🚀 go-pugleaf

 At the end of this submission you will find a solution to the
 problem relating to the minimum separation between the numerator or
 denominator of a fraction and the fraction bar, a question raised and
 discussed in several postings last month. I would like
 to begin with a more general problem which provides the tool
 used for that solution.

 PART ONE:

 To measure typeset text from a math list in TeX, the material is
 placed in an \hbox. One must then specify the style of
 the text's "free state" inside the box. The horizontal dimension in
 the free state is affected by "actions at infinity" (glue) and is
 intrinsically uncertain; the height and depth are determined
 "quasi-locally", and are fixed once TeX decides on the current style.
 However, if an item is to be set in a cramped style, absent a TeX
 command \cramp{}, how does one measure its height and depth?

 I have also felt a need for \cramp for aesthetic reasons when there
 is a repetition of an expression close by the original in a line of
 text, once in an uncramped style, and once in the corresponding
 cramped style. As well, there is a not insignificant amount of space
 that can be saved in a document with multiple occurrences of, e.g.,
 ${\x}^{{\x}^{{\x}^{\cdot^{\cdot^{\cdot^{\x}}}}}}$, if this could
 be set in a cramped style.

 I have most of the standard TeX references, but I haven't come across
 any mention of the possibility of obtaining a cramped style "on
 demand". Thus there may be some general interest in the preliminary
 version of my <cramped.sty> macro, and I append it below.

 (I also haven't come across any discussion of a "feature" which turns
 cramping off, viz., prefix the material which TeX would normally
 cramp by an explicit style reminder. Compare, for example,
 $\sqrt{X^2+Y^2}$ with $\sqrt{\textstyle X^2+Y^2}$. However, I doubt
 that this is an intended feature since:
 i) TeX is normally insensitive to redundant style commands, and from
 the user's point of view, this is the way it should be; and,
 ii) cramping will be turned off in midstream by an explicit style
 command (e.g., $\sqrt{X^2+\textstyle Y^2} has X^2 in cramped style
 and Y^2 in uncramped style).
 I am completely out of my depth when it comes to understanding the
 details of The Program, and please, no flames (not even for the
 mixing of metaphors!), but it seems to me that this "feature"
 is a result of the cramped_style calculation in section 702 of The
 Program being ignored while section 703 updates only size-dependent
 quantities.
 Although I reserve my rights to priority for the "finder's fee"
 mentioned in the Preface to The Program, I would like to hear the
 opinions of gurus on the net.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% cramped.sty    Preliminary version           July 21,1992
%
% Enables the typesetting of material in the cramped styles
% D', T', S' and SS'.
%
% \cramp{<foo.bar>} will set <foo.bar> in the cramped version C' of
% the current style C.
%
% \Cramp{\...style}{<foo.bar>} (where you specify the current style)
% has been left accessible.
%
%
% Author: Michael Herschorn
%         Departmentof Mathematics and Statistics
%         McGill University
%         805 Sherbrooke Street West               Phone: 514-398-3825
%         Montreal QC  CANADA  H3A 2K6               FAX: 514-398-3899
%         email: herschor@gauss.math.mcgill.ca
%
% Please email me your comments and suggestions as well as informing
% me directly of any bugs you may find.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\catcode`@=11
\def\cramp{\mathpalette\Cramp}%
\def\Cramp#1#2{\setbox0=\hbox{$\m@th#1\cr@mp{#2}$}
     \Ch@p#1 \dimen2=\the\ht0 \advance\dimen2 -\dimen0
     \ht0=\the\dimen2 \box0}
\def\cr@mp{\kern-\the\nulldelimiterspace\radical"380380}%
\def\Ch@p#1{\ifx#1\displaystyle
    \dimen0=\the\fontdimen5\textfont2
    \advance\dimen0 -.75\dimen0
    \advance\dimen0\the\fontdimen8\textfont3
  \else
%%    \ifx#1\textstyle
    \dimen0=\the\fontdimen8\textfont3
    \advance\dimen0 .25\dimen0
 \fi}%%%%%%%%% If you have "variations" in family 3,
%%%%%% comment out the preceding line and "uncomment"
%%%%%% the 7 lines beginning with "%%".
%%  \else
%%    \ifx#1\scriptstyle\dimen0=\the\fontdimen8\scriptfont3
%%    \advance\dimen0 .25\dimen0
%%  \else \dimen0=\the\fontdimen8\scriptscriptfont3
%%    \advance\dimen0 .25\dimen0
%%  \fi\fi\fi}
\catcode`@=12
\endinput
%%
% end of file cramped.sty
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


 PART TWO:



 Michael Downes <MJD@MATH.AMS.COM> (22 Jun 1992, Info-TeX@shsu.edu)
 wrote:

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+Dan Luecking wrote (19 Jun 1992, Info-TeX@shsu.edu):
+
+> Michael Barr posted a query a while back about the separation
+> between the numerator/denominator of a fraction and the fraction bar.
+>
+> I have not seen a response, so having tried to understand the
+> TeXbook on this point and made some headway in doing so I thought
+> I might discuss the problem.
+
+> ... I have run into it several times and would like to know if
+> anyone has defined a fraction making command to deal with it. One
+> way is to insert a 0 width rule extending below the numerator and
+> above the denominator, but I do not see a good to make this
+> automatic. A way I have considered, but rejected, is to increase the
+> relevant parameter, (something like \fontdimen8\textfont3=1pt
+> instead of 0.4pt) and then define all \frac's using \above (in which
+> the thickness of the rule is explicitly given). The problem is that
+> default_rule_thickness is used in several internal calculations
+> including the positioning of sub/superscripts.
+>
+> Does anybody have anything to say about this?
+
+  I have observed the same problem in Amer. Math. Soc. publications,
+  although the examples I've happened to notice involved letters with
+  descenders such as p and q, rather than vertical bars. Your analysis
+  of the problem corresponds closely to what I found in some cursory
+  investigations.
+
+  The problem seems to be slightly worse than you suggest---or maybe you
+  just didn't bother to amplify on results of your experiments: The most
+  significant drawback of the rejected solution of setting \fontdimen 8
+  \textfont 3 to a larger value and using \above instead of \over is
+  not the internal calculations you mention but the fact that the fraction
+  bar thickness and the space around it are always dependent, even in
+  \above, so that if we say $1\above.4pt 2$ the space around the
+  fraction bar is calculated from the .4pt, not from \fontdimen 8 of
+  \textfont 3, and we're back where we started from.
+
+> ... Is it worthwhile suggesting
+> that a major macro package try do something about it? Have any done
+> anything?
+
+  If my understanding of the problem is correct, it's impossible for a
+  macro package to provide any reasonable solution. The only reasonable
+  solution is to change TeX, the program. (And my definition of
+  `reasonable' is pretty extreme when it comes to macro writing; I have
+  done or contemplated doing things at the macro level that any
+  `reasonable' programmer would consider unreasonable.)
+
+> Is there any real problem at higher resolutions (i.e., does
+> everything look OK printed at 1000dpi)?
+
+  In my opinion, there is not enough space around the fraction bar in
+  the problematic combinations (textstyle fractions, vertical bars,
+  letters with descenders), even at high resolution. I'd rather have
+  some control to adjust the space slightly.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


 The Grand Wizard made his design choices a.e. optimally, and I would
 stick with his implementation of 15d in Appendix G of The  TeXbook.
 Apparent differences between the letter-base levels in the numerators
 and in the denominators of fractions in the same line spoil the
 appearance of the line; shifting the numerator up or the denominator
 down in a fraction with adequate clearance so as to match the letter-
 base levels of another fraction in the line introduces ugly white
 space.  The more I understand the problem, the better I like the GW's
 balancing act.

 Nevertheless, as pointed out by Dan Luecking, Berthold K.P. Horn and
 Michael Downes, it would have been nice to have a free parameter to
 adjust in special circumstances. ("Just so you shouldn't need to
 use it", as the Yiddish expression puts it.)

 Once we have <cramped.sty>, it becomes a simple hack to solve this
 problem (and other similar ones). I wrote (and attach a copy of)
 <spreadfraction.sty> as an illustration. (The coding is not elegant,
 but it works.) It should be used only when absolutely necessary,
 and even then, it should be used sparingly.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% spreadfraction.sty        Preliminary version           July 21,1992
%
% If used in a non-cramped context, \xfrac{#1}{#2}{#3} produces
% {#1\over #2} exactly as TeX would, except that the minimum clearance
% that will be tolerated between the numerator or denominator of this
% fraction and the bar line is #3 times the default rule thickness.
% (See Rule 15d in Appendix G of The TeXbook. TeX hardwires in
% #3=3 in \displaystyle and #3=1 in the other styles; spreadfraction.sty
% allows the use of arbitrary values. In other words, .phi. is replaced
% by #3 times the default rule thickness, everything else is kept
% constant.)
%
% In a cramped context, i.e., if the entire fraction is a Rad atom
% (under a root sign), an Over atom (from \overline), in the denominator
% of another fraction, is a subscript, or is a superscript to an
% entity in cramped style, \xfracc{#1}{#2}{#3} should be used instead
% of \xfrac{#1}{#2}{#3}. (I may automate the selection of \xfracc in a
% future version; since spreadfraction.sty is intended for touch-up use,
% I don't consider automation a priority (and it may be like putting a
% tiara on a baglady!)).
%
% \Xfrac{\...style}{}{}{} (identical to \xfrac{}{}{}, except
% that you specify the style) and \Xfracc{\...style}{}{}{} (DITTO),
% have been left accessible to the user.
%
% To obtain a version of spreadfraction in which the minimum clearances
% are #3 times the hardwired ones, declare \DEKtrue after inputting
% spreadfraction.sty
%
%
% Author: Michael Herschorn
%         Departmentof Mathematics and Statistics
%         McGill University
%         805 Sherbrooke Street West               Phone: 514-398-3825
%         Montreal QC  CANADA  H3A 2K6               FAX: 514-398-3899
%         email: herschor@gauss.math.mcgill.ca
%
% Please email me your comments and suggestions as well as informing
% me directly of any bugs you may find.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newif\ifDEK
\input cramped.sty
\catcode`@=11
\def\xfrac#1#2#3{\M@thpalette\Xfrac{#1}{#2}{#3}}
\def\M@thpalette#1#2#3#4{%
       \mathchoice{#1\displaystyle{#2}{#3}{#4}}%
        {#1\textstyle{#2}{#3}{#4}}%
        {#1\scriptstyle{#2}{#3}{#4}}%
        {#1\scriptscriptstyle{#2}{#3}{#4}}}
\def\Xfrac#1#2#3#4{\ifx#1\displaystyle
       \def\C@rr@ntF@nt{\textfont}
        \let\@djSt@le=\textstyle
         \L@rgeS@ze#1{#2}{#3}{#4}
          \else
           \ifx#1\textstyle \def\C@rr@ntF@nt{\textfont}
            \let\@djSt@le=\scriptstyle
             \N@rmalS@ze#1{#2}{#3}{#4}
          \else
           \ifx#1\scriptstyle
            \def\C@rr@ntF@nt{\scriptfont}
             \let\@djSt@le=\scriptscriptstyle
              \N@rmalS@ze#1{#2}{#3}{#4}
          \else
            \def\C@rr@ntF@nt{\scriptscriptfont}
             \let\@djSt@le=\scriptscriptstyle
              \N@rmalS@ze#1{#2}{#3}{#4}
           \fi
         \fi
       \fi}
\def\L@rgeS@ze#1#2#3#4{\dimen1=\the\fontdimen8\textfont2
              \dimen2=\the\fontdimen11\textfont2
              \N@mer@tor#1{#2}{#3}{#4}}
\def\N@rmalS@ze#1#2#3#4{%
              \dimen1=\the\fontdimen9\C@rr@ntF@nt2
              \dimen2=\the\fontdimen12\C@rr@ntF@nt2
              \N@mer@tor#1{#2}{#3}{#4}}
\def\N@mer@tor#1#2#3#4{%
              \dimen3=\the\fontdimen22\C@rr@ntF@nt2
              \dimen4=\the\fontdimen8\C@rr@ntF@nt3
              \dimen5=#4\dimen4
       \ifDEK\ifx#1\displaystyle\advance\dimen5 2\dimen5
           \else
         \fi
           \else
          \fi
              \advance\dimen4 -.5\dimen4
              \setbox1=\hbox{$\m@th\@djSt@le{#2}$}
              \dimen0=\dimen1 \advance\dimen0 -\the\dp1
              \advance\dimen0 -\dimen3
              \advance\dimen0 -\dimen4
       \ifdim\dimen5>\dimen0 \advance\dimen1 \dimen5
              \advance\dimen1 -\dimen0
              \D@nomin@tor#1{#3}
           \else
              \D@nomin@tor#1{#3}
        \fi}
\def\D@nomin@tor#1#2{%
              \setbox2=\hbox{$\m@th
              \@djSt@le\Cramp\@djSt@le{#2}$}
              \dimen0=\dimen2 \advance\dimen0 \dimen3
              \advance\dimen0 -\the\ht2
              \advance\dimen0 -\dimen4
        \ifdim\dimen5>\dimen0 \advance\dimen2 \dimen5
              \advance\dimen2 -\dimen0 \Qu@ti@nt#1
            \else
             \Qu@ti@nt#1
       \fi}
\def\Qu@ti@nt#1{%
         \ifdim\wd1<\wd2 \dimen9=\the\wd2
               \setbox3=\hbox to \the\dimen9{%
               $\m@th\hfil\box1\hfil$}
               \setbox1=\box3  \dimen8=\dimen9
         \else \dimen9=\the\wd1
               \setbox3=\hbox to \the\dimen9{%
               $\m@th\hfil\box2\hfil$}
               \setbox2=\box3
         \fi
               \advance\dimen1 -\the\dp1
               \advance\dimen1 -\dimen3
               \advance\dimen1 -\dimen4
               \dimen6=\dimen2 \advance\dimen6 -\the\ht2
               \advance\dimen6 \dimen3
               \advance\dimen6 -\dimen4
               \kern\the\nulldelimiterspace
               \raise-\the\dimen2 \hbox to\the\dimen9{%
               \vbox{\box1\kern\the\dimen1\hrule
               \kern\the\dimen6\box2}}
               \kern\the\nulldelimiterspace}
\def\Xfracc#1#2#3#4{\ifx#1\displaystyle
        \let\@djSt@l@=\textstyle
         \G@toResult{#1}{#2}{#3}{#4}
      \else
        \ifx#1\textstyle \let\@djSt@l@=\scriptstyle
         \G@toResult{#1}{#2}{#3}{#4}
      \else
           \let\@djSt@l@=\scriptscriptstyle
            \G@toResult{#1}{#2}{#3}{#4}
           \fi
        \fi}
\def\G@toResult#1#2#3#4{%
           \Xfrac#1{\Cramp\@djSt@l@{#2}}{#3}{#4}}
\def\xfracc#1#2#3{\mathchoice
      {\Xfrac\displaystyle{\Cramp\textstyle{#1}}{#2}{#3}}%
      {\Xfrac\textstyle{\Cramp\scriptstyle{#1}}{#2}{#3}}%
      {\Xfrac\scriptstyle{\Cramp%
       \scriptscriptstyle{#1}}{#2}{#3}}%
      {\Xfrac\scriptscriptstyle{\Cramp%
          \scriptscriptstyle{#1}}{#2}{#3}}}
\catcode`@=12
\endinput
%%
%% end spreadfraction.sty
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 I would appreciate comments and feedback by email and shall
 summarize and post.


 Michael Herschorn

***********************************************************************

 Michael Herschorn                  email:herschor@gauss.math.mcgill.ca
 Department of Mathematics and Statistics             fax: 514-398-3899
 McGill University                                  phone: 514-398-3825
 805 Sherbrooke Street West
 Montreal QC CANADA  H3A 2K6

***********************************************************************