[Tex/LaTex] Reformat \frac depending on numerator/denominator size difference

fractions

Is it possible to detect cases where the second argument of \frac is much smaller than the first argument, and split it? Namely (I'm using a new macro name \wfrac to simplify things),

\wfrac{x}{2} % => \frac{x}{2}
\wfrac{x+y}{2} % => \frac{1}{2}\left(x+y\right)
\wfrac{\sum_{i=1}^{N} i^3}{x+y+z+t}
% => \frac{1}{x+y+z+t} \left(\sum_{i=1}^{N} i^3\right)
% => or \left(\sum_{i=1}^{N} i^3\right) / \left(x+y+z+t\right)
\wfrac{\sum_{i=1}^{N} i^3}{\sum_{i=1}^{N} i}
% => \frac{...}{...} (unchanged)

Feel free to tell me that this is a bad idea.

EDIT: As David notes in his comment, it is not clear what I define as bigger/smaller. The third example in particular features a case where the numerator is taller, but the denominator is wider. he also notes that \wfrac{x}{y} could become (x)/(y). I am building the requirements as I write the question, so feel free to tweak them at your will.

\wfrac{x}{y} should become one of \frac{x}{y}, or \frac{1}{y}(x), or (x)/(y), with ()/ scaling as appropriate. Here is one scheme that could give nice results. Denote hx, hy the heights (plus depths) of x and y; wx, wy their widths; h0 and w0 customizable thresholds for what is considered "small", for instance the dimensions of the formula $a+b$ , but probably the threshold should depend on the \mathstyle.

If hy < h0 and wy < w0,
1. If hx < h0 and wx < max(w0, 2 * wy), use \frac{x}{y}.
2. Otherwise use \frac{1}{y}(x).
Otherwise use (x)/(y).

Best Answer

I though this was an interesting question, so I gave it a try.
Let me start off with the result:

main.tex:

\documentclass{article}
\usepackage{wfrac}

\begin{document}
\setmaxeq{a+b} % Reference equation for size

\[ \wfrac{x}{2} \]
\[ \wfrac{x+y}{2} \]
\[ \wfrac{\sum_{i=1}^{N} i^3}{x+y+z+t} \]
\[ \wfrac{\sum_{i=1}^{N} i^3}{\sum_{i=1}^{N} i} \]

\setmax{34pt}{10pt} % Manual maximum size

\[ \wfrac{\sum_{i=1}^{N} i^3}{\sum_{i=1}^{N} i} \]

\setmaxeq{x} % Reference equation again

\[ \wfrac{x}{x} \]
\[ \efrac{x}{x} \]
\end{document}

And it looks like this:

result

Options:

text: compare to the equation in \textstyle.
display: compare to the equation in \displaystyle.
none: compare to the equation in whatever style is currently active.
noparen: no parentheses
small: small maximum height and width
big: big maximum height and width
huge: huge maximum height and width
lparen: set left paren, eg: lparen=\left[
rparen: set right paren, eg: rparen={\right]}
div: set division mark, eg: div=\div

Commands

\wfrac: fraction following 'less' rules.
\efrac: fraction following 'less or equal' rules.
\setmax: set the maximum size.
\setmaxeq: set the maximum size using a reference equation.
\getmax: get the maximum size.
\getsize: get the size from an equation.
\setparen: set the parentheses, takes two arguments.
\setdiv: set the division mark.

And here is wfrac.sty:

\NeedsTeXFormat{LaTeX2e}[1994/12/01]
\ProvidesPackage{wfrac}[2012/11/03 v1.01 intelligent fractions]

% Lengths and widths
\newdimen \wfrac@hx
\newdimen \wfrac@hy
\newdimen \wfrac@hmax
\newdimen \wfrac@wx
\newdimen \wfrac@wy
\newdimen \wfrac@wmax
\newdimen \wfrac@wmaxcalc

% Parenthesis and division mark
\newcommand*{\wfrac@lparen}{\left(}
\newcommand*{\wfrac@rparen}{\right)}
\newcommand*{\wfrac@div}{\middle/}

% Options
\IfFileExists{xkeyval.sty}{
    \RequirePackage{xkeyval}
    \DeclareOptionX{lparen}{\renewcommand*{\wfrac@lparen}{##1}}
    \DeclareOptionX{rparen}{\renewcommand*{\wfrac@rparen}{##1}}
    \DeclareOptionX{div}{\renewcommand*{\wfrac@div}{##1}}
}{
    \let\DeclareOptionX\DeclareOption
    \let\ExecuteOptionsX\ExecuteOptions
    \let\ProcessOptionsX\ProcessOptions
}

\DeclareOptionX{text}{\edef\wfrac@style{\textstyle}}
\DeclareOptionX{display}{\edef\wfrac@style{\displaystyle}}
\DeclareOptionX{none}{\edef\wfrac@style{}}
\DeclareOptionX{noparen}{\renewcommand*{\wfrac@lparen}{}\renewcommand*{\wfrac@rparen}{}}
\DeclareOptionX{small}{\wfrac@wmax = 25pt \wfrac@hmax = 10pt}
\DeclareOptionX{big}{\wfrac@wmax = 50pt \wfrac@hmax = 50pt}
\DeclareOptionX{huge}{\wfrac@wmax = 100pt \wfrac@hmax = 100pt}
\ExecuteOptionsX{text,small}
\ProcessOptionsX\relax

% Fraction variations
\newcommand*{\wfrac@Afrac}[2]{\left. \wfrac@lparen #1 \wfrac@rparen \wfrac@div \wfrac@lparen #2 \wfrac@rparen \right.}
\newcommand*{\wfrac@Bfrac}[2]{\frac{1}{#2}\wfrac@lparen #1 \wfrac@rparen}
\newcommand*{\wfrac@Cfrac}[2]{\frac{#1}{#2}}

% Main commands
\newcommand*\setparen[2]{
    \renewcommand*{\wfrac@lparen}{#1}
    \renewcommand*{\wfrac@rparen}{#2}
}

\newcommand*\setdiv[1]{
    \renewcommand*{\wfrac@div}{#1}
}

\newcommand*\setmax[2]{
    \wfrac@wmax = #1
    \wfrac@hmax = #2
}

\newcommand*\setmaxeq[1]{
    \settowidth{\wfrac@wmax}{$ \wfrac@style #1 $}
    \settoheight{\wfrac@hmax}{$ \wfrac@style #1 $}
}

\newcommand*\getmax{
    \the\wfrac@wmax $\times$ \the\wfrac@hmax
}

\newcommand*\getsize[1]{
    \settowidth{\wfrac@wx}{$ \wfrac@style #1 $}
    \settoheight{\wfrac@hx}{$ \wfrac@style #1 $}
    \the\wfrac@wx $\times$ \the\wfrac@hx
}

\newcommand*\wfrac[2]{
    \settowidth{\wfrac@wx}{$ \wfrac@style #1 $}
    \settowidth{\wfrac@wy}{$ \wfrac@style #2 $}
    \settoheight{\wfrac@hx}{$ \wfrac@style #1 $}
    \settoheight{\wfrac@hy}{$ \wfrac@style #2 $}    
    % max(w0, 2 * wy)
    \ifdim \wfrac@wmax < 2\wfrac@wy
        \wfrac@wmaxcalc = 2\wfrac@wy
    \else
        \wfrac@wmaxcalc = \wfrac@wmax
    \fi
    %
    \ifdim \wfrac@hy < \wfrac@hmax
        \ifdim \wfrac@wy < \wfrac@wmax
            \ifdim \wfrac@hx < \wfrac@hmax
                \ifdim \wfrac@wx < \wfrac@wmaxcalc
                    \let\wfrac@frac=\wfrac@Cfrac
                \else
                    \let\wfrac@frac=\wfrac@Bfrac
                \fi
            \else
                \let\wfrac@frac=\wfrac@Bfrac
            \fi 
        \else
            \let\wfrac@frac=\wfrac@Afrac
        \fi
    \else
        \let\wfrac@frac=\wfrac@Afrac
    \fi
    %
    \wfrac@frac{#1}{#2}
}

\newcommand*\efrac[2]{
    \settowidth{\wfrac@wx}{$ \wfrac@style #1 $}
    \settowidth{\wfrac@wy}{$ \wfrac@style #2 $}
    \settoheight{\wfrac@hx}{$ \wfrac@style #1 $}
    \settoheight{\wfrac@hy}{$ \wfrac@style #2 $}    
    % max(w0, 2 * wy)
    \ifdim \wfrac@wmax < 2\wfrac@wy
        \wfrac@wmaxcalc = 2\wfrac@wy
    \else
        \wfrac@wmaxcalc = \wfrac@wmax
    \fi
    %
    \ifdim \wfrac@hy > \wfrac@hmax
        \let\wfrac@frac=\wfrac@Afrac
    \else
        \ifdim \wfrac@wy > \wfrac@wmax
            \let\wfrac@frac=\wfrac@Afrac
        \else
            \ifdim \wfrac@hx > \wfrac@hmax
                \let\wfrac@frac=\wfrac@Bfrac
            \else
                \ifdim \wfrac@wx > \wfrac@wmaxcalc
                    \let\wfrac@frac=\wfrac@Bfrac
                \else
                    \let\wfrac@frac=\wfrac@Cfrac
                \fi
            \fi 
        \fi
    \fi
    %
    \wfrac@frac{#1}{#2}
}

\endinput

Related Solutions

[Tex/LaTex] Improved kerning in fractions

One way to shift your numerators in an italic context would be

 {\hss\tiny\kern\fontdimen\@ne\font#2\/\kern-\fontdimen\@ne\font\hss}\cr%        Emit numerator [#2]

as for safe ways of looking ahead and adding space or not, you might want to look at latex's definition of \DeclareTextFontCommand which defines \textit and friends to add italic correction at the start or end, depending.

[Tex/LaTex] Aligned numerator and denominator in \frac or any other variant

As illustrated in the highest-voted answer to Align denominator of fraction to left, \hfill will do the trick.

Here's a macro, \myfrac, that puts the \hfill either in the numerator or the denominator as necessary

\newcommand{\myfrac}[2]{%
    \setbox0\hbox{$#1$}        % put the numerator in box0
    \dimen0=\wd0               % measure box0
    \setbox1\hbox{$#2$}        % put the denominator in box1
    \dimen1=\wd1               % measure box1
    \ifdim\wd0<\wd1            % if box0 is narrower than box1
        \dfrac{#1\hfill}{#2}   % put \hfill in the numerator
    \else                      
        \dfrac{#1}{#2\hfill}   % otherwise put \hfill in the denominator
    \fi
}

screenshot

Complete MWE

\documentclass{article}
\usepackage{amsmath}

\newcommand{\myfrac}[2]{%
    \setbox0\hbox{$#1$}        % put the numerator in box0
    \dimen0=\wd0               % measure box0
    \setbox1\hbox{$#2$}        % put the denominator in box1
    \dimen1=\wd1               % measure box1
    \ifdim\wd0<\wd1            % if box0 is narrower than box1
        \dfrac{#1\hfill}{#2}   % put \hfill in the numerator
    \else                      
        \dfrac{#1}{#2\hfill}   % otherwise put \hfill in the denominator
    \fi
}
\begin{document}
\begin{itemize}
    \item[Original] $\dfrac{(x_i - x)^{N+1}}{(N+1)!}$
    \item[Test 1] $\myfrac{(x_i - x)^{N+1}}{(N+1)!}$
    \item[Test 2] $\myfrac{(N+1)!}{(x_i - x)^{N+1}}$
\end{itemize}
\end{document}