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% Copyright 2019 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{Mathematical Expressions}
\label{pgfmath-syntax}
The easiest way of using \pgfname's mathematical engine is to provide a
mathematical expression given in familiar infix notation, for example,
|1cm+4*2cm/5.5| or |2*3+3*sin(30)|. This expression can be parsed by the
mathematical engine and the result can be placed in a dimension register, a
counter, or a macro.
It should be noted that all calculations must not exceed $\pm16383.99999$ at
\emph{any} point, because the underlying computations rely on \TeX{}
dimensions. This means that many of the underlying computations are necessarily
approximate and, in addition, not very fast. \TeX{} is, after all, a
typesetting language and not ideally suited to relatively advanced mathematical
operations. However, it is possible to change the computations as described in
Section~\ref{pgfmath-reimplement}.
In the present section, the high-level macros for parsing an expression are
explained first, then the syntax for expression is explained.
\subsection{Parsing Expressions}
\label{pgfmath-registers}
\label{pgfmath-parsing}
\subsubsection{Commands}
The \todosp{why 2 labels?}basic command for invoking the parser of \pgfname's
mathematical engine is the following:
\begin{command}{\pgfmathparse\marg{expression}}
This macro parses \meta{expression} and returns the result without units in
the macro |\pgfmathresult|.
\example |\pgfmathparse{2pt+3.5pt}| will set |\pgfmathresult| to the text
|5.5|.
In the following, the special properties of this command are explained. The
exact syntax of mathematical expressions is explained in Sections
\ref{pgfmath-operators} and~\ref{pgfmath-functions}.
%
\begin{itemize}
\item The result stored in the macro |\pgfmathresult| is a decimal
\emph{without units}. This is true regardless of whether the
\meta{expression} contains any unit specification. All numbers with
units are converted to points first. See
Section~\ref{pgfmath-units} for details on units.
\item The parser will recognize \TeX{} registers and box dimensions, so
|\mydimen|, |0.5\mydimen|, |\wd\mybox|, |0.5\dp\mybox|,
|\mycount\mydimen| and so on can be parsed.
\item The $\varepsilon$-TeX\ extensions |\dimexpr|, |\numexpr|,
|\glueexpr|, and |\muexpr| are recognized and evaluated. The values
they result in will be used in the further evaluation, as if you
had put |\the| before them.
\item Parenthesis can be used to change the order of the evaluation.
\item Various functions are recognized, so it is possible to parse
|sin(.5*pi r)*60|, which means ``the sine of $0.5$ times $\pi$
radians, multiplied by 60''. The argument of functions can be any
expression.
\item Scientific notation in the form |1.234e+4| is recognized (but the
restriction on the range of values still applies). The exponent
symbol can be upper or lower case (i.e., |E| or |e|).
\item An integer with a zero-prefix (excluding, of course zero itself),
is interpreted as an octal number and is automatically converted to
base 10.
\item An integer with prefix |0x| or |0X| is interpreted as a
hexadecimal number and is automatically converted to base 10.
Alphabetic digits can be in uppercase or lowercase.
\item An integer with prefix |0b| or |0B| is interpreted as a binary
number and is automatically converted to base 10.
\item An expression (or part of an expression) surrounded with double
quotes (i.e., the character |"|) will not be evaluated. Obviously
this should be used with great care.
\end{itemize}
\end{command}
\begin{command}{\pgfmathqparse\marg{expression}}
This macro is similar to |\pgfmathparse|: it parses \meta{expression} and
returns the result in the macro |\pgfmathresult|. It differs in two
respects. Firstly, |\pgfmathqparse| does not parse functions, scientific
notation, the prefixes for binary octal, or hexadecimal numbers, nor does
it accept the special use of |"|, |?| or |:| characters. Secondly, numbers
in \meta{expression} \emph{must} specify a \TeX{} unit (except in such
instances as |0.5\pgf@x|), which greatly simplifies the problem of parsing
real numbers. As a result of these restrictions |\pgfmathqparse| is about
twice as fast as |\pgfmathparse|. Note that the result will still be a
number without units.
\end{command}
\begin{command}{\pgfmathpostparse}
At the end of the parse this command is executed, allowing some custom
action to be performed on the result of the parse. When this command is
executed, the macro |\pgfmathresult| will hold the result of the parse (as
always, without units). The result of the custom action should be used to
redefine |\pgfmathresult| appropriately. By default, this command is
equivalent to |\relax|. This differs from previous versions, where, if the
parsed expression contained no units, the result of the parse was scaled
according to the value in |\pgfmathresultunitscale| (which by default was
|1|).
This scaling can be turned on again using:
|\let\pgfmathpostparse=\pgfmathscaleresult|. Note, however that by scaling
the result, the base conversion functions will not work, and the |"|
character should not be used to quote parts of an expression.
\end{command}
Instead of the |\pgfmathparse| macro you can also use wrapper commands, whose
usage is very similar to their cousins in the \calcname{} package. The only
difference is that the expressions can be any expression that is handled by
|\pgfmathparse|. For all of the following commands, if \meta{expression} starts
with |+|, no parsing is done and a simple assignment or increment is done using
normal \TeX\ assignments or increments. This will be orders of magnitude faster
than calling the parser.
The effect of the following commands is always local to the current \TeX\
scope.
\begin{command}{\pgfmathsetlength\marg{register}\marg{expression}}
Basically, this command sets the length of the \TeX{} \meta{register} to
the value specified by \meta{expression}. However, there is some fine
print:
First, in case \meta{expression} starts with a |+|, a simple \TeX\
assignment is done. In particular, \meta{register} can be a glue register
and \meta{expression} be something like |+1pt plus 1fil| and the
\meta{register} will be assigned the expected value.
Second, when the \meta{expression} does not start with |+|, it is first
parsed using |\pgfmathparse|, resulting in a (dimensionless) value
|\pgfmathresult|. Now, if the parser encountered the unit |mu| somewhere in
the expression, it assumes that \meta{register} is a |\muskip| register and
will try to assign to \meta{register} the value |\pgfmathresult| followed
by |mu|. Otherwise, in case |mu| was not encountered, it is assumed that
\meta{register} is a dimension register or a glue register and we assign
|\pgfmathresult| followed by |pt| to it.
The net effect of the above is that you can write things like
%
\begin{codeexample}[]
\muskipdef\mymuskip=0
\pgfmathsetlength{\mymuskip}{1mu+3*4mu} \the\mymuskip
\end{codeexample}
%
\begin{codeexample}[]
\dimendef\mydimen=0
\pgfmathsetlength{\mydimen}{1pt+3*4pt} \the\mydimen
\end{codeexample}
%
\begin{codeexample}[]
\skipdef\myskip=0
\pgfmathsetlength{\myskip}{1pt+3*4pt} \the\myskip
\end{codeexample}
One thing that will \emph{not} work is
|\pgfmathsetlength{\myskip}{1pt plus 1fil}| since the parser does not
support fill's. You can, however, use the |+| notation in this case:
%
\begin{codeexample}[]
\skipdef\myskip=0
\pgfmathsetlength{\myskip}{+1pt plus 1fil} \the\myskip
\end{codeexample}
%
\end{command}
\begin{command}{\pgfmathaddtolength\marg{register}\marg{expression}}
Adds the value of \meta{expression} to the \TeX{} \meta{register}. All of
the special consideration mentioned for |\pgfmathsetlength| also apply here
in the same way.
\end{command}
\begin{command}{\pgfmathsetcount\marg{count register}\marg{expression}}
Sets the value of the \TeX{} \meta{count register}, to the \emph{truncated}
value specified by \meta{expression}.
\end{command}
\begin{command}{\pgfmathaddtocount\marg{count register}\marg{expression}}
Adds the \emph{truncated} value of \meta{expression} to the \TeX{}
\meta{count register}.
\end{command}
\begin{command}{\pgfmathsetcounter\marg{counter}\marg{expression}}
Sets the value of the \LaTeX{} \meta{counter} to the \emph{truncated} value
specified by \meta{expression}.
\end{command}
\begin{command}{\pgfmathaddtocounter\marg{counter}\marg{expression}}
Adds the \emph{truncated} value of \meta{expression} to \meta{counter}.
\end{command}
\begin{command}{\pgfmathsetmacro\marg{macro}\marg{expression}}
Defines \meta{macro} as the value of \meta{expression}. The result is a
decimal without units.
\end{command}
\begin{command}{\pgfmathsetlengthmacro\marg{macro}\marg{expression}}
Defines \meta{macro} as the value of \meta{expression} \LaTeX{} \emph{in
points}.
\end{command}
\begin{command}{\pgfmathtruncatemacro\marg{macro}\marg{expression}}
Defines \meta{macro} as the truncated value of \meta{expression}.
\end{command}
\subsubsection{Considerations Concerning Units}
\label{pgfmath-units}
As was explained earlier, the parser commands like |\pgfmathparse| will always
return a result without units in it and all dimensions that have a unit like
|10pt| or |1in| will first be converted to \TeX\ points (|pt|) and, then, the
unit is dropped.
Sometimes it is useful, nevertheless, to find out whether an expression or not.
For this, you can use the following commands:
{\let\ifpgfmathunitsdeclared\relax
\begin{command}{\ifpgfmathunitsdeclared}
After a call of |\pgfmathparse| this if will be true exactly if
some unit was encountered in the expression. It is always set
globally in each call.
Note that \emph{any} ``mentioning'' of a unit inside an
expression will set this \TeX-if to true. In particular, even an
expressionlike |2pt/1pt|, which arguably should be considered
``scalar'' or ``unit-free'' will still have this \TeX-if set to
true. However, see the |scalar| function for a way to change
this.
\end{command}
}
\begin{math-function}{scalar(\mvar{value})}
\mathcommand
This function is the identity function on its input, but it will reset the
\TeX-if |\ifpgfmathunitsdeclared|. Thus, it can be used to indicate that
the given \meta{value} should be considered as a ``scalar'' even when it
contains units; but note that it will work even when the \meta{value} is a
string or something else. The only effect of this function is to clear the
unit declaration.
%
\begin{codeexample}[]
\pgfmathparse{scalar(1pt/2pt)} \pgfmathresult\
\ifpgfmathunitsdeclared with \else without \fi unit
\end{codeexample}
Note, however, that this command (currently) really just clears the \TeX-if
as the input is scanned from left-to-right. Thus, even if there is a use of
a unit before the |scalar| function is used, the \TeX-if will be cleared:
%
\begin{codeexample}[]
\pgfmathparse{1pt+scalar(1pt)} \pgfmathresult\
\ifpgfmathunitsdeclared with \else without \fi unit
\end{codeexample}
The other way round, a use of a unit after the |scalar| function will set
the units once more.
%
\begin{codeexample}[]
\pgfmathparse{scalar(1pt)+1pt} \pgfmathresult\
\ifpgfmathunitsdeclared with \else without \fi unit
\end{codeexample}
For these reasons, you should use the function only on the outermost level
of an expression.
A typical use of this function is the following:
%
\begin{codeexample}[preamble={\usetikzlibrary{calc,quotes}}]
\tikz{
\coordinate["$A$"] (A) at (2,2);
\coordinate["$B$" below] (B) at (0,0);
\coordinate["$C$" below] (C) at (3,0);
\draw (A) -- (B) -- (C) -- cycle;
\path
let \p1 =($(A)-(B)$), \p2 =($(A)-(C)$),
\n1 = {veclen(\x1,\y1)}, \n2 = {veclen(\x2,\y2)}
in coordinate ["$D$" below] (D) at ($ (B)!scalar(\n1/(\n1+\n2))!(C) $);
\draw (A) -- (D);
}
\end{codeexample}
%
\end{math-function}
A special kind of units are \TeX's ``math units'' (|mu|). It will be treated as
if |pt| had been used, but you can check whether an expression contained a math
unit using the following:
%
{\let\ifpgfmathmathunitsdeclared\relax
\begin{command}{\ifpgfmathmathunitsdeclared}
This \TeX-if is similar to |\ifpgfmathunitsdeclared|, but it
is only set when the unit |mu| is encountered at least
once. In this case, |\ifpgfmathunitsdeclared| will \emph{also}
be set to true. The |scalar| function has no effect on this \TeX-if.
\end{command}
}
\subsection{Syntax for Mathematical Expressions: Operators}
The syntax for the expressions recognized by |\pgfmathparse| and friends is
rather straightforward. Let us start with the operators.
\label{pgfmath-operators}
The following operators (presented in the context in which they are used) are
recognized:
%
\begin{math-operator}{+}{infix}{add}
Adds \mvar{x} to \mvar{y}.
\end{math-operator}
\begin{math-operator}{-}{infix}{subtract}
Subtracts \mvar{y} from \mvar{x}.
\end{math-operator}
\begin{math-operator}{-}{prefix}{neg}
Reverses the sign of \mvar{x}.
\end{math-operator}
\begin{math-operator}{*}{infix}{multiply}
Multiplies \mvar{x} by \mvar{y}.
\end{math-operator}
\begin{math-operator}{/}{infix}{divide}
Divides \mvar{x} by \mvar{y}. An error will result if \mvar{y} is 0, or if
the result of the division is too big for the mathematical engine. Please
remember when using this command that accurate (and reasonably quick)
division of real numbers that are not integers is particularly tricky in
\TeX.
\end{math-operator}
\begin{math-operator}{\char`\^}{infix}{pow}
Raises \mvar{x} to the power \mvar{y}.
\end{math-operator}
\begin{math-operator}{\protect\exclamationmarktext}{postfix}{factorial}
Calculates the factorial of \mvar{x}.
\end{math-operator}
\begin{math-operator}{r}{postfix}{deg}
Converts \mvar{x} to degrees (\mvar{x} is assumed to be in radians). This
operator has the same precedence as multiplication.
\end{math-operator}
\begin{math-operators}{?}{:}{conditional}{ifthenelse}
|?| and |:| are special operators which can be used as a shorthand for |if|
\mvar{x} |then| \mvar{y} |else| \mvar{z} inside the parser. The expression
\mvar{x} is taken to be true if it evaluates to any non-zero value.
\end{math-operators}
\begin{math-operator}{==}{infix}{equal}
Returns |1| if \mvar{x}$=$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{>}{infix}{greater}
Returns |1| if \mvar{x}$>$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{<}{infix}{less}
Returns |1| if \mvar{x}$<$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{\protect\exclamationmarktext=}{infix}{notequal}
Returns |1| if \mvar{x}$\neq$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{>=}{infix}{notless}
Returns |1| if \mvar{x}$\geq$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{<=}{infix}{notgreater}
Returns |1| if \mvar{x}$\leq$\mvar{y}, |0| otherwise.
\end{math-operator}
\begin{math-operator}{{\char`\&}{\char`\&}}{infix}{and}
Returns |1| if both \mvar{x} and \mvar{y} evaluate to some non-zero value.
Both arguments are evaluated.
\end{math-operator}
{
\catcode`\|=12
\begin{math-operator}[no index]{||}{infix}{or}
\index{*pgfmanualvbarvbarr@\protect\texttt{\protect\pgfmanualvbarvbar} math operator}%
\index{Math operators!*pgfmanualvbarvbar@\protect\texttt{\protect\pgfmanualvbarvbar}}%
Returns {\tt 1} if either \mvar{x} or \mvar{y} evaluate to some non-zero
value.
\end{math-operator}
}
\begin{math-operator}{\protect\exclamationmarktext}{prefix}{not}
Returns |1| if \mvar{x} evaluates to zero, |0| otherwise.
\end{math-operator}
\begin{math-operators}{(}{)}{group}{}
These operators act in the usual way, that is, to control the order in
which operators are executed, for example, |(1+2)*3|. This includes the
grouping of arguments for functions, for example, |sin(30*10)| or
|mod(72,3)| (the comma character is also treated as an operator).
Parentheses for functions with one argument are not always necessary,
|sin 30| (note the space) is the same as |sin(30)|. However, functions have
the highest precedence so, |sin 30*10| is the same as |sin(30)*10|.
\end{math-operators}
\begin{math-operators}{\char`\{}{\char`\}}{array}{}
These operators are used to process array-like structures (within an
expression these characters do not act like \TeX{} grouping tokens). The
\meta{array specification} consists of comma separated elements, for
example, |{1, 2, 3, 4, 5}|. Each element in the array will be evaluated as
it is parsed, so expressions can be used. In addition, an element of an
array can be an array itself, allowing multiple dimension arrays to be
simulated: |{1, {2,3}, {4,5}, 6}|. When storing an array in a macro, do not
forget the surrounding braces: |\def\myarray{{1,2,3}}| not
|\def\myarray{1,2,3}|.
%
\begin{codeexample}[]
\def\myarray{{1,"two",2+1,"IV","cinq","sechs",sin(\i*5)*14}}
\foreach \i in {0,...,6}{\pgfmathparse{\myarray[\i]}\pgfmathresult, }
\end{codeexample}
%
\end{math-operators}
\chardef\lbrack=`\[
\chardef\rbrack=`\]
\begin{math-operators}{\lbrack}{\rbrack}{array access}{array}
|[| and |]| are two operators used in one particular circumstance: to
access an array (specified using the |{| and |}| operators) using the index
\mvar{x}. Indexing starts from zero, so, if the index is greater than, or
equal to, the number of values in the array, an error will occur, and zero
will be returned.
%
\begin{codeexample}[]
\def\myarray{{7,-3,4,-9,11}}
\pgfmathparse{\myarray[3]} \pgfmathresult
\end{codeexample}
If the array is defined to have multiple dimensions, then the array access
operators can be immediately repeated.
%
\begin{codeexample}[]
\def\print#1{\pgfmathparse{#1}\pgfmathresult}
\def\identitymatrix{{{1,0,0},{0,1,0},{0,0,1}}}
\tikz[x=0.5cm,y=0.5cm]\foreach \i in {0,1,2} \foreach \j in {0,1,2}
\node at (\j,-\i) [anchor=base] {\print{\identitymatrix[\i][\j]}};
\end{codeexample}
%
\end{math-operators}
\begin{math-operators}{\char`\"}{\char`\"}{group}{}
These operators are used to quote \mvar{x}. However, as every expression is
expanded with |\edef| before it is parsed, macros (e.g., font commands like
|\tt| or |\Huge|) may need to be ``protected'' from this expansion (e.g.,
|\noexpand\Huge|). Ideally, you should avoid such macros anyway. Obviously,
these operators should be used with great care as further calculations are
unlikely to be possible with the result.
%
\begin{codeexample}[]
\def\x{5}
\foreach \y in {0,10}{
\pgfmathparse{\x > \y ? "\noexpand\Large Bigger" : "\noexpand\tiny smaller"}
\x\ is \pgfmathresult\ than \y.
}
\end{codeexample}
%
\end{math-operators}
\subsection{Syntax for Mathematical Expressions: Functions}
\label{pgfmath-functions}
The following functions are recognized:
\medskip
\def\mathlink#1{\hyperlink{math:#1}{\tt#1}}
\begin{tikzpicture}
\foreach \f [count=\i from 0] in
{abs,acos,add,and,array,asin,atan,atan2,bin,ceil,cos,
cosec,cosh,cot,deg,depth,div,divide,e,equal,factorial, false,
floor,frac,gcd,greater,height,hex,Hex,int,ifthenelse,iseven,isodd,isprime,
less,ln,log10,log2,max,min,mod,Mod,multiply,
neg,not,notequal,notgreater,notless,
oct,or,pi,pow,rad,rand,random,real,rnd,round,
scalar,sec,sign,sin,sinh,sqrt,subtract,tan,tanh,true, veclen,width}
\node [anchor=base west] at ({int(\i/12)*2.5cm},{-mod(\i,12)*1.1*\baselineskip}) {\mathlink{\f}};
\end{tikzpicture}
\bigskip
Each function has a \pgfname{} command associated with it (which is also shown
with the function below). In general, the command is simply the name of the
function prefixed with |\pgfmath|, for example, |\pgfmathadd|, but there are
some notable exceptions.
\subsubsection{Basic arithmetic functions}
\label{pgfmath-functions-basic}
\begin{math-function}{add(\mvar{x},\mvar{y})}
\mathcommand
Adds $x$ and $y$.
%
\begin{codeexample}[]
\pgfmathparse{add(75,6)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{subtract(\mvar{x},\mvar{y})}
\mathcommand
Subtract $y$ from $x$.
%
\begin{codeexample}[]
\pgfmathparse{subtract(75,6)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{neg(\mvar{x})}
\mathcommand
This returns $-\mvar{x}$.
%
\begin{codeexample}[]
\pgfmathparse{neg(50)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{multiply(\mvar{x},\mvar{y})}
\mathcommand
Multiply $x$ by $y$.
%
\begin{codeexample}[]
\pgfmathparse{multiply(75,6)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{divide(\mvar{x},\mvar{y})}
\mathcommand
Divide $x$ by $y$.
%
\begin{codeexample}[]
\pgfmathparse{divide(75,6)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{div(\mvar{x},\mvar{y})}
\mathcommand
Divide $x$ by $y$ and return the integer part of the result.
%
\begin{codeexample}[]
\pgfmathparse{div(75,9)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{factorial(\mvar{x})}
\mathcommand
Return \mvar{x}!.
%
\begin{codeexample}[]
\pgfmathparse{factorial(5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{sqrt(\mvar{x})}
\mathcommand
Calculates $\sqrt{\textrm{\mvar{x}}}$.
%
\begin{codeexample}[]
\pgfmathparse{sqrt(10)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{sqrt(8765.432)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{pow(\mvar{x},\mvar{y})}
\mathcommand
Raises \mvar{x} to the power \mvar{y}. For greatest accuracy, \mvar{y}
should be an integer. If \mvar{y} is not an integer, the actual calculation
will be an approximation of $e^{y \ln(x)}$.
%
\begin{codeexample}[]
\pgfmathparse{pow(2,7)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{e}
\mathcommand
Returns the value 2.718281828.
%
{
\catcode`\^=7
\begin{codeexample}[]
\pgfmathparse{(e^2-e^-2)/2} \pgfmathresult
\end{codeexample}
}
\end{math-function}
\begin{math-function}{exp(\mvar{x})}
\mathcommand
{ \catcode`\^=7
Maclaurin series for $e^x$.
}
\begin{codeexample}[]
\pgfmathparse{exp(1)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{exp(2.34)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{ln(\mvar{x})}
\mathcommand
{ \catcode`\^=7
An approximation for $\ln(\textrm{\mvar{x}})$. This uses an algorithm of
Rouben Rostamian, and coefficients suggested by Alain Matthes.
}
\begin{codeexample}[]
\pgfmathparse{ln(10)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{ln(exp(5))} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{log10(\mvar{x})}
\mathcommand[logten(\mvar{x})]
An approximation for $\log_{10}(\textrm{\mvar{x}})$.
%
\begin{codeexample}[]
\pgfmathparse{log10(100)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{log2(\mvar{x})}
\mathcommand[logtwo(\mvar{x})]
An approximation for $\log_2(\textrm{\mvar{x}})$.
%
\begin{codeexample}[]
\pgfmathparse{log2(128)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{abs(\mvar{x})}
\mathcommand
Evaluates the absolute value of $x$.
%
\begin{codeexample}[]
\pgfmathparse{abs(-5)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{-abs(4*-3)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{mod(\mvar{x},\mvar{y})}
\mathcommand
This evaluates \mvar{x} modulo \mvar{y}, using truncated division. The sign
of the result is the same as the sign of
$\frac{\textrm{\mvar{x}}}{\textrm{\mvar{y}}}$.
%
\begin{codeexample}[]
\pgfmathparse{mod(20,6)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{mod(-100,30)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{Mod(\mvar{x},\mvar{y})}
\mathcommand
This evaluates \mvar{x} modulo \mvar{y}, using floored division. The sign
of the result is never negative.
%
\begin{codeexample}[]
\pgfmathparse{Mod(-100,30)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{sign(\mvar{x})}
\mathcommand
Returns the sign of $x$.
%
\begin{codeexample}[]
\pgfmathparse{sign(-5)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{sign(0)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{sign(5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\subsubsection{Rounding functions}
\label{pgfmath-functions-rounding}
\begin{math-function}{round(\mvar{x})}
\mathcommand
Rounds \mvar{x} to the nearest integer. It uses ``asymmetric half-up''
rounding. So |1.5| is rounded to |2|, but |-1.5| is rounded to |-2|
(\emph{not} |-1|).
%
\begin{codeexample}[]
\pgfmathparse{round(32.5/17)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{round(398/12)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{floor(\mvar{x})}
\mathcommand
Rounds \mvar{x} down to the nearest integer.
%
\begin{codeexample}[]
\pgfmathparse{floor(32.5/17)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{floor(398/12)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{floor(-398/12)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{ceil(\mvar{x})}
\mathcommand
Rounds \mvar{x} up to the nearest integer.
%
\begin{codeexample}[]
\pgfmathparse{ceil(32.5/17)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{ceil(398/12)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{ceil(-398/12)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{int(\mvar{x})}
\mathcommand
Returns the integer part of \mvar{x}.
%
\begin{codeexample}[]
\pgfmathparse{int(32.5/17)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{frac(\mvar{x})}
\mathcommand
Returns the fractional part of \mvar{x}.
%
\begin{codeexample}[]
\pgfmathparse{frac(32.5/17)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{real(\mvar{x})}
\mathcommand
Ensures \mvar{x} contains a decimal point.
%
\begin{codeexample}[]
\pgfmathparse{real(4)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\subsubsection{Integer arithmetics functions}
\label{pgfmath-functions-integerarithmetics}
\begin{math-function}{gcd(\mvar{x},\mvar{y})}
\mathcommand
Computes the greatest common divider of \mvar{x} and \mvar{y}.
%
\begin{codeexample}[]
\pgfmathparse{gcd(42,56)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{isodd(\mvar{x})}
\mathcommand
Returns |1| if the integer part of \mvar{x} is odd. Otherwise, returns |0|.
%
\begin{codeexample}[]
\pgfmathparse{isodd(2)} \pgfmathresult,
\pgfmathparse{isodd(3)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{iseven(\mvar{x})}
\mathcommand
Returns |1| if the integer part of \mvar{x} is even. Otherwise, returns |0|.
%
\begin{codeexample}[]
\pgfmathparse{iseven(2)} \pgfmathresult,
\pgfmathparse{iseven(3)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{isprime(\mvar{x})}
\mathcommand
Returns |1| if the integer part of \mvar{x} is prime. Otherwise, returns |0|.
%
\begin{codeexample}[]
\pgfmathparse{isprime(1)} \pgfmathresult,
\pgfmathparse{isprime(2)} \pgfmathresult,
\pgfmathparse{isprime(31)} \pgfmathresult,
\pgfmathparse{isprime(64)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\subsubsection{Trigonometric functions}
\label{pgfmath-functions-trigonometric}
\begin{math-function}{pi}
\mathcommand
Returns the value $\pi=3.141592654$.
%
\begin{codeexample}[]
\pgfmathparse{pi} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{pi r} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{rad(\mvar{x})}
\mathcommand
Convert \mvar{x} to radians. \mvar{x} is assumed to be in degrees.
%
\begin{codeexample}[]
\pgfmathparse{rad(90)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{deg(\mvar{x})}
\mathcommand
Convert \mvar{x} to degrees. \mvar{x} is assumed to be in radians.
%
\begin{codeexample}[]
\pgfmathparse{deg(3*pi/2)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{sin(\mvar{x})}
\mathcommand
%
Sine of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{sin(60)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{sin(pi/3 r)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{cos(\mvar{x})}
\mathcommand
Cosine of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{cos(60)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{cos(pi/3 r)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{tan(\mvar{x})}
\mathcommand
Tangent of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{tan(45)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfmathparse{tan(2*pi/8 r)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{sec(\mvar{x})}
\mathcommand
Secant of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{sec(45)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{cosec(\mvar{x})}
\mathcommand
Cosecant of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{cosec(30)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{cot(\mvar{x})}
\mathcommand
Cotangent of \mvar{x}. By employing the |r| operator, \mvar{x} can be in
radians.
%
\begin{codeexample}[]
\pgfmathparse{cot(15)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{asin(\mvar{x})}
\mathcommand
Arcsine of \mvar{x}. The result is in degrees and in the range $\pm90^\circ$.
%
\begin{codeexample}[]
\pgfmathparse{asin(0.7071)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{acos(\mvar{x})}
\mathcommand
Arccosine of \mvar{x} in degrees. The result is in the range $[0^\circ,180^\circ]$.
%
\begin{codeexample}[]
\pgfmathparse{acos(0.5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{atan(\mvar{x})}
\mathcommand
Arctangent of $x$ in degrees.
%
\begin{codeexample}[]
\pgfmathparse{atan(1)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{atan2(\mvar{y},\mvar{x})}
\mathcommand[atantwo(\mvar{y},\mvar{x})]
Arctangent of $y\div x$ in degrees. This also takes into account the
quadrants.
%
\begin{codeexample}[]
\pgfmathparse{atan2(-4,3)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{key}{/pgf/trig format=\mchoice{deg,rad} (initially deg)}
Allows to define whether trigonometric math functions (i.e.\ all in this
subsection) operate with degrees or with radians.
%
\begin{codeexample}[]
\pgfmathparse{cos(45)} \pgfmathresult
\end{codeexample}
\begin{codeexample}[]
\pgfkeys{/pgf/trig format=rad}
\pgfmathparse{cos(pi/2)} \pgfmathresult
\end{codeexample}
The initial configuration |trig format=deg| is the base of \pgfname: almost
all of it is based on degrees.
Specifying |trig format=rad| is most useful for data visualization where
the angles are typically given in radians. However, it is applied to all
trigonometric functions for which the option applies, including any drawing
instructions which operate on angles.
%
\begin{codeexample}[]
\begin{tikzpicture}
\draw[-stealth]
(0:1) -- (45:1) -- (90:1) -- (135:1) -- (180:1);
\draw[-stealth,trig format=rad,red]
(pi:1) -- (5/4*pi:1) -- (6/4*pi:1) -- (7/4*pi:1) -- (2*pi:1);
\end{tikzpicture}
\end{codeexample}
\paragraph{Warning:}
At the time of this writing, this feature is ``experimental''. Please
handle it with care: there may be path instructions or libraries in
\pgfname\ which rely on |trig format=deg|. The intended usage of
|trig format=rad| is for local scopes -- and as option for data
visualization.
\end{key}
\subsubsection{Comparison and logical functions}
\label{pgfmath-functions-comparison}
\begin{math-function}{equal(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if $\mvar{x}=\mvar{y}$ and |0| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{equal(20,20)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{greater(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if $\mvar{x}>\mvar{y}$ and |0| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{greater(20,25)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{less(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if $\mvar{x}<\mvar{y}$ and |0| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{greater(20,25)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{notequal(\mvar{x},\mvar{y})}
\mathcommand
This returns |0| if $\mvar{x}=\mvar{y}$ and |1| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{notequal(20,25)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{notgreater(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if $\mvar{x}\leq\mvar{y}$ and |0| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{notgreater(20,25)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{notless(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if $\mvar{x}\geq\mvar{y}$ and |0| otherwise.
%
\begin{codeexample}[]
\pgfmathparse{notless(20,25)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{and(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if \mvar{x} and \mvar{y} both evaluate to non-zero values.
Otherwise |0| is returned.
%
\begin{codeexample}[]
\pgfmathparse{and(5>4,6>7)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{or(\mvar{x},\mvar{y})}
\mathcommand
This returns |1| if either \mvar{x} or \mvar{y} evaluate to non-zero
values. Otherwise |0| is returned.
%
\begin{codeexample}[]
\pgfmathparse{or(5>4,6>7)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{not(\mvar{x})}
\mathcommand
This returns |1| if $\mvar{x}=0$, otherwise |0|.
%
\begin{codeexample}[]
\pgfmathparse{not(true)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{ifthenelse(\mvar{x},\mvar{y},\mvar{z})}
\mathcommand
This returns \mvar{y} if \mvar{x} evaluates to some non-zero value,
otherwise \mvar{z} is returned.
%
\begin{codeexample}[]
\pgfmathparse{ifthenelse(5==4,"yes","no")} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{true}
\mathcommand
This evaluates to |1|.
%
\begin{codeexample}[]
\pgfmathparse{true ? "yes" : "no"} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{false}
\mathcommand
This evaluates to |0|.
%
\begin{codeexample}[]
\pgfmathparse{false ? "yes" : "no"} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\subsubsection{Pseudo-random functions}
\label{pgfmath-functions-random}
\begin{math-function}{rnd}
\mathcommand
Generates a pseudo-random number between $0$ and $1$ with a uniform
distribution.
%
\begin{codeexample}[pre={\pgfmathsetseed{1}}]
\foreach \x in {1,...,10}{\pgfmathparse{rnd}\pgfmathresult, }
\end{codeexample}
%
\end{math-function}
\begin{math-function}{rand}
\mathcommand
Generates a pseudo-random number between $-1$ and $1$ with a uniform
distribution.
%
\begin{codeexample}[pre={\pgfmathsetseed{1}}]
\foreach \x in {1,...,10}{\pgfmathparse{rand}\pgfmathresult, }
\end{codeexample}
%
\end{math-function}
\begin{math-function}{random(\opt{\mvar{x},\mvar{y}})}
\mathcommand
This function takes zero, one or two arguments. If there are zero
arguments, a uniform random number between $0$ and $1$ is generated. If
there is one argument \mvar{x}, a random integer between $1$ and \mvar{x}
is generated. Finally, if there are two arguments, a random integer between
\mvar{x} and \mvar{y} is generated. If there are no arguments, the
\pgfname{} command should be called as follows: |\pgfmathrandom{}|.
%
\begin{codeexample}[pre={\pgfmathsetseed{1}}]
\foreach \x in {1,...,10}{\pgfmathparse{random()}\pgfmathresult, }
\end{codeexample}
\begin{codeexample}[pre={\pgfmathsetseed{1}}]
\foreach \x in {1,...,10}{\pgfmathparse{random(100)}\pgfmathresult, }
\end{codeexample}
\begin{codeexample}[pre={\pgfmathsetseed{1}}]
\foreach \x in {1,...,10}{\pgfmathparse{random(232,762)}\pgfmathresult, }
\end{codeexample}
%
\end{math-function}
\subsubsection{Base conversion functions}
\label{pgfmath-functions-base}
\begin{math-function}{hex(\mvar{x})}
\mathcommand
Convert \mvar{x}{} (assumed to be an integer in base 10) to a hexadecimal
representation, using lower case alphabetic digits. No further calculation
will be possible with the result.
%
\begin{codeexample}[]
\pgfmathparse{hex(65535)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{Hex(\mvar{x})}
\mathcommand
Convert \mvar{x}{} (assumed to be an integer in base 10) to a hexadecimal
representation, using upper case alphabetic digits. No further calculation
will be possible with the result.
%
\begin{codeexample}[]
\pgfmathparse{Hex(65535)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{oct(\mvar{x})}
\mathcommand
Convert \mvar{x}{} (assumed to be an integer in base 10) to an octal
representation. No further calculation should be attempted with the result,
as the parser can only process numbers converted to base 10.
%
\begin{codeexample}[]
\pgfmathparse{oct(63)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{bin(\mvar{x})}
\mathcommand
Convert \mvar{x}{} (assumed to be an integer in base 10) to a binary
representation. No further calculation should be attempted with the result,
as the parser can only process numbers converted to base 10.
%
\begin{codeexample}[]
\pgfmathparse{bin(185)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\subsubsection{Miscellaneous functions}
\label{pgfmath-functions-misc}
\begin{math-function}{min(\mvar{x$_1$},\mvar{x$_2$},\ldots,\mvar{x$_n$})}
\mathcommand[min({\mvar{x$_1$},\mvar{x$_2$},\ldots},{\ldots,\mvar{x$_{n-1}$},\mvar{x$_n$}})]
Return the minimum value from \mvar{x$_1$}\ldots\mvar{x$_n$}. For
historical reasons, the command |\pgfmathmin| takes two arguments, but each
of these can contain an arbitrary number of comma separated values.
%
\begin{codeexample}[]
\pgfmathparse{min(3,4,-2,250,-8,100)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{max(\mvar{x$_1$},\mvar{x$_2$},\ldots,\mvar{x$_n$})}
\mathcommand[max({\mvar{x$_1$},\mvar{x$_2$},\ldots},{\ldots,\mvar{x$_{n-1}$},\mvar{x$_n$}})]
Return the maximum value from \mvar{x$_1$}\ldots\mvar{x$_n$}. Again, for
historical reasons, the command |\pgfmathmax| takes two arguments, but each
of these can contain an arbitrary number of comma separated values.
%
\begin{codeexample}[]
\pgfmathparse{max(3,4,-2,250,-8,100)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{veclen(\mvar{x},\mvar{y})}
\mathcommand
Calculates $\sqrt{\left(\textrm{\mvar{x}}^2+\textrm{\mvar{y}}^2\right)}$.
This uses a polynomial approximation, based on ideas of Rouben Rostamian
%
\begin{codeexample}[]
\pgfmathparse{veclen(12,5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{array(\mvar{x},\mvar{y})}
\mathcommand
This accesses the array \mvar{x} at the index \mvar{y}. The array must
begin and end with braces (e.g., |{1,2,3,4}|) and array indexing starts at
|0|.
%
\begin{codeexample}[]
\pgfmathparse{array({9,13,17,21},2)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
The following hyperbolic functions were adapted from code suggested by Martin
Heller:
\begin{math-function}{sinh(\mvar{x})}
\mathcommand
The hyperbolic sine of \mvar{x}
%
\begin{codeexample}[]
\pgfmathparse{sinh(0.5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{cosh(\mvar{x})}
\mathcommand
The hyperbolic cosine of \mvar{x}
%
\begin{codeexample}[]
\pgfmathparse{cosh(0.5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{tanh(\mvar{x})}
\mathcommand
The hyperbolic tangent of \mvar{x}
%
\begin{codeexample}[]
\pgfmathparse{tanh(0.5)} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{width("\mvar{x}")}
\mathcommand
Return the width of a \TeX{} (horizontal) box containing \mvar{x}. The
quote characters are necessary to prevent \mvar{x}{} from being parsed. It
is important to remember that any expression is expanded with |\edef|
before being parsed, so any macros (e.g., font commands like |\tt| or
|\Huge|) will need to be ``protected'' (e.g., |\noexpand\Huge| is usually
sufficient).
%
\begin{codeexample}[]
\pgfmathparse{width("Some Lovely Text")} \pgfmathresult
\end{codeexample}
Note that results of this method are provided in points.
\end{math-function}
\begin{math-function}{height("\mvar{x}")}
\mathcommand
Return the height of a box containing \mvar{x}.
%
\begin{codeexample}[]
\pgfmathparse{height("Some Lovely Text")} \pgfmathresult
\end{codeexample}
%
\end{math-function}
\begin{math-function}{depth("\mvar{x}")}
\mathcommand
Returns the depth of a box containing \mvar{x}.
%
\begin{codeexample}[]
\pgfmathparse{depth("Some Lovely Text")} \pgfmathresult
\end{codeexample}
%
\end{math-function}