%%% A set of macros for various transformations of TeX boxes.
%%% (bases on plain and pdfeTeX primitives)
%
% Version: 2.3
% Author: Pawe/l Jackowski (P.Jackowski@gust.org.pl)
% Public Domain
%
% The macro provides a bunch of TeX box transformations. It was initially
% inspired by trans.tex (BOP, bop@bop.com.pl), remade to work with pdf(e)TeX.
%
% Files:
%     pdf-trans.tex
%     example.tex
%
% History:
%     05.2004, v1.0b
%         * first embryo
%     11.2004, v1.0
%         * modified, released
%     08.2005, v2.0
%         * general rework;
%             - fraction-based trigonometry
%             - pdfliteral dimensions rounding
%             - some other small changes
%     09.2005, v2.1
%         * some adds;
%             - \setbpround <0..4> sets pdf dimen rounding precision
%             - \boxgobble <box> macro swallows <box> and returns \hbox{}
%     10.2005, v2.2
%         * rework;
%             - \bboxtrans vs \cboxtrans switches
%             - \boxext* stuff; yet another scaling approach
%
%     10.2013, v2.3
%         * nasty bug in \roundbponce; a sign was dropped when the number
%           had no integer part (-0.99234 was rounded to 0.99). In current
%           implementation small negative numbers might be rounded to -0.0
%           (minus sign remains)
%         * \roundbponce renamed to \round:bp:once
%

\def\starttrans{%
 \xdef\endtrans{%
  \catcode`\noexpand\@=\the\catcode`\@
  \catcode`\noexpand\:=\the\catcode`\:
 }\catcode`\@=11 \catcode`\:=11 }

\starttrans

% Each transformation macro defined below expands to (h)box. Thus, can be used
% whenever (h)box can. Each must be followed by the box that is to be
% transformed. This scheme allows to cumulate transformations, that are applied
% from right to left. For instance,
%    \boxrotatebb{30}\boxscale{300}\hbox{Aqq}
% first enlarges  boxed `Aqq' three times, then rotates it 30 degrees. Thanks
% to \transboxcheck trick (see below), \copy and \box can be used as well as
% \hbox, \vbox, \vtop. However, the consequence is that the transformed box
% can't be void (\boxinfo definition is an exception here). \transbox register
% contains the box being transformed.

\newbox\transbox

% Whenever we perform relative dimension scaling, a number provided as a macro
% parameter is divided by \transfactor. Default is percentage (ie. 75 -> 0.75).
% One may notice, that we often perform the same operation (ie. division)
% several times instead of defining \transfactor as a float-like string. In
% example, if there are two dimens, \X and \Y, and both are to be scaled by
% 0.23, we always say `\numexpr\X*23/\transfactor' and
% `\numexpr\Y*23/\transfactor' instead of 0.23\X 0.23\Y. In some cases it gives
% significant preciseness improvement.

\newcount\transfactor
\transfactor=100

% plus some internal variables

\newdimen\trans:dim
\newdimen\trans:dim:a
\newdimen\trans:dim:b
\newdimen\trans:dim:c
\newdimen\trans:dim:d

\newcount\trans:count

\def\trans:def{}
\def\trans:def:a{}
\def\trans:def:b{}
\def\trans:def:c{}
\def\trans:def:d{}

% Here the main trick starts. Each macro usually expands to \hbox\bgroup...,
% defines \transboxtodo procedure, assigns the following box, then performs
% \transboxtodo and ends with \egroup. If the box being transformed is generic
% `\hbox{', TeX finishes the assignment just after opening `{', while the box being
% assigned (\transbox) is still void. Thus, we check \ifvoid\transbox, and if
% so, we put \transboxtodo on \aftergroup stack to let TeX finish box
% assignment. Obviously it crashes, if one tries to transform a void box.
% So, if you see 'Missing } inserted' error, you are probably trying to transform
% a void box. You can use \boxinfo to check the situation.

\def\transboxini{%
 \afterassignment\transboxcheck
 \setbox\transbox}

\def\transboxcheck{%
 \ifvoid\transbox
  \expandafter\aftergroup
 \fi\transboxtodo}

% When some macro changes box dimensions directly (i.e. \ht\transbox=0pt),
% we may need to be sure that transformed box behaves like \hbox.

\def\transhboxini{%
 \afterassignment\transhboxcheck
 \setbox\transbox}

\def\transhboxcheck{%
 \ifvoid\transbox
  \expandafter\aftergroup\expandafter\transhboxwrap
 \else
  \expandafter\transhboxwrap
 \fi}

\def\transhboxwrap{%
 \ifvbox\transbox
  \setbox\transbox\hbox{\box\transbox}%
 \fi
 \transboxtodo}

\long\def\transboxdef#1\transboxend{%
 \bgroup\def\transboxtodo{#1\egroup}\transboxini}

\long\def\transhboxdef#1\transboxend{%
 \bgroup\def\transboxtodo{#1\egroup}\transhboxini}

% Sometimes we need to keep bounding box untouched and transform box content
% only (or vice versa). Each transformation that affects both the bounding box
% (via resizing \ht|\wd|\dp) and the box content itself (via \pdfliteral)
% checks \iftransbbox and \iftranscbox flags. No practical reasons to affect
% \boxrotatebb by this feature, however.

\newif\iftransbbox
\newif\iftranscbox
\transbboxtrue
\transcboxtrue

\long\def\transbcboxdef#1\transbboxdef#2\transcboxdef#3\transboxend{%
 \bgroup\edef\transboxtodo{%
  \unexpanded{#1}%
  \iftransbbox\unexpanded{#2}\fi
  \iftranscbox\unexpanded{#3}\else\box\transbox\fi
 \egroup}\transhboxini}

\def\bboxtranson{%
 \hbox\bgroup\transbboxtrue
  \def\transboxtodo{\box\transbox\egroup}\transboxini}

\def\bboxtransoff{%
 \hbox\bgroup\transbboxfalse
  \def\transboxtodo{\box\transbox\egroup}\transboxini}

\def\cboxtranson{%
 \hbox\bgroup\transcboxtrue
  \def\transboxtodo{\box\transbox\egroup}\transboxini}

\def\cboxtransoff{%
 \hbox\bgroup\transcboxfalse
  \def\transboxtodo{\box\transbox\egroup}\transboxini}

\long\def\bboxtrans#1{\iftranscbox\cboxtransoff#1\cboxtranson\else#1\fi}
\long\def\cboxtrans#1{\iftransbbox\bboxtransoff#1\bboxtranson\else#1\fi}

% User-defined operation on \transbox. Remember, that \transboxcheck assumes
% initially void \transbox, so #1 should contain something like \box\tranbox,
% \unvbox\tranbox, \setbox\transbox\voidb@x etc.

\def\hboxtrans{\hbox\transboxdef}
\def\vboxtrans{\vbox\transboxdef}
\def\vtoptrans{\vtop\transboxdef}
\let\boxtrans\hboxtrans

% flip

\def\boxflipx{%
 \hbox\transboxdef
  \savebp\trans:def\wd\transbox
  \pdfliteral{q -1 0 0 1 \trans:def\space 0 cm}%
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxflipy{%
 \hbox\transboxdef
  \savebp\trans:def\wd\transbox
  \pdfliteral{q 1 0 0 -1 0 \tobp{\ht\transbox-\dp\transbox} cm}%
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxflipxy{%
 \hbox\transboxdef
  \savebp\trans:def\wd\transbox
  \pdfliteral{q -1 0 0 -1 \trans:def\space \tobp{\ht\transbox-\dp\transbox} cm}%
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\let\boxflip\boxflipxy

% \boxflipy plus \ht\transbox <-> \dp\transbox exchange

\def\boxflipbase{%
 \hbox\transbcboxdef
 \transbboxdef
  \trans:dim=\ht\transbox
  \ht\transbox=\dp\transbox
  \dp\transbox=\trans:dim
 \transcboxdef
  \pdfliteral{q 1 0 0 -1 0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

% scale by factor

\def\boxscalexy#1#2{%
 \hbox\transbcboxdef
 \transbboxdef
  \wd\transbox=\dimexpr\wd\transbox*(#1)/\transfactor\relax
  \ht\transbox=\dimexpr\ht\transbox*(#2)/\transfactor\relax
  \dp\transbox=\dimexpr\dp\transbox*(#2)/\transfactor\relax
 \transcboxdef
  \edef\trans:def:a{\fdivide{#1}\transfactor}%
  \edef\trans:def:b{\fdivide{#2}\transfactor}%
  \pdfliteral{q \trans:def:a\space 0 0 \trans:def:b\space 0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxscale#1{\boxscalexy{#1}{#1}}
\def\boxscalex#1{\boxscalexy{#1}\transfactor}
\def\boxscaley#1{\boxscalexy\transfactor{#1}}
\let\boxscalez\boxscale

% scale to dimen

\def\boxscaleto#1#2#3#4{%
 \hbox\transbcboxdef
  \trans:dim:a=\dimexpr#1\relax
  \trans:dim:b=\dimexpr#2\relax
  \trans:dim:c=\dimexpr#3\relax
  \trans:dim:d=\dimexpr#4\relax
 \transbboxdef
  \wd\transbox=\dimexpr\wd\transbox*\trans:dim:a/\trans:dim:b\relax
  \ht\transbox=\dimexpr\ht\transbox*\trans:dim:c/\trans:dim:d\relax
  \dp\transbox=\dimexpr\dp\transbox*\trans:dim:c/\trans:dim:d\relax
 \transcboxdef
  \edef\trans:def:a{\fdivide\trans:dim:a\trans:dim:b}%
  \edef\trans:def:b{\fdivide\trans:dim:c\trans:dim:d}%
  \pdfliteral{q \trans:def:a\space 0 0 \trans:def:b\space 0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxscalexyto#1#2{\boxscaleto{#1}{\wd\transbox}{#2}{\ht\transbox+\dp\transbox}}
\def\boxscalexto#1{\boxscaleto{#1}{\wd\transbox}{1sp}{1sp}}
\def\boxscaleyto#1{\boxscaleto{1sp}{1sp}{#1}{\ht\transbox+\dp\transbox}}
\def\boxscalehtto#1{\boxscaleto{1sp}{1sp}{#1}{\ht\transbox}}
\def\boxscaledpto#1{\boxscaleto{1sp}{1sp}{#1}{\dp\transbox}}
\let\boxscalewdto\boxscalexto

% scale \wd|\ht|\dp to dimen, others uniformly

\def\boxuniscaleto#1#2{%
 \hbox\transbcboxdef
  \trans:dim:a=\dimexpr#1\relax
  \trans:dim:b=\dimexpr#2\relax
 \transbboxdef
  \wd\transbox=\dimexpr\wd\transbox*\trans:dim:a/\trans:dim:b\relax
  \ht\transbox=\dimexpr\ht\transbox*\trans:dim:a/\trans:dim:b\relax
  \dp\transbox=\dimexpr\dp\transbox*\trans:dim:a/\trans:dim:b\relax
 \transcboxdef
  \edef\trans:def:a{\fdivide\trans:dim:a\trans:dim:b}%
  \pdfliteral{q \trans:def:a\space 0 0 \trans:def:a\space 0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxuniscalexto#1{\boxuniscaleto{#1}{\wd\transbox}}
\def\boxuniscaleyto#1{\boxuniscaleto{#1}{\ht\transbox+\dp\transbox}}
\def\boxuniscalehtto#1{\boxuniscaleto{#1}{\ht\transbox}}
\def\boxuniscaledpto#1{\boxuniscaleto{#1}{\dp\transbox}}
\let\boxuniscalewdto\boxuniscalexto

% yet another scaling approach; extend \wd|\ht|\dp do dimen, scale accordingly

\def\boxextscaleto#1#2#3{%
 \hbox\transbcboxdef
  \trans:dim:a=\wd\transbox
  \trans:dim:b=\dimexpr#1\relax
  \trans:dim:c=\dimexpr\ht\transbox+\dp\transbox\relax
  \trans:dim:d=\dimexpr#2+#3\relax
  \trans:dim=\dp\transbox
 \transbboxdef
  \wd\transbox=\trans:dim:b
  \ht\transbox=\dimexpr#2\relax
  \dp\transbox=\dimexpr#3\relax
 \transcboxdef
  \edef\trans:def:a{\fdivide\trans:dim:b\trans:dim:a}%
  \edef\trans:def:b{\fdivide\trans:dim:d\trans:dim:c}%
  \savebp\trans:def:c\dimexpr-\dp\transbox+(\dp\transbox+\ht\transbox)*\trans:dim/\trans:dim:c\relax
  \pdfliteral{q \trans:def:a\space 0 0 \trans:def:b\space 0 \trans:def:c\space cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxextscale#1#2#3{\boxextscaleto{\wd\transbox+#1}{\ht\transbox+#2}{\dp\transbox+#3}}

% resizing; the result may be different for hboxes and vboxes;
% since we have \boxextscaleto plus \iftransbbox and \iftranscbox flags,
% the following two macros are obsolete by now

\def\boxresizeto#1#2#3{%
 \hbox\transboxdef
  \ifx\relax#1\relax\else \wd\transbox=\dimexpr#1\relax \fi
  \ifx\relax#2\relax\else \ht\transbox=\dimexpr#2\relax \fi
  \ifx\relax#3\relax\else \dp\transbox=\dimexpr#3\relax \fi
  \box\transbox
 \transboxend}

\def\boxresize#1#2#3{%
 \hbox\transboxdef
  \ifx\relax#1\relax\else \wd\transbox=\dimexpr\wd\transbox+#1\relax \fi
  \ifx\relax#2\relax\else \ht\transbox=\dimexpr\ht\transbox+#2\relax \fi
  \ifx\relax#3\relax\else \dp\transbox=\dimexpr\dp\transbox+#3\relax \fi
  \box\transbox
 \transboxend}

% extents; to keep things consistent for negative extents, \dp\transbox becomes 0pt

\def\boxextents#1#2#3#4{% <left> <right> <top> <bottom>
 \hbox\transboxdef
  \kern\dimexpr#1\relax
   \vbox{%
    \kern\dimexpr#3\relax
    \box\transbox
    \kern\dimexpr#4\relax
   }%
  \kern\dimexpr#2\relax
 \transboxend}

\def\boxhextent#1#2{\boxextents{#1}{#2}\z@\z@}
\def\boxvextent#1#2{\boxextents\z@\z@{#1}{#2}}
\def\boxextent#1{\boxextents{#1}{#1}{#1}{#1}}

% and yet another approach; append extents to box content

\def\boxexts#1#2#3#4{% <left> <right> <top> <bottom>
 \hbox\transboxdef
  \trans:dim:a=\wd\transbox
  \trans:dim:b=\dimexpr\trans:dim:a+#1+#2\relax
  \trans:dim:c=\dimexpr\ht\transbox+\dp\transbox\relax
  \trans:dim:d=\dimexpr\trans:dim:c+#3+#4\relax
  \edef\trans:def:a{\fdivide\trans:dim:b\trans:dim:a}%
  \edef\trans:def:b{\fdivide\trans:dim:d\trans:dim:c}%
  \savebp\trans:def:c-\dimexpr#1\relax
  \savebp\trans:def:d\dimexpr-#4+(#3+#4)*\dp\transbox/\trans:dim:c\relax
  \pdfliteral{q \trans:def:a\space 0 0 \trans:def:b\space
                \trans:def:c\space \trans:def:d\space cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxhext#1#2{\boxexts{#1}{#2}\z@\z@}
\def\boxvext#1#2{\boxexts\z@\z@{#1}{#2}}
\def\boxext#1{\boxexts{#1}{#1}{#1}{#1}}

% raw translation

\def\boxtranslate#1#2{%
 \hbox\transboxdef
  \pdfliteral{q 1 0 0 1 \tobp{#1} \tobp{#2} cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

% heuristic rotation by 90 degrees (left and right variants)

\def\boxrevolveleft{%
 \hbox\transbcboxdef
  \trans:dim:a=\wd\transbox
  \trans:dim:b=\ht\transbox
 \transbboxdef
  \wd\transbox=\dimexpr\ht\transbox+\dp\transbox\relax
  \ht\transbox=\trans:dim:a
  \dp\transbox=\z@
 \transcboxdef
  \pdfliteral{q 0 1 -1 0 \tobp{\trans:dim:b} 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxrevolveright{%
 \hbox\transbcboxdef
  \trans:dim:a=\wd\transbox
  \trans:dim:b=\dp\transbox
 \transbboxdef
  \wd\transbox=\dimexpr\ht\transbox+\dp\transbox\relax
  \ht\transbox=\trans:dim:a
  \dp\transbox=\z@
 \transcboxdef
  \pdfliteral{q 0 -1 1 0 \tobp{\trans:dim:b} \tobp{\trans:dim:a} cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\let\boxrevolvepi\boxflipxy

% clockwise rotation relative to base point

\def\boxrotate#1{%
 \hbox\transboxdef
  \floatsincos\trans:def:a\trans:def:b{#1}%
  \pdfliteral{q \trans:def:b\space
                \negbp\trans:def:a\space % clockwise
                \trans:def:a\space       % vs anti clockwise
                \trans:def:b\space
                0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

% clockwise rotation relative to base point translated (x,y)

\def\boxrotatexy#1#2#3{%
 \hbox\transboxdef
  \floatsincos\trans:def:a\trans:def:b{#1}%
  \savebp\trans:def:c\dimexpr#2\relax
  \savebp\trans:def:d\dimexpr#3\relax
  \pdfliteral{q \trans:def:b\space
                \negbp\trans:def:a\space
                \trans:def:a\space
                \trans:def:b\space
                \trans:def:c\space
                \trans:def:d\space cm
                1 0 0 1 \negbp\trans:def:c\space
                        \negbp\trans:def:d\space cm}% ?
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\def\boxrotatec#1{\boxrotatexy{#1}{\wd\transbox/2}{(\ht\transbox-\dp\transbox)/2}}
\def\boxrotatell#1{\boxrotatexy{#1}\z@{-\dp\transbox}}
\def\boxrotatelr#1{\boxrotatexy{#1}{\wd\transbox}{-\dp\transbox}}
\def\boxrotateul#1{\boxrotatexy{#1}\z@{\ht\transbox}}
\def\boxrotateur#1{\boxrotatexy{#1}{\wd\transbox}{\ht\transbox}}

% bbox-wise rotation (left and right variants)

\newif\ifbboxright

\def\box:rotate:bb#1{%
 \trans:dim:a=\wd\transbox
 \trans:dim:b=\ht\transbox
 \trans:dim:c=\dp\transbox
 \trans:dim:d=\dimexpr\ht\transbox+\dp\transbox\relax
 \trans:count=\reducetrigangle{#1}\fractperiod\relax
 \ifcase\fracttrigfourth\trans:count\relax
  \fr@ct:sin:cos:i\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr\fr@ct:mul\trans:dim:a\trans:def:b
                      +\fr@ct:mul\trans:dim:d\trans:def:a\relax
  \ht\transbox=\dimexpr\fr@ct:mul\trans:dim:b\trans:def:b\relax
  \dp\transbox=\dimexpr\fr@ct:mul\trans:dim:a\trans:def:a
                      +\fr@ct:mul\trans:dim:c\trans:def:b\relax
  \savebp\trans:def:c=\dimexpr\fr@ct:mul\trans:dim:c\trans:def:a\relax
 \or
  \fr@ct:sin:cos:ii\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr-\fr@ct:mul\trans:dim:a\trans:def:b
                       +\fr@ct:mul\trans:dim:d\trans:def:a\relax
  \ht\transbox=\dimexpr-\fr@ct:mul\trans:dim:c\trans:def:b\relax
  \dp\transbox=\dimexpr-\fr@ct:mul\trans:dim:b\trans:def:b
                        +\fr@ct:mul\trans:dim:a\trans:def:a\relax
  \savebp\trans:def:c=\dimexpr-\fr@ct:mul\trans:dim:a\trans:def:b
                              +\fr@ct:mul\trans:dim:c\trans:def:a\relax
 \or
  \fr@ct:sin:cos:iii\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr-\fr@ct:mul\trans:dim:a\trans:def:b
                       -\fr@ct:mul\trans:dim:d\trans:def:a\relax
  \ht\transbox=\dimexpr-\fr@ct:mul\trans:dim:a\trans:def:a
                       -\fr@ct:mul\trans:dim:c\trans:def:b\relax
  \dp\transbox=\dimexpr-\fr@ct:mul\trans:dim:b\trans:def:b\relax
  \savebp\trans:def:c=\dimexpr-\fr@ct:mul\trans:dim:a\trans:def:b
                              -\fr@ct:mul\trans:dim:b\trans:def:a\relax
 \or
  \fr@ct:sin:cos:iv\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr\fr@ct:mul\trans:dim:a\trans:def:b
                      -\fr@ct:mul\trans:dim:d\trans:def:a\relax
  \ht\transbox=\dimexpr\fr@ct:mul\trans:dim:b\trans:def:b
                      -\fr@ct:mul\trans:dim:a\trans:def:a\relax
  \dp\transbox=\dimexpr\fr@ct:mul\trans:dim:c\trans:def:b\relax
  \savebp\trans:def:c=\dimexpr-\fr@ct:mul\trans:dim:b\trans:def:a\relax
 \fi
 \ifbboxright
  \trans:dim:d=\dimexpr\fr@ct:mul\trans:dim:a\trans:def:a\relax
  \ht\transbox=\dimexpr\ht\transbox+\trans:dim:d\relax
  \dp\transbox=\dimexpr\dp\transbox-\trans:dim:d\relax
  \savebp\trans:def:d\trans:dim:d
 \else
  \def\trans:def:d{0}%
 \fi
 \edef\trans:def:a{\fr@ct:div\trans:def:a}%
 \edef\trans:def:b{\fr@ct:div\trans:def:b}%
 \pdfliteral{q \trans:def:b\space
               \negbp\trans:def:a\space
               \trans:def:a\space
               \trans:def:b\space
               \trans:def:c\space
               \trans:def:d\space cm}%
 \savebp\trans:def=\wd\transbox
 \box\transbox
 \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}}

\def\boxrotatebbl#1{%
 \hbox\transboxdef
  \bboxrightfalse
  \box:rotate:bb{#1}%
 \transboxend}

\def\boxrotatebbr#1{%
 \hbox\transboxdef
  \bboxrighttrue
  \box:rotate:bb{#1}%
 \transboxend}

% the default is left variant

\bboxrightfalse

\def\boxrotatebb#1{%
 \hbox\transboxdef
  \box:rotate:bb{#1}%
 \transboxend}

% slanting; \boxslantx{<x>}\boxslanty{<y>} is NOT equivalent to \boxslantxy{<x>}{<y>}

\def\boxslant#1#2{%
 \hbox\transboxdef
  \fractsincos\trans:def:a\trans:def:b{#1}%
  \fractsincos\trans:def:c\trans:def:d{#2}%
  \edef\trans:def:a{\fdivide\trans:def:a\trans:def:b}%
  \edef\trans:def:c{\fdivide\trans:def:c\trans:def:d}%
  \pdfliteral{q 1 \trans:def:c\space \trans:def:a\space 1 0 0 cm}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

\let\boxslantxy\boxslant
\def\boxslantx#1{\boxslant{#1}{0}} % not \z@!!
\def\boxslanty#1{\boxslant{0}{#1}}

% bounding box wise slanting (left and right variant)

\def\box:slant:bb#1#2{%
 \trans:dim:a=\wd\transbox
 \trans:dim:b=\ht\transbox
 \trans:dim:c=\dp\transbox
 \trans:dim:d=\dimexpr\ht\transbox+\dp\transbox\relax
 \trans:count=\reducetrigangle{#1}{2*\fractfourth}\relax
 \ifcase\fracttrigfourth\trans:count\relax
  \fr@ct:sin:cos:i\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr\trans:dim:a+\trans:dim:d*\trans:def:a/\trans:def:b\relax
  \savebp\trans:def:c=\dimexpr\trans:dim:c*\trans:def:a/\trans:def:b\relax
 \or
  \fr@ct:sin:cos:ii\trans:def:a\trans:def:b\trans:count
  \wd\transbox=\dimexpr\trans:dim:a-\trans:dim:d*\trans:def:a/\trans:def:b\relax
  \savebp\trans:def:c=\dimexpr-\trans:dim:b*\trans:def:a/\trans:def:b\relax
 \fi
 \edef\trans:def{\fdivide\trans:def:a\trans:def:b}%
 \trans:count=\reducetrigangle{#2}{2*\fractfourth}\relax
 \ifcase\fracttrigfourth\trans:count\relax
  \fr@ct:sin:cos:i\trans:def:a\trans:def:b\trans:count
  \ht\transbox=\dimexpr\trans:dim:b+\trans:dim:a*\trans:def:a/\trans:def:b\relax
 \or
  \fr@ct:sin:cos:ii\trans:def:a\trans:def:b\trans:count
  \dp\transbox=\dimexpr\trans:dim:c-\trans:dim:a*\trans:def:a/\trans:def:b\relax
 \fi
 \ifbboxright
  \trans:dim:d=\dimexpr-\trans:dim:a*\trans:def:a/\trans:def:b\relax
  \ht\transbox=\dimexpr\ht\transbox+\trans:dim:d\relax
  \dp\transbox=\dimexpr\dp\transbox-\trans:dim:d\relax
  \savebp\trans:def:d\trans:dim:d
 \else
  \def\trans:def:d{0}%
 \fi
 \edef\trans:def:a{\fdivide\trans:def:a\trans:def:b}%
 \pdfliteral{q 1
               \trans:def:a\space
               \trans:def\space
               1
               \trans:def:c\space
               \trans:def:d\space cm}%
 \savebp\trans:def=\wd\transbox
 \box\transbox
 \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}}

\def\boxslantbbl#1#2{%
 \hbox\transboxdef
  \bboxrightfalse
  \box:slant:bb{#1}{#2}%
 \transboxend}

\def\boxslantbbr#1#2{%
 \hbox\transboxdef
  \bboxrighttrue
  \box:slant:bb{#1}{#2}%
 \transboxend}

\def\boxslantbb#1#2{%
 \hbox\transboxdef
  \box:slant:bb{#1}{#2}%
 \transboxend}

\def\boxslantbbx#1{\boxslantbb{#1}{0}} % not \z@!!
\def\boxslantbby#1{\boxslantbb{0}{#1}}
\def\boxslantbbry#1{\boxslantbbr{0}{#1}}
\def\boxslantbbly#1{\boxslantbbl{0}{#1}}

% on-fly conversion to XObject (\pdfxform)

\def\boxxform{%
 \hbox\transboxdef
  \immediate\pdfxform\transbox
  \pdfrefxform\pdflastxform
 \transboxend}

\def\boxxformspec#1\boxxform{%
 \hbox\transboxdef
  \immediate\pdfxform#1\transbox
  \pdfrefxform\pdflastxform
 \transboxend}

% Some previous version of pdftrans had \boxclip defined as \boxpath{}{W n}. This didn't
% work properly since \boxpath restores the graphic state before placing the box itself
% (see below).

% clipping

\def\boxclip{%
 \hbox\transboxdef
  \savebp\trans:def\wd\transbox
  \pdfliteral{q 0 \tobp{-\dp\transbox} \trans:def\space
              \tobp{\ht\transbox+\dp\transbox} re W n}%
  \box\transbox
  \pdfliteral{Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

% up and down

\def\boxraise#1{%
 \hbox\transboxdef
  \raise\dimexpr#1\relax\box\transbox
 \transboxend}

\def\boxlower#1{%
 \hbox\transboxdef
  \lower\dimexpr#1\relax\box\transbox
 \transboxend}

\let\boxbaselineup\boxlower
\let\boxbaselinedown\boxraise

\def\boxbaselineat#1{% baseline at #1 of the total box vdim
 \hbox\transboxdef
  \lower\dimexpr(\ht\transbox+\dp\transbox)*(#1)/\transfactor-\dp\transbox\relax
  \box\transbox
 \transboxend}

\def\boxmoveleft#1{%
 \vbox\transboxdef
  \moveleft\dimexpr#1\relax\box\transbox
 \transboxend}

\def\boxmoveright#1{%
 \vbox\transboxdef
  \moveright\dimexpr#1\relax\box\transbox
 \transboxend}

% rule-like spec of the box size (risky hack)

\def\b@x:rule#1#2#{%
 #1\transboxdef
  \setbox0\hbox{\vrule width\wd\transbox height\ht\transbox depth\dp\transbox #2}%
  \wd\transbox=\wd0
  \ht\transbox=\ht0
  \dp\transbox=\dp0
  \box\transbox
 \transboxend#1}

\def\hboxr{\b@x:rule\hbox}
\def\vboxr{\b@x:rule\vbox}
\def\vtopr{\b@x:rule\vtop}
\let\boxr\hboxr

% reboxing; restore box natural dimensions (without shrink or stretch).

\def\revbox{%
 \vbox\transboxdef
  \unvbox\transbox
 \transboxend}

\def\revtop{%
 \vtop\transboxdef
  \unvbox\transbox
 \transboxend}

\def\rehbox{%
 \hbox\transboxdef
  \unhbox\transbox
 \transboxend}

% just wrap

\def\hboxwrap{%
 \hbox\transboxdef
  \box\transbox
 \transboxend}

\def\vboxwrap{%
 \vbox\transboxdef
  \box\transbox
 \transboxend}

\def\vtopwrap{%
 \vtop\transboxdef
  \box\transbox
 \transboxend}

% show boxes

\def\boxshow#1#2#3{%
 \hbox\transboxdef
  \savebp\trans:def:a\wd\transbox
  \savebp\trans:def:b\dimexpr\ht\transbox+\dp\transbox\relax
  \savebp\trans:def:c\dp\transbox
  \box\transbox
  \pdfliteral{%
  q #1
  1 0 0 1 \negbp\trans:def:a\space 0 cm
  0 \negbp\trans:def:c\space \trans:def:a\space \trans:def:b\space re S
  #2
  0 0 m
  \trans:def:a\space 0 l S #3 Q}%
 \transboxend}

\def\boxsh{\boxshow{0 0 1 RG}{0 0 .8 RG [2 2]1 d}{}}

% box painted according to graphic state parameters (ie. on layer).

\def\boxgs#1#2{%
 \hbox\transboxdef
  \pdfliteral{q #1}%
  \savebp\trans:def\wd\transbox
  \box\transbox
  \pdfliteral{#2 Q 1 0 0 1 \trans:def\space 0 cm}%
 \transboxend}

% markers around the box

\def\boxmarkers#1#2#3{%
 \hbox\transboxdef
  \copy\transbox
  \trans:dim:a=\dimexpr#1\relax
  \trans:dim:b=\dimexpr#2\relax
  \pdfliteral{q #3}%
  \savebp\trans:def-\dp\transbox
  \box:markers:h
  \savebp\trans:def\ht\transbox
  \box:markers:h
  \savebp\trans:def-\wd\transbox
  \box:markers:v
  \savebp\trans:def\z@
  \box:markers:v
  \pdfliteral{S Q}%
  \setbox\transbox\box\voidb@x
 \transboxend}

\def\box:markers:h{%
  \savebp\trans:def:a\trans:dim:a
  \savebp\trans:def:b\trans:dim:b
  \pdfliteral{%
   \trans:def:a\space\trans:def\space m \trans:def:b\space\trans:def\space l}%
  \savebp\trans:def:a\dimexpr-\wd\transbox-\trans:dim:a\relax
  \savebp\trans:def:b\dimexpr-\wd\transbox-\trans:dim:b\relax
  \pdfliteral{%
   \trans:def:a\space\trans:def\space m \trans:def:b\space\trans:def\space l}}

\def\box:markers:v{%
  \savebp\trans:def:a\dimexpr-\dp\transbox-\trans:dim:a\relax
  \savebp\trans:def:b\dimexpr-\dp\transbox-\trans:dim:b\relax
  \pdfliteral{%
   \trans:def\space \trans:def:a\space m \trans:def\space \trans:def:b\space l}%
  \savebp\trans:def:a\dimexpr\ht\transbox+\trans:dim:a\relax
  \savebp\trans:def:b\dimexpr\ht\transbox+\trans:dim:b\relax
  \pdfliteral{%
   \trans:def\space \trans:def:a\space m \trans:def\space \trans:def:b\space l}}

% for sake of compatibility...

\def\boxm#1#2{\boxmarkers{#1}{#2}{}}
\let\boxmarks\boxmarkers

% simple phantom

\def\boxphantom{%
 \hbox\transboxdef
  \hbox to\wd\transbox
   {\vrule width\z@ height\ht\transbox depth\dp\transbox\hss}%
 \transboxend}

% simple smash

\def\boxsmash{%
 \hbox\transhboxdef
  \wd\transbox=\z@
  \ht\transbox=\z@
  \dp\transbox=\z@
  \box\transbox
 \transboxend}

\def\hboxsmash{%
 \hbox\transhboxdef
  \wd\transbox=\z@
  \box\transbox
 \transboxend}

\def\vboxsmash{%
 \vbox\transhboxdef
  \ht\transbox=\z@
  \dp\transbox=\z@
  \box\transbox
 \transboxend}

\def\boxgobble{%
 \hbox\transboxdef
%  \global\setbox\transbox=\box\voidb@x
 \transboxend}

% say something about the following box (this one can be followed void box)

\def\box:about#1{%
 \hbox\bgroup
  \def\transboxtodo{%
   \trans:dim:a=\wd\transbox
   \trans:dim:b=\ht\transbox
   \trans:dim:c=\dp\transbox
   \box\transbox
   \hbox to\z@{\hss
    \hbox to\trans:dim:a{\hss
     \lower\trans:dim:c\vbox to\z@{\vss
      \vbox to\dimexpr\trans:dim:b+\trans:dim:c{\vss\tt
       #1\vbox{\vskip1ex
        \halign{\hskip1ex plus 1fil####&####\hskip1ex plus 1fil\cr
         \trans:def\span\cr
         wd & \the\trans:dim:a\cr
         ht & \the\trans:dim:b\cr
         dp & \the\trans:dim:c\cr}%
       \vskip1ex
       }%
      \vss}%
     }%
    \hss}%
   }%
   \egroup}\def\trans:def{}\def\trans:def:a{}\box:@bout}

\def\box:@bout#1{%
 \ifcase
  \ifx#1\hbox 0 \else
  \ifx#1\vbox 1 \else
  \ifx#1\vtop 2 \else
  \ifx#1\box  3 \else
  \ifx#1\copy 4 \else 5 \fi\fi\fi\fi\fi
 \edef\trans:def{\trans:def\string\hbox}\expandafter\transboxini\or
 \edef\trans:def{\trans:def\string\vbox}\expandafter\transboxini\or
 \edef\trans:def{\trans:def\string\vtop}\expandafter\transboxini\or
 \edef\trans:def{\trans:def\string\box}\expandafter\boxabout:register\or
 \edef\trans:def{\trans:def\string\copy}\expandafter\boxabout:register\or
 \ifx#1\trans:def:a\errmessage{`#1' is not a box}\fi\let\trans:def:a#1% endless loop otherwise
 \edef\trans:def{\trans:def\string#1->}\expandafter\expandafter\expandafter\box:@bout\fi
 #1}

\def\boxabout:register#1{%
 \let\trans:def:a#1%
 \afterassignment\boxabout:r@gister\trans:count}

\def\boxabout:r@gister{%
 \edef\trans:def{\trans:def\the\trans:count\space
  (\ifvoid\trans:count void\else
   \ifhbox\trans:count hbox\else
   \ifvbox\trans:count vbox\fi\fi\fi)}%
 \afterassignment\transboxtodo
 \setbox\transbox\trans:def:a\trans:count}

\def\boxinfo{\box:about{\boxpath{.3 w 0 G 0 0 .3 0 k}{B}}}
\def\boxabout#1{\box:about{\boxgs{#1}{}}}

% paint box path as a background of the box

\def\boxpath#1#2{%
 \hbox\transboxdef
  \savebp\trans:def:a\wd\transbox
  \savebp\trans:def:b\dimexpr\ht\transbox+\dp\transbox\relax
  \savebp\trans:def:c\dp\transbox
  \pdfliteral{q #1
              0
              \negbp\trans:def:c\space
              \trans:def:a\space
              \trans:def:b\space re #2 Q}%
  \box\transbox
 \transboxend}

\def\boxroundpath#1#2#3{%
 \hbox\transboxdef
  \savebp\trans:def:a\wd\transbox
  \savebp\trans:def:b\ht\transbox
  \savebp\trans:def:c\dp\transbox
  \trans:dim:d=\dimexpr#1\relax
  \savebp\trans:def:d\trans:dim:d
  \pdfliteral{q #2}%
  \pdfliteral{0 \trans:def:d\space m}%
  \savebp\trans:def\dimexpr\ht\transbox-\trans:dim:d \relax
  \pdfliteral{%
   0 \trans:def\space l 0 \trans:def:b\space \trans:def:d\space \trans:def:b\space y}%
  \savebp\trans:def\dimexpr\wd\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def\space \trans:def:b\space l}%
  \savebp\trans:def\dimexpr\ht\transbox-\trans:dim:d \relax
  \pdfliteral{%
   \trans:def:a\space \trans:def:b\space \trans:def:a\space \trans:def\space y}%
  \savebp\trans:def\dimexpr\dp\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def:a\space \negbp\trans:def\space l}%
  \savebp\trans:def\dimexpr\wd\transbox-\trans:dim:d \relax
  \pdfliteral{%
   \trans:def:a\space \negbp\trans:def:c\space \trans:def\space \negbp\trans:def:c\space y
   \trans:def:d\space \negbp\trans:def:c\space l}%
  \savebp\trans:def\dimexpr\dp\transbox-\trans:dim:d \relax
  \pdfliteral{%
   0 \negbp\trans:def:c\space 0 \negbp\trans:def\space y 0 \trans:def:d\space l}%
  \pdfliteral{h #3 Q}%
  \box\transbox
 \transboxend}

\def\boxedgypath#1#2#3{%
 \hbox\transboxdef
  \savebp\trans:def:a\wd\transbox
  \savebp\trans:def:b\ht\transbox
  \savebp\trans:def:c\dp\transbox
  \trans:dim:d=\dimexpr#1\relax
  \savebp\trans:def:d\trans:dim:d
  \pdfliteral{q #2}%
  \pdfliteral{0 \trans:def:d\space m}%
  \savebp\trans:def\dimexpr\ht\transbox-\trans:dim:d \relax
  \pdfliteral{0 \trans:def\space l \trans:def:d\space \trans:def:b\space l}%
  \savebp\trans:def\dimexpr\wd\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def\space \trans:def:b\space l}%
  \savebp\trans:def\dimexpr\ht\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def:a\space \trans:def\space l}%
  \savebp\trans:def\dimexpr\dp\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def:a\space \negbp\trans:def\space l}%
  \savebp\trans:def\dimexpr\wd\transbox-\trans:dim:d \relax
  \pdfliteral{\trans:def\space \negbp\trans:def:c\space l
   \trans:def:d\space \negbp\trans:def:c\space l}%
  \savebp\trans:def\dimexpr\dp\transbox-\trans:dim:d \relax
  \pdfliteral{0 \negbp\trans:def\space l 0 \trans:def:d\space l}%
  \pdfliteral{h #3 Q}%
  \box\transbox
 \transboxend}

% obsolete

\let\boxsquarepath\boxedgypath

%%% ARITHMETIC

% some shortcuts

\def\expandnumberafter#1#2{\expandafter#1\expandafter{\number#2}}

\def\expandtwonumbersafter#1#2#3{%
 \expandafter#1\expandafter
 {\number#2\expandafter}\expandafter
 {\number#3}}

\def\expandthreenumbersafter#1#2#3#4{%
 \expandafter#1\expandafter
 {\number#2\expandafter}\expandafter
 {\number#3\expandafter}\expandafter
 {\number#4}}

\def\expandnumexprafter#1#2{\expandafter#1\expandafter{\number\numexpr#2}}

\def\expandtwonumexprafter#1#2#3{%
 \expandafter#1\expandafter
 {\number\numexpr#2\expandafter}\expandafter
 {\number\numexpr#3}}

\def\expandthreenumexprafter#1#2#3#4{%
 \expandafter#1\expandafter
 {\number\numexpr#2\expandafter}\expandafter
 {\number\numexpr#3\expandafter}\expandafter
 {\number\numexpr#4}}

\def\expanddimexprafter#1#2{\expandafter#1\expandafter{\the\dimexpr#2}}

% Whenever we write a dimen into PDF code, we need to convert it from TeX units
% to Postscript big points. We handle that as precise as possible, using the
% fact that eTeX handles 64bit numbers as temporary results of expressions such
% as A*B/C. A is dimension in points, B is \pt:f@ctor and C is \bp:f@ctor. Note
% that factor 100pt/100bp is more precise than 1pt/1bp or 10pt/10bp, but NOT
% less precise than 1000pt/1000bp. Thus, 100 is the optimum.

\edef\pt:f@ctor{\number\dimexpr100pt} % NOT \dimexpr100\p@!
\edef\bp:f@ctor{\number\dimexpr100bp} %

\begingroup
 \catcode`\P=12
 \catcode`\T=12
 \lccode`P=`p
 \lccode`T=`t
 \lowercase{\gdef\with@ut:pt#1PT{#1}}
\endgroup

\def\withoutpt{\expandafter\with@ut:pt}
\def\big:p@ints#1#2{#1\the\dimexpr#2*\pt:f@ctor/\bp:f@ctor\relax}
\def\negbp#1{\withoutpt\the\dimexpr-#1pt\relax}

% If we assume that \pdfdecimaldigits never exceeds possible range (0..4),
% we can implement PDF dimens rounding in the following way:

\def\t@bp{\big:p@ints\withoutpt}

\def\roundbp#1{%
 \expandafter\expandafter
 \csname r@und:bp:\romannumeral\pdfdecimaldigits\expandafter\endcsname
 \expandafter\with@ut:pt\the\dimexpr(#1)*\pt:f@ctor/\bp:f@ctor\relax0000\relax}

\def\r@und:bp:    #1.#2#3\relax{\number\numexpr#1#2/10\relax}
\def\r@und:bp:i   #1.#2#3#4\relax{\round:bp:once{#1}{#2#3}\relax}
\def\r@und:bp:ii  #1.#2#3#4#5\relax{\round:bp:once{#1}{#2#3#4}\relax}
\def\r@und:bp:iii #1.#2#3#4#5#6\relax{\round:bp:once{#1}{#2#3#4#5}\relax}
\def\r@und:bp:iv  #1.#2#3#4#5#6#7\relax{\round:bp:once{#1}{#2#3#4#5#6}\relax}

% To speed-up things one may say

\def\setbpround#1{% 0..4
 \edef\roundbp##1{%
  \unexpanded{\expandafter\expandafter\expandafter}\expandafter\noexpand
  \csname r@und:bp:\romannumeral#1\endcsname
  \unexpanded{\expandafter\with@ut:pt\the}%
  \dimexpr(##1)*\unexpanded{\pt:f@ctor/\bp:f@ctor}\relax0000\relax}}

\def\round:bp:once#1#2{%
 %\number\numexpr#1\ifnum#1<0-\else+\fi % 15.10.2013: wrong! that drops minus sign in case -0.xxxx
 %(\m@ne+\expandafter\r@und:bp:once\number\numexpr1#2/10\relax}
 \ifnum#11<0-\number\numexpr-\else\number\numexpr\fi
 #1+(\m@ne+\expandafter\r@und:bp:once\number\numexpr1#2/10\relax}

\def\r@und:bp:once#1#2\relax{#1)\relax\ifnum#2>0.#2\fi}

\def\savebp#1{%
 \def\s@vebp{%
  \edef#1{\tobp{\bp:dim@n}}}%
 \afterassignment\s@vebp\bp:dim@n}

\newdimen\bp:dim@n

\def\disablebpround{\let\tobp\t@bp}
\def\enablebpround{\let\tobp\roundbp}

%\disablebpround
\enablebpround

% Lets play with basic arithmetic operations. To make things consistent, each
% function expands to \numexpr|\dimexpr, even if could be easily expanded to
% digits. This approach ensures predictable behaviour whenever a \function is
% followed by \expandafter or \relax. To avoid evaluating the same expressions
% twice or more, each function expands its parameters before performing the
% final operation. This scheme makes temporary macros reusable.

% absolute value

\def\absint{\expandnumexprafter\absoluteint}
\def\absoluteint#1{\numexpr\ifnum#1<\z@-\fi#1}
\def\absdim{\expanddimexprafter\absolutedim}
\def\absolutedim#1{\dimexpr\ifdim#1<\z@-\fi#1}

% Various approaches to integer division:
% floor(a/b) -- the largest integer LOWER than a/b
%  ceil(a/b) -- the lowest integer HIGHER than a/b
%   int(a/b) -- the integer part (floor for a/b>=0 and ceil for a/b<0)
%  nint(a/b) -- rounding (the nearest integer)

\def\expanddivisionafter#1#2#3{%
 \expandnumexprafter#1{#2/#3}{#2}{#3}}

\def\divfloor{\expandtwonumexprafter\dividefloor}
\def\dividefloor{\expanddivisionafter\divide:fl@@r}
\def\divide:fl@@r#1#2#3{%
 \numexpr#1%
 \ifcase\ifnum#2<0 \ifnum#3<0 1 \else 0 \fi
        \else      \ifnum#3<0 1 \else 0 \fi \fi
 \ifnum\numexpr#1*#3>#2-\@ne\fi\or
 \ifnum\numexpr#1*#3<#2-\@ne\fi\fi}

\def\divceil{\expandtwonumexprafter\divideceil}
\def\divideceil{\expanddivisionafter\divide:c@il}
\def\divide:c@il#1#2#3{%
 \numexpr#1%
 \ifcase\ifnum#2<0 \ifnum#3<0 1 \else 0 \fi
        \else      \ifnum#3<0 1 \else 0 \fi \fi
 \ifnum\numexpr#1*#3<#2+\@ne\fi\or
 \ifnum\numexpr#1*#3>#2+\@ne\fi\fi}

\def\divint{\expandtwonumexprafter\divideint}
\def\divideint{\expanddivisionafter\divide:int}
\def\divide:int#1#2#3{%
 \numexpr#1%
 \ifcase\ifnum#2<0 \ifnum#3<0 3 \else 1 \fi
        \else      \ifnum#3<0 2 \else 0 \fi \fi
 \ifnum\numexpr#1*#3>#2-\@ne\fi\or
 \ifnum\numexpr#1*#3<#2+\@ne\fi\or
 \ifnum\numexpr#1*#3>#2+\@ne\fi\or
 \ifnum\numexpr#1*#3<#2-\@ne\fi\fi}

\def\divnint{\expandtwonumexprafter\dividenint}
\def\dividenint#1#2{\numexpr#1/#2}

% modulo

\def\mod{\expandtwonumexprafter\modulo}
\def\modulo{\expanddivisionafter\do:m@dulo}
\def\do:m@dulo#1#2#3{\numexpr#2-#3*\divide:fl@@r{#1}{#2}{#3}\relax}

% If we don't divide (by) negative numbers, the following macros works
% a bit faster.

\def\divfloorpos{\expandtwonumexprafter\dividefloorpos}
\def\dividefloorpos{\expanddivisionafter\divide:fl@@r:pos}
\def\divide:fl@@r:pos#1#2#3{\numexpr#1\ifnum\numexpr#1*#3>#2-\@ne\fi}

\def\divceilpos{\expandtwonumexprafter\divideceilpos}
\def\divideceilpos{\expanddivisionafter\divide:c@il:pos}
\def\divide:c@il:pos#1#2#3{\numexpr#1\ifnum\numexpr#1*#3<#2+\@ne\fi}

\let\divintpos\divfloorpos
\let\divideintegerpos\dividefloorpos

\def\modpos{\expandtwonumexprafter\modulopos}
\def\modulopos{\expanddivisionafter\modulo:p@s}
\def\modulo:p@s#1#2#3{\numexpr#2-#3*\divide:fl@@r:pos{#1}{#2}{#3}\relax}

% One sticky problem in all the division related macros above... We always
% check if some rounding is present using the formula
%     \divident/\divisor*\divisor <=> \divident
% It works fine if one say (2\maxdimen)/\maxdimen, but arithmetic overflow
% occurs for (2\maxdimen)/(\maxdimen+1) or (2\maxdimen)/2. Thus, can be used
% for reasonably small numbers.

% Having an integer division we can round float-like strings

\def\floatround#1{\divnint{\dimexpr#1pt}\p@}
\def\floatfloor#1{\divfloor{\dimexpr#1pt}\p@}
\def\floatceil#1{\divceil{\dimexpr#1pt}\p@}
\def\floatint#1{\divint{\dimexpr#1pt}\p@}
\def\floatnint#1{\divnint{\dimexpr#1pt}\p@}

% Now lets implement integer by integer division with float-like result. The
% following approach is quite fast and precise enough for most practical
% purposes, but resulting floats are limited to \maxdimen expressed in points
% (16383.99998).

\def\fdivide{\expandtwonumexprafter\flo@t:divide}
\def\flo@t:divide#1#2{\withoutpt\the\dimexpr\numexpr#1*\p@/#2\relax sp\relax}

% Note, that \fdivide produce a float-like string that can be used as a factor
% preceding a dimen (i.e \hsize=\fdivide{2}{7}\hsize). For sake of preciseness
% however, internally we always use \dimexpr\hsize*2/7 in such cases (see
% pdftrans.tex).

% Yet another approach to division with float-like result. \divfloat produce a
% float-like string with fixed precision. There is still numeric overflow risk
% mentioned above. No preciseness and result limitation, however.

\def\divfloat{% <divisor> <divident> <preciseness>
 \expandthreenumexprafter\dividefloat}

\def\dividefloat#1#2#3{%
 \expandnumberafter\divide:flo@t % \absoluteint returns \numexpr
 {\absoluteint{\divideint{#1}{#2}}}{#1}{#2}{#3}}

\def\divide:flo@t#1#2#3{% it is not enough to check the sign of #1
 \ifnum#2<0 \ifnum#3>0 -\fi\else
 \ifnum#2>0 \ifnum#3<0 -\fi\fi\fi
 #1.\expandthreenumexprafter\divide:fl@@t
 {#1}{\absoluteint{#2}}{\absoluteint{#3}}}

\def\divide:fl@@t#1#2#3{%
 \expandnumexprafter\divide:flo@t:modulo{#2-#1*#3}{#3}}

\def\divide:flo@t:modulo#1#2{%
 \ifnum#1<214748365
  \expandtwonumbersafter\divide:flo@t:result{#10}{#2\expandafter}%
 \else
  \expandtwonumexprafter\divide:flo@t:modulo{#1/2}{#2/2\expandafter}%
 \fi}

\def\divide:flo@t:result#1#2#3{%
 \ifnum#3>1
  \expandtwonumexprafter\divide:flo@t:repeat
  {#3-\@ne}{\dividefloorpos{#1}{#2}\expandafter}%
 \else
  \number\divide:flo@t:last{#1}{#2}\relax
  \expandafter\gobbletwo
 \fi{#1}{#2}}

\def\divide:flo@t:repeat#1#2#3#4{%
 #2\expandnumexprafter\divide:flo@t:modulo{#3-#2*#4}{#4}{#1}}

\newcount\floatprecision
\floatprecision=6

\def\roundlast{\let\divide:flo@t:last\dividenint}
\def\floorlast{\let\divide:flo@t:last\divideint}
\roundlast

% One may say \let\tobp\roundfixedbp to enable alternative (fixed) rounding.

\def\tofixedbp#1{\divfloat{\dimexpr#1}\b@\floatprecision}
\def\roundfixedbp#1{\divfloat{\dimexpr#1}\b@\pdfdecimaldigits}

% And now comes a real challenge -- trigonometry. In the first approach, values
% of trigonometric functions were predefined for angles of range 0..90. Real
% angle values were not supported. Now things are a bit slower, but much more
% precise. In particular, we handle real angles values. The trigonometry
% implementation is excerpted from trans.tex and originally was inspired by
% mf.web. The clue is that any angle value (not only integer) can be
% represented as a series of predefined ,,coins''. Trigonometric functions can
% be then recursively calculated as follows:
%
% sin(a+b) = sin(a)cos(b) + sin(b)cos(a)
% cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
%
% Big thanks for BOP team for encourage.

\def\fractdegree#1{\numexpr16*\dimexpr#1pt}           % 2^20sp = 16pt = 1degree
\edef\fractfactor{\number\numexpr\maxdimen+\@ne}      % 2^30
\edef\fractfourth{\number\numexpr90*\fractdegree\@ne} % 90 degrees
\edef\fractperiod{\number\numexpr4*\fractfourth}      % 360 degrees

\def\reducefractangle#1{% reduce to 0..#2
 \expandnumberafter\reduce:fr@ct:angle{\fractdegree{#1}}}

\def\reduce:fr@ct:angle#1#2{%
 \ifnum#1<0
  \numexpr#2-\modpos{-#1}{#2}\relax
 \else
  \modpos{#1}{#2}%
 \fi}

% For sake of backward compatibility we leave a hook for integer angles

\def\reduceintangle#1#2{%
 \expandtwonumexprafter\reduce:int:@ngle{#1}{#2/\fractdegree\@ne}}

\def\reduce:int:@ngle#1#2{\fractdegree{\reduce:fr@ct:angle{#1}{#2}}}

\def\enablefractangle{\let\reducetrigangle\reducefractangle}
\def\disablefractangle{\let\reducetrigangle\reduceintangle}
\enablefractangle

% some macro shortcuts

\def\fracttrigfourth#1{% returns 0..3 (quarter)
 \dividefloorpos{#1}\fractfourth}

\def\fr@ct:mul#1#2{#1*#2/\fractfactor}
\def\fr@ct:div#1{\fdivide{#1}\fractfactor}

% constant fractions

\def\fr@ct:angle#1{\ifcase\numexpr#1\relax
62914560\or % 60
47185920\or % 45
31457280\or % 30
16777216\or % 2^4
 8388608\or % 2^3
 4194304\or % 2^2
 2097152\or % 2^1
 1048576\or % 2^0
  524288\or % 2^-1
  262144\or % 2^-2
  131072\or % 2^-3
   65536\or % 2^-4
   32768\or % 2^-5
   16384\or % 2^-6
    8192\or % 2^-7
    4096\or % 2^-8
    2048\or % 2^-9
    1024\or % 2^-10
     512\or % 2^-11
     256\or % 2^-12
     128\or % 2^-13
      64\or % 2^-14
      32\or % 2^-15
      16\or % 2^-16
       8\or % 2^-17
       4\or % 2^-18
       2\or % 2^-19
       1\fi}% 2^-20

\def\fr@ct:sin#1{\ifcase\numexpr#1\relax
929887697\or % 60
759250125\or % 45
536870912\or % 30
295963357\or % 2^4
149435979\or % 2^3
 74900443\or % 2^2
 37473049\or % 2^1
 18739379\or % 2^0
  9370046\or % 2^-1
  4685068\or % 2^-2
  2342539\or % 2^-3
  1171270\or % 2^-4
   585635\or % 2^-5
   292818\or % 2^-6
   146409\or % 2^-7
    73204\or % 2^-8
    36602\or % 2^-9
    18301\or % 2^-10
     9151\or % 2^-11
     4575\or % 2^-12
     2288\or % 2^-13
     1144\or % 2^-14
      572\or % 2^-15
      286\or % 2^-16
      143\or % 2^-17
       71\or % 2^-18
       36\or % 2^-19
       18\fi}% 2^-20

\def\fr@ct:cos#1{\ifcase\numexpr#1\relax
 536870912\or % 60
 759250125\or % 45
 929887697\or % 30
1032146887\or % 2^4
1063292242\or % 2^3
1071126243\or % 2^2
1073087729\or % 2^1
1073578288\or % 2^0
1073700939\or % 2^-1
1073731603\or % 2^-2
1073739269\or % 2^-3
1073741185\or % 2^-4
1073741664\or % 2^-5
1073741784\or % 2^-6
1073741814\or % 2^-7
1073741822\or % 2^-8
1073741823\or % 2^-9
1073741824\or % 2^-10
1073741824\or % 2^-11
1073741824\or % 2^-12
1073741824\or % 2^-13
1073741824\or % 2^-14
1073741824\or % 2^-15
1073741824\or % 2^-16
1073741824\or % 2^-17
1073741824\or % 2^-18
1073741824\or % 2^-19
1073741824\fi}% 2^-20

% I like \ifcase...\or...\fi acting as arrays because of its readability.
% For sake of speed however, we say:

\trans:count=0
\loop
 \expandafter\edef\csname
   fractangle:\the\trans:count\endcsname{\fr@ct:angle\trans:count}
 \expandafter\edef\csname
   fractsinvalue:\the\trans:count\endcsname{\fr@ct:sin\trans:count}
 \expandafter\edef\csname
   fractcosvalue:\the\trans:count\endcsname{\fr@ct:cos\trans:count}
 \ifnum\trans:count<27
 \advance\trans:count by1
\repeat
\def\fr@ct:angle#1{\csname fractangle:\number#1\endcsname}
\def\fr@ct:sin#1{\csname fractsinvalue:\number#1\endcsname}
\def\fr@ct:cos#1{\csname fractcosvalue:\number#1\endcsname}

% Here the main loop starts.

\def\fracttrig#1{% <angle> -> <sine> <cosine> <command>
 \expandnumexprafter\fr@ct:trig{\reducetrigangle{#1}\fractperiod}}

\def\fr@ct:trig#1{%
 \csname fr@ct:trig:\romannumeral\fracttrigfourth{#1}+\@ne\endcsname
 {#1}}

\def\fr@ct:trig:i#1#2#3{% <angle> <sine> <cosine>
 \expandthreenumexprafter\fr@ct:trig:cont{#1}{#2}{#3}\z@}

\def\fr@ct:trig:ii#1#2#3{%
 \expandthreenumexprafter\fr@ct:trig:cont{#1-\fractfourth}{#3}{-#2}\z@}

\def\fr@ct:trig:iii#1#2#3{%
 \expandthreenumexprafter\fr@ct:trig:cont{#1-2*\fractfourth}{-#2}{-#3}\z@}

\def\fr@ct:trig:iv#1#2#3{%
 \expandthreenumexprafter\fr@ct:trig:cont{#1-3*\fractfourth}{-#3}{#2}\z@}

\def\fr@ct:trig:cont#1#2#3#4{% <angle> <sine> <cosine> <index>
 \ifcase
  \ifnum#1>0 \ifnum#1<\fr@ct:angle{#4} 0 \else 1 \fi \else 2 \fi
  \expandafter\fr@ct:trig:cont\expandafter
  {\number#1\expandafter}\expandafter
  {\number#2\expandafter}\expandafter
  {\number#3\expandafter}\expandafter
  {\number\numexpr#4+\@ne\expandafter}\or
  \expandafter\fr@ct:trig:cont\expandafter
  {\number\numexpr#1-\fr@ct:angle{#4}\expandafter}\expandafter
  {\number\numexpr\fr@ct:mul{#2}{\fr@ct:cos{#4}}%
                 +\fr@ct:mul{#3}{\fr@ct:sin{#4}}\expandafter}\expandafter
  {\number\numexpr\fr@ct:mul{#3}{\fr@ct:cos{#4}}%
                 -\fr@ct:mul{#2}{\fr@ct:sin{#4}}\expandafter}\expandafter
  {\number\numexpr#4+\@ne\expandafter}\or
  \fracttrigend{#2}{#3}\fi}

% ...and finally, execute the next command with two first parameters being
% scaled sine and cosine.

\def\fracttrigend#1#2\fi#3{\fi#3{#1}{#2}}

\def\fractsincos#1#2#3{\fracttrig{#3}\z@\fractfactor\fr@ct:sin:cos#1#2}
\def\fr@ct:sin:cos#1#2#3#4{\def#3{#1}\def#4{#2}}

% In pdftrans we use the following shortcuts

\def\fr@ct:sin:cos:i#1#2#3{\fr@ct:trig:i{#3}\z@\fractfactor\fr@ct:sin:cos#1#2}
\def\fr@ct:sin:cos:ii#1#2#3{\fr@ct:trig:ii{#3}\z@\fractfactor\fr@ct:sin:cos#1#2}
\def\fr@ct:sin:cos:iii#1#2#3{\fr@ct:trig:iii{#3}\z@\fractfactor\fr@ct:sin:cos#1#2}
\def\fr@ct:sin:cos:iv#1#2#3{\fr@ct:trig:iv{#3}\z@\fractfactor\fr@ct:sin:cos#1#2}

\def\floatsincos#1#2#3{\fracttrig{#3}\z@\fractfactor\flo@t:sin:c@s#1#2}
\def\flo@t:sin:c@s#1#2#3#4{%
 \edef#3{\fr@ct:div{#1}}%
 \edef#4{\fr@ct:div{#2}}}

\endtrans
\endinput
