summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/l3kernel/l3fp.dtx
blob: d4f0a435ec99fbb794971f4ff7d756fb2a4bd869 (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
% \iffalse meta-comment
%
%% File: l3fp.dtx Copyright (C) 2011-2012 The LaTeX3 Project
%%
%% It may be distributed and/or modified under the conditions of the
%% LaTeX Project Public License (LPPL), either version 1.3c of this
%% license or (at your option) any later version.  The latest version
%% of this license is in the file
%%
%%    http://www.latex-project.org/lppl.txt
%%
%% This file is part of the "l3kernel bundle" (The Work in LPPL)
%% and all files in that bundle must be distributed together.
%%
%% The released version of this bundle is available from CTAN.
%%
%% -----------------------------------------------------------------------
%%
%% The development version of the bundle can be found at
%%
%%    http://www.latex-project.org/svnroot/experimental/trunk/
%%
%% for those people who are interested.
%%
%%%%%%%%%%%
%% NOTE: %%
%%%%%%%%%%%
%%
%%   Snapshots taken from the repository represent work in progress and may
%%   not work or may contain conflicting material!  We therefore ask
%%   people _not_ to put them into distributions, archives, etc. without
%%   prior consultation with the LaTeX3 Project Team.
%%
%% -----------------------------------------------------------------------
%%
%
%<*driver|package>
\RequirePackage{l3bootstrap}
\GetIdInfo$Id: l3fp.dtx 4521 2013-07-09 11:45:31Z joseph $
  {L3 Floating points}
%</driver|package>
%<*driver>
\documentclass[full]{l3doc}
\usepackage{amsmath}
\begin{document}
  \DocInput{\jobname.dtx}
\end{document}
%</driver>
% \fi
%
% ^^A need to provide this inside the file:
%
% \providecommand\nan{\texttt{NaN}}
%
%
% \title{^^A
%   The \textsf{l3fp} package: floating points^^A
%   \thanks{This file describes v\ExplFileVersion,
%     last revised \ExplFileDate.}^^A
% }
%
% \author{^^A
%  The \LaTeX3 Project\thanks
%    {^^A
%      E-mail:
%        \href{mailto:latex-team@latex-project.org}
%          {latex-team@latex-project.org}^^A
%    }^^A
% }
%
% \date{Released \ExplFileDate}
%
% \maketitle
%
% \begin{documentation}
%
% A decimal floating point number is one which is stored as a significand and a
% separate exponent.  The module implements expandably a wide set of
% arithmetic, trigonometric, and other operations on decimal floating point
% numbers, to be used within floating point expressions.  Floating point
% expressions support the following operations with their usual
% precedence.
% \begin{itemize}
%   \item Basic arithmetic: addition $x+y$, subtraction $x-y$,
%     multiplication $x*y$, division $x/y$, and parentheses.
%   \item Comparison operators: $x\mathop{\mathtt{<}}y$,
%     $x\mathop{\mathtt{<=}}y$, $x\mathop{\mathtt{>?}}y$,
%     $x\mathop{\mathtt{!=}}y$ \emph{etc.}
%   \item Boolean logic: negation $\mathop{!}x$, conjunction
%     $x\mathop{\&\&}y$, disjunction $x\mathop{\vert\vert}y$, ternary
%     operator $x\mathop{?}y\mathop{:}z$.
%   \item Exponentials: $\exp x$, $\ln x$, $x^y$.
%   \item Trigonometry: $\sin x$, $\cos x$, $\tan x$, $\cot x$, $\sec
%     x$, $\csc x$.
%   \item [\emph{(not yet)}] Inverse trigonometric functions:
%     $\operatorname{asin} x$, $\operatorname{acos} x$,
%     $\operatorname{atan} x$, $\operatorname{acot} x$,
%     $\operatorname{asec} x$, $\operatorname{acsc} x$.
%   \item [\emph{(not yet)}] Hyperbolic functions and their inverse
%     functions: $\sinh x$, $\cosh x$, $\tanh x$, $\coth x$,
%     $\operatorname{sech} x$, $\operatorname{csch}$, and
%     $\operatorname{asinh} x$, $\operatorname{acosh} x$,
%     $\operatorname{atanh} x$, $\operatorname{acoth} x$,
%     $\operatorname{asech} x$, $\operatorname{acsch} x$.
%   \item Extrema: $\max(x,y,\ldots)$, $\min(x,y,\ldots)$,
%     $\operatorname{abs}(x)$.
%   \item Rounding functions: $\operatorname{round}(x,n)$ round to
%     closest, $\operatorname{round0}(x,n)$ round towards zero,
%     $\operatorname{round\pm}(x,n)$ round towards $\pm\infty$.  And
%     \emph{(not yet)} modulo, and \enquote{quantize}.
%   \item Constants: \texttt{pi}, \texttt{deg} (one degree in radians).
%   \item Dimensions, automatically expressed in points, \emph{e.g.},
%     \texttt{pc} is $12$.
%   \item Automatic conversion (no need for \cs{\meta{type}_use:N}) of
%     integer, dimension, and skip variables to floating points,
%     expressing dimensions in points and ignoring the stretch and
%     shrink components of skips.
% \end{itemize}
% Floating point numbers can be given either explicitly (in a form such
% as |1.234e-34|, or |-.0001|), or as a stored floating point variable,
% which is automatically replaced by its current value.  See
% section~\ref{sec:l3fp:fp-floats} for a description of what a floating point is,
% section~\ref{sec:l3fp:fp-precedence} for details about how an expression is
% parsed, and section~\ref{sec:l3fp:fp-operations} to know what the various
% operations do.  Some operations may raise exceptions (error messages),
% described in section~\ref{sec:l3fp:fp-exceptions}.
%
% An example of use could be the following.
% \begin{verbatim}
%   \LaTeX{} can now compute: $ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3}
%   = \ExplSyntaxOn \fp_to_decimal:n {sin 3.5 /2 + 2e-3} $.
% \end{verbatim}
% But in all fairness, this module is mostly meant as an underlying tool
% for higher-level commands.  For example, one could provide a function
% to typeset nicely the result of floating point computations.
% \begin{verbatim}
%   \usepackage{xparse, siunitx}
%   \ExplSyntaxOn
%   \NewDocumentCommand { \calcnum } { m }
%     { \num { \fp_to_scientific:n {#1} } }
%   \ExplSyntaxOff
%   \calcnum { 2 pi * sin ( 2.3 ^ 5 ) }
% \end{verbatim}
%
% \section{Creating and initialising floating point variables}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp001]
%   {\fp_new:N, \fp_new:c}
%   \begin{syntax}
%     \cs{fp_new:N} \meta{fp~var}
%   \end{syntax}
%   Creates a new \meta{fp~var} or raises an error if the name is
%   already taken. The declaration is global. The \meta{fp~var} will
%   initially be $+0$.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp001]
%   {\fp_const:Nn, \fp_const:cn}
%   \begin{syntax}
%     \cs{fp_const:Nn} \meta{fp~var} \Arg{floating point expression}
%   \end{syntax}
%   Creates a new constant \meta{fp~var} or raises an error if the name
%   is already taken. The \meta{fp~var} will be set globally equal to
%   the result of evaluating the \meta{floating point expression}.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp001]
%   {\fp_zero:N, \fp_zero:c, \fp_gzero:N, \fp_gzero:c}
%   \begin{syntax}
%     \cs{fp_zero:N} \meta{fp~var}
%   \end{syntax}
%   Sets the \meta{fp~var} to~$+0$.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp001]
%   {\fp_zero_new:N, \fp_zero_new:c, \fp_gzero_new:N, \fp_gzero_new:c}
%   \begin{syntax}
%     \cs{fp_zero_new:N} \meta{fp~var}
%   \end{syntax}
%   Ensures that the \meta{fp~var} exists globally
%   by applying \cs{fp_new:N} if necessary, then applies
%   \cs{fp_(g)zero:N} to leave the \meta{fp~var} set to zero.
% \end{function}
%
% \section{Setting floating point variables}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp002]
%   {\fp_set:Nn, \fp_set:cn, \fp_gset:Nn, \fp_gset:cn}
%   \begin{syntax}
%     \cs{fp_set:Nn} \meta{fp~var} \Arg{floating point expression}
%   \end{syntax}
%   Sets \meta{fp~var} equal to the result of computing the
%   \meta{floating point expression}.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp002]
%   {
%     \fp_set_eq:NN , \fp_set_eq:cN , \fp_set_eq:Nc , \fp_set_eq:cc ,
%     \fp_gset_eq:NN, \fp_gset_eq:cN, \fp_gset_eq:Nc, \fp_gset_eq:cc
%   }
%   \begin{syntax}
%     \cs{fp_set_eq:NN} \meta{fp~var_1} \meta{fp~var_2}
%   \end{syntax}
%   Sets the floating point variable \meta{fp~var_1} equal to the current
%   value of \meta{fp~var_2}.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp002]
%   {\fp_add:Nn, \fp_add:cn, \fp_gadd:Nn, \fp_gadd:cn}
%   \begin{syntax}
%     \cs{fp_add:Nn} \meta{fp~var} \Arg{floating point expression}
%   \end{syntax}
%   Adds the result of computing the \meta{floating point expression} to
%   the \meta{fp~var}.
% \end{function}
%
% \begin{function}[updated = 2012-05-08, tested = m3fp002]
%   {\fp_sub:Nn, \fp_sub:cn, \fp_gsub:Nn, \fp_gsub:cn}
%   \begin{syntax}
%     \cs{fp_sub:Nn} \meta{fp~var} \Arg{floating point expression}
%   \end{syntax}
%   Subtracts the result of computing the \meta{floating point
%     expression} from the \meta{fp~var}.
% \end{function}
%
% \section{Using floating point numbers}
%
% \begin{function}[EXP, added = 2012-05-08, updated = 2012-07-08,
%   tested = m3fp-convert003]{\fp_eval:n}
%   \begin{syntax}
%     \cs{fp_eval:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and expresses the
%   result as a decimal number with~$16$ significant figures and no
%   exponent.  Leading or trailing zeros may be inserted to compensate
%   for the exponent.  Non-significant trailing zeros are trimmed, and
%   integers are expressed without a decimal separator.  The values
%   $\pm\infty$ and \nan{} trigger an \enquote{invalid operation}
%   exception.  This function is identical to \cs{fp_to_decimal:n}.
% \end{function}
%
% \begin{function}[EXP, added = 2012-05-08, updated = 2012-07-08,
%   tested = m3fp-convert003]
%   {\fp_to_decimal:N, \fp_to_decimal:c, \fp_to_decimal:n}
%   \begin{syntax}
%     \cs{fp_to_decimal:N} \meta{fp~var}
%     \cs{fp_to_decimal:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and expresses the
%   result as a decimal number with $16$ significant figures and no
%   exponent.  Leading or trailing zeros may be inserted to compensate
%   for the exponent.  Non-significant trailing zeros are trimmed, and
%   integers are expressed without a decimal separator.  The values
%   $\pm\infty$ and \nan{} trigger an \enquote{invalid operation}
%   exception.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-07-08, tested = m3fp-convert003]
%   {\fp_to_dim:N, \fp_to_dim:c, \fp_to_dim:n}
%   \begin{syntax}
%     \cs{fp_to_dim:N} \meta{fp~var}
%     \cs{fp_to_dim:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and expresses the
%   result as a dimension (in \texttt{pt}) suitable for use in dimension
%   expressions.  The output is identical to \cs{fp_to_decimal:n}, with
%   an additional trailing \texttt{pt}.  In particular, the result may
%   be outside the range $[- 2^{14} + 2^{-17}, 2^{14} - 2^{-17}]$ of
%   valid \TeX{} dimensions, leading to overflow errors if used as a
%   dimension.  The values $\pm\infty$ and \nan{} trigger an
%   \enquote{invalid operation} exception.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-07-08, tested = m3fp-convert003]
%   {\fp_to_int:N, \fp_to_int:c, \fp_to_int:n}
%   \begin{syntax}
%     \cs{fp_to_int:N} \meta{fp~var}
%     \cs{fp_to_int:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression}, and rounds the
%   result to the closest integer, with ties rounded to an even integer.
%   The result may be outside the range $[- 2^{31} + 1, 2^{31} - 1]$ of
%   valid \TeX{} integers, triggering \TeX{} errors if used in an
%   integer expression.  The values $\pm\infty$ and \nan{} trigger
%   an \enquote{invalid operation} exception.
% \end{function}
%
% \begin{function}[EXP, added = 2012-05-08, updated = 2012-07-08,
%   tested = m3fp-convert003]
%   {\fp_to_scientific:N, \fp_to_scientific:c, \fp_to_scientific:n}
%   \begin{syntax}
%     \cs{fp_to_scientific:N} \meta{fp~var}
%     \cs{fp_to_scientific:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and expresses the
%   result in scientific notation with $16$ significant figures:
%   \begin{quote}
%     \meta{optional \texttt{-}}\meta{digit}\texttt{.}\meta{15 digits}\texttt{e}\meta{optional sign}\meta{exponent}
%   \end{quote}
%   The leading \meta{digit} is non-zero except in the case of $\pm 0$.
%   The values $\pm\infty$ and \nan{} trigger an \enquote{invalid
%     operation} exception.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-07-08, tested = m3fp-convert003]
%   {\fp_to_tl:N, \fp_to_tl:c, \fp_to_tl:n}
%   \begin{syntax}
%     \cs{fp_to_tl:N} \meta{fp~var}
%     \cs{fp_to_tl:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and expresses the
%   result in (almost) the shortest possible form.  Numbers in the
%   ranges $(0,10^{-3})$ and $[10^{16},\infty)$ are expressed in
%   scientific notation with trailing zeros trimmed (see
%   \cs{fp_to_scientific:n}).  Numbers in the range $[10^{-3},10^{16})$
%   are expressed in a decimal notation without exponent, with trailing
%   zeros trimmed, and no decimal separator for integer values (see
%   \cs{fp_to_decimal:n}.  Negative numbers start with |-|.  The
%   special values $\pm 0$, $\pm \inf$ and \nan{} are rendered as
%   |0|, |-0|, \texttt{inf}, \texttt{-inf}, and \texttt{nan}
%   respectively.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-07-08, tested = m3fp-convert003]
%   {\fp_use:N, \fp_use:c}
%   \begin{syntax}
%     \cs{fp_use:N} \meta{fp~var}
%   \end{syntax}
%   Inserts the value of the \meta{fp~var} into the input stream as a
%   decimal number with $16$ significant figures and no exponent.
%   Leading or trailing zeros may be inserted to compensate for the
%   exponent.  Non-significant trailing zeros are trimmed.  Integers are
%   expressed without a decimal separator.  The values $\pm\infty$ and
%   \nan{} trigger an \enquote{invalid operation} exception.  This
%   function is identical to \cs{fp_to_decimal:N}.
% \end{function}
%
% \section{Floating point conditionals}
%
% \begin{function}[EXP, pTF, updated = 2012-05-08, tested = m3fp002]
%   {\fp_if_exist:N, \fp_if_exist:c}
%   \begin{syntax}
%     \cs{fp_if_exist_p:N} \meta{fp~var}
%     \cs{fp_if_exist:NTF} \meta{fp~var} \Arg{true code} \Arg{false code}
%   \end{syntax}
%   Tests whether the \meta{fp~var} is currently defined.  This does not
%   check that the \meta{fp~var} really is a floating point variable.
% \end{function}
%
% \begin{function}[EXP, pTF, updated = 2012-05-08,
%   tested = m3fp-logic001]{\fp_compare:nNn, \fp_compare:n}
%   \begin{syntax}
%     \cs{fp_compare_p:nNn} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2}
%     \cs{fp_compare:nNnTF} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2} \Arg{true code} \Arg{false code}
%     \cs{fp_compare_p:n} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \}
%     \cs{fp_compare:nTF} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \} \Arg{true code} \Arg{false code}
%   \end{syntax}
%   Compares the \meta{fpexpr_1} and the \meta{fpexpr_2}, and returns
%   \texttt{true} if the \meta{relation} is obeyed.  Two floating point
%   numbers $x$ and $y$ may obey four mutually exclusive relations:
%   $x<y$, $x=y$, $x>y$, or $x$ and $y$ are not ordered.  The latter
%   case occurs exactly when one of the operands is \nan{}, and
%   this relations is denoted by the symbol |?|.  The \texttt{nNn}
%   functions support the \meta{relations} |<|, |=|, |>|, and |?|.  The
%   \texttt{n} functions support as a \meta{relation} any non-empty string of
%   those four symbols, plus optional leading |!| (which negate the
%   \meta{relation}), with the restriction that the \meta{relation} may
%   not start with |?|.  Common choices of \meta{relation} include |>=|
%   (greater or equal), |!=| (not equal), |!?| (comparable).  Note that
%   a \nan{} is distinct from any value, even another
%   \nan{}, hence $x=x$ is not true for a \nan{}.
%   Since a \nan{} is not comparable to any floating point, to test if a
%   value is \nan{}, one can use the following, where~$0$ is an
%   arbitrary floating point.
%   \begin{verbatim}
%     \fp_compare:nNnTF { <value> } ? { 0 }
%       { } % <value> is nan
%       { } % <value> is not nan
%   \end{verbatim}
% \end{function}
%
% \section{Floating point expression loops}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_do_until:nNnn}
%   \begin{syntax}
%      \cs{fp_do_until:nNnn} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2} \Arg{code}
%   \end{syntax}
%   Places the \meta{code} in the input stream for \TeX{} to process,
%   and then evaluates the relationship between the two \meta{floating
%     point expressions} as described for \cs{fp_compare:nNnTF}.  If the
%   test is \texttt{false} then the \meta{code} will be inserted into
%   the input stream again and a loop will occur until the
%   \meta{relation} is \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_do_while:nNnn}
%   \begin{syntax}
%      \cs{fp_do_while:nNnn} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2} \Arg{code}
%   \end{syntax}
%   Places the \meta{code} in the input stream for \TeX{} to process,
%   and then evaluates the relationship between the two \meta{floating
%     point expressions} as described for \cs{fp_compare:nNnTF}.  If the
%   test is \texttt{true} then the \meta{code} will be inserted into the
%   input stream again and a loop will occur until the \meta{relation}
%   is \texttt{false}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_until_do:nNnn}
%   \begin{syntax}
%      \cs{fp_until_do:nNnn} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2} \Arg{code}
%   \end{syntax}
%   Evaluates the relationship between the two \meta{floating point
%     expressions} as described for \cs{fp_compare:nNnTF}, and then
%   places the \meta{code} in the input stream if the \meta{relation} is
%   \texttt{false}.  After the \meta{code} has been processed by \TeX{}
%   the test will be repeated, and a loop will occur until the test is
%   \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_while_do:nNnn}
%   \begin{syntax}
%      \cs{fp_while_do:nNnn} \Arg{fpexpr_1} \meta{relation} \Arg{fpexpr_2} \Arg{code}
%   \end{syntax}
%   Evaluates the relationship between the two \meta{floating point
%     expressions} as described for \cs{fp_compare:nNnTF}, and then
%   places the \meta{code} in the input stream if the \meta{relation} is
%   \texttt{true}.  After the \meta{code} has been processed by \TeX{}
%   the test will be repeated, and a loop will occur until the test is
%   \texttt{false}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_do_until:nn}
%   \begin{syntax}
%      \cs{fp_do_until:nn} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \} \Arg{code}
%   \end{syntax}
%   Places the \meta{code} in the input stream for \TeX{} to process,
%   and then evaluates the relationship between the two \meta{floating
%     point expressions} as described for \cs{fp_compare:nTF}.  If the
%   test is \texttt{false} then the \meta{code} will be inserted into
%   the input stream again and a loop will occur until the
%   \meta{relation} is \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_do_while:nn}
%   \begin{syntax}
%      \cs{fp_do_while:nn} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \} \Arg{code}
%   \end{syntax}
%   Places the \meta{code} in the input stream for \TeX{} to process,
%   and then evaluates the relationship between the two \meta{floating
%     point expressions} as described for \cs{fp_compare:nTF}.  If the
%   test is \texttt{true} then the \meta{code} will be inserted into the
%   input stream again and a loop will occur until the \meta{relation}
%   is \texttt{false}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_until_do:nn}
%   \begin{syntax}
%      \cs{fp_until_do:nn} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \} \Arg{code}
%   \end{syntax}
%   Evaluates the relationship between the two \meta{floating point
%     expressions} as described for \cs{fp_compare:nTF}, and then places
%   the \meta{code} in the input stream if the \meta{relation} is
%   \texttt{false}.  After the \meta{code} has been processed by \TeX{}
%   the test will be repeated, and a loop will occur until the test is
%   \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP, added = 2012-08-16, tested = m3fp-logic003]
%   {\fp_while_do:nn}
%   \begin{syntax}
%      \cs{fp_while_do:nn} \{ \meta{fpexpr_1} \meta{relation} \meta{fpexpr_2} \} \Arg{code}
%   \end{syntax}
%   Evaluates the relationship between the two \meta{floating point
%     expressions} as described for \cs{fp_compare:nTF}, and then places
%   the \meta{code} in the input stream if the \meta{relation} is
%   \texttt{true}.  After the \meta{code} has been processed by \TeX{}
%   the test will be repeated, and a loop will occur until the test is
%   \texttt{false}.
% \end{function}
%
% \section{Some useful constants, and scratch variables}
%
% \begin{variable}[added = 2012-05-08]{\c_zero_fp, \c_minus_zero_fp}
%   Zero, with either sign.
% \end{variable}
% 
% \begin{variable}[added = 2012-05-08]{\c_one_fp}
%   One as an \texttt{fp}: useful for comparisons in some places.
% \end{variable}
%
% \begin{variable}[added = 2012-05-08]{\c_inf_fp, \c_minus_inf_fp}
%   Infinity, with either sign.  These can be input directly in a
%   floating point expression as \texttt{inf} and \texttt{-inf}.
% \end{variable}
%
% \begin{variable}[updated = 2012-05-08]{\c_e_fp}
%   The value of the base of the natural logarithm, $\mathrm{e} = \exp(1)$.
% \end{variable}
%
% \begin{variable}[updated = 2012-05-08]{\c_pi_fp}
%   The value of $\pi$.  This can be input directly in a floating point
%   expression as \texttt{pi}.  The value is rounded in a slightly odd
%   way, to ensure for instance that \texttt{sin(pi)} yields an exact $0$.
% \end{variable}
%
% \begin{variable}[added = 2012-05-08]{\c_one_degree_fp}
%   The value of $1^{\circ}$ in radians.  Multiply an angle given in
%   degrees by this value to obtain a result in radians, suitable to be
%   used for trigonometric functions.  Within floating point
%   expressions, this can be accessed as \texttt{deg}.  Note that
%   \texttt{180 deg = pi} exactly.
% \end{variable}
%
% \begin{variable}{\l_tmpa_fp, \l_tmpb_fp}
%   Scratch floating points for local assignment. These are never used by
%   the kernel code, and so are safe for use with any \LaTeX3-defined
%   function. However, they may be overwritten by other non-kernel
%   code and so should only be used for short-term storage.
% \end{variable}
%
% \begin{variable}{\g_tmpa_fp, \g_tmpb_fp}
%   Scratch floating points for global assignment. These are never used by
%   the kernel code, and so are safe for use with any \LaTeX3-defined
%   function. However, they may be overwritten by other non-kernel
%   code and so should only be used for short-term storage.
% \end{variable}
%
% \section{Floating point exceptions}
% \label{sec:l3fp:fp-exceptions}
%
% \emph{The functions defined in this section are experimental, and
%   their functionality may be altered or removed altogether.}
%
% \enquote{Exceptions} may occur when performing some floating point
% operations, such as \texttt{0 / 0}, or \texttt{10 ** 1e9999}.  The
% \textsc{IEEE} standard defines $5$ types of exceptions.
% \begin{itemize}
% \item \emph{Overflow} occurs whenever the result of an operation is
%   too large to be represented as a normal floating point number.  This
%   results in $\pm \infty$.
% \item \emph{Underflow} occurs whenever the result of an operation is
%   too close to $0$ to be represented as a normal floating point
%   number.  This results in $\pm 0$.
% \item \emph{Invalid operation} occurs for operations with no defined
%   outcome, for instance $0/0$, or $\sin(\infty)$, and almost any
%   operation involving a \nan{}.  This normally results in a \nan{},
%   except for conversion functions whose target type does not have a
%   notion of \nan{} (\emph{e.g.}, \cs{fp_to_dim:n}).
% \item \emph{Division by zero} occurs when dividing a non-zero number
%   by $0$, or when evaluating \emph{e.g.}, $\ln(0)$ or $\cot(0)$.  This
%   results in $\pm\infty$.
% \item \emph{Inexact} occurs whenever the result of a computation is
%   not exact, in other words, almost always.  At the moment, this
%   exception is entirely ignored in \LaTeX3.
% \end{itemize}
% To each exception is associated a \enquote{flag}, which can be either
% \emph{on} or \emph{off}.  By default, the \enquote{invalid operation}
% exception triggers an (expandable) error, and raises the corresponding
% flag.  Other exceptions only raise the corresponding flag.  The state
% of the flag can be tested
% and modified.  The behaviour when an exception occurs can be modified
% (using \cs{fp_trap:nn}) to either produce an error and turn the flag
% on, or only turn the flag on, or do nothing at all.
%
% \begin{function}[EXP, pTF, added = 2012-08-08, tested = m3fp-traps001]
%   {\fp_if_flag_on:n}
%   \begin{syntax}
%     \cs{fp_if_flag_on_p:n} \Arg{exception}
%     \cs{fp_if_flag_on:nTF} \Arg{exception} \Arg{true code} \Arg{false code}
%   \end{syntax}
%   Tests if the flag for the \meta{exception} is on, which normally
%   means the given \meta{exception} has occurred.
%   \emph{This function is experimental, and may be altered or removed.}
% \end{function}
%
% \begin{function}[added = 2012-08-08, tested = m3fp-traps001]
%   {\fp_flag_off:n}
%   \begin{syntax}
%     \cs{fp_flag_off:n} \Arg{exception}
%   \end{syntax}
%   Locally turns off the flag which indicates whether the
%   \meta{exception} has occurred.
%   \emph{This function is experimental, and may be altered or removed.}
% \end{function}
%
% \begin{function}[EXP, added = 2012-08-08, tested = m3fp-traps001]
%   {\fp_flag_on:n}
%   \begin{syntax}
%     \cs{fp_flag_on:n} \Arg{exception}
%   \end{syntax}
%   Locally turns on the flag to indicate (or pretend) that the
%   \meta{exception} has occurred.  Note that this function is
%   expandable: it is used internally by \pkg{l3fp} to signal when
%   exceptions do occur.
%   \emph{This function is experimental, and may be altered or removed.}
% \end{function}
%
% \begin{function}[added = 2012-07-19, updated = 2012-08-08,
%   tested = m3fp-traps001]{\fp_trap:nn}
%   \begin{syntax}
%     \cs{fp_trap:nn} \Arg{exception} \Arg{trap type}
%   \end{syntax}
%   All occurrences of the \meta{exception} (\texttt{invalid_operation},
%   \texttt{division_by_zero}, \texttt{overflow}, or \texttt{underflow})
%   within the current group are treated as \meta{trap type}, which can
%   be
%   \begin{itemize}
%     \item \texttt{none}: the \meta{exception} will be entirely
%       ignored, and leave no trace;
%     \item \texttt{flag}: the \meta{exception} will turn the
%       corresponding flag on when it occurs;
%     \item \texttt{error}: additionally, the \meta{exception} will halt
%       the \TeX{} run and display some information about the current
%       operation in the terminal.
%   \end{itemize}
%   \emph{This function is experimental, and may be altered or removed.}
% \end{function}
%
% \section{Viewing floating points}
%
% \begin{function}[added = 2012-05-08, updated = 2012-08-14,
%   tested = m3fp002]{\fp_show:N, \fp_show:c, \fp_show:n}
%   \begin{syntax}
%     \cs{fp_show:N} \meta{fp~var}
%     \cs{fp_show:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and displays the
%   result in the terminal.
% \end{function}
%
% \section{Floating point expressions}
%
% \subsection{Input of floating point numbers} \label{sec:l3fp:fp-floats}
%
% We support four types of floating point numbers:
% \begin{itemize}
%   \item $\pm 0.d_1d_2\ldots{}d_{16} \cdot 10^{n}$, a normal floating
%     point number, with $d_i\in [0,9]$, $d_1\neq 0$, and $\lvert n\rvert
%     \leq \ExplSyntaxOn \int_use:N \c__fp_max_exponent_int$;
%   \item $\pm 0$, zero, with a given sign;
%   \item $\pm \infty$, infinity, with a given sign;
%   \item \nan{}, is \enquote{not a number}, and can be either quiet
%     or signalling (\emph{not yet}: this distinction is currently
%     unsupported);
%   \item [\emph{(not yet)}] subnormal numbers $\pm 0.d_1d_2\ldots{}d_{16}
%     \cdot 10^{-\ExplSyntaxOn\int_use:N \c__fp_max_exponent_int}$ with
%     $d_1=0$.
% \end{itemize}
% Normal floating point numbers are stored in base $10$, with $16$
% significant figures.
%
% On input, a normal floating point number consists of:
% \begin{itemize}
%   \item \meta{sign}: a possibly empty string of |+| and |-| characters;
%   \item \meta{significand}: a non-empty string of digits together with zero
%     or one dot;
%   \item \meta{exponent} optionally: the character |e|, followed by a
%     possibly empty string of |+|~and~|-| tokens, and a non-empty string
%     of digits.
% \end{itemize}
% The sign of the resulting number is |+| if \meta{sign} contains an
% even number of |-|, and |-| otherwise, hence, an empty \meta{sign}
% denotes a non-negative input.  The stored significand is obtained from
% \meta{significand} by omitting the decimal separator and leading zeros,
% and rounding to $16$ significant digits, filling with trailing zeros
% if necessary.  In particular, the value stored is exact if the input
% \meta{significand} has at most $16$ digits.  The stored \meta{exponent}
% is obtained by combining the input \meta{exponent} ($0$ if absent)
% with a shift depending on the position of the significand and the number
% of leading zeros.
%
% A special case arises if the resulting \meta{exponent} is either too
% large or too small for the floating point number to be
% represented.  This results either in an overflow (the number is then
% replaced by $\pm\infty$), or an underflow (resulting in $\pm 0$).
%
% The result is thus $\pm 0$ if and only if \meta{significand} contains no
% non-zero digit (\emph{i.e.}, consists only in~|0| characters, and an
% optional |.| character), or if there is an underflow.  Note that a
% single dot is currently a valid floating point number, equal to~$+0$,
% but that is not guaranteed to remain true.
%
% Special numbers are input as follows:
% \begin{itemize}
%   \item \texttt{inf} represents $+\infty$, and can be preceded by any
%     \meta{sign}, yielding $\pm\infty$ as appropriate.
%   \item \texttt{nan} represents a (quiet) non-number.  It can be
%     preceded by any sign, but that will be ignored.
%   \item Any unrecognizable string triggers an error, and produces a
%     \nan{}.
% \end{itemize}
%
% Note that~|e-1| is not a representation of $10^{-1}$, because it could
% be mistaken with the difference of \enquote{\texttt{e}} and $1$.  This
% is consistent with several other programming languages.  However, in
% order to avoid confusions, |e-1| is not considered to be this
% difference either.  To input the base of natural logarithms, use
% \texttt{exp(1)} or \cs{c_e_fp}.
%
% \subsection{Precedence of operators}
% \label{sec:l3fp:fp-precedence}
%
% We list here all the operations supported in floating point
% expressions, in order of decreasing precedence: operations listed
% earlier bind more tightly than operations listed below them.
% \begin{itemize}
%   \item Implicit multiplication by juxtaposition (\texttt{2pi}, \emph{etc}).
%   \item Function calls (\texttt{sin}, \texttt{ln}, \emph{etc}).
%   \item Binary |**| and |^| (right associative).
%   \item Unary |+|, |-|, |!|.
%   \item Binary |*|, |/| and |%|.
%   \item Binary |+| and |-|.
%   \item Comparisons |>=|, |!=|, |<?|, \emph{etc}.
%   \item Logical \texttt{and}, denoted by |&&|.
%   \item Logical \texttt{or}, denoted by \verb+||+.
%   \item Ternary operator |?:| (right associative).
% \end{itemize}
% The precedence of operations can be overridden using parentheses.
% In particular, those precedences imply that
% \begin{align*}
%   \mathtt{sin 2pi} & = \sin(2\pi) = 0, \\
%   \mathtt{2\char`\^2max(3,4)} & = 2^{2 \max(3,4)} = 256.
% \end{align*}
% Functions are called on the value of their argument, contrarily to
% \TeX{} macros.
%
% \subsection{Operations} \label{sec:l3fp:fp-operations}
%
% We now present the various operations allowed in floating point
% expressions, from the lowest precedence to the highest.  When used as
% a truth value, a floating point expression is \texttt{false} if it is
% $\pm 0$, and \texttt{true} otherwise, including when it is \nan{}.
%
% \begin{function}[tested = m3fp-logic002]{?:}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} |?| \meta{operand_2} |:| \meta{operand_3} \}
%   \end{syntax}
%   The ternary operator |?:| results in \meta{operand_2} if
%   \meta{operand_1} is true, and \meta{operand_3} if it is false (equal to
%   $\pm 0$).  All three \meta{operands} are evaluated in all cases.  The
%   operator is right associative, hence
%   \begin{verbatim}
%     \fp_eval:n
%       {
%         1 + 3 > 4 ? 1 :
%         2 + 4 > 5 ? 2 :
%         3 + 5 > 6 ? 3 : 4
%       }
%   \end{verbatim}
%   first tests whether $1 + 3 > 4$; since this isn't true, the branch
%   following |:| is taken, and $2 + 4 > 5$ is compared; since this is
%   true, the branch before |:| is taken, and everything else is
%   (evaluated then) ignored.  That allows testing for various cases in
%   a concise manner, with the drawback that all computations are made
%   in all cases.
% \end{function}
%
% \begingroup \catcode`\|=12
% \begin{function}[tested = m3fp-logic002]{TWO BARS}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} \texttt{||} \meta{operand_2} \}
%   \end{syntax}
%   If \meta{operand_1} is true (non-zero), use that value, otherwise the
%   value of \meta{operand_2}.  Both \meta{operands} are evaluated in all
%   cases.
% \end{function}
% \endgroup
%
% \begingroup \catcode`\&=12
% \begin{function}[tested = m3fp-logic002]{&&}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} \texttt{&&} \meta{operand_2} \}
%   \end{syntax}
%   If \meta{operand_1} is false (equal to $\pm 0$), use that value,
%   otherwise the value of \meta{operand_2}.  Both \meta{operands} are
%   evaluated in all cases.
% \end{function}
% \endgroup
%
% \begin{function}[tested = m3fp-logic001]{\<, =, >, ?}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} \meta{comparison} \meta{operand_2} \}
%   \end{syntax}
%   The \meta{comparison} consists of a non-empty string of |<|, |=|,
%   |>|, and |?|, optionally preceded by |!|.  It may not start with
%   |?|.  This evaluates to $+1$ if the \meta{comparison} between the
%   \meta{operand_1} and \meta{operand_2} is true, and $+0$ otherwise.
% \end{function}
%
% \begin{function}[tested = m3fp-basics001]{+, -}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} |+| \meta{operand_2} \}
%     \cs{fp_eval:n} \{ \meta{operand_1} |-| \meta{operand_2} \}
%   \end{syntax}
%   Computes the sum or the difference of its two \meta{operands}.  The
%   \enquote{invalid operation} exception occurs for $\infty-\infty$.
%   \enquote{Underflow} and \enquote{overflow} occur when appropriate.
% \end{function}
%
% \begin{function}[tested = {m3fp-basics002, m3fp-basics003}]{*, /}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} |*| \meta{operand_2} \}
%     \cs{fp_eval:n} \{ \meta{operand_1} |/| \meta{operand_2} \}
%   \end{syntax}
%   Computes the product or the ratio of its two \meta{operands}.  The
%   \enquote{invalid operation} exception occurs for $\infty/\infty$,
%   $0/0$, or $0*\infty$.  \enquote{Division by zero} occurs when
%   dividing a finite non-zero number by $\pm 0$.  \enquote{Underflow}
%   and \enquote{overflow} occur when appropriate.
% \end{function}
%
% \begin{function}[tested = m3fp-basics004]{+, -, !}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |+| \meta{operand} \}
%     \cs{fp_eval:n} \{ |-| \meta{operand} \}
%     \cs{fp_eval:n} \{ |!| \meta{operand} \}
%   \end{syntax}
%   The unary |+| does nothing, the unary |-| changes the sign of the
%   \meta{operand}, and |!| \meta{operand} evaluates to $1$ if
%   \meta{operand} is false and $0$ otherwise (this is the \texttt{not}
%   boolean function).  Those operations never raise exceptions.
% \end{function}
%
% \begingroup\catcode`\^=12
% \begin{function}[tested = m3fp-expo001]{**, ^}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ \meta{operand_1} |**| \meta{operand_2} \}
%     \cs{fp_eval:n} \{ \meta{operand_1} |^| \meta{operand_2} \}
%   \end{syntax}
%   Raises \meta{operand_1} to the power \meta{operand_2}.  This
%   operation is right associative, hence \texttt{2 ** 2 ** 3} equals
%   $2^{2^{3}} = 256$.  The \enquote{invalid operation} exception occurs
%   if \meta{operand_1} is negative or $-0$, and \meta{operand_2} is not
%   an integer, unless the result is zero (in that case, the sign is
%   chosen arbitrarily to be $+0$).  \enquote{Division by zero} occurs
%   when raising $\pm 0$ to a strictly negative power.
%   \enquote{Underflow} and \enquote{overflow} occur when appropriate.
% \end{function}
% \endgroup
%
% \begin{function}[tested = m3fp-basics004]{abs}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |abs(| \meta{fpexpr} |)| \}
%   \end{syntax}
%   Computes the absolute value of the \meta{fpexpr}.  This function
%   does not raise any exception beyond those raised when computing its
%   operand \meta{fpexpr}.  See also \cs{fp_abs:n}.
% \end{function}
%
% \begin{function}[tested = m3fp-expo001]{exp}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |exp(| \meta{fpexpr} |)| \}
%   \end{syntax}
%   Computes the exponential of the \meta{fpexpr}.  \enquote{Underflow}
%   and \enquote{overflow} occur when appropriate.
% \end{function}
%
% \begin{function}[tested = m3fp-expo001]{ln}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |ln(| \meta{fpexpr} |)| \}
%   \end{syntax}
%   Computes the natural logarithm of the \meta{fpexpr}.  Negative
%   numbers have no (real) logarithm, hence the \enquote{invalid
%     operation} is raised in that case, including for $\ln(-0)$.
%   \enquote{Division by zero} occurs when evaluating $\ln(+0) =
%   -\infty$.  \enquote{Underflow} and \enquote{overflow} occur when
%   appropriate.
% \end{function}
%
% \begin{function}[tested = m3fp-logic002]{max, min}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |max(| \meta{fpexpr_1} |,| \meta{fpexpr_2} |,| \ldots{} |)| \}
%     \cs{fp_eval:n} \{ |min(| \meta{fpexpr_1} |,| \meta{fpexpr_2} |,| \ldots{} |)| \}
%   \end{syntax}
%   Evaluates each \meta{fpexpr} and computes the largest (smallest) of
%   those.  If any of the \meta{fpexpr} is a \nan{}, the result is
%   \nan{}.  Those operations do not raise exceptions.
% \end{function}
%
% \begin{function}[tested = {m3fp-round001, m3fp-round002}]
%   {round, round0, round+, round-}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |round| \meta{option} |(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |round| \meta{option} |(| \meta{fpexpr_1} , \meta{fpexpr_2} |)| \}
%   \end{syntax}
%   Rounds \meta{fpexpr_1} to \meta{fpexpr_2} places.  When
%   \meta{fpexpr_2} is omitted, it is assumed to be $0$, \emph{i.e.},
%   \meta{fpexpr_1} is rounded to an integer.  The \meta{option}
%   controls the rounding direction:
%   \begin{itemize}
%     \item by default, the operation rounds to the closest allowed number
%       (rounding ties to even);
%     \item with |0|, the operation rounds towards $0$, \emph{i.e.}, truncates;
%     \item with |+|, the operation rounds towards $+\infty$;
%     \item with |-|, the operation rounds towards $-\infty$.
%   \end{itemize}
%   If \meta{fpexpr_2} does not yield an integer less than $10^{8}$ in
%   absolute value, then an \enquote{invalid operation} exception is
%   raised.  \enquote{Overflow} may occur if the result is infinite
%   (this cannot happen unless $\meta{fpexpr_2}\string<-9984$).
% \end{function}
%
% \begin{function}[tested = m3fp-trig001]{sin, cos, tan, cot, csc, sec}
%   \begin{syntax}
%     \cs{fp_eval:n} \{ |sin(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |cos(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |tan(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |cot(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |csc(| \meta{fpexpr} |)| \}
%     \cs{fp_eval:n} \{ |sec(| \meta{fpexpr} |)| \}
%   \end{syntax}
%   Computes the sine, cosine, tangent, cotangent, cosecant, or secant
%   of the \meta{fpexpr}.  The trigonometric functions are undefined for
%   an argument of $\pm\infty$, leading to the \enquote{invalid
%     operation} exception.  Additionally, evaluating tangent,
%   cotangent, cosecant, or secant at one of their poles leads to a
%   \enquote{division by zero} exception.  \enquote{Underflow} and
%   \enquote{overflow} occur when appropriate.
% \end{function}
%
% \begin{variable}[tested = m3fp-parse001]{inf, nan}
%   The special values $+\infty$, $-\infty$, and \nan{} are represented
%   as \texttt{inf}, \texttt{-inf} and \texttt{nan} (see \cs{c_inf_fp},
%   \cs{c_minus_inf_fp} and \cs{c_nan_fp}).
% \end{variable}
%
% \begin{variable}[tested = m3fp-parse001]{pi}
%   The value of $\pi$ (see \cs{c_pi_fp}).
% \end{variable}
%
% \begin{variable}[tested = m3fp-parse001]{deg}
%   The value of $1^{\circ}$ in radians (see \cs{c_one_degree_fp}).
% \end{variable}
%
% \begin{variable}[tested = m3fp-parse001]
%   {em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, sp}
%   \newcommand{\unit}[1]{\text{\texttt{#1}}}
%   Those units of measurement are equal to their values in \unit{pt},
%   namely
%   \begin{align*}
%     1 \unit{in} & = 72.27 \unit{pt} \\
%     1 \unit{pt} & = 1 \unit{pt} \\
%     1 \unit{pc} & = 12 \unit{pt} \\
%     1 \unit{cm} & = \frac{1}{2.54} \unit{in} = 28.45275590551181 \unit{pt} \\
%     1 \unit{mm} & = \frac{1}{25.4} \unit{in} = 2.845275590551181 \unit{pt} \\
%     1 \unit{dd} & = 0.376065 \unit{mm} = 1.07000856496063 \unit{pt} \\
%     1 \unit{cc} & = 12 \unit{dd} = 12.84010277952756 \unit{pt} \\
%     1 \unit{nd} & = 0.375 \unit{mm} = 1.066978346456693 \unit{pt} \\
%     1 \unit{nc} & = 12 \unit{nd} = 12.80374015748031 \unit{pt} \\
%     1 \unit{bp} & = \frac{1}{72} \unit{in} = 1.00375 \unit{pt} \\
%     1 \unit{sp} & = 2^{-16} \unit{pt} = 1.52587890625e-5 \unit{pt}.
%   \end{align*}
%   The values of the (font-dependent) units \unit{em} and \unit{ex} are
%   gathered from \TeX{} when the surrounding floating point expression
%   is evaluated.
% \end{variable}
%
% \begin{variable}[tested = m3fp-parse001]{true, false}
%   Other names for $1$ and $+0$.
% \end{variable}
%
% \begin{function}[EXP, added = 2012-05-08, tested = m3fp-convert002]
%   {\dim_to_fp:n}
%   \begin{syntax}
%     \cs{dim_to_fp:n} \Arg{dimexpr}
%   \end{syntax}
%   Expands to an internal floating point number equal to the value of
%   the \meta{dimexpr} in \texttt{pt}.  Since dimension expressions are
%   evaluated much faster than their floating point equivalent,
%   \cs{dim_to_fp:n} can be used to speed up parts of a computation
%   where a low precision is acceptable.
% \end{function}
%
% \begin{function}[EXP, added = 2012-05-14, updated = 2012-07-08,
%   tested = m3fp-convert003]{\fp_abs:n}
%   \begin{syntax}
%     \cs{fp_abs:n} \Arg{floating point expression}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} as described for
%   \cs{fp_eval:n} and leaves the absolute value of the result in the
%   input stream.  This function does not raise any exception beyond
%   those raised when evaluating its argument.  Within floating point
%   expressions, |abs()| can be used.
% \end{function}
%
% \begin{function}[EXP, added = 2012-09-26, tested = m3fp-convert003]
%   {\fp_max:nn, \fp_min:nn}
%   \begin{syntax}
%     \cs{fp_max:nn} \Arg{fp expression 1} \Arg{fp expression 2}
%   \end{syntax}
%   Evaluates the \meta{floating point expressions} as described for
%   \cs{fp_eval:n} and leaves the resulting larger (\texttt{max}) or smaller
%   (\texttt{min}) value in the input stream.  This function does not raise any
%   exception beyond those raised when evaluating its argument.  Within
%   floating point expressions, |max()| and |min()| can be used.
% \end{function}
%
% \section{Disclaimer and roadmap}
%
% The package may break down if:
% \begin{itemize}
%   \item the escape character is either a digit, or an underscore,
%   \item the \tn{uccodes} are changed: the test for whether a character
%     is a letter actually tests if the upper-case code of the character
%     is between A and Z.
% \end{itemize}
%
% The following need to be done. I'll try to time-order the items.
% \begin{itemize}
%   \item Decide what exponent range to consider.
%   \item Change the internal representation of fp, by replacing braced
%     groups of $4$ digits by delimited arguments.  Also consider
%     changing the fp structure a bit to allow using
%     \cs{pdftex_strcmp:D} to compare (not in \LuaTeX{}: it is too
%     slow)?
%   \item Modulo and remainder, and rounding functions |quantize|,
%     |quantize0|, |quantize+|, |quantize-|, |quantize=|, |round=|.
%     Should the modulo also be provided as (catcode 12) |%|?
%   \item \cs{fp_format:nn} \Arg{fpexpr} \Arg{format}, but what should
%     \meta{format} be?  More general pretty printing?
%   \item Add |and|, |or|, |xor|?  Perhaps under the names \texttt{all},
%     \texttt{any}, and \texttt{xor}?
%   \item Add \texttt{csc} and \texttt{sec}.
%   \item Add $\log(x,b)$ for logarithm of $x$ in base $b$.
%   \item \texttt{hypot} (Euclidean length) and 
%     $\operatorname{atan}(x,y) = \operatorname{atan}(x/y)$,
%     also called \texttt{atan2} in other math packages.
%     Cartesian-to-polar transform.  Other inverse trigonometric functions
%     \texttt{acos}, \texttt{asin}, \texttt{atan} (one and two arguments).
%     Also \texttt{asec}, \texttt{acsc}?
%   \item Hyperbolic functions \texttt{cosh}, \texttt{sinh}, \texttt{tanh}.
%   \item Inverse hyperbolics.
%   \item Base conversion, input such as \texttt{0xAB.CDEF}.
%   \item Random numbers (pgfmath provides |rnd|, |rand|, |random|), with
%     seed reset at every \cs{fp_set:Nn}.
%   \item Factorial (not with |!|), gamma function.
%   \item Improve coefficients of the \texttt{sin} and \texttt{tan}
%     series.
%   \item Treat upper and lower case letters identically in
%     identifiers, and ignore underscores.
%   \item Parse $-3<-2<-1$ as it should, not $(-3<-2)<-1$.
%   \item Add an |array(1,2,3)| and |i=complex(0,1)|.
%   \item Provide an experimental |map| function?  Perhaps easier to
%     implement if it is a single character, |@sin(1,2)|?
%   \item Provide \cs{fp_if_nan:nTF}, and an |isnan| function?
% \end{itemize}
% \pkg{Pgfmath} also provides box-measurements (depth, height, width), but
% boxes are not possible expandably.
%
% Bugs. (Exclamation points mark important bugs.)
% \begin{itemize}
% \item[!] Some functions are not monotonic when they should.  For
%   instance, $\sin(1-10^{-16})$ is wrongly greater than $\sin(1)$.
%   \item Add exceptions to |?:|, |!<=>?|, |&&|, \verb"||", and |!|.
%   \item |round| should accept any integer as its second argument.
%   \item Logarithms of numbers very close to $1$ are inaccurate.
%   \item \texttt{tan} and \texttt{cot} give very slightly wrong results
%     for arguments near $10^{-8}$.
%   \item When rounding towards $-\infty$, |\dim_to_fp:n {0pt}| should
%     return $-0$, not $+0$.
%   \item The result of $(\pm0)+(\pm0)$ should depend on the rounding
%     mode.
%   \item \texttt{0e9999999999} gives a \TeX{} \enquote{number too
%       large} error.
%   \item Conversion to integers with \cs{fp_to_int:n} does not check
%     for overflow.
%   \item Subnormals are not implemented.
%   \item |max(-inf)| will lose any information attached to this |-inf|.
%   \item The overflow trap receives the wrong argument in
%     \pkg{l3fp-expo} (see |exp(1e5678)| in \file{m3fp-traps001}).
% \end{itemize}
%
% Possible optimizations/improvements.
% \begin{itemize}
%   \item Optimize argument reduction for trigonometric functions: we
%     don't need $6\times 4$ digits here, only $4\times 4$.
%   \item In subsection~\ref{sec:l3fp:fp-floats}, write a grammar.
%   \item Fix the |TWO BARS| business with the index.
%   \item It would be nice if the \texttt{parse} auxiliaries for each
%     operation were set up in the corresponding module, rather than
%     centralizing in \pkg{l3fp-parse}.
%   \item Some functions should get an |_o| ending to indicate that they
%     expand after their result.
%   \item More care should be given to distinguish expandable/restricted
%     expandable (auxiliary and internal) functions.
%   \item The code for the \texttt{ternary} set of functions is ugly.
%   \item There are many |~| missing in the doc to avoid bad line-breaks.
%   \item The algorithm for computing the logarithm of the significand
%     could be made to use a $5$ terms Taylor series instead of $10$
%     terms by taking $c = 2000/(\lfloor 200x\rfloor +1) \in [10,95]$
%     instead of $c\in [1,10]$.  Also, it would then be possible to
%     simplify the computation of $t$, using methods similar to
%     \cs{__fp_fixed_div_to_float:ww}.  However, we would then have to
%     hard-code the logarithms of $44$ small integers instead of $9$.
%   \item Improve notations in the explanations of the division
%     algorithm (\pkg{l3fp-basics}).
%   \item Understand and document \cs{__fp_basics_pack_weird_low:NNNNw}
%     and \cs{__fp_basics_pack_weird_high:NNNNNNNNw} better.  Move the
%     other \texttt{basics_pack} auxiliaries to \pkg{l3fp-aux} under a
%     better name.
%   \item Find out if underflow can really occur for trigonometric
%     functions, and redoc as appropriate.
%   \item Add bibliography.  Some of Kahan's articles, some previous
%     \TeX{} fp packages, the international standards,\ldots{}
%   \item Also take into account the \enquote{inexact} exception?
% \end{itemize}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{l3fp} implementation}
%
%    \begin{macrocode}
%<*package>
%    \end{macrocode}
%
%    \begin{macrocode}
\ProvidesExplPackage
  {\ExplFileName}{\ExplFileDate}{\ExplFileVersion}{\ExplFileDescription}
\__expl_package_check:
%    \end{macrocode}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \end{implementation}
%
% \PrintIndex