pgfplotsutilifdatelessthan@true
			\else
				\ifnum#2>#5\relax
					\pgfplotsutilifdatelessthan@false
				\else
					\ifnum#3<#6\relax
						\pgfplotsutilifdatelessthan@true
					\else
						\pgfplotsutilifdatelessthan@false
					\fi
				\fi
			\fi
		\fi
	\fi
}%
\long\def\pgfplotsutilifdategreaterthan#1-#2-#3\cmp #4-#5-#6\relax#7#8{%
	\long\def\pgfplotsutilifdatelessthan@true{#7}%
	\long\def\pgfplotsutilifdatelessthan@false{#8}%
	\ifnum#1>#4\relax
		\pgfplotsutilifdatelessthan@true
	\else
		\ifnum#1<#4\relax
			\pgfplotsutilifdatelessthan@false
		\else
			\ifnum#2>#5\relax
				\pgfplotsutilifdatelessthan@true
			\else
				\ifnum#2<#5\relax
					\pgfplotsutilifdatelessthan@false
				\else
					\ifnum#3>#6\relax
						\pgfplotsutilifdatelessthan@true
					\else
						\pgfplotsutilifdatelessthan@false
					\fi
				\fi
			\fi
		\fi
	\fi
}%
\pgfkeys{%
	% /pgfplots/iflessthan{#1}{#2}{#3}{#4}:
	% #1: arg1 (a macro containing the first argument)
	% #2: arg2 (a macro containing the second argument)
	% #3: code to invoke in case arg1<arg2
	% #4: code to invoke in case arg1>=arg2
	% the iflessthan method will be invoked within local scopes.
	/pgfplots/fixed </.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{\ifdim##1pt<##2pt\relax##3\else ##4\fi},%
	},
	/pgfplots/fixed <,
	/pgfplots/fixed >/.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{\ifdim##1pt>##2pt\relax##3\else ##4\fi},%
	},
	/pgfplots/int </.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{\ifnum##1<##2\relax##3\else ##4\fi},%
	},
	/pgfplots/int >/.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{\ifnum##1>##2\relax##3\else ##4\fi},%
	},
	/pgfplots/float </.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\pgfmathfloatparsenumber{##1}%
			\let\pgfplots@iflt@arga=\pgfmathresult
			%
			\pgfmathfloatparsenumber{##2}%
			\let\pgfplots@iflt@argb=\pgfmathresult
			%
			\pgfmathfloatlessthan@{\pgfplots@iflt@arga}{\pgfplots@iflt@argb}%
			\ifpgfmathfloatcomparison
				##3%
			\else
				##4%
			\fi
		}%
	},%
	/pgfplots/float >/.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\pgfmathfloatparsenumber{##1}%
			\let\pgfplots@iflt@arga=\pgfmathresult
			%
			\pgfmathfloatparsenumber{##2}%
			\let\pgfplots@iflt@argb=\pgfmathresult
			%
			\pgfmathfloatgreaterthan@{\pgfplots@iflt@arga}{\pgfplots@iflt@argb}%
			\ifpgfmathfloatcomparison
				##3%
			\else
				##4%
			\fi
		}%
	},%
	% 
	% compares YYYY-MM-DD < YYYY-MM-DD
	/pgfplots/date </.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\edef\pgfplotsarray@ltdate{##1\noexpand\cmp ##2}%
			\expandafter\pgfplotsutilifdatelessthan\pgfplotsarray@ltdate\relax{##3}{##4}%
		}%
	},%
	/pgfplots/date >/.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\edef\pgfplotsarray@ltdate{##1\noexpand\cmp ##2}%
			\expandafter\pgfplotsutilifdategreaterthan\pgfplotsarray@ltdate\relax{##3}{##4}%
		}%
	},%
	%
	/pgfplots/string </.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\t@pgfplots@toka=\expandafter{##1}%
			\t@pgfplots@tokb=\expandafter{##2}%
			\edef\pgfplots@loc@TMPa{{\the\t@pgfplots@toka}{\the\t@pgfplots@tokb}}%
			\expandafter\pgfplotsutilstrcmp\pgfplots@loc@TMPa
			\if1\pgfplotsretval ##3\else ##4\fi
		}%
	},%
	/pgfplots/string >/.style={%
		/pgfplots/iflessthan/.code args={##1##2##3##4}{%
			\t@pgfplots@toka=\expandafter{##1}%
			\t@pgfplots@tokb=\expandafter{##2}%
			\edef\pgfplots@loc@TMPa{{\the\t@pgfplots@toka}{\the\t@pgfplots@tokb}}%
			\expandafter\pgfplotsutilstrcmp\pgfplots@loc@TMPa
			\if2\pgfplotsretval ##3\else ##4\fi
		}%
	},%
}

% defines \pgfplotsretval to be
%
% \ifpgfpicture #1\else #2\fi
%
% and handles the '/pgfplots/invoke before crossref tikzpicture' key.
% 
% #1 first argument to \ifpgfpicture
% #2 second argument to \ifpgfpicture
\def\pgfplots@assemble@ref@picture#1#2{%
	\begingroup
	\pgfkeysgetvalue{/pgfplots/invoke before crossref tikzpicture}\pgfplots@loc@TMPa
	\pgfkeysgetvalue{/pgfplots/invoke after crossref tikzpicture}\pgfplots@loc@TMPb
	\toks0={#1}%
	\toks1=\expandafter{\pgfplots@loc@TMPa}%
	\toks2={#2}%
	\toks3=\expandafter{\pgfplots@loc@TMPb}%
	\xdef\pgfplots@glob@TMPa{%
		\noexpand\ifpgfpicture
			\the\toks0 %
		\noexpand\else
			\the\toks1
			\the\toks2
			\the\toks3
		\noexpand\fi
	}%
	\endgroup
	\let\pgfplotsretval=\pgfplots@glob@TMPa
}%

\def\pgfplots@auxwrite@latex#1{%
	\if@filesw
		\immediate\write\@auxout{#1}%
	\fi
}%
\pgfutil@IfUndefined{if@filesw}{%
	\def\pgfplots@auxwrite#1{}%
}{%
	\let\pgfplots@auxwrite=\pgfplots@auxwrite@latex
}%

% sorts three elements.
%
% #1,#2,#3 are three macros containing the input.
% On output, these macros are redefined to represent the sorted range.
% The values of these macros must be a <dimen>, followed by anything
% else (but no \relax please)
%
% Example:
% \def\A{10pt furthervalues}
% \def\B{4pt value of b}
% \def\C{1pt value of c}
% \pgfplotsutilsortthree\A\B\C
% ->
% \A={1pt value of c}
% \B={4pt value of b}
% \C={10pt furthervalues}
%
% It uses insertion sort.
\def\pgfplotsutilsortthree#1#2#3{%
	\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#1\relax
	\afterassignment\pgfplots@gobble@until@relax \pgf@xb=#2\relax
	\ifdim\pgf@xa>\pgf@xb
		\let\pgfplots@loc@TMPa=#2%
		\let#2=#1%
		\let#1=\pgfplots@loc@TMPa%
	\fi
	% first two are sorted.
	%
	\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#2\relax
	\afterassignment\pgfplots@gobble@until@relax \pgf@xb=#3\relax
	\let\pgfplots@loc@TMPa=#3%
	\ifdim\pgf@xa>\pgf@xb
		\let#3=#2%
		\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#1\relax
		\ifdim\pgf@xa>\pgf@xb
			\let#2=#1%
			\let#1=\pgfplots@loc@TMPa%
		\else
			\let#2=\pgfplots@loc@TMPa%
		\fi
	\fi
	% ok.
}%

% Like \pgfplotsutilsortthree, but for four elements (of the same
% form, see \pgfplotsutilsortthree).
%
% It uses insertion sort.
\def\pgfplotsutilsortfour#1#2#3#4{%
	\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#1\relax
	\afterassignment\pgfplots@gobble@until@relax \pgf@xb=#2\relax
	\ifdim\pgf@xa>\pgf@xb
		\let\pgfplots@loc@TMPa=#2%
		\let#2=#1%
		\let#1=\pgfplots@loc@TMPa%
	\fi
	% first two are sorted.
	%
	\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#2\relax
	\afterassignment\pgfplots@gobble@until@relax \pgf@xb=#3\relax
	\let\pgfplots@loc@TMPa=#3%
	\ifdim\pgf@xa>\pgf@xb
		\let#3=#2%
		\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#1\relax
		\ifdim\pgf@xa>\pgf@xb
			\let#2=#1%
			\let#1=\pgfplots@loc@TMPa%
		\else
			\let#2=\pgfplots@loc@TMPa%
		\fi
	\fi
	% ok, first three are sorted.
	%
	\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#3\relax
	\afterassignment\pgfplots@gobble@until@relax \pgf@xb=#4\relax
	\let\pgfplots@loc@TMPa=#4%
	\ifdim\pgf@xa>\pgf@xb
		\let#4=#3%
		\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#2\relax
		\ifdim\pgf@xa>\pgf@xb
			\let#3=#2%
			\afterassignment\pgfplots@gobble@until@relax \pgf@xa=#1\relax
			\ifdim\pgf@xa>\pgf@xb
				\let#2=#1%
				\let#1=\pgfplots@loc@TMPa%
			\else
				\let#2=\pgfplots@loc@TMPa%
			\fi
		\else
			\let#3=\pgfplots@loc@TMPa%
		\fi
	\fi
	% ok, first four are sorted.
}%

% A variant of \pgfutil@ifnextchar which allows the '#' character
% inside of either '#2' or '#3'.
%
% It is slightly more expensive and relies on \t@pgfplots@tokc
\long\def\pgfplotsutil@ifnextchar#1#2#3{%
  \let\pgfplotsutil@reserved@d=#1%
  \t@pgfplots@tokc={#2}%
  \edef\pgfplotsutil@reserved@a{\the\t@pgfplots@tokc}%
  \t@pgfplots@tokc={#3}%
  \def\pgfplotsutil@reserved@b{\the\t@pgfplots@tokc}%
  \futurelet\pgfplotsutil@let@token\pgfplotsutil@ifnch}
\def\pgfplotsutil@ifnch{%
  \ifx\pgfplotsutil@let@token\pgfplotsutil@sptoken
    \let\pgfplotsutil@reserved@c\pgfplotsutil@xifnch
  \else
    \ifx\pgfplotsutil@let@token\pgfplotsutil@reserved@d
      \let\pgfplotsutil@reserved@c\pgfplotsutil@reserved@a
    \else
      \let\pgfplotsutil@reserved@c\pgfplotsutil@reserved@b
    \fi
  \fi
  \pgfplotsutil@reserved@c}
{%
  \def\:{\global\let\pgfplotsutil@sptoken= } \:
  \def\:{\pgfplotsutil@xifnch} \expandafter\gdef\: {\futurelet\pgfplotsutil@let@token\pgfplotsutil@ifnch}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% basic 3d linear algebra utils.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Converts a 3d vectors to string.
% #1: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the result vector.
% #3: the argument for \pgfplotscoordmath (such as 'default')
% 
\def\pgfplotsmathvectortostring#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectorcompwiseop@unary\pgfplots@loc@TMPa\pgfplots@EOI{#2}{{tostring}}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Reads a 3d vector's coordinates.
% #1: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the result vector.
% #2: the argument for \pgfplotscoordmath (such as 'default')
% 
% Example:
% \pgfplotsmathvectorfromstring{1,4,5}{default}
% will parse each component
\def\pgfplotsmathvectorfromstring#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectorcompwiseop@unary\pgfplots@loc@TMPa\pgfplots@EOI{#2}{{parsenumber}}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Takes two 3d vectors and subtracts them.
% #1: a vector which expands to the form 'x,y,z'
% #2: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the difference vector.
% #3: the argument for \pgfplotscoordmath (such as 'default')
% 
\def\pgfplotsmathvectorsubtract#1#2#3{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorcompwiseop@binary\pgfplots@loc@TMPa\pgfplots@EOI{#3}{subtract}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsmathvectorcompwiseop@binary#1,#2,#3:#4,#5,#6\pgfplots@EOI#7#8{%
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\def\Bx{#4}%
	\def\By{#5}%
	\def\Bz{#6}%
	\pgfplotscoordmath{#7}{op}{#8}{{\Ax}{\Bx}}%
	\let\pgfplots@loc@TMPa=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{#8}{{\Ay}{\By}}%
	\let\pgfplots@loc@TMPb=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{#8}{{\Az}{\Bz}}%
	\let\pgfplots@loc@TMPc=\pgfmathresult
	\edef\pgfplotsretval{\pgfplots@loc@TMPa,\pgfplots@loc@TMPb,\pgfplots@loc@TMPc}%
}%

% #5: an operation ENCLOSED IN BRACES!
% like '{tostring}' or '{op}{<someunaryop>}'
\def\pgfplotsmathvectorcompwiseop@unary#1,#2,#3\pgfplots@EOI#4#5{%
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\pgfplotscoordmath{#4}#5{\Ax}%
	\let\pgfplots@loc@TMPa=\pgfmathresult
	\pgfplotscoordmath{#4}#5{\Ay}%
	\let\pgfplots@loc@TMPb=\pgfmathresult
	\pgfplotscoordmath{#4}#5{\Az}%
	\let\pgfplots@loc@TMPc=\pgfmathresult
	\edef\pgfplotsretval{\pgfplots@loc@TMPa,\pgfplots@loc@TMPb,\pgfplots@loc@TMPc}%
}%

% Takes two 3d vectors and adds them.
% #1: a vector which expands to the form 'x,y,z'
% #2: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the added vector.
% #3: the argument for \pgfplotscoordmath (such as 'default')
% 
\def\pgfplotsmathvectoradd#1#2#3{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorcompwiseop@binary\pgfplots@loc@TMPa\pgfplots@EOI{#3}{add}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Takes two 3d vectors and applies a component-wise operation.
% #1: a vector which expands to the form 'x,y,z'
% #2: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain a boolean vector with three
% entries.
% #3: a scalar operation which takes two arguments (such as "add", "subtract", "equal")
% #4 the argument for \pgfplotscoordmath (such as 'default')
% 
% Example: \pgfplotsmathvectorcompwise{1,1,1}{0,1,0}{add}{pgfbasic}
% -> \pgfplotsretval={1.0,2.0,1.0}
\def\pgfplotsmathvectorcompwise#1#2#3#4{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorcompwiseop@binary\pgfplots@loc@TMPa\pgfplots@EOI{#4}{#3}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Defines \pgfplotsretval to x+y+z
% #1: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the result scalar.
% #2: the argument for \pgfplotscoordmath (such as 'default')
% 
% Example:
% \pgfplotsmathvectorsum{1,4,5}{default}
% \pgfplotsretval = 10 (in that format)
\def\pgfplotsmathvectorsum#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectorsum@\pgfplots@loc@TMPa\pgfplots@EOI{#2}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

\def\pgfplotsmathvectorsum@#1,#2,#3\pgfplots@EOI#4{%
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\pgfplotscoordmath{#4}{op}{add}{{\Ax}{\Ay}}%
	\let\pgfplotsretval=\pgfmathresult
	\pgfplotscoordmath{#4}{op}{add}{{\pgfplotsretval}{\Az}}%
	\let\pgfplotsretval=\pgfmathresult
}%

% Computes the euclidean scalar product of two 3d vectors #1 and #2 in math
% format #3.
\def\pgfplotsmathvectorscalarproduct#1#2#3{%
	\pgfplotsmathvectorscaleindividually{#1}{#2}{#3}%
	\pgfplotsmathvectorsum{\pgfplotsretval}{#3}%
}%

% takes a vector #1 and defines three new macros 
% \csname #2x\endcsname, 
% \csname #2y\endcsname, 
% \csname #2z\endcsname, 
% with the components.
% #3: the math format
\def\pgfplotsmathvectortocomponents#1#2#3{%
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectortocomponents@\pgfplots@loc@TMPa\pgfplots@EOI{#3}{#2}%
}%
\def\pgfplotsmathvectortocomponents@#1,#2,#3\pgfplots@EOI#4#5{%
	\expandafter\def\csname #5x\endcsname{#1}%
	\expandafter\def\csname #5y\endcsname{#2}%
	\expandafter\def\csname #5z\endcsname{#3}%
}%


% Takes two 3d vectors and computes their cross product.
% #1: a vector which expands to the form 'x,y,z'
% #2: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the result vector.
% #3: the argument for \pgfplotscoordmath (such as 'default')
%
\def\pgfplotsmathvectorcrossprod#1#2#3{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorcrossprod@\pgfplots@loc@TMPa\pgfplots@EOI{#3}%
%\message{\string\pgfplotsmathvectorcrossprod{#1}{#2} = \pgfplotsretval^^J}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsmathvectorcrossprod@#1,#2,#3:#4,#5,#6\pgfplots@EOI#7{%
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\def\Bx{#4}%
	\def\By{#5}%
	\def\Bz{#6}%
	%
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ay}{\Bz}}\let\AyBz=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Az}{\By}}\let\AzBy=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Az}{\Bx}}\let\AzBx=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ax}{\Bz}}\let\AxBz=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ax}{\By}}\let\AxBy=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ay}{\Bx}}\let\AyBx=\pgfmathresult
	%
	\pgfplotscoordmath{#7}{op}{subtract}{{\AyBz}{\AzBy}}\let\pgfplots@loc@TMPa=\pgfmathresult%
	\pgfplotscoordmath{#7}{op}{subtract}{{\AzBx}{\AxBz}}\let\pgfplots@loc@TMPb=\pgfmathresult%
	\pgfplotscoordmath{#7}{op}{subtract}{{\AxBy}{\AyBx}}\let\pgfplots@loc@TMPc=\pgfmathresult%
	\edef\pgfplotsretval{\pgfplots@loc@TMPa,\pgfplots@loc@TMPb,\pgfplots@loc@TMPc}%
}%

% Takes two 3d vectors and defines a new vector (Ax*Bx,Ay*By,Az*Bz)
% #1: a vector which expands to the form 'x,y,z'
% #2: a vector which expands to the form 'x,y,z'
% on output, \pgfplotsretval will contain the result vector.
% #3: the argument for \pgfplotscoordmath (such as 'default')
%
\def\pgfplotsmathvectorscaleindividually#1#2#3{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorscaleindividually@\pgfplots@loc@TMPa\pgfplots@EOI{#3}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsmathvectorscaleindividually@#1,#2,#3:#4,#5,#6\pgfplots@EOI#7{%
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\def\Bx{#4}%
	\def\By{#5}%
	\def\Bz{#6}%
	%
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ax}{\Bx}}\let\Cx=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Ay}{\By}}\let\Cy=\pgfmathresult
	\pgfplotscoordmath{#7}{op}{multiply}{{\Az}{\Bz}}\let\Cz=\pgfmathresult
	%
	\edef\pgfplotsretval{\Cx,\Cy,\Cz}%
}%

% #1 a vector as x,y,z
% #2 the math format of its components
\def\pgfplotspointfromcsvvector#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotspointfromcsvvector@\pgfplots@loc@TMPa\pgfplots@EOI{#2}
	\endgroup
}%
\def\pgfplotspointfromcsvvector@#1,#2,#3\pgfplots@EOI#4{%
	\pgfplotscoordmath{#4}{tofixed}{#1}\let\Ax=\pgfmathresult
	\pgfplotscoordmath{#4}{tofixed}{#2}\let\Ay=\pgfmathresult
	\pgfplotscoordmath{#4}{tofixed}{#3}\let\Az=\pgfmathresult
	\pgfplotsqpointxyz\Ax\Ay\Az
	\pgf@process{}%
}%

% #1 a vector as (x,y,z)
% #2 a coordmath value
% defines \pgfplotsretval to contain the vector length
\def\pgfplotsmathvectorlength#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectorlength@\pgfplots@loc@TMPa\pgfplots@EOI{#2}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsmathvectorlength@#1,#2,#3\pgfplots@EOI#4{%
	\begingroup
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	%
	\pgfplotscoordmath{#4}{op}{multiply}{{\Ax}{\Ax}}\let\Axx=\pgfmathresult
	\pgfplotscoordmath{#4}{op}{multiply}{{\Ay}{\Ay}}\let\Ayy=\pgfmathresult
	\pgfplotscoordmath{#4}{op}{multiply}{{\Az}{\Az}}\let\Azz=\pgfmathresult
	%
	\pgfplotscoordmath{#4}{op}{add}{{\Axx}{\Ayy}}%
	\pgfplotscoordmath{#4}{op}{add}{{\Azz}{\pgfmathresult}}%
	\pgfplotscoordmath{#4}{op}{sqrt}{{\pgfmathresult}}%
	\let\pgfplotsretval=\pgfmathresult
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% #1 a vector as (x,y,z)
% #2 a scale
% #3 a coordmath value
% defines \pgfplotsretval to contain the scaled vector
\def\pgfplotsmathvectorscale#1#2#3{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1:#2}%
	\expandafter\pgfplotsmathvectorscale@\pgfplots@loc@TMPa\pgfplots@EOI{#3}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%
\def\pgfplotsmathvectorscale@#1,#2,#3:#4\pgfplots@EOI#5{%
	\begingroup
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\pgfplotscoordmath{#5}{parsenumber}{#4}\let\S\pgfmathresult
	%
	\pgfplotscoordmath{#5}{op}{multiply}{{\Ax}{\S}}\let\Ax=\pgfmathresult
	\pgfplotscoordmath{#5}{op}{multiply}{{\Ay}{\S}}\let\Ay=\pgfmathresult
	\pgfplotscoordmath{#5}{op}{multiply}{{\Az}{\S}}\let\Az=\pgfmathresult
	%
	\edef\pgfplotsretval{\Ax,\Ay,\Az}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%

% Takes an input vector #1 = x,y,z and applies the inverse data scale
% trafo.
%
% The result is \pgfplotsretval; it will be in '#2' coordmath for
% every component.
\def\pgfplotsmathvectordatascaletrafoinverse#1#2{%
	\begingroup
	\edef\pgfplots@loc@TMPa{#1}%
	\expandafter\pgfplotsmathvectordatascaletrafoinverse@\pgfplots@loc@TMPa\pgfplots@EOI{#2}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}
\def\pgfplotsmathvectordatascaletrafoinverse@#1,#2,#3\pgfplots@EOI#4{%
	\begingroup
	\def\Ax{#1}%
	\def\Ay{#2}%
	\def\Az{#3}%
	\ifpgfplots@apply@datatrafo@x
		\pgfplotscoordmath{x}{datascaletrafo inverse}{\Ax}\let\Ax=\pgfmathresult
	\fi
	\ifpgfplots@apply@datatrafo@y
		\pgfplotscoordmath{y}{datascaletrafo inverse}{\Ay}\let\Ay=\pgfmathresult
	\fi
	\ifpgfplots@curplot@threedim
		\ifpgfplots@apply@datatrafo@z
			\pgfplotscoordmath{z}{datascaletrafo inverse}{\Az}\let\Az=\pgfmathresult
		\fi
	\fi
	\pgfplotsmathvectorfromstring{\Ax,\Ay,\Az}{#4}%
	\pgfmath@smuggleone\pgfplotsretval
	\endgroup
}%


{
\catcode`\"=12
\gdef\pgfplotsDQ{"}%
\catcode`\|=12
\gdef\pgfplotsVERTBAR{|}%
\catcode`\#=12
\gdef\pgfplotsHASH{#}%
\catcode`\%=12 \gdef\pgfplotsPERCENT{%}}

\pgfutil@IfUndefined{directlua}{}{%
	\directlua{require(\pgfplotsDQ pgfplots.lua\pgfplotsDQ)}%
}%

% swaps the content of two macros
\def\pgfplotsutil@swap#1#2{%
	\let\pgfplotsutil@swap@=#1%
	\let#1=#2%
	\let#2=\pgfplotsutil@swap@
}%

\input pgfplotsliststructure.code.tex
\input pgfplotsliststructureext.code.tex
\input pgfplotsarray.code.tex
\input pgfplotsmatrix.code.tex
\input pgfplotstableshared.code.tex
\input pgfplotsdeque.code.tex
\input pgfplotsbinary.code.tex
\input pgfplotsutil.verb.code.tex

\usepgfplotslibrary{surfshading}

\endinput