% Copyright 2013 by Mark Wibrow
%
% 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{Math Library}
\label{section-library-math}

\begin{tikzlibrary}{math}
  This library defines a simple mathematical language
  to define simple functions and perform sequences of basic mathematical 
  operations.
\end{tikzlibrary}


\subsection{Overview}

  \pgfname\ and \tikzname\ both use the \pgfname\ mathematical engine
  which provides many commands for parsing expressions. Unfortunately
  the \pgfname\ math engine is somewhat cumbersome for long sequences 
  of mathematical 
  operations, particularly when assigning values to multiple variables.
  The \tikzname\ |calc| library
  provides some additional ``convenience'' operations for doing calculations
  (particularly with coordinates), but this can only be used inside
  \tikzname\ path commands.
  
  This |math| library provides a means to
  perform sequences of mathematical operations in a more `user friendly'
  manner than the \pgfname\ math engine. In addition, the coordinate calculations of the |calc| library 
  can be accessed (provided it is loaded).
  %
  However as the |math| library uses the \pgfname\ math engine -- which uses pure \TeX\ 
  to perform all its calculations -- it is subject to the same speed
  and accuracy limitations. It is worth bearing this in mind, before 
  trying to implement algorithms requiring intensive and highly
  accurate computation. You can, of course use the |fp| or the |fpu|
  libraries to increase the accuracy (but not necessarily the speed)
  of computations.
  
  For most purposes, the features provided by this library are accessed
  using the following command:
  
\begin{command}{\tikzmath\texttt{\{}\meta{statements}\texttt{\}}}

  This command process  a series of \meta{statements} which can 
  represent assignments, function definitions, conditional evaluation,
  and iterations. It provides, in effect,  a miniature mathematical
  language to perform basic mathematical operations.
  Perhaps the most important thing to remember is that \emph{every 
  statement should end with a semi-colon}. This is
  likely to be the most common reason why the |\tikzmath| command fails.

\begin{codeexample}[]
\tikzmath{
  % Adapted from http://www.cs.northwestern.edu/academics/courses/110/html/fib_rec.html
  function fibonacci(\n) {
    if \n == 0 then {
      return 0;
    } else {
       return fibonacci2(\n, 0, 1);
     };
  };
  function fibonacci2(\n, \p, \q) {
    if \n == 1 then {
      return \q;
    } else {
      return fibonacci2(\n-1, \q, \p+\q);
    };
  };
  int \f, \i;
  for \i in {0,1,...,20}{
    \f = fibonacci(\i);
    print {\f, };
  };
}
\end{codeexample}

\end{command}

In addition to this command the following key is provided:
  
\begin{key}{/tikz/evaluate={\meta{statements}}}
  This key simply executes |\tikzmath{|\meta{statements}|}|.
  
\begin{codeexample}[]
\tikz[x=0.25cm,y=0.25cm,
  evaluate={
    int \i, \j;
    for \i in {0,...,10}{
      for \j in {0,...,10}{
        \a{\i,\j} = (\i+\j)*5;
      };
    };
  }
]
\foreach \i in {0,...,10}
  \foreach \j in {0,...,10}
    \fill [red!\a{\i,\j}!yellow]  (\i,\j) rectangle ++(1, 1);

\end{codeexample}


\end{key}


  The following sections describe the miniature language that this
  library provides and   can be used in the |\tikzmath| command and the |evaluate| key.
  The language consists only of simple keywords and expressions but
  the mini-parser allows you to format code in a reasonably versatile
  way (much like the |tikz| parser) except that 
  \emph{all the keywords must be followed by at least one space}.
  This is the second most important thing to remember (after 
  remembering to insert semi-colons at the end of every statement).


\subsection{Assignment}
  
  In the simplest case, you will want to evaluate an expression
  and assign it to a macro, or a \TeX\ count or dimension register.
  In this case, use of the |math| library is straightforward:

\begin{codeexample}[]
\newcount\mycount
\newdimen\mydimen
\tikzmath{
  \a = 4*5+6;
  \b = sin(30)*4;
  \mycount = log10(2048) / log10(2);
  \mydimen = 15^2;
}
\a, \b, \the\mycount, \the\mydimen
\end{codeexample}


  In addition, \TeX-macros (\emph{not} \TeX\ registers)
  can be suffixed with an index, similar to
  indices in mathematical notation, for example, $x_1$, $x_2$, $x_3$:
  
\begin{codeexample}[]
\tikzmath{
  \x1 = 3+4; \x2 = 30+40; \x3 = 300+400;
}
\x1, \x2, \x3
\end{codeexample}

  The index does not have to be a number. By using braces |{}|,
  more sophisticated indices can be created:
  
\begin{codeexample}[]
\tikzmath{
  \c{air} = 340; \c{water} = 1435; \c{steel} = 6100;
}
\foreach \medium in {air,steel}{The speed of sound in \medium\ is \c{\medium} m/s. }
\end{codeexample}

  You should not, however, try to mix indexed and non-indexed variables. Once
  an assignment is made using an index, the |math| library expects
  all instances of the variable on the right hand side of an assignment 
  to be followed by an index. This effect is reversed if you
  subsequently make an assignment to the variable without an index: the |math| library
  (or to be precise the \pgfname\ math-engine)
  will then ignore any index following the variable on the right hand side
  of an assignment.

  In some cases, you may wish to assign a value or expression to a variable
  without evaluating it with the \pgfname\ math-engine. In this case, you
  can use the following keyword:

\begin{math-keyword}{{let} \meta{variable} \texttt{=} \meta{expression}\texttt{;}}
  
  This keyword assigns \meta{expression} to \meta{variable}
  without evaluation. The \meta{expression} is however fully
  expanded using |\edef|. Any spaces preceding \meta{expression}
  are removed, but any trailing spaces (before the semi-colon)
  are included.

\begin{codeexample}[]
\tikzmath{
  let \x = (5*4)+1;
  let \c1 = blue; 
}
\x, ``\c1''
\end{codeexample}
\end{math-keyword}
  
\subsection{Integers, ``Real'' Numbers, and Coordinates}
  
  By default, assignments are made by evaluating expressions
  using the \pgfname\ math-engine and 
  results  are usually returned as number with a decimal point (unless you
  are assigning to a count register or use the |int| function).
  %
  As this is not always desirable, the |math| library allows
  variables -- which \emph{must} be \TeX\ macros -- to be `declared'
  as being a particular `type'. The library recognizes three types:
  integers (numbers without a decimal point),
  real numbers (numbers with a decimal point\footnote{Strictly speaking, due to the 
  finite range and precision of \TeX\ numerical capabilities, the term ``real'' is
  not correct.}), and coordinates.
  
  To declare a variable as being one of the three types,
  you  can use the keywords shown below. It is important to remember
  that by telling the |math| library you want it to do a
  particular assignment for a variable, it will also do the same 
  assignment when the variable is indexed.
  
\begin{codeexample}[]
\tikzmath{
  integer \x;
  \x1 = 3+4; \x2 = 30+40; \x3 = 300+400;
}
\x1, \x2, \x3
\end{codeexample}


%  But, if you want integer results without using a count register or the
%  |int| function, you can use a keyword to indicate this:

\begin{math-keyword}{{integer} \meta{variable}\opt{\texttt{,} 
\meta{additional variables}}\texttt{;}}

  The |integer| keyword indicates that assignments to the \meta{variable} or 
  the comma separated list of \meta{additional variables} should be 
  truncated (not rounded) to integers. The variables should be ordinary 
  macros -- \emph{not} \TeX\ registers. In addition the variables
  should \emph{not} be indexed. 

\begin{codeexample}[]
\tikzmath{ 
   integer \x, \y, \z; 
   \x = 4*5+6; 
   \y = sin(30)*4;
   \z = log10(512) / log10(2);
   print {$x=\x$, $y=\y$, $z=\z$}; 
}
\end{codeexample}

\end{math-keyword}

\begin{math-keyword}{{int} \meta{variable}\opt{\texttt{,} 
\meta{additional variables}}\texttt{;}}
  Short version of the |integer| keyword.
\end{math-keyword}

  Having declared a variable as an integer, the |math| library will 
  continue to assign only integers to that variable within the 
  current \TeX\ scope. If you wish to assign non-integer (i.e., \emph{real})
  numbers to the same variable, the following keyword can be used.

\begin{math-keyword}{{real} \meta{variable}\opt{\texttt{,} 
\meta{additional variables}}\texttt{;}}
  The |real| keyword ensures that assignments \meta{variable}
  (and \meta{additional variables}) will not be truncated to integers.
\end{math-keyword}
  
  In order to take advantage of |math| library interface to the
  |calc| library  you must indicate that a variable is to be assigned 
  coordinates, using the following keyword.
  
\begin{math-keyword}{{coordinate} 
\meta{variable}\opt{\texttt{,} 
\meta{additional variables}}\texttt{;}}%

  This keyword enables \tikzname-style coordinates such as |(2cm,3pt)| or
  |(my node.east)| to be parsed and
  assigned to \meta{variable} in the form $x,y$, which can then be used
  in a |tikzpicture|:

\begin{codeexample}[]
\tikzmath{
   coordinate \c;
   \c = (45:10pt);
}
\tikz\draw (0,0) -- (\c);
\end{codeexample}
  
  If the \tikzname\ |calc| library is loaded,
  coordinate calculations can be performed; 
  the coordinate expression does not have to be
  surrounded by |($|\ldots|$)|.
  
\begin{codeexample}[]
\tikzmath{
   coordinate \c, \d;
   \c = (-1,2)+(1,-1);
   \d = (4,1)-(2,-1);
}
\tikz\draw (\c) -- (\d);
\end{codeexample}
  
  In addition to assigning the $x$ and $y$ coordinates to \meta{variable}
  (possibly with an optional index), two further
  variables are defined. The first takes the name of \meta{variable}
  (e.g., |\c|) suffixed with |x| (i.e., |\cx|) and is assigned the
  $x$ coordinate of |\c|.  
   The second takes the name of \meta{variable}
    suffixed with |y| (i.e., |\cy|) and is assigned the
    $y$ coordinate of |\c|.  

\begin{codeexample}[]
\tikzmath{
   coordinate \c;
   \c1 = (30:20pt);
   \c2 = (210:20pt);
}
\tikz\draw (\cx1,\cy1) -- (\cx2,\cy1) -- (\cx2,\cy2) -- (\cx1,\cy2);
\end{codeexample}
  
\end{math-keyword}

%\begin{math-keyword}{{point} 
%\meta{variable}\opt{\texttt{,} 
%\meta{additional variables}}\texttt{;}}%
%
%  The |point| keyword is a synonym for the |coordinate| keyword
%  and performs the same function.
%\end{math-keyword}

  
\subsection{Repeating Things}

\begin{math-keyword}{{for} \meta{variable} \texttt{in 
\{}\meta{list}\texttt{\}\{}\meta{expressions}\texttt{\};}}
 
   This is a ``trimmed down'' version of the |\foreach| command available
   as part of \pgfname\ and \tikzname, but cannot currently be used
   outside of the |\tikzmath| command.
   It is important to note the following:
   
\begin{itemize}

\item
  Every value in \meta{list} is evaluated using the \pgfname\ mathematical 
  engine. However, if an item in \meta{list} contains a comma, it \emph{must} be 
  surrounded
  by braces, for example, |{mod(5, 2)}|.
  
\begin{codeexample}[]
\tikzmath{
  int \x, \v;
  \v=1;
  for \x in {1,...,{random(3,10)}}{
     \v=\v*2;
  };
  print {$x=\x, v=\v$};
} 
\end{codeexample}

\item
  Because each item is evaluated, you cannot 
  use \tikzname\ coordinates in \meta{list}.

\item
  Only single variable assignment is supported.
  
\item
  The ``dots notation'' (e.\,g., |1,2,...,9|) can be used in \meta{list},
  but is not as sophisticated as the \pgfname\ |\foreach| command.
  In particular, contextual replacement is not possible.  

\item
  Assignments that occur in the loop body \emph{are not scoped}. 
  They last beyond the body of each iteration and the end of the |for| 
  statement. This includes the values assigned to the \meta{variable}.
  
\begin{codeexample}[]
\tikzmath{
  int \x, \y;
  \y = 0;
  for \x1 in {1,...,5}{
    for \x2 in {10,20,...,50}{
      \y = \y+\x1*\x2; 
    };
  };
}
$x_1=\x1, x_2=\x2, y=\y$
\end{codeexample}





\end{itemize}

\end{math-keyword}




  
  


\subsection{Branching Statements}

  Sometimes you may wish to execute different statements
  depending on the value of an expression. In this
  case the following keyword can be used:

\begin{math-keyword}{{if} \meta{condition} \texttt{then \{}\meta{if-non-zero-statements}\texttt{\};}}

  This keyword executes \meta{if-non-zero-statements} if the expression in \meta{condition}
  evaluates to any value other than zero.

\end{math-keyword}

\begin{math-keyword}{{if} \meta{condition} \texttt{then \{}\meta{if-non-zero-statements}\texttt{\}} \texttt{else} \texttt{\{}\meta{if-zero-statements}\texttt{\}}\texttt{;}}

  This keyword executes \meta{if-non-zero-statements} if the expression in \meta{condition}
  evaluates to any value other than zero and the \meta{if-zero-statements} are executed if
  the expression in \meta{condition} evaluates to zero.

\begin{codeexample}[]
  \begin{tikzpicture}
  \tikzmath{
    int \x;
    for \k in {0,10,...,350}{
      if \k>260 then { let \c = orange; } else { 
        if \k>170 then { let \c = blue; } else {
          if \k>80 then { let \c = red; } else {
            let \c = green; }; }; };
      {
        \path [fill=\c!50, draw=\c] (\k:0.5cm) -- (\k:1cm) -- 
          (\k+5:1cm) -- (\k+5:0.5cm) -- cycle;
      };
    };
  }
  \end{tikzpicture}
\end{codeexample}
\end{math-keyword}

\subsection{Declaring Functions}

  You can add functions by using the following keywords:

\begin{math-keyword}{{function} \meta{name}\texttt{(}\meta{arguments}\texttt{) \{} \meta{definition} \texttt{\};}}

  This keyword works much like the |declare function| provided by the
  \pgfname\ math-engine.
  The function \meta{name} can be any name that is not already a
  function name in the current scope. The list of \meta{arguments} are 
  comma separated \TeX\ macros such as |\x|, or |\y| (it is not
  possible to declare functions that take variable numbers of arguments). 
  If the function
  takes no arguments then the parentheses need not be used.
  It is very important to note that the arrays that the
  \pgfname\ math engine supports 
  \emph{cannot currently be passed as arguments to functions}.
  
  
  The function \meta{definition} should be a sequence of statements
  that can be parsed by the |\tikzmath| command and 
  should use the commands specified in the \meta{arguments}.  
  The |return| keyword (described below) should be used to indicate
  the value returned by the function.
  %
  Although \meta{definition} can take any statements accepted by
  |\tikzmath|, it is not advisable try to define functions inside other 
  functions.
  
\begin{codeexample}[]
\tikzmath{
  function product(\x,\y) {
    return \x*\y;
  };
  int \i, \i, \k;
  \i = random(1,10);
  \j = random(20, 40);
  \k = product(\i, \j);
  print { $\i\times \j = \k$ };
}
\end{codeexample}
  
\end{math-keyword}

\begin{math-keyword}{{return} \meta{expression}\texttt{;}}
  
  This keyword should be used as the last executed statement 
  in a function definition to indicate the value that should be 
  returned.
  
\end{math-keyword}


\subsection{Executing Code Outside the Parser}
  
  Sometimes you may wish to do ``something'' outside the parser,
  perhaps display some intermediate result or execute some code.
  In this case
  you have two options. Firstly, the following keyword can be used:
    
\begin{math-keyword}{{print} \texttt{\{}\meta{code}\texttt{\};}}
  
  Execute \meta{code} immediately. This is intended as convenience 
  keyword for displaying information in a document (analogous to
  the |print| command in real programming languages).
  The \meta{code} is executed inside a \TeX\ group.
  
\begin{codeexample}[]
\tikzmath{
  int \x, \y, \z;
  \x = random(2, 5);
  for \y in {0,...,6}{
    \z = \x^\y;
    print {$\x^\y=\z$, };
  };
}
\end{codeexample}

\end{math-keyword}
  
  Secondly, if a statement begins with  a brace |{|, then 
  everything up to the closing brace |}| is collected
  and executed (the closing brace \emph{must} 
  be followed by a semi-colon).
  Like the |print| keyword, the contents of the braces 
  is executed inside a \TeX\ group.
  Unlike the |print| keyword, the brace notation can be used in
  functions so that |tikz| path commands can be safely executed 
  inside a |tikzpicture|.
  
\begin{codeexample}[]
\begin{tikzpicture}
\draw [help lines] grid (3,2);
\tikzmath{
  coordinate \c;
  for \x in {0,10,...,360}{
    \c = (1.5cm, 1cm) + (\x:1cm and 0.5cm);
    { \fill (\c) circle [radius=1pt]; };
  };
}
\end{tikzpicture}
\end{codeexample}






\endinput
