% 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,dim,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} \begin{math-function}{dim(\mvar{x})} \mathcommand This returns the dimension of the array \mvar{x}. The array must begin and end with braces (e.g., |{1,2,3,4}|). % \begin{codeexample}[] \pgfmathparse{dim({1,2,3,4})} \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}