A fully expandable conversion to hexadecimal numbers

The TeX primitive \number will convert anything that can be converted to a number into a decimal number and what's more, it is expandable. That is \edef\foo{\number\count37} will define \foo to be whatever the value of count register 37 is. It turns out that using ε-TeX's \numexpr, it is possible to write a fully expandable macro \hexnumber that expands to the (positive) hexadecimal that is its argument. It's not quite as good as \number because it is a macro and not a primitive, but it is the best we can do without modifying TeX. To be fully explicit about what is going on here, I've opted to write this in plain TeX (with \numexpr, of course). First, we just need some boilerplate code that compares two numbers. (After I wrote this, I remembered that the LaTeX package etoolbox has a similar macro—in fact, it is implemented in exactly the same way, but that's not really surprising as it's the right way to do it—and so I renamed my macro to match etoolbox.)

\catcode`@11
\def\@firstoftwo#1#2{#1}
\def\@secondoftwo#1#2{#2}
\def\ifnumcomp#1#2#3{%
        \ifnum\numexpr#1\relax#2\numexpr#3\relax
                \expandafter\@firstoftwo
        \else
                \expandafter\@secondoftwo
        \fi
}

The first line is the same as LaTeX's \makeatletter. The next two are standard LaTeX definitions. Next, we need a way to divide two integers and truncate the result. TeX's primitive \divide does exactly this, but it is not expandable. \numexpr can do division; however, it rounds rather than truncates, so we need to handle that ourselves. In particular, if x<round(x/y)*y, then the division was rounded up so we need to subtract 1.

\def\truncdiv#1#2{%
        \ifnumcomp{#1}<{(#1)/(#2)*(#2)}{%
                \numexpr(#1)/(#2)-1%
        }{%
                \numexpr(#1)/(#2)%
        }%
}

This brings us to the main macro.

\def\hexnumber#1{%
        \ifnumcomp{#1}<0{}{\hn@i{#1}{}}%
}

If the argument is negative, expand to nothing. This is similar to the primitive \romannumeral. Otherwise call our first helper function. (Technically, there is no calling going on, just expansion.)

\def\hn@i#1#2{%
        \ifnumcomp{#1}<{16}
        {%
                \hn@digit{#1}#2%
        }{%
                \expandafter\hn@ii\expandafter{%
                        \the\numexpr\truncdiv{#1}{16}%
                }{#1}{#2}%
        }%
}

This macro takes two arguments. The first is the number to convert to hex and the second is the string converted so far. When \hexnumber calls this, it passes its argument as the first argument and an empty argument for the second. If the number to convert is less than 16, convert it to hex followed by what has already been converted. Otherwise, we call the second helper macro where the first argument is the decimal representation of our value to convert divided by 16. (The \expandafters are to prevent large \numexpr expressions from piling up.) The second and third arguments are our value to convert and string converted. Our final helper macro looks complicated, but isn't terribly.

\def\hn@ii#1#2#3{%
        \expandafter\hn@i\expandafter{%
                \number\numexpr#1\expandafter\expandafter\expandafter
                \expandafter\expandafter\expandafter\expandafter}%
                \expandafter\expandafter\expandafter\expandafter
                \expandafter\expandafter\expandafter{%
                        \hn@digit{(#2)-16*(#1)}#3}%
}

Again, we don't really need quite so many \expandafters, but they keep the second (resp. third) argument to \hn@i (resp. \hn@ii) from piling up. This is essentially just expanding to \hn@i{#1}{\hn@digit{(#2)-16*(#1)}#3} except that both arguments are fully expanded. Finally, we need to be able to turn a number 0≤x<16 into a single hex digit. One little caveat is that to act like \number, we need all of the output to have category code 12. To that end, we start a new group, change the catcodes, and then globally define \hn@digit.

\begingroup
\catcode`012\catcode`112\catcode`212\catcode`312\catcode`412
\catcode`512\catcode`612\catcode`712\catcode`812\catcode`912
\catcode`A12\catcode`B12\catcode`C12\catcode`D12\catcode`E12
\catcode`F12
\gdef\hn@digit#1{%
        \ifcase\numexpr#1\relax 0%
        \or \expandafter 1%
        \or \expandafter 2%
        \or \expandafter 3%
        \or \expandafter 4%
        \or \expandafter 5%
        \or \expandafter 6%
        \or \expandafter 7%
        \or \expandafter 8%
        \or \expandafter 9%
        \or \expandafter A%
        \or \expandafter B%
        \or \expandafter C%
        \or \expandafter D%
        \or \expandafter E%
        \or \expandafter F%
        \fi
}
\endgroup

And that's all there is to it!

\vspace* is broken

As LaTeX tutorials are quick to point out, \vspace{x} can disappear at the top or bottom of pages. This is because each \vspace expands to two \vskips which are discardable. Thus, if you write \vspace{3in} near the bottom of the page, the next paragraph will start at the top of the next page.

Of course, if the goal is to start writing the text below the top of the page, then \vspace will not work. The solution, or so say the tutorials, is to use \vspace* instead. This, we're led to believe will always insert that much space. Want to start writing 1 inch from the top margin? No problem, use \vspace*{1in}. The trouble is, that's not quite true. It turns out that TeX has a parameter called \topskip which is related to the amount of glue TeX puts at the top of the page. It's goal is to make the baseline of the first box on the main vertical list of each page start at the same place.

Consider the first line of a page consisted of a single x and the first line of the next page consisted of a single X. We'd like the first line of text on each page to line up, even though they have different heights. So that's exactly what TeX does. If h is the height of the first line of text on the page and t is the value of \topskip, then TeX inserts max {t - h, 0} glue before the first box on the page.

So what does that have to do with \vspace*? The answer lies in how LaTeX implements \vspace*. It is implemented as a 0 height \hrule (because \hrule is not discardable), a penalty to prevent page breaks, and then two \vskips (the second is for 0 pt, presumably so that \unskip doesn't remove it). There is also some code to deal with \vspace* in horizontal mode as well as some bookkeeping involving \prevdepth that isn't important here. Because it starts off with an \hrule, TeX inserts \topskip glue. In practice, this means that \vspace*{0pt} will add vertical space at the top of a page. This is undesirable. TeX by Topic suggests that one use \hbox{}\kern-\topskip when one desires material start at the very top of the page. Obviously, this solution is not workable for \vspace* since it will insert -\topskip space every time. A better solution is to save \topskip, set it equal to 0 pt, perform the space and then restore it. Something like the following.

\begingroup
\topskip0pt
\vspace*{3in}%
\endgroup

Here, we're using TeX's grouping facility to keep the assignment local to the group. A better way is to define a new macro, \Vspace, that performs all of the relevant code. Most of this is copied from the definition of \@vspacer—a helper macro for \vspace*.

\newskip\Vspace@topskip
\providecommand\Vspace[1]{%
        \Vspace@topskip\topskip
        \topskip\z@skip
        \ifvmode
                \dimen@\prevdepth
                \hrule\@height\z@
                \nobreak
                \vskip\glueexpr#1\relax
                \vskip\z@skip
                \prevdepth\dimen@
        \else
                \@bsphack
                \vadjust{%
                        \@restorepar
                        \hrule\@height\z@
                        \nobreak
                        \vskip#1%
                        \vskip\z@skip
                }%
                \@esphack
        \fi
        \topskip\Vspace@topskip
}

This macro can be used wherever \vspace* was used and it will always give exactly the space in the argument. Of course, this might not always be desired. In particular, if you want to start 4 lines down from the top of the page, then \vspace*{4\baselineskip} will do exactly that and keep all lines on various pages aligned. Using \Vspace to get the same space would require \Vspace{4\baselineskip + \topskip}.

Sieve of Eratosthenes

By popular demand, here is the Sieve of Eratosthenes written in Plain TeX.

\newcount\cm\cm100
\newcount\ci
\newcount\cj
\def\prime{prime}
\def\composite{composite}
\def\set#1#2{\global\expandafter\let\csname s\number#1\endcsname#2}
\def\get#1{\csname s\number#1\endcsname}
\def\doprime{%
        \set\ci\prime
        \cj\ci
        \loop
                \advance\cj\ci
                \ifnum\cj<\cm
                        \set\cj\composite
        \repeat
}
\ci2
\loop\ifnum\ci<\cm
        \expandafter\ifx\csname s\number\ci\endcsname\relax
                \begingroup
                \doprime
                \endgroup
        \fi
        \advance\ci1
\repeat
% Print them
\ci2
\loop\ifnum\ci<\cm
        \number\ci: \get\ci\endgraf
        \advance\ci1
\repeat
\bye

Replacing \begingroup with \let\outerbody\body and \endgroup with \let\body\outerbody, you can test even more numbers due to the difference in hash space and save space. Of course, you can test twice as many if you don't bother testing the even numbers.

\newcount\cm\cm300000
\newcount\ci
\newcount\cj
\def\prime{prime}
\def\composite{composite}
\def\set#1#2{\global\expandafter\let\csname s\number#1\endcsname#2}
\def\get#1{\csname s\number#1\endcsname}
\def\doprime{%
        \set\ci\prime
        \cj\ci
        \loop
                \advance\cj\ci
                \advance\cj\ci
                \ifnum\cj<\cm
                        \ifodd\cj
                                \set\cj\composite
                        \fi
        \repeat
}
\set2\prime
\ci3
\loop\ifnum\ci<\cm
        \expandafter\ifx\csname s\number\ci\endcsname\relax
                \let\outerbody\body
                \doprime
                \let\body\outerbody
        \fi
        \advance\ci2
\repeat
% Print them
2: \prime\par
\ci3
\loop\ifnum\ci<\cm
        \number\ci: \ifodd\ci\get\ci\else\composite\fi\endgraf
        \advance\ci1
\repeat
\bye

Hendrik Vogt pointed out that Knuth wrote a different prime computing function in TeX that appears in The TeXbook on page 218. Thanks!

LaTeX's architecture

Someone asked about the architecture of LaTeX. After avoiding the question for a day, I decided to answer. Since it's possible the question will be closed and possibly deleted, I figured I'd reproduce the relevant portion of my answer here. Briefly, LaTeX consists of several layers. The lowest layer is the three hundred or so TeX primitives such as \hbox or \vskip. On top of this, there is the LaTeX kernel. This is a set of macros such as \documentclass, \usepackage, or \begin. The documented source of the kernel is documented in the source2e document. Each document written in LaTeX begins (more or less) with \documentclass{foo} which loads the file foo.cls, for example article.cls or book.cls. This file is another set of macros, this time written using a combination of TeX primitives and macros from the LaTeX kernel. These class files can also load another class to extend it or modify the macros from that class. It can also load packages which are up next. LaTeX packages—that is, files ending in .sty (and their related files)—are additional sets of macros written using any combination of macros from lower levels. Some are specific to particular document classes, others are more general. These packages can be loaded using \usepackage from user documents, or \RequirePackage from classes and other packages. At the highest level, the users' documents use a class file and optionally packages. These documents contain macros written using any combination of TeX primitives and class- and package-defined macros as well as the prose, verse, and mathematics to be typeset. There is a lot more detail that one could go into regarding things like fonts and class and package options, but this isn't so "brief" after all.

Private bibliographies

I have two versions of my cv: one for public consumption and one which contains extra private bibliography entries for work that is in submission. Obviously these are mostly the same so I had one one that had a \newif\ifpublic and then later included the commands to for the private bibliography entries \unless\ifpublic ... \fi. Of course, the source still leaked information about the papers in submission which made it so that I could not distribute the source. The solution to this problem lies with \InputIfFileExists. So, for example, one can write

\InputIfFileExists{private}{}{}

which will \input{private} if private.tex exists. This is pretty good except that now we have to have 2 additional files, both private.tex and the private bibliography file, call it private.bib. Here, we can use a feature of BibTeX to reduce the number of extra files. Since BibTeX ignores all text outside of @{...} and TeX ignores everything in a file after \endinput, we can have the necessary LaTeX code for including the private bibliography at the start of private.bib and end the code with \endinput before the BibTeX entries begin. Then the \InputIfFileExists can be changed to

\InputIfFileExists{private.bib}

Now the source for the cv can be distributed. If a public version of the cv needs to be generated, then one can insert another \endinput at the start of private.bib or temporarily move private.bib aside while building.

Computing the maximum width of two boxes

On occasion, one has cause to compare two boxes and compute the maximum of the two widths. If the two boxes are 0 and 2, then the standard way to do that is something like

\ifdim\wd0>\wd2
    \dimen0 \wd0
\else
    \dimen0 \wd2
\fi

Of course, this works perfectly well. But sometimes, you want something that's a little less easy to read. Thus, I propose

\dimen0 \wd\ifdim\wd0>\wd2 0 \else2 \fi

The spaces around the numbers are to keep TeX from reading more input looking for additional digits.

http://tex.stackexchange.com/

New stack exchange site for TeX and LaTeX. I've probably spent too much time there. I've been meaning to write more here...and I will soon.