summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/tools/calc.dtx
blob: 0c57e6e00fdf7385a7d1a4e233f11965fe0484c5 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
% \iffalse meta-comment
%
% Copyright 1993-2014
%
% The LaTeX3 Project and any individual authors listed elsewhere
% in this file.
%
% This file is part of the Standard LaTeX `Tools Bundle'.
% -------------------------------------------------------
%
% It may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3c
% of this license or (at your option) any later version.
% The latest version of this license is in
%    http://www.latex-project.org/lppl.txt
% and version 1.3c or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% The list of all files belonging to the LaTeX `Tools Bundle' is
% given in the file `manifest.txt'.
%
% \fi
% \iffalse
%% File `calc.dtx'.
%% Copyright (C) 1992--1995
%%          Kresten Krab Thorup and Frank Jensen.
%% Copyright (C) 1997--2007
%%          Kresten Krab Thorup, Frank Jensen and the LaTeX3 Project.
%%
%% The original authors (fj@hugin.dk and  krab@daimi.aau.dk) have
%% contributed this package to the LaTeX distribution.
%% Problems with this package should now be sent using latexbug.tex to
%% the normal LaTeX bug report address.
%
%<*dtx>
          \ProvidesFile{calc.dtx}
%</dtx>
%<package>\NeedsTeXFormat{LaTeX2e}
%<package>\ProvidesPackage{calc}
%<driver> \ProvidesFile{calc.drv}
% \fi
%         \ProvidesFile{calc.dtx}
          [2014/10/28 v4.3 Infix arithmetic (KKT,FJ)]
%
% \iffalse
%<*driver>
\documentclass{ltxdoc}
\EnableCrossrefs
\RecordChanges
\usepackage{calc}
\begin{document}
\DocInput{calc.dtx}
\end{document}
%</driver>
% \fi
%
% \GetFileInfo{calc.dtx}
% \CheckSum{676}
%
% \title{The \texttt{calc} package\\Infix notation
%        arithmetic in \LaTeX\thanks{We thank Frank Mittelbach for his
%        valuable comments and suggestions which have greatly improved
%        this package.}}
% \author{Kresten Krab Thorup, Frank Jensen (and Chris Rowley)}
% \date{\filedate}
%
% \maketitle
%
% \changes{v4.0d}{1997/11/08}
%    {Contributed to tools distribution}
% \changes{v4.1a}{1998/06/07}
%    {Added text sizes: CAR}
% \changes{v4.1a}{1998/06/07}
%    {Attempt to make user-syntax robust: CAR}
%
% \newenvironment{calc-syntax}
%    {\par
%     \parskip\medskipamount
%     \def\is{\ \hangindent3\parindent$\longrightarrow$~}%
%     \def\alt{\ $\vert$~}%
%     \rightskip 0pt plus 1fil
%     \def\<##1>{\mbox{\NormalSpaces$\langle$##1\/$\rangle$}}%
%     \IgnoreSpaces\obeyspaces%
% }{\par\vskip\parskip}
% {\obeyspaces\gdef\NormalSpaces{\let =\space}\gdef\IgnoreSpaces{\def {}}}
%
% \def\<#1>{$\langle$#1\/$\rangle$}%
% \def\s#1{\ensuremath{[\![#1]\!]}}
% \def\savecode#1{\hbox{${}_{\hookrightarrow[#1]}$}}
% \def\gassign{\Leftarrow}
% \def\lassign{\leftarrow}
%
% \begin{abstract}
% The \texttt{calc} package reimplements the \LaTeX\ commands
% |\setcounter|, |\addtocounter|, |\setlength|, and |\addtolength|.
% Instead of a simple value, these commands now accept an infix
% notation expression.
% \end{abstract}
%
% \section{Introduction}
%
% Arithmetic in \TeX\ is done using low-level operations such as
% |\advance| and |\multiply|.  This may be acceptable when developing
% a macro package, but it is not an acceptable interface for the
% end-user.
%
% This package introduces proper infix notation arithmetic which is
% much more familiar to most people.  The infix notation is more
% readable and easier to modify than the alternative: a sequence of
% assignment and arithmetic instructions.  One of the arithmetic
% instructions (|\divide|) does not even have an equivalent in
% standard \LaTeX.
%
% The infix expressions can be used in arguments to macros (the
% \texttt{calc} package doesn't employ category code changes to
% achieve its goals).\footnote{However, it therefore assumes that the
%    category codes of the special characters, such as \texttt{(*/)}
%    in its syntax do not change.}
%
% \section{Informal description}
%
% Standard \LaTeX\ provides the following set of commands to
% manipulate counters and lengths \cite[pages 194 and~216]{latexman}.
% \begin{itemize}
% \item[]\hskip-\leftmargin
%    |\setcounter{|\textit{ctr}|}{|\textit{num}|}| sets the
%    value of the counter \textit{ctr} equal to (the value of)
%    \textit{num}.  (Fragile)
% \item[]\hskip-\leftmargin
%    |\addtocounter{|\textit{ctr}|}{|\textit{num}|}|
%    increments the value of the counter \textit{ctr} by (the
%    value of) \textit{num}.  (Fragile)
%
% \item[]\hskip-\leftmargin
%    |\setlength{|\textit{cmd}|}{|\textit{len}|}| sets the value of
%    the length command \textit{cmd} equal to (the value of) \textit{len}.
%    (Robust)
% \item[]\hskip-\leftmargin
%    |\addtolength{|\textit{cmd}|}{|\textit{len}|}| sets the value of
%    the length command \textit{cmd} equal to its current value plus
%    (the value of) \textit{len}.  (Robust)
% \end{itemize}
% (The |\setcounter| and |\addtocounter| commands have global effect,
% while the |\setlength| and |\addtolength| commands obey the normal
% scoping rules.)  In standard \LaTeX, the arguments to these commands
% must be simple values.  The \texttt{calc} package extends these
% commands to accept infix notation expressions, denoting values of
% appropriate types.  Using the \texttt{calc} package, \textit{num} is
% replaced by \<integer expression>, and \textit{len} is replaced by
% \<glue expression>.  The formal syntax of \<integer expression> and
% \<glue expression> is given below.
%
% In addition to these commands to explicitly set a length, many \LaTeX\
% commands take a length argument. After loading this package, most of
% these commands will accept a  \<glue expression>. This includes
% the optional width argument of |\makebox|, the width argument of
% |\parbox|, |minipage|, and a |tabular| |p|-column, and many similar
% constructions. (This package does not redefine any of these commands,
% but they are defined by default to read their arguments by |\setlength|
% and so automatically benefit from the enhanced |\setlength| command
% provided by this package.)
%
% In the following, we shall use standard \TeX\ terminology.  The
% correspondence between \TeX\ and \LaTeX\ terminology is as follows:
% \LaTeX\ counters correspond to \TeX's count registers; they hold
% quantities of type \<number>.  \LaTeX\ length commands correspond to
% \TeX's dimen (for rigid lengths) and skip (for rubber lengths)
% registers; they hold quantities of types \<dimen> and \<glue>,
% respectively.
%
% \TeX\ gives us primitive operations to perform arithmetic on registers as
% follows:
% \begin{itemize}
%   \item addition and subtraction on all types of quantities without
%       restrictions;
%   \item multiplication and division by an \emph{integer} can be
%       performed on a register of any type;
%   \item multiplication by a \emph{real} number (i.e., a number with a
%       fractional part) can be performed on a register of any type,
%       but the stretch and shrink components of a glue quantity are
%       discarded.
% \end{itemize}
% The \texttt{calc} package uses these \TeX\ primitives but provides a
% more user-friendly notation for expressing the arithmetic.
%
% An expression is formed of numerical quantities (such as explicit
% constants and \LaTeX\ counters and length commands) and binary
% operators (the tokens `\texttt{+}', `\texttt{-}', `\texttt{*}', and
% `\texttt{/}' with their usual meaning) using the familiar infix
% notation; parentheses may be used to override the usual precedences
% (that multiplication/division have higher precedence than
% addition/subtraction).
%
% Expressions must be properly typed.  This means, e.g., that a dimen
% expression must be a sum of dimen terms: i.e., you cannot say
% `\texttt{2cm+4}' but `\texttt{2cm+4pt}' is valid.
%
% In a dimen term, the dimension part must come first; the same holds
% for glue terms.  Also, multiplication and division by non-integer
% quantities require a special syntax; see below.
%
% Evaluation of subexpressions at the same level of precedence
% proceeds from left to right.  Consider a dimen term such as
% ``\texttt{4cm*3*4}''.  First, the value of the factor \texttt{4cm} is
% assigned to a dimen register, then this register is multiplied
% by~$3$ (using |\multiply|), and, finally, the register is multiplied
% by~$4$ (again using |\multiply|).  This also explains why the
% dimension part (i.e., the part with the unit designation) must come
% first; \TeX\ simply doesn't allow untyped constants to be assigned
% to a dimen register.
%
% The \texttt{calc} package also allows multiplication and division by
% real numbers.  However, a special syntax is required: you must use
% |\real{|\<decimal constant>|}|\footnote{Actually, instead of
% \<decimal constant>, the more general \<optional signs>\<factor> can
% be used.  However, that doesn't add any extra expressive power to
% the language of infix expressions.} or
% |\ratio{|\<dimen expression>|}{|\<dimen expression>|}| to denote a
% real value to be used for multiplication/division.  The first form has
% the obvious meaning, and the second form denotes the number obtained
% by dividing the value of the first expression by the value of the
% second expression.
%
%    A later addition to the package (in June 1998) allows an additional
%    method of specifying a factor of type dimen by setting some text
%    (in LR-mode) and measuring its dimensions: these are denoted as
%    follows.
%\begin{quote}
%    |\widthof{|\<text>|}|\quad
%    |\heightof{|\<text>|}|\quad
%    |\depthof{|\<text>|}|
%\end{quote}
%    These calculate the natural sizes of the \<text> in exactly the
%    same way as is done for the commands |\settowidth| etc.~on
%    Page~216 of the manual~\cite{latexman}.
%    In August 2005 the package was further extended to provide the command
%\begin{quote}
%    |\totalheightof{|\<text>|}|
%\end{quote}
%    This command does exactly what you'd expect from its name.
%    Additionally the package also provides the command
%\begin{quote}
%    |\settototalheight{|\<cmd>|}{|\<text>|}|
%\end{quote}
%
%
%    Note that there is a small difference in the usage of these two
%    methods of accessing text dimensions.  After
%    |\settowidth{\txtwd}{Some text}| you can use:
%\begin{verbatim}
%    \setlength{\parskip}{0.68\textwd}
%\end{verbatim}
%    whereas using the more direct access to the width of the text
%    requires the longer form for multiplication, thus:
%\begin{verbatim}
%    \setlength{\parskip}{\widthof{Some text} * \real{0.68}}
%\end{verbatim}
%
% \TeX\ discards the stretch and shrink components of glue when glue
% is multiplied by a real number.  So, for example,
%\begin{verbatim}
%    \setlength{\parskip}{3pt plus 3pt * \real{1.5}}
%\end{verbatim}
% will set the paragraph separation to 4.5pt with no stretch or
% shrink.  Incidentally, note how spaces can be used to enhance
% readability. When \TeX\ is scanning for a \<number> etc.\ it is
% common to terminate the scanning with a space token or by inserting
% \cs{relax}. As of version~4.3 \textsf{calc} allows \cs{relax} tokens
% to appear in places where they would usually be used for terminating
% \TeX's scanning. In short this is just before any of \texttt{+-*/)}
% or at the end of the expression being evaluated.
%
% When \TeX\ performs arithmetic on integers, any fractional part of
% the results are discarded.  For example,
%\begin{verbatim}
%    \setcounter{x}{7/2}
%    \setcounter{y}{3*\real{1.6}}
%    \setcounter{z}{3*\real{1.7}}
%\end{verbatim}
% will assign the value~$3$ to the counter~\texttt{x}, the value~$4$
% to~\texttt{y}, and the value~$5$ to~\texttt{z}.  This truncation
% also applies to \emph{intermediate} results in the sequential
% computation of a composite expression; thus, the following command
%\begin{verbatim}
%    \setcounter{x}{3 * \real{1.6} * \real{1.7}}
%\end{verbatim}
% will assign~$6$ to~\texttt{x}.
%
% As an example of the use of |\ratio|, consider the problem of
% scaling a figure to occupy the full width (i.e., |\textwidth|) of
% the body of a page.  Assume that the original dimensions of the
% figure are given by the dimen (length) variables, |\Xsize| and
% |\Ysize|.  The height of the scaled figure can then be expressed by
%\begin{verbatim}
%    \setlength{\newYsize}{\Ysize*\ratio{\textwidth}{\Xsize}}
%\end{verbatim}
%
%
%
% Another new feature introduced in August 2005 was $\max$ and $\min$
% operations with associated macros
%\begin{quote}
%    |\maxof{|\<\textit{type} expression>|}{|\<\textit{type} expression>|}|
%    \\
%    |\minof{|\<\textit{type} expression>|}{|\<\textit{type} expression>|}|
%\end{quote}
% When \textit{type} is either \meta{glue} or \meta{dimen} these macros
% are allowed only as part of addition or subtraction but when
% \textit{type} is \meta{integer} they can also be used when
% multiplying and dividing. In the latter case they follow the
% same syntax rules as |\ratio| and |\real| which means they must come
% after the |*| or the |/|. Thus
%\begin{verbatim}
%  \setcounter{x}{3*\maxof{4+5}{3*4}+\minof{2*\real{1.6}}{5-1}}
%\end{verbatim}
% will assign $3\times\max(9,12)+\min(3,4)=39$ to |x|. Similarly
%\begin{verbatim}
%  \setlength{\parindent}{%
%    \minof{3pt}{\parskip}*\real{1.5}*\maxof{2*\real{1.6}}{2-1}}
%\end{verbatim}
% will assign $\min(13.5\textrm{pt},4.5\cs{parskip})$ to \cs{parindent}
%
%
%
% \section{Formal syntax}
%
% The syntax is described by the following set of rules.
% Note that the definitions of \<number>, \<dimen>, \<glue>,
% \<decimal constant>, and \<plus or minus> are
% as in Chapter~24 of The \TeX book~\cite{texbook}; and \<text>
% is LR-mode material, as in the manual~\cite{latexman}.
% We use \textit{type} as a meta-variable, standing for
% `integer', `dimen', and `glue'.\footnote{This version of the
% \texttt{calc} package doesn't support evaluation of muglue expressions.}
%
% \begin{calc-syntax}
%    \<\textit{type} expression>^^A
%       \is  \<\textit{type} term>^^A
%       \alt \<\textit{type} expression> \<plus or minus> \<\textit{type} term>
%
%    \<\textit{type} term>^^A
%       \is  \<\textit{type} term> \<\textit{type} scan stop>
%       \alt \<\textit{type} factor>^^A
%       \alt \<\textit{type} term> \<multiply or divide> \<integer>^^A
%       \alt \<\textit{type} term> \<multiply or divide> \<real number>^^A
%       \alt \<\textit{type} term> \<multiply or divide>^^A
%            \<$\max$ or $\min$ integer>^^A
%
%    \<\textit{type} scan stop>^^A
%       \is  \<empty>^^A
%       \alt \<optional space>^^A
%       \alt |\relax|
%
%    \<\textit{type} factor>^^A
%       \is  \<\textit{type}>^^A
%       \alt \<text dimen factor>^^A
%       \alt \<$\max$ or $\min$ \textit{type}>^^A
%       \alt |(|$_{12}$ \<\textit{type} expression> |)|$_{12}$
%
%    \<integer> \is \<number>
%
%    \<$\max$ or $\min$ \textit{type}> \is \<$\max$ or $\min$ command>^^A
%               |{| \<\textit{type} expression> |}|^^A
%               |{| \<\textit{type} expression> |}|
%
%    \<$\max$ or $\min$ command> \is |\maxof|^^A
%                                \alt |\minof|
%
%    \<text dimen factor>^^A
%       \is  \<text dimen command>|{| \<text> |}|
%
%    \<text dimen command>^^A
%       \is   |\widthof|^^A
%       \alt  |\heightof|^^A
%       \alt  |\depthof|^^A
%       \alt  |\totalheightof|^^A
%
%    \<multiply or divide>^^A
%       \is  |*|$_{12}$^^A
%       \alt |/|$_{12}$
%
%    \<real number>^^A
%       \is  |\ratio{| \<dimen expression> |}{| \<dimen expression> |}|^^A
%       \alt |\real{| \<optional signs> \<decimal constant> |}|
%
%    \<plus or minus>^^A
%       \is  |+|$_{12}$^^A
%       \alt |-|$_{12}$
%
%    \<decimal constant>^^A
%       \is  |.|$_{12}$^^A
%       \alt |,|$_{12}$^^A
%       \alt \<digit> \<decimal constant>^^A
%       \alt \<decimal constant> \<digit>
%
%    \<digit>^^A
%       \is  |0|$_{12}$^^A
%       \alt |1|$_{12}$^^A
%       \alt |2|$_{12}$^^A
%       \alt |3|$_{12}$^^A
%       \alt |4|$_{12}$^^A
%       \alt |5|$_{12}$^^A
%       \alt |6|$_{12}$^^A
%       \alt |7|$_{12}$^^A
%       \alt |8|$_{12}$^^A
%       \alt |9|$_{12}$
%
%    \<optional signs>^^A
%       \is  \<optional spaces>^^A
%       \alt \<optional signs> \<plus or minus> \<optional spaces>
%
% \end{calc-syntax}
%
%
% Relying heavily on \TeX\ to do the underlying assignments, it is
% only natural for \texttt{calc} to simulate \TeX's parsing machinery
% for these quantities. Therefore it a)~imposes the same restrictions
% on the catcode of syntax characters as \TeX\ and b)~tries to expand
% its argument fully. a)~means that implicit characters for the tokens
% |*|$_{12}$, |/|$_{12}$, |(|$_{12}$, and |)|$_{12}$ will not
% work\footnote{e\TeX\ also assumes these catcodes when parsing a
% \cs{numexpr}, \cs{dimexpr}, \cs{glueexpr}, or \cs{muglueexpr} and
% does not allow implicit characters.} but because of~b), the
% expansion should allow you to use macros that expand to explicit
% syntax characters.
%
%
% \StopEventually{
% \begin{thebibliography}{1}
%    \bibitem{texbook}
%       \textsc{D. E. Knuth}.
%       \newblock \textit{The \TeX{}book} (Computers \& Typesetting Volume A).
%       \newblock Addison-Wesley, Reading, Massachusetts, 1986.
%    \bibitem{latexman}
%       \textsc{L. Lamport}.
%       \newblock \textit{\LaTeX, A Document Preparation System.}
%       \newblock Addison-Wesley, Reading, Massachusetts, Second
%       edition 1994/1985.
% \end{thebibliography}
% \PrintChanges
% }
%
% \section{The evaluation scheme}
% \label{evaluation:scheme}
%
% In this section, we shall for simplicity consider only expressions
% containing `$+$' (addition) and `$*$' (multiplication) operators.
% It is trivial to add subtraction and division.
%
% An expression $E$ is a sum of terms: $T_1+\cdots+T_n$; a term is a
% product of factors: $F_1*\cdots*F_m$; a factor is either a simple
% numeric quantity~$f$ (like \<number> as described in the \TeX book),
% or a parenthesized expression~$(E')$.
%
% Since the \TeX\ engine can only execute arithmetic operations in a
% machine-code like manner, we have to find a way to translate the
% infix notation into this `instruction set'.
%
% Our goal is to design a translation scheme that translates~$X$ (an
% expression, a term, or a factor) into a sequence of \TeX\ instructions
% that does the following [Invariance Property]: correctly
% evaluates~$X$, leaves the result in a global register~$A$ (using a
% global assignment), and does not perform global assignments to the
% scratch register~$B$; moreover, the code sequence must be balanced
% with respect to \TeX\ groups.  We shall denote the code sequence
% corresponding to~$X$ by \s{X}.
%
% In the replacement code specified below, we use the following
% conventions:
% \begin{itemize}
%    \item $A$ and $B$ denote registers; all assignments to~$A$ will
%       be global, and all assignments to~$B$ will be local.
%    \item ``$\gassign$'' means global assignment to the register on
%       the lhs.
%    \item ``$\lassign $'' means local assignment to the register on
%       the lhs.
%    \item ``\savecode C'' means ``save the code~$C$ until the current
%       group (scope) ends, then execute it.''  This corresponds to
%       the \TeX-primitive |\aftergroup|.
%    \item ``$\{$'' denotes the start of a new group, and ``$\}$''
%       denotes the end of a group.
% \end{itemize}
%
% Let us consider an expression $T_1+T_2+\cdots+T_n$.  Assuming that
% \s{T_k} ($1\le k\le n$) attains the stated goal, the following code
% clearly attains the stated goal for their sum:
% \begin{eqnarray*}
% \s{T_1+T_2+\cdots+T_n}&\Longrightarrow&
%       \{\,\s{T_1}\,\} \; B\lassign A \quad
%       \{\,\s{T_2}\,\} \; B\lassign B+A \\
%       &&\qquad \ldots \quad \{\,\s{T_n}\,\} \; B\lassign B+A
%                       \quad A\gassign B
% \end{eqnarray*}
% Note the extra level of grouping enclosing each of \s{T_1}, \s{T_2},
% \ldots,~\s{T_n}.  This will ensure that register~$B$, used to
% compute the sum of the terms, is not clobbered by the intermediate
% computations of the individual terms.  Actually, the group
% enclosing~\s{T_1} is unnecessary, but it turns out to be simpler if
% all terms are treated the same way.
%
% The code sequence ``$\{\,\s{T_2}\,\}\;B\lassign B+A$'' can be translated
% into the following equivalent code sequence:
% ``$\{\savecode{B\lassign B+A}\,\s{T_2}\,\}$''.  This observation turns
% out to be the key to the implementation: The ``$\savecode{B\lassign
% B+A}$'' is generated \emph{before} $T_2$ is translated, at the same
% time as the `$+$' operator between $T_1$ and~$T_2$ is seen.
%
% Now, the specification of the translation scheme is straightforward:
% \begin{eqnarray*}
%    \s{f}&\Longrightarrow&A\gassign f\\[\smallskipamount]
%    \s{(E')}&\Longrightarrow&\s{E'}\\[\smallskipamount]
%    \s{T_1+T_2+\cdots+T_n}&\Longrightarrow&
%       \{\savecode{B\lassign A}\,\s{T_1}\,\} \quad
%       \{\savecode{B\lassign B+A}\,\s{T_2}\,\} \\
%       &&\qquad \ldots \quad \{\savecode{B\lassign B+A}\,\s{T_n}\,\}
%                       \quad A\gassign B
%                       \\[\smallskipamount]
%    \s{F_1*F_2*\cdots*F_m}&\Longrightarrow&
%       \{\savecode{B\lassign A}\,\s{F_1}\,\} \quad
%       \{\savecode{B\lassign B*A}\,\s{F_2}\,\}\\
%       &&\qquad \ldots \quad \{\savecode{B\lassign B*A}\,\s{F_m}\,\}
%                       \quad A\gassign B
% \end{eqnarray*}
% By structural induction, it is easily seen that the stated property
% is attained.
%
% By inspection of this translation scheme, we see that we have to
% generate the following code:
% \begin{itemize}
%    \item we must generate ``$\{\savecode{B\lassign
%       A}\{\savecode{B\lassign A}$'' at the left border of an
%       expression (i.e., for each left parenthesis and the implicit
%       left parenthesis at the beginning of the whole expression);
%    \item we must generate ``$\}A\gassign B\}A\gassign B$'' at the
%       right border of an expression (i.e., each right parenthesis
%       and the implicit right parenthesis at the end of the full
%       expression);
%    \item `\texttt{*}' is replaced by ``$\}\{\savecode{B\lassign
%       B*A}$'';
%    \item `\texttt{+}' is replaced by
%       ``$\}A\gassign B\}\{\savecode{B\lassign
%       B+A}\{\savecode{B\lassign A}$'';
%    \item when we see (expect) a numeric quantity, we insert the
%    assignment code ``$A\gassign$'' in front of the quantity and let
%    \TeX\ parse it.
% \end{itemize}
%
% \section{Implementation}
%
% For brevity define
% \begin{calc-syntax}
%    \<numeric> \is \<number> \alt \<dimen> \alt \<glue> \alt \<muglue>
% \end{calc-syntax}
% So far we have ignored the question of how to determine the type of
% register to be used in the code.  However, it is easy to see that
% (1)~`$*$' always initiates an \<integer factor>, (2)~all
% \<numeric>s in an expression, except those which are part of an
% \<integer factor>, are of the same type as the whole expression, and
% all \<numeric>s in an \<integer factor> are \<number>s.
%
% We have to ensure that $A$ and~$B$ always have an appropriate type
% for the \<numeric>s they manipulate.  We can achieve this by having
% an instance of $A$ and~$B$ for each type.  Initially, $A$~and~$B$
% refer to registers of the proper type for the whole expression.
% When an \<integer factor> is expected, we must change $A$ and~$B$ to
% refer to integer type registers.  We can accomplish this by
% including instructions to change the type of $A$ and~$B$ to integer
% type as part of the replacement code for~`$*$; if we append such
% instructions to the replacement code described above, we also ensure
% that the type-change is local (provided that the type-changing
% instructions only have local effect).  However, note that the
% instance of~$A$ referred to in $\savecode{B\lassign B*A}$ is the
% integer instance of~$A$.
%
% We shall use |\begingroup| and |\endgroup| for the open-group and
% close-group characters.  This avoids problems with spacing in math
% (as pointed out to us by Frank Mittelbach).
%
% \subsection{Getting started}
%
% Now we have enough insight to do the actual implementation in \TeX.
% First, we announce the macro package.\footnote{Code moved to top of file}
%    \begin{macrocode}
%<*package>
%\NeedsTeXFormat{LaTeX2e}
%\ProvidesPackage{calc}[\filedate\space\fileversion]
%    \end{macrocode}
%
% \subsection{Assignment macros}
%
% \begin{macro}{\calc@assign@generic}
% \changes{v4.2}{2005/08/06}{Removed a few redundant \cs{expandafter}s}
% The |\calc@assign@generic| macro takes four arguments: (1~and~2) the
% registers to be used
% for global and local manipulations, respectively; (3)~the lvalue
% part; (4)~the expression to be evaluated.
%
% The third argument (the lvalue) will be used as a prefix to a
% register that contains the value of the specified expression (the
% fourth argument).
%
% In general, an lvalue is anything that may be followed by a variable
% of the appropriate type.  As an example, |\linepenalty| and
% |\global\advance\linepenalty| may both be followed by an \<integer
% variable>.
%
% The macros described below refer to the registers by the names
% |\calc@A| and |\calc@B|; this is accomplished by
% |\let|-assignments.
%
% As discovered in Section~\ref{evaluation:scheme}, we have to
% generate code as
% if the expression is parenthesized.  As described below,
% |\calc@open| is the macro that replaces a left parenthesis by its
% corresponding \TeX\ code sequence.  When the scanning process sees
% the exclamation point, it generates an |\endgroup| and stops.  As we
% recall from Section~\ref{evaluation:scheme}, the correct expansion
% of a right
% parenthesis is ``$\}A\gassign B\}A\gassign B$''.  The remaining
% tokens of this expansion are inserted explicitly, except that the
% last assignment has been replaced by the lvalue part (i.e.,
% argument~|#3| of |\calc@assign@generic|) followed by |\calc@B|.
%    \begin{macrocode}
\def\calc@assign@generic#1#2#3#4{\let\calc@A#1\let\calc@B#2%
    \calc@open(#4!%
    \global\calc@A\calc@B\endgroup#3\calc@B}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\calc@assign@count}
% \begin{macro}{\calc@assign@dimen}
% \begin{macro}{\calc@assign@skip}
% We need three instances of the |\calc@assign@generic| macro,
% corresponding to the types \<integer>, \<dimen>, and \<glue>.
%    \begin{macrocode}
\def\calc@assign@count{\calc@assign@generic\calc@Acount\calc@Bcount}
\def\calc@assign@dimen{\calc@assign@generic\calc@Adimen\calc@Bdimen}
\def\calc@assign@skip{\calc@assign@generic\calc@Askip\calc@Bskip}
%    \end{macrocode}
% \end{macro}\end{macro}\end{macro}
% These macros each refer to two registers, one
% to be used globally and one to be used locally.
% We must allocate these registers.
%    \begin{macrocode}
\newcount\calc@Acount   \newcount\calc@Bcount
\newdimen\calc@Adimen   \newdimen\calc@Bdimen
\newskip\calc@Askip     \newskip\calc@Bskip
%    \end{macrocode}
%
% \subsection{The \LaTeX\ interface}
%
% \begin{macro}{\setcounter}
% \begin{macro}{\addtocounter}
% \changes{v4.2}{2005/08/06}
%  {Fix to make \cs{addtocounter} work with \texttt{amstext}}
% \begin{macro}{\steptocounter}
% \changes{v4.2}{2005/08/06}
%  {Avoid redundant processing. PR/3795}
% \begin{macro}{\setlength}
% \begin{macro}{\addtolength}
% As promised, we redefine the following standard \LaTeX\ commands:
% |\setcounter|,
% |\addtocounter|, |\setlength|, and |\addtolength|.
%    \begin{macrocode}
\def\setcounter#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}%
   {\calc@assign@count{\global\csname c@#1\endcsname}{#2}}}
%    \end{macrocode}
%    \begin{macrocode}
  \def\addtocounter#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}%
    {\calc@assign@count{\global\advance\csname c@#1\endcsname}{#2}}}%
%    \end{macrocode}
% We also fix \cs{stepcounter} to not go through the whole \texttt{calc}
% process.
%    \begin{macrocode}
  \def\stepcounter#1{\@ifundefined {c@#1}%
    {\@nocounterr {#1}}%
    {\global\advance\csname c@#1\endcsname \@ne
    \begingroup
      \let\@elt\@stpelt \csname cl@#1\endcsname
    \endgroup}}%
%    \end{macrocode}
% If the \texttt{amstext} package is loaded we must add the
% |\iffirstchoice@| switch as well. We patch the commands this
% way since it's good practice when we know how many arguments they take.
%    \begin{macrocode}
\@ifpackageloaded{amstext}{%
 \expandafter\def\expandafter\stepcounter
    \expandafter#\expandafter1\expandafter{%
    \expandafter\iffirstchoice@\stepcounter{#1}\fi
 }
 \expandafter\def\expandafter\addtocounter
    \expandafter#\expandafter1\expandafter#\expandafter2\expandafter{%
    \expandafter\iffirstchoice@\addtocounter{#1}{#2}\fi
 }
}{}
%    \end{macrocode}
%    \begin{macrocode}
\DeclareRobustCommand\setlength{\calc@assign@skip}
\DeclareRobustCommand\addtolength[1]{\calc@assign@skip{\advance#1}}
%    \end{macrocode}
% (|\setlength| and |\addtolength| are robust according to
% \cite{latexman}.)
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \subsection{The scanner}
%
% We evaluate expressions by explicit scanning of characters.  We do
% not rely on active characters for this.
%
% The scanner consists of two parts, |\calc@pre@scan| and
% |\calc@post@scan|; |\calc@pre@scan| consumes left parentheses, and
% |\calc@post@scan| consumes binary operator, |\real|, |\ratio|, and
% right parenthesis tokens.
%
% \begin{macro}{\calc@pre@scan}
% \begin{macro}{\@calc@pre@scan}
% \changes{v4.2}{2005/08/06}
%    {Added macro and force expansion}
%
%    Note that this is called at least once on every use of calc
%    processing, even when none of the extended syntax is present; it
%    therefore needs to be made very efficient.
%
%    It reads the initial part of expressions, until some \<text dimen
%    factor> or \<numeric> is seen; in fact, anything not explicitly
%    recognized here is taken to be a \<numeric> of some sort as this
%    allows unary
%   `\texttt{+}' and unary `\texttt{-}' to be treated easily and
%    correctly\footnote{In the few contexts where signs are allowed:
%    this could, I think, be extended (CAR).} but means that anything
%    illegal will simply generate a \TeX-level error, often a
%    reasonably comprehensible one!
%
%    The |\romannumeral-`\a| part is a little trick which forces expansion
%    in case |#1| is a normal macro, something that occurs from time to
%    time. A conditional test inside will possibly leave a trailing
%    \cs{fi} but this remnant is removed later when \cs{calc@post@scan}
%    performs the same trick.
%
%    The many |\expandafter|s are needed to efficiently end the nested
%    conditionals so that |\calc@textsize| and |\calc@maxmin@addsub| can
%    process their argument.
% \changes{v4.1a}{1998/06/07}
%    {Added code for text sizes: CAR}
% \changes{v4.1b}{1998/07/07}
%    {Correction to  ifx true case}
% \changes{v4.2}{2005/08/06}
%    {Added \cs{maxof} and \cs{minof} operations}
%    \begin{macrocode}
\def\calc@pre@scan#1{%
  \expandafter\@calc@pre@scan\romannumeral-`\a#1}
\def\@calc@pre@scan#1{%
  \ifx(#1%
    \expandafter\calc@open
  \else
    \ifx\widthof#1%
      \expandafter\expandafter\expandafter\calc@textsize
    \else
      \ifx\maxof#1%
        \expandafter\expandafter\expandafter\expandafter
        \expandafter\expandafter\expandafter\calc@maxmin@addsub
      \else
        \calc@numeric% no \expandafter needed for this one.
      \fi
    \fi
  \fi
  #1}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@open}
% \begin{macro}{\calc@initB}
% |\calc@open| is used when there is a left parenthesis right ahead.
% This parenthesis is replaced by \TeX\ code corresponding to the code
% sequence ``$\{\savecode{B\lassign A}\{\savecode{B\lassign A}$''
% derived in Section~\ref{evaluation:scheme}.  Finally,
% |\calc@pre@scan| is
% called again.
%    \begin{macrocode}
\def\calc@open({\begingroup\aftergroup\calc@initB
   \begingroup\aftergroup\calc@initB
   \calc@pre@scan}
\def\calc@initB{\calc@B\calc@A}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{\calc@numeric}
% |\calc@numeric| assigns the following value to |\calc@A| and then
% transfers control to |\calc@post@scan|.
%    \begin{macrocode}
\def\calc@numeric{\afterassignment\calc@post@scan \global\calc@A}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\widthof}
% \begin{macro}{\heightof}
% \begin{macro}{\depthof}
% \changes{v4.1a}{1998/06/07}
%    {Added macros: CAR}
% \begin{macro}{\totalheightof}
% \changes{v4.2}{2005/08/06}
%    {Added macro}
% \changes{v4.2}{2005/08/06}
%    {Added informative message for reserved macros}
%
%    These do not need any particular definition when they are scanned
%    so, for efficiency and robustness, we make them all equivalent to
%    the same harmless (I hope) unexpandable command.\footnote{If this
%    level of safety is not needed then the code can be sped up:
%    CAR.}  Thus the test in |\@calc@pre@scan| finds any of them.
%
%    As we have to check for these commands explicitly we must ensure
%    that our definition wins. Using \cs{newcommand} gives an error when
%    loading \texttt{calc} and may be mildly surprising. This should be
%    a little more informative.
%    \begin{macrocode}
\@for\reserved@a:=widthof,heightof,depthof,totalheightof,maxof,minof\do
{\@ifundefined{\reserved@a}{}{%
  \PackageError{calc}{%
  The\space calc\space package\space reserves\space the\space
  command\space name\space `\@backslashchar\reserved@a'\MessageBreak
  but\space it\space has\space already\space been\space defined\space
  with\space the\space meaning\MessageBreak
  `\expandafter\meaning\csname\reserved@a\endcsname'.\MessageBreak
  This\space original\space definition\space will\space be\space lost}%
  {If\space you\space need\space a\space command\space with\space
  this\space definition,\space you\space must\space use\space a\space
  different\space name.}}%
}
\let\widthof\ignorespaces
\let\heightof\ignorespaces
\let\depthof\ignorespaces
\let\totalheightof\ignorespaces
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@textsize}
% \changes{v4.1a}{1998/06/07}
%    {Added macro: CAR}
% \changes{v4.1a}{1998/06/07}
%    {Added macro: CAR}
% \changes{v4.2}{2005/08/06}
%    {Extended macro with \cs{totalheightof}}
%    The presence of the above four commands invokes this code, where
%    we must distinguish them from each other.
%    This implementation is somewhat optimized by using low-level
%    code from the commands |\settowidth|, etc.\footnote{It is based on
%    suggestions by Donald Arseneau and David Carlisle.}
%
%    Within the text argument we must restore the normal meanings of
%    the four user-level commands since arbitrary material can appear
%    in here, including further uses of calc.
%    \begin{macrocode}
\def\calc@textsize #1#2{%
  \begingroup
    \let\widthof\wd
    \let\heightof\ht
    \let\depthof\dp
    \def\totalheightof{\ht\dp}%
%    \end{macrocode}
% We must expand the argument one level if it's \cs{totalheightof}
% and it doesn't hurt the other three.
%    \begin{macrocode}
    \expandafter\@settodim\expandafter{#1}%
      {\global\calc@A}%
      {%
       \let\widthof\ignorespaces
       \let\heightof\ignorespaces
       \let\depthof\ignorespaces
       \let\totalheightof\ignorespaces
       #2}%
  \endgroup
  \calc@post@scan}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\calc@post@scan}
% \begin{macro}{\@calc@post@scan}
% \changes{v4.2}{2005/08/06}{Added macro and force expansion}
% \changes{v4.3}{2007/08/22}{Discard terminating \cs{relax} tokens and
%   avoid extra error message from \cs{calc@next}}
% The macro |\calc@post@scan| is called right after a value has been
% read.  At this point, a binary operator, a sequence of right
% parentheses, an optional \cs{relax}, and the end-of-expression mark
% (`|!|') is allowed.\footnote{Is \texttt{!} a good choice, CAR?}
% Depending on our findings, we call a suitable macro to generate the
% corresponding \TeX\ code (except when we detect the
% end-of-expression marker: then scanning ends, and control is
% returned to |\calc@assign@generic|).
%
% This macro may be optimized by selecting a different order of
% |\ifx|-tests.  The test for `\texttt{!}' (end-of-expression) is
% placed first as it will always be performed: this is the only test
% to be performed if the expression consists of a single \<numeric>.
% This ensures that documents that do not use the extra expressive
% power provided by the \texttt{calc} package only suffer a minimum
% slowdown in processing time.
%    \begin{macrocode}
\def\calc@post@scan#1{%
 \expandafter\@calc@post@scan\romannumeral-`\a#1}
\def\@calc@post@scan#1{%
  \ifx#1!\let\calc@next\endgroup \else
    \ifx#1+\let\calc@next\calc@add \else
      \ifx#1-\let\calc@next\calc@subtract \else
        \ifx#1*\let\calc@next\calc@multiplyx \else
          \ifx#1/\let\calc@next\calc@dividex \else
            \ifx#1)\let\calc@next\calc@close \else
              \ifx#1\relax\let\calc@next\calc@post@scan \else
                \def\calc@next{\calc@error#1}%
              \fi
            \fi
          \fi
        \fi
      \fi
    \fi
  \fi
  \calc@next}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@add}
% \begin{macro}{\calc@subtract}
% \begin{macro}{\calc@generic@add}
% \begin{macro}{\calc@addAtoB}
% \begin{macro}{\calc@subtractAfromB}
% The replacement code for the binary operators `\texttt{+}' and
% `\texttt{-}' follow a common pattern; the only difference is the
% token that is stored away by |\aftergroup|.  After this replacement
% code, control is transferred to |\calc@pre@scan|.
%    \begin{macrocode}
\def\calc@add{\calc@generic@add\calc@addAtoB}
\def\calc@subtract{\calc@generic@add\calc@subtractAfromB}
\def\calc@generic@add#1{\endgroup\global\calc@A\calc@B\endgroup
   \begingroup\aftergroup#1\begingroup\aftergroup\calc@initB
   \calc@pre@scan}
\def\calc@addAtoB{\advance\calc@B\calc@A}
\def\calc@subtractAfromB{\advance\calc@B-\calc@A}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
%    \begin{macro}{\real}
%    \begin{macro}{\ratio}
%    \begin{macro}{\calc@ratio@x}
%    \begin{macro}{\calc@real@x}
%    The multiplicative operators, `\texttt{*}' and `\texttt{/}', may be
%    followed by a |\real|, |\ratio|, |\minof|, or |\maxof|  token. The
%    last two of these control sequences are defined by \texttt{calc} as
%    they are needed by the scanner for addition or subtraction while the
%    first two are not defined (at least not by the \texttt{calc}
%    package); this,
%    unfortunately, leaves them highly non-robust.  We therefore
%    equate them to |\relax| but only if they have not already been
%    defined\footnote{Suggested code from David Carlisle.}
%    (by some other package: dangerous but possible!); this
%    will also make them appear to be undefined to a \LaTeX{} user
%    (also possibly dangerous).
% \changes{v4.1a}{1998/06/07}
%    {Added macro set-ups to make them robust but undefined: CAR}
%    \begin{macrocode}
\ifx\real\@undefined\let\real\relax\fi
\ifx\ratio\@undefined\let\ratio\relax\fi
%    \end{macrocode}
%    In order to test for |\real| or |\ratio|, we define these
%    two.\footnote{May not need the extra names, CAR?}
%    \begin{macrocode}
\def\calc@ratio@x{\ratio}
\def\calc@real@x{\real}
%    \end{macrocode}
%    \end{macro}
%    \end{macro}
%    \end{macro}
%    \end{macro}
% \begin{macro}{\calc@multiplyx}
% \changes{v4.2}{2005/08/06}
%    {Added $\protect\max$ and $\protect\min$ operations}
% \begin{macro}{\calc@dividex}
% \changes{v4.2}{2005/08/06}
%    {Added $\protect\max$ and $\protect\min$ operations}
% Test which operator followed |*| or |/|. If none followed it's just
% a standard multiplication or division.
%    \begin{macrocode}
\def\calc@multiplyx#1{\def\calc@tmp{#1}%
  \ifx\calc@tmp\calc@ratio@x \let\calc@next\calc@ratio@multiply \else
    \ifx\calc@tmp\calc@real@x \let\calc@next\calc@real@multiply \else
      \ifx\maxof#1\let\calc@next\calc@maxmin@multiply \else
        \let\calc@next\calc@multiply
      \fi
    \fi
  \fi
  \calc@next#1}
\def\calc@dividex#1{\def\calc@tmp{#1}%
  \ifx\calc@tmp\calc@ratio@x \let\calc@next\calc@ratio@divide \else
    \ifx\calc@tmp\calc@real@x \let\calc@next\calc@real@divide \else
      \ifx\maxof#1\let\calc@next\calc@maxmin@divide \else
        \let\calc@next\calc@divide
      \fi
    \fi
  \fi
  \calc@next#1}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@multiply}
% \begin{macro}{\calc@divide}
% \begin{macro}{\calc@generic@multiply}
% \begin{macro}{\calc@multiplyBbyA}
% \begin{macro}{\calc@divideBbyA}
% The binary operators `\texttt{*}' and `\texttt{/}' also insert code
% as determined above.  Moreover, the meaning of |\calc@A| and
% |\calc@B| is changed as factors following a multiplication and
% division operator always have integer type; the original meaning of
% these macros will be restored when the factor has been read and
% evaluated.
%    \begin{macrocode}
\def\calc@multiply{\calc@generic@multiply\calc@multiplyBbyA}
\def\calc@divide{\calc@generic@multiply\calc@divideBbyA}
\def\calc@generic@multiply#1{\endgroup\begingroup
   \let\calc@A\calc@Acount \let\calc@B\calc@Bcount
   \aftergroup#1\calc@pre@scan}
\def\calc@multiplyBbyA{\multiply\calc@B\calc@Acount}
\def\calc@divideBbyA{\divide\calc@B\calc@Acount}
%    \end{macrocode}
% Since the value to use in the multiplication/division operation is
% stored in the |\calc@Acount| register, the |\calc@multiplyBbyA| and
% |\calc@divideBbyA| macros use this register.
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@close}
% |\calc@close| generates code for a right parenthesis (which was
% derived to be ``$\}A\gassign B\}A\gassign B$'' in
% Section~\ref{evaluation:scheme}).  After this code, the control is
% returned to
% |\calc@post@scan| in order to look for another right parenthesis or
% a binary operator.
%    \begin{macrocode}
\def\calc@close
   {\endgroup\global\calc@A\calc@B
    \endgroup\global\calc@A\calc@B
    \calc@post@scan}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Calculating a ratio}
%
% \begin{macro}{\calc@ratio@multiply}
% \begin{macro}{\calc@ratio@divide}
% When |\calc@post@scan| encounters a |\ratio| control sequence, it hands
% control to one of the macros |\calc@ratio@multiply| or |\calc@ratio@divide|,
% depending on the preceding character. Those macros both forward the
% control to the macro |\calc@ratio@evaluate|, which performs two steps: (1) it
% calculates the ratio, which is saved in the global macro token
% |\calc@the@ratio|; (2) it makes sure that the value of |\calc@B| will be
% multiplied by the ratio as soon as the current group ends.
%
% The following macros call |\calc@ratio@evaluate| which multiplies
% |\calc@B| by the ratio, but |\calc@ratio@divide| flips the arguments
% so that the `opposite' fraction is actually evaluated.
%    \begin{macrocode}
\def\calc@ratio@multiply\ratio{\calc@ratio@evaluate}
\def\calc@ratio@divide\ratio#1#2{\calc@ratio@evaluate{#2}{#1}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{\calc@Ccount}
% \begin{macro}{\calc@numerator}
% \begin{macro}{\calc@denominator}
% We shall need two registers for temporary usage in the
% calculations.  We can save one register since we can reuse
% |\calc@Bcount|.
%    \begin{macrocode}
\newcount\calc@Ccount
\let\calc@numerator=\calc@Bcount
\let\calc@denominator=\calc@Ccount
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\calc@ratio@evaluate}
% Here is the macro that handles the actual evaluation of ratios.  The
% procedure is
% this: First, the two expressions are evaluated and coerced to
% integers. The whole procedure is enclosed in a group to be able to
% use the registers |\calc@numerator| and |\calc@denominator| for temporary
% manipulations.
%    \begin{macrocode}
\def\calc@ratio@evaluate#1#2{%
   \endgroup\begingroup
      \calc@assign@dimen\calc@numerator{#1}%
      \calc@assign@dimen\calc@denominator{#2}%
%    \end{macrocode}
% Here we calculate the ratio.  First, we check for negative numerator
% and/or denominator; note that \TeX\ interprets two minus signs the
% same as a plus sign.  Then, we calculate the integer part.
% The minus sign(s), the integer part, and a decimal point, form the
% initial expansion of the |\calc@the@ratio| macro.
%    \begin{macrocode}
      \gdef\calc@the@ratio{}%
      \ifnum\calc@numerator<0 \calc@numerator-\calc@numerator
         \gdef\calc@the@ratio{-}%
      \fi
      \ifnum\calc@denominator<0 \calc@denominator-\calc@denominator
         \xdef\calc@the@ratio{\calc@the@ratio-}%
      \fi
      \calc@Acount\calc@numerator
      \divide\calc@Acount\calc@denominator
      \xdef\calc@the@ratio{\calc@the@ratio\number\calc@Acount.}%
%    \end{macrocode}
% Now we generate the digits after the decimal point, one at a time.
% When \TeX\ scans these digits (in the actual multiplication
% operation), it forms a fixed-point number with 16~bits for
% the fractional part.  We hope that six digits is sufficient, even
% though the last digit may not be rounded correctly.
%    \begin{macrocode}
      \calc@next@digit \calc@next@digit \calc@next@digit
      \calc@next@digit \calc@next@digit \calc@next@digit
   \endgroup
%    \end{macrocode}
% Now we have the ratio represented (as the expansion of the global
% macro |\calc@the@ratio|) in the syntax \<decimal constant>
% \cite[page~270]{texbook}.  This is fed to |\calc@multiply@by@real|
% that will
% perform the actual multiplication.  It is important that the
% multiplication takes place at the correct grouping level so that the
% correct instance of the $B$ register will be used.  Also note that
% we do not need the |\aftergroup| mechanism in this case.
%    \begin{macrocode}
   \calc@multiply@by@real\calc@the@ratio
   \begingroup
   \calc@post@scan}
%    \end{macrocode}
% \end{macro}
% The |\begingroup| inserted before the |\calc@post@scan| will be
% matched by the |\endgroup| generated as part of the replacement of a
% subsequent binary operator or right parenthesis.
% \begin{macro}{\calc@next@digit}
%    \begin{macrocode}
\def\calc@next@digit{%
      \multiply\calc@Acount\calc@denominator
      \advance\calc@numerator -\calc@Acount
      \multiply\calc@numerator 10
      \calc@Acount\calc@numerator
      \divide\calc@Acount\calc@denominator
      \xdef\calc@the@ratio{\calc@the@ratio\number\calc@Acount}}
%    \end{macrocode}
% \end{macro}
% \begin{macro}{\calc@multiply@by@real}
% In the following code, it is important that we first assign the
% result to a dimen register.  Otherwise, \TeX\ won't allow us to
% multiply with a real number.
%    \begin{macrocode}
\def\calc@multiply@by@real#1{\calc@Bdimen #1\calc@B \calc@B\calc@Bdimen}
%    \end{macrocode}
% (Note that this code wouldn't work if |\calc@B| were a muglue
% register.  This is the real reason why the \texttt{calc} package
% doesn't support muglue expressions.  To support muglue expressions
% in full, the |\calc@multiply@by@real| macro must use a muglue register
% instead of |\calc@Bdimen| when |\calc@B| is a muglue register;
% otherwise, a dimen register should be used.  Since integer
% expressions can appear as part of a muglue expression, it would be
% necessary to determine the correct register to use each time a
% multiplication is made.)
% \end{macro}
%
% \subsection{Multiplication by real numbers}
%
% \begin{macro}{\calc@real@multiply}
% \begin{macro}{\calc@real@divide}
% This is similar to the |\calc@ratio@evaluate| macro above, except that
% it is considerably simplified since we don't need to calculate the
% factor explicitly.
%    \begin{macrocode}
\def\calc@real@multiply\real#1{\endgroup
   \calc@multiply@by@real{#1}\begingroup
   \calc@post@scan}
\def\calc@real@divide\real#1{\calc@ratio@evaluate{1pt}{#1pt}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{$\max$ and $\min$ operations}
%
% \begin{macro}{\maxof}
% \begin{macro}{\minof}
% \changes{v4.2}{2005/08/06}
%    {Added macros}
% With version 4.2, the $\max$ and $\min$ operators were
% added to \texttt{calc}. The user functions for them are \cs{maxof} and
% \cs{minof} respectively.
% These macros are internally similar to \cs{widthof} etc.\ in that they
% are unexpandable and easily recognizable by the scanner.
%    \begin{macrocode}
\let\maxof\@@italiccorr
\let\minof\@@italiccorr
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
%
% \begin{macro}{\calc@Cskip}
% \begin{macro}{\ifcalc@count@}
% The $\max$ and $\min$ operations take two arguments so we need an extra
% \<skip> register. We also add a switch for determining when to perform
% a \<skip> or a \<count> assignment.
%    \begin{macrocode}
\newskip\calc@Cskip
\newif\ifcalc@count@
%    \end{macrocode}
% \end{macro}
% \end{macro}
%  \begin{macro}{\calc@maxmin@addsub}
%  \begin{macro}{\calc@maxmin@generic}
% \changes{v4.2}{2005/08/06}{Macros added}
% When doing addition or subtraction with a $\max$ or $\min$ operator, we
% first check if |\calc@A| is a \<count> register or not and then set the
% switch. Then call the real function which sets |\calc@A| to the desired
% value and continue as usual with |\calc@post@scan|.
%    \begin{macrocode}
\def\calc@maxmin@addsub#1#2#3{\begingroup
  \ifx\calc@A\calc@Acount%
    \calc@count@true
  \else
    \calc@count@false
  \fi
  \calc@maxmin@generic#1{#2}{#3}%
  \endgroup
  \calc@post@scan
}
%    \end{macrocode}
% Check the switch and do either \<count> or \<skip> assignments. Note that
% |\maxof| and |\minof| are not set to |>| and |<| until after the
% assignments, which ensures we can nest them without problems. Then set
% |\calc@A| to the correct one.
%    \begin{macrocode}
\def\calc@maxmin@generic#1#2#3{%
  \begingroup
    \ifcalc@count@
      \calc@assign@count\calc@Ccount{#2}%
      \calc@assign@count\calc@Bcount{#3}%
      \def\minof{<}\def\maxof{>}%
      \global\calc@A\ifnum\calc@Ccount#1\calc@Bcount
        \calc@Ccount\else\calc@Bcount\fi
    \else
      \calc@assign@skip\calc@Cskip{#2}%
      \calc@assign@skip\calc@Bskip{#3}%
      \def\minof{<}\def\maxof{>}%
      \global\calc@A\ifdim\calc@Cskip#1\calc@Bskip
        \calc@Cskip\else\calc@Bskip\fi
    \fi
  \endgroup
}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\calc@maxmin@divmul}
% \begin{macro}{\calc@maxmin@multiply}
% \begin{macro}{\calc@maxmin@divide}
% \changes{v4.2}{2005/08/06}{Macros added}
% When doing division or multiplication we must be using \<count> registers
% so we set the switch. Other than that it is almost business as usual when
% multiplying or dividing. |#1| is the instruction to either multiply or
% divide |\calc@B| by |\calc@A|, |#2| is either |\maxof| or |\minof| which
% is waiting in the input stream and |#3| and |#4| are the calc expressions.
% We end it all as usual by calling |\calc@post@scan|.
%    \begin{macrocode}
\def\calc@maxmin@divmul#1#2#3#4{%
  \endgroup\begingroup
  \calc@count@true
  \aftergroup#1%
  \calc@maxmin@generic#2{#3}{#4}%
  \endgroup\begingroup
  \calc@post@scan
}
%    \end{macrocode}
% The two functions called when seeing a |*| or a |/|.
%    \begin{macrocode}
\def\calc@maxmin@multiply{\calc@maxmin@divmul\calc@multiplyBbyA}
\def\calc@maxmin@divide  {\calc@maxmin@divmul\calc@divideBbyA}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \section{Reporting errors}
% \begin{macro}{\calc@error}
% \changes{v4.0d}{1997/11/08}
%    {Use \cs{PackageError} for error messages (DPC)}
% \changes{v4.0e}{1997/11/11}
%    {typo fixed}
% If |\calc@post@scan| reads a character that is not one of `\texttt{+}',
% `\texttt{-}', `\texttt{*}', `\texttt{/}', or `\texttt{)}', an error
% has occurred, and this is reported to the user.  Violations in the
% syntax of \<numeric>s will be detected and reported by \TeX.
% \changes{v4.1a}{1998/06/07}
%    {Improved, I hope, error message: CAR}
%    \begin{macrocode}
\def\calc@error#1{%
   \PackageError{calc}%
     {`#1' invalid at this point}%
     {I expected to see one of: + - * / )}}
%    \end{macrocode}
% \end{macro}
%
% \section{Other additions}
% \begin{macro}{\@settodim}
% \changes{v4.2}{2005/08/06}
%    {Changed kernel macro}
% \begin{macro}{\settototalheight}
% \changes{v4.2}{2005/08/06}
%    {Added macro}
% The kernel macro \cs{@settodim} is changed so that it runs through a list
% containing \cs{ht}, \cs{wd}, and \cs{dp} and than advance the length
% one step at a time. We just have to use a scratch register in case the
% user decides to put in a \cs{global} prefix on the length register.
% A search on the internet confirmed that some people do that kind of thing.
%    \begin{macrocode}
\def\@settodim#1#2#3{%
  \setbox\@tempboxa\hbox{{#3}}%
  \dimen@ii=\z@
  \@tf@r\reserved@a #1\do{%
  \advance\dimen@ii\reserved@a\@tempboxa}%
  #2=\dimen@ii
  \setbox\@tempboxa\box\voidb@x}
%    \end{macrocode}
% Now the user level macro is straightforward.
%    \begin{macrocode}
\def\settototalheight{\@settodim{\ht\dp}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% That's the end of the package.
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \Finale
\endinput