-
Notifications
You must be signed in to change notification settings - Fork 178
/
l3fp-aux.dtx
1089 lines (1089 loc) · 39.2 KB
/
l3fp-aux.dtx
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
% \iffalse meta-comment
%
%% File: l3fp-aux.dtx Copyright(C) 2011-2017 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
%
% https://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 development version of the bundle can be found at
%
% https://github.com/latex3/latex3
%
% for those people who are interested.
%
%<*driver>
\documentclass[full,kernel]{l3doc}
\begin{document}
\DocInput{\jobname.dtx}
\end{document}
%</driver>
% \fi
%
% \title{^^A
% The \textsf{l3fp-aux} package\\ Support for floating points^^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 2017/12/05}
%
% \maketitle
%
% \begin{documentation}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{l3fp-aux} implementation}
%
% \begin{macrocode}
%<*initex|package>
% \end{macrocode}
%
% \begin{macrocode}
%<@@=fp>
% \end{macrocode}
%
% ^^A todo: make sanitize and pack more homogeneous between modules.
%
% \subsection{Internal representation}
%
% Internally, a floating point number \meta{X} is a
% token list containing
% \begin{quote}
% \cs{s_@@} \cs{@@_chk:w} \meta{case} \meta{sign} \meta{body} |;|
% \end{quote}
% Let us explain each piece separately.
%
% Internal floating point numbers are used in expressions,
% and in this context are subject to \texttt{f}-expansion. They must
% leave a recognizable mark after \texttt{f}-expansion, to prevent the
% floating point number from being re-parsed. Thus, \cs{s_@@}
% is simply another name for \tn{relax}.
%
% When used directly without an accessor function, floating points
% should produce an error: this is the role of \cs{@@_chk:w}. We could
% make floating point variables be protected to prevent them from
% expanding under \texttt{x}-expansion, but it seems more convenient to
% treat them as a subcase of token list variables.
%
% The (decimal part of the) IEEE-754-2008 standard requires the format
% to be able to represent special floating point numbers besides the
% usual positive and negative cases. We distinguish the various
% possibilities by their \meta{case}, which is a single digit:
% \begin{itemize}
% \item[0] zeros: |+0| and |-0|,
% \item[1] \enquote{normal} numbers (positive and negative),
% \item[2] infinities: |+inf| and |-inf|,
% \item[3] quiet and signalling \texttt{nan}.
% \end{itemize}
% The \meta{sign} is |0| (positive) or |2| (negative),
% except in the case of \texttt{nan}, which have $\meta{sign} = 1$.
% This ensures that changing the \meta{sign} digit to $2-\meta{sign}$
% is exactly equivalent to changing the sign of the number.
%
% Special floating point numbers have the form
% \begin{quote}
% \cs{s_@@} \cs{@@_chk:w} \meta{case} \meta{sign} \cs[no-index]{s_@@_\ldots} |;|
% \end{quote}
% where \cs[no-index]{s_@@_\ldots} is a scan mark carrying information about how the
% number was formed (useful for debugging).
%
% Normal floating point numbers ($\meta{case} = 1$) have the form
% \begin{quote}
% \cs{s_@@} \cs{@@_chk:w} 1 \meta{sign} \Arg{exponent}
% \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} |;|
% \end{quote}
% Here, the \meta{exponent} is an integer, between
% $-\ExplSyntaxOn\int_use:N\c__fp_minus_min_exponent_int$ and
% $\ExplSyntaxOn\int_use:N\c__fp_max_exponent_int$. The body consists
% in four blocks of exactly $4$ digits,
% $0000 \leq \meta{X_i} \leq 9999$, and the floating point is
% \[
% (-1)^{\meta{sign}/2} \meta{X_1}\meta{X_2}\meta{X_3}\meta{X_4}\cdot 10^{\meta{exponent}-16}
% \]
% where we have concatenated the $16$ digits. Currently, floating point numbers are normalized such that
% the \meta{exponent} is minimal, in other words, $1000 \leq \meta{X_1} \leq 9999$.
%
% \begin{table}\centering
% \caption{Internal representation of floating point numbers.}
% \label{tab:fp-convert-special}
% \begin{tabular}{ll}
% \toprule
% \multicolumn{1}{c}{Representation} & Meaning \\
% \midrule
% 0 0 \cs[no-index]{s_@@_\ldots} \texttt{;} & Positive zero. \\
% 0 2 \cs[no-index]{s_@@_\ldots} \texttt{;} & Negative zero. \\
% 1 0 \Arg{exponent} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \texttt{;}
% & Positive floating point. \\
% 1 2 \Arg{exponent} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4} \texttt{;}
% & Negative floating point. \\
% 2 0 \cs[no-index]{s_@@_\ldots} \texttt{;} & Positive infinity. \\
% 2 2 \cs[no-index]{s_@@_\ldots} \texttt{;} & Negative infinity. \\
% 3 1 \cs[no-index]{s_@@_\ldots} \texttt{;} & Quiet \texttt{nan}. \\
% 3 1 \cs[no-index]{s_@@_\ldots} \texttt{;} & Signalling \texttt{nan}. \\
% \bottomrule
% \end{tabular}
% \end{table}
%
% Calculations are done in base $10000$, \emph{i.e.} one myriad.
%
% \subsection{Using arguments and semicolons}
%
% \begin{macro}[EXP]{\@@_use_none_stop_f:n}
% This function removes an argument (typically a digit) and replaces
% it by \cs{exp_stop_f:}, a marker which stops \texttt{f}-type
% expansion.
% \begin{macrocode}
\cs_new:Npn \@@_use_none_stop_f:n #1 { \exp_stop_f: }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_use_s:n, \@@_use_s:nn}
% Those functions place a semicolon after one or two arguments
% (typically digits).
% \begin{macrocode}
\cs_new:Npn \@@_use_s:n #1 { #1; }
\cs_new:Npn \@@_use_s:nn #1#2 { #1#2; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
% {\@@_use_none_until_s:w, \@@_use_i_until_s:nw, \@@_use_ii_until_s:nnw}
% Those functions select specific arguments among a set of arguments
% delimited by a semicolon.
% \begin{macrocode}
\cs_new:Npn \@@_use_none_until_s:w #1; { }
\cs_new:Npn \@@_use_i_until_s:nw #1#2; {#1}
\cs_new:Npn \@@_use_ii_until_s:nnw #1#2#3; {#2}
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_reverse_args:Nww}
% Many internal functions take arguments delimited by semicolons, and
% it is occasionally useful to swap two such arguments.
% \begin{macrocode}
\cs_new:Npn \@@_reverse_args:Nww #1 #2; #3; { #1 #3; #2; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_rrot:www}
% Rotate three arguments delimited by semicolons. This is the inverse
% (or the square) of the Forth primitive |ROT|, hence the name.
% \begin{macrocode}
\cs_new:Npn \@@_rrot:www #1; #2; #3; { #2; #3; #1; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_use_i:ww, \@@_use_i:www}
% Many internal functions take arguments delimited by semicolons, and
% it is occasionally useful to remove one or two such arguments.
% \begin{macrocode}
\cs_new:Npn \@@_use_i:ww #1; #2; { #1; }
\cs_new:Npn \@@_use_i:www #1; #2; #3; { #1; }
% \end{macrocode}
% \end{macro}
%
% \subsection{Constants, and structure of floating points}
%
% \begin{macro}{\s_@@, \@@_chk:w}
% Floating points numbers all start with \cs{s_@@} \cs{@@_chk:w},
% where \cs{s_@@} is equal to the \TeX{} primitive \tn{relax}, and
% \cs{@@_chk:w} is protected. The rest of the floating point number
% is made of characters (or \tn{relax}). This ensures that nothing
% expands under \texttt{f}-expansion, nor under \texttt{x}-expansion.
% However, when typeset, \cs{s_@@} does nothing, and \cs{@@_chk:w} is
% expanded. We define \cs{@@_chk:w} to produce an error.
% \begin{macrocode}
\__scan_new:N \s_@@
\cs_new_protected:Npn \@@_chk:w #1 ;
{
\__msg_kernel_error:nnx { kernel } { misused-fp }
{ \fp_to_tl:n { \s_@@ \@@_chk:w #1 ; } }
}
% \end{macrocode}
% \end{macro}
%
% \begin{macro}{\s_@@_mark, \s_@@_stop}
% Aliases of \cs{tex_relax:D}, used to terminate expressions.
% \begin{macrocode}
\__scan_new:N \s_@@_mark
\__scan_new:N \s_@@_stop
% \end{macrocode}
% \end{macro}
%
% \begin{macro}
% {
% \s_@@_invalid, \s_@@_underflow, \s_@@_overflow,
% \s_@@_division, \s_@@_exact
% }
% A couple of scan marks used to indicate where special floating point
% numbers come from.
% \begin{macrocode}
\__scan_new:N \s_@@_invalid
\__scan_new:N \s_@@_underflow
\__scan_new:N \s_@@_overflow
\__scan_new:N \s_@@_division
\__scan_new:N \s_@@_exact
% \end{macrocode}
% \end{macro}
%
% \begin{variable}
% {\c_zero_fp, \c_minus_zero_fp, \c_inf_fp, \c_minus_inf_fp, \c_nan_fp}
% The special floating points. We define the floating points here as \enquote{exact}.
% \begin{macrocode}
\tl_const:Nn \c_zero_fp { \s_@@ \@@_chk:w 0 0 \s_@@_exact ; }
\tl_const:Nn \c_minus_zero_fp { \s_@@ \@@_chk:w 0 2 \s_@@_exact ; }
\tl_const:Nn \c_inf_fp { \s_@@ \@@_chk:w 2 0 \s_@@_exact ; }
\tl_const:Nn \c_minus_inf_fp { \s_@@ \@@_chk:w 2 2 \s_@@_exact ; }
\tl_const:Nn \c_nan_fp { \s_@@ \@@_chk:w 3 1 \s_@@_exact ; }
% \end{macrocode}
% \end{variable}
%
% \begin{variable}{\c_@@_prec_int, \c_@@_half_prec_int, \c_@@_block_int}
% The number of digits of floating points.
% \begin{macrocode}
\int_const:Nn \c_@@_prec_int { 16 }
\int_const:Nn \c_@@_half_prec_int { 8 }
\int_const:Nn \c_@@_block_int { 4 }
% \end{macrocode}
% \end{variable}
%
% \begin{variable}{\c_@@_myriad_int}
% Blocks have $4$~digits so this integer is useful.
% \begin{macrocode}
\int_const:Nn \c_@@_myriad_int { 10000 }
% \end{macrocode}
% \end{variable}
%
% \begin{variable}{\c_@@_minus_min_exponent_int, \c_@@_max_exponent_int}
% Normal floating point numbers have an exponent between $-$
% \texttt{minus_min_exponent} and \texttt{max_exponent} inclusive.
% Larger numbers are rounded to $\pm\infty$. Smaller numbers are
% rounded to $\pm 0$. It would be more natural to define a
% \texttt{min_exponent} with the opposite sign but that would waste
% one \TeX{} count.
% \begin{macrocode}
\int_const:Nn \c_@@_minus_min_exponent_int { 10000 }
\int_const:Nn \c_@@_max_exponent_int { 10000 }
% \end{macrocode}
% \end{variable}
%
% \begin{variable}{\c_@@_max_exp_exponent_int}
% If a number's exponent is larger than that, its exponential
% overflows/underflows.
% \begin{macrocode}
\int_const:Nn \c_@@_max_exp_exponent_int { 5 }
% \end{macrocode}
% \end{variable}
%
% \begin{variable}{\c_@@_overflowing_fp}
% A floating point number that is bigger than all normal floating
% point numbers. This replaces infinities when converting to formats
% that do not support infinities.
% \begin{macrocode}
\tl_const:Nx \c_@@_overflowing_fp
{
\s_@@ \@@_chk:w 1 0
{ \int_eval:n { \c_@@_max_exponent_int + 1 } }
{1000} {0000} {0000} {0000} ;
}
% \end{macrocode}
% \end{variable}
%
% \begin{macro}[EXP]{\@@_zero_fp:N, \@@_inf_fp:N}
% In case of overflow or underflow, we have to output
% a zero or infinity with a given sign.
% \begin{macrocode}
\cs_new:Npn \@@_zero_fp:N #1
{ \s_@@ \@@_chk:w 0 #1 \s_@@_underflow ; }
\cs_new:Npn \@@_inf_fp:N #1
{ \s_@@ \@@_chk:w 2 #1 \s_@@_overflow ; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_exponent:w}
% For normal numbers, the function expands to the exponent, otherwise
% to $0$. This is used in \pkg{l3str-format}.
% \begin{macrocode}
\cs_new:Npn \@@_exponent:w \s_@@ \@@_chk:w #1
{
\if_meaning:w 1 #1
\exp_after:wN \@@_use_ii_until_s:nnw
\else:
\exp_after:wN \@@_use_i_until_s:nw
\exp_after:wN 0
\fi:
}
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_neg_sign:N}
% When appearing in an integer expression or after \cs{__int_value:w},
% this expands to the sign opposite to |#1|, namely $0$ (positive) is
% turned to $2$ (negative), $1$ (\texttt{nan}) to $1$, and $2$ to $0$.
% \begin{macrocode}
\cs_new:Npn \@@_neg_sign:N #1
{ \__int_eval:w 2 - #1 \__int_eval_end: }
% \end{macrocode}
% \end{macro}
%
% \subsection{Overflow, underflow, and exact zero}
%
%^^A todo: the sign of exact zeros should depend on the rounding mode.
%
% \begin{macro}[EXP]{\@@_sanitize:Nw, \@@_sanitize:wN}
% \begin{macro}[EXP]{\@@_sanitize_zero:w}
% Expects the sign and the exponent in some order, then the
% significand (which we don't touch). Outputs the corresponding
% floating point number, possibly underflowed to $\pm 0$ or overflowed
% to $\pm\infty$. The functions \cs{@@_underflow:w} and
% \cs{@@_overflow:w} are defined in \pkg{l3fp-traps}.
% \begin{macrocode}
\cs_new:Npn \@@_sanitize:Nw #1 #2;
{
\if_case:w
\if_int_compare:w #2 > \c_@@_max_exponent_int 1 ~ \else:
\if_int_compare:w #2 < - \c_@@_minus_min_exponent_int 2 ~ \else:
\if_meaning:w 1 #1 3 ~ \fi: \fi: \fi: 0 ~
\or: \exp_after:wN \@@_overflow:w
\or: \exp_after:wN \@@_underflow:w
\or: \exp_after:wN \@@_sanitize_zero:w
\fi:
\s_@@ \@@_chk:w 1 #1 {#2}
}
\cs_new:Npn \@@_sanitize:wN #1; #2 { \@@_sanitize:Nw #2 #1; }
\cs_new:Npn \@@_sanitize_zero:w \s_@@ \@@_chk:w #1 #2 #3;
{ \c_zero_fp }
% \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{Expanding after a floating point number}
%
% \begin{macro}[EXP]{\@@_exp_after_o:w}
% \begin{macro}[EXP]{\@@_exp_after_f:nw}
% \begin{syntax}
% \cs{@@_exp_after_o:w} \meta{floating point}
% \cs{@@_exp_after_f:nw} \Arg{tokens} \meta{floating point}
% \end{syntax}
% Places \meta{tokens} (empty in the case of \cs{@@_exp_after_o:w})
% between the \meta{floating point} and the \meta{more tokens}, then
% hits those tokens with either \texttt{o}-expansion (one
% \cs{exp_after:wN}) or \texttt{f}-expansion, and leaves the floating
% point number unchanged.
%
% We first distinguish normal floating points, which have a significand,
% from the much simpler special floating points.
% \begin{macrocode}
\cs_new:Npn \@@_exp_after_o:w \s_@@ \@@_chk:w #1
{
\if_meaning:w 1 #1
\exp_after:wN \@@_exp_after_normal:nNNw
\else:
\exp_after:wN \@@_exp_after_special:nNNw
\fi:
{ }
#1
}
\cs_new:Npn \@@_exp_after_f:nw #1 \s_@@ \@@_chk:w #2
{
\if_meaning:w 1 #2
\exp_after:wN \@@_exp_after_normal:nNNw
\else:
\exp_after:wN \@@_exp_after_special:nNNw
\fi:
{ \exp:w \exp_end_continue_f:w #1 }
#2
}
% \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_exp_after_special:nNNw}
% \begin{syntax}
% \cs{@@_exp_after_special:nNNw} \Arg{after} \meta{case} \meta{sign} \meta{scan mark} |;|
% \end{syntax}
% Special floating point numbers are easy to jump over since they
% contain few tokens.
% \begin{macrocode}
\cs_new:Npn \@@_exp_after_special:nNNw #1#2#3#4;
{
\exp_after:wN \s_@@
\exp_after:wN \@@_chk:w
\exp_after:wN #2
\exp_after:wN #3
\exp_after:wN #4
\exp_after:wN ;
#1
}
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_exp_after_normal:nNNw}
% For normal floating point numbers, life is slightly harder, since we
% have many tokens to jump over. Here it would be slightly better if
% the digits were not braced but instead were delimited arguments (for
% instance delimited by |,|). That may be changed some day.
% \begin{macrocode}
\cs_new:Npn \@@_exp_after_normal:nNNw #1 1 #2 #3 #4#5#6#7;
{
\exp_after:wN \@@_exp_after_normal:Nwwwww
\exp_after:wN #2
\__int_value:w #3 \exp_after:wN ;
\__int_value:w 1 #4 \exp_after:wN ;
\__int_value:w 1 #5 \exp_after:wN ;
\__int_value:w 1 #6 \exp_after:wN ;
\__int_value:w 1 #7 \exp_after:wN ; #1
}
\cs_new:Npn \@@_exp_after_normal:Nwwwww
#1 #2; 1 #3 ; 1 #4 ; 1 #5 ; 1 #6 ;
{ \s_@@ \@@_chk:w 1 #1 {#2} {#3} {#4} {#5} {#6} ; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_exp_after_array_f:w}
% \begin{macro}[EXP]{\@@_exp_after_stop_f:nw}
% \begin{syntax}
% \cs{@@_exp_after_array_f:w}
% \meta{fp_1} |;|
% \ldots{}
% \meta{fp_n} |;|
% \cs{s_@@_stop}
% \end{syntax}
% \begin{macrocode}
\cs_new:Npn \@@_exp_after_array_f:w #1
{
\cs:w @@_exp_after \@@_type_from_scan:N #1 _f:nw \cs_end:
{ \@@_exp_after_array_f:w }
#1
}
\cs_new_eq:NN \@@_exp_after_stop_f:nw \use_none:nn
% \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{Packing digits}
%
% When a positive integer |#1| is known to be less than $10^8$, the
% following trick splits it into two blocks of $4$ digits, padding
% with zeros on the left.
% \begin{verbatim}
% \cs_new:Npn \pack:NNNNNw #1 #2#3#4#5 #6; { {#2#3#4#5} {#6} }
% \exp_after:wN \pack:NNNNNw
% \__int_value:w \__int_eval:w 1 0000 0000 + #1 ;
% \end{verbatim}
% The idea is that adding $10^8$ to the number ensures that it has
% exactly $9$ digits, and can then easily find which digits correspond
% to what position in the number. Of course, this can be modified
% for any number of digits less or equal to~$9$ (we are limited by
% \TeX{}'s integers). This method is very heavily relied upon in
% \texttt{l3fp-basics}.
%
% More specifically, the auxiliary inserts |+ #1#2#3#4#5 ; {#6}|, which
% allows us to compute several blocks of $4$ digits in a nested manner,
% performing carries on the fly. Say we want to compute $1\,2345 \times
% 6677\,8899$. With simplified names, we would do
% \begin{verbatim}
% \exp_after:wN \post_processing:w
% \__int_value:w \__int_eval:w - 5 0000
% \exp_after:wN \pack:NNNNNw
% \__int_value:w \__int_eval:w 4 9995 0000
% + 12345 * 6677
% \exp_after:wN \pack:NNNNNw
% \__int_value:w \__int_eval:w 5 0000 0000
% + 12345 * 8899 ;
% \end{verbatim}
% The \cs{exp_after:wN} triggers |\__int_value:w \__int_eval:w|, which
% starts a first computation, whose initial value is $- 5\,0000$ (the
% \enquote{leading shift}). In that computation appears an
% \cs{exp_after:wN}, which triggers the nested computation
% |\__int_value:w \__int_eval:w| with starting value $4\,9995\,0000$ (the
% \enquote{middle shift}). That, in turn, expands \cs{exp_after:wN}
% which triggers the third computation. The third computation's value
% is $5\,0000\,0000 + 12345 \times 8899$, which has $9$ digits. Adding
% $5\cdot 10^{8}$ to the product allowed us to know how many digits to
% expect as long as the numbers to multiply are not too big; it
% also works to some extent with negative results. The \texttt{pack}
% function puts the last $4$ of those $9$ digits into a brace group,
% moves the semi-colon delimiter, and inserts a |+|, which combines the
% carry with the previous computation. The shifts nicely combine into
% $5\,0000\,0000 / 10^{4} + 4\,9995\,0000 = 5\,0000\,0000$. As long as
% the operands are in some range, the result of this second computation
% has $9$ digits. The corresponding \texttt{pack} function,
% expanded after the result is computed, braces the last $4$ digits, and
% leaves |+| \meta{5 digits} for the initial computation. The
% \enquote{leading shift} cancels the combination of the other shifts,
% and the |\post_processing:w| takes care of packing the last few
% digits.
%
% Admittedly, this is quite intricate. It is probably the key in making
% \pkg{l3fp} as fast as other pure \TeX{} floating point units despite
% its increased precision. In fact, this is used so much that we
% provide different sets of packing functions and shifts, depending on
% ranges of input.
%
% \begin{macro}[EXP]{\@@_pack:NNNNNw}
% \begin{variable}
% {
% \c_@@_trailing_shift_int ,
% \c_@@_middle_shift_int ,
% \c_@@_leading_shift_int ,
% }
% This set of shifts allows for computations involving results in the
% range $[-4\cdot 10^{8}, 5\cdot 10^{8}-1]$. Shifted values all have
% exactly $9$ digits.
% \begin{macrocode}
\int_const:Nn \c_@@_leading_shift_int { - 5 0000 }
\int_const:Nn \c_@@_middle_shift_int { 5 0000 * 9999 }
\int_const:Nn \c_@@_trailing_shift_int { 5 0000 * 10000 }
\cs_new:Npn \@@_pack:NNNNNw #1 #2#3#4#5 #6; { + #1#2#3#4#5 ; {#6} }
% \end{macrocode}
% \end{variable}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_pack_big:NNNNNNw}
% \begin{variable}
% {
% \c_@@_big_trailing_shift_int ,
% \c_@@_big_middle_shift_int ,
% \c_@@_big_leading_shift_int ,
% }
% This set of shifts allows for computations involving results in the
% range $[-5\cdot 10^{8}, 6\cdot 10^{8}-1]$ (actually a bit more).
% Shifted values all have exactly $10$ digits. Note that the upper
% bound is due to \TeX{}'s limit of $2^{31}-1$ on integers. The
% shifts are chosen to be roughly the mid-point of $10^{9}$ and
% $2^{31}$, the two bounds on $10$-digit integers in \TeX{}.
% \begin{macrocode}
\int_const:Nn \c_@@_big_leading_shift_int { - 15 2374 }
\int_const:Nn \c_@@_big_middle_shift_int { 15 2374 * 9999 }
\int_const:Nn \c_@@_big_trailing_shift_int { 15 2374 * 10000 }
\cs_new:Npn \@@_pack_big:NNNNNNw #1#2 #3#4#5#6 #7;
{ + #1#2#3#4#5#6 ; {#7} }
% \end{macrocode}
% \end{variable}
% \end{macro}
%
% ^^A \@@_pack_Bigg:NNNNNNw = \@@_pack_big:NNNNNNw ?
% \begin{macro}[EXP]{\@@_pack_Bigg:NNNNNNw}
% \begin{variable}
% {
% \c_@@_Bigg_trailing_shift_int ,
% \c_@@_Bigg_middle_shift_int ,
% \c_@@_Bigg_leading_shift_int ,
% }
% This set of shifts allows for computations with results in the
% range $[-1\cdot 10^{9}, 147483647]$; the end-point is $2^{31} - 1 -
% 2\cdot 10^{9} \simeq 1.47\cdot 10^{8}$. Shifted values all have
% exactly $10$ digits.
% \begin{macrocode}
\int_const:Nn \c_@@_Bigg_leading_shift_int { - 20 0000 }
\int_const:Nn \c_@@_Bigg_middle_shift_int { 20 0000 * 9999 }
\int_const:Nn \c_@@_Bigg_trailing_shift_int { 20 0000 * 10000 }
\cs_new:Npn \@@_pack_Bigg:NNNNNNw #1#2 #3#4#5#6 #7;
{ + #1#2#3#4#5#6 ; {#7} }
% \end{macrocode}
% \end{variable}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_pack_twice_four:wNNNNNNNN}
% \begin{syntax}
% \cs{@@_pack_twice_four:wNNNNNNNN} \meta{tokens} |;| \meta{$\geq 8$ digits}
% \end{syntax}
% Grabs two sets of $4$ digits and places them before the semi-colon
% delimiter. Putting several copies of this function before a
% semicolon packs more digits since each takes the digits
% packed by the others in its first argument.
% \begin{macrocode}
\cs_new:Npn \@@_pack_twice_four:wNNNNNNNN #1; #2#3#4#5 #6#7#8#9
{ #1 {#2#3#4#5} {#6#7#8#9} ; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_pack_eight:wNNNNNNNN}
% \begin{syntax}
% \cs{@@_pack_eight:wNNNNNNNN} \meta{tokens} |;| \meta{$\geq 8$ digits}
% \end{syntax}
% Grabs one set of $8$ digits and places them before the semi-colon
% delimiter as a single group. Putting several copies of this
% function before a semicolon packs more digits since each
% takes the digits packed by the others in its first argument.
% \begin{macrocode}
\cs_new:Npn \@@_pack_eight:wNNNNNNNN #1; #2#3#4#5 #6#7#8#9
{ #1 {#2#3#4#5#6#7#8#9} ; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
% {
% \@@_basics_pack_low:NNNNNw,
% \@@_basics_pack_high:NNNNNw,
% \@@_basics_pack_high_carry:w
% }
% Addition and multiplication of significands are done in two steps:
% first compute a (more or less) exact result, then round and pack
% digits in the final (braced) form. These functions take care of the
% packing, with special attention given to the case where rounding has
% caused a carry. Since rounding can only shift the final digit by
% $1$, a carry always produces an exact power of $10$. Thus,
% \cs{@@_basics_pack_high_carry:w} is always followed by four times
% |{0000}|.
%
% This is used in \pkg{l3fp-basics} and \pkg{l3fp-extended}.
% \begin{macrocode}
\cs_new:Npn \@@_basics_pack_low:NNNNNw #1 #2#3#4#5 #6;
{ + #1 - 1 ; {#2#3#4#5} {#6} ; }
\cs_new:Npn \@@_basics_pack_high:NNNNNw #1 #2#3#4#5 #6;
{
\if_meaning:w 2 #1
\@@_basics_pack_high_carry:w
\fi:
; {#2#3#4#5} {#6}
}
\cs_new:Npn \@@_basics_pack_high_carry:w \fi: ; #1
{ \fi: + 1 ; {1000} }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
% {
% \@@_basics_pack_weird_low:NNNNw,
% \@@_basics_pack_weird_high:NNNNNNNNw
% }
% This is used in \pkg{l3fp-basics} for additions and
% divisions. Their syntax is confusing, hence the name.
% \begin{macrocode}
\cs_new:Npn \@@_basics_pack_weird_low:NNNNw #1 #2#3#4 #5;
{
\if_meaning:w 2 #1
+ 1
\fi:
\__int_eval_end:
#2#3#4; {#5} ;
}
\cs_new:Npn \@@_basics_pack_weird_high:NNNNNNNNw
1 #1#2#3#4 #5#6#7#8 #9; { ; {#1#2#3#4} {#5#6#7#8} {#9} }
% \end{macrocode}
% \end{macro}
%
% \subsection{Decimate (dividing by a power of 10)}
%
% ^^A begin[todo]
% \begin{macro}[EXP]{\@@_decimate:nNnnnn}
% \begin{syntax}
% \cs{@@_decimate:nNnnnn} \Arg{shift} \meta{f_1}
% ~~\Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4}
% \end{syntax}
% Each \meta{X_i} consists in $4$ digits exactly,
% and $1000\leq\meta{X_1}<9999$. The first argument determines
% by how much we shift the digits. \meta{f_1} is called as follows:
% \begin{syntax}
% \meta{f_1} \meta{rounding} \Arg{X'_1} \Arg{X'_2} \meta{extra-digits} |;|
% \end{syntax}
% where $0\leq\meta{X'_i}<10^{8}-1$ are $8$ digit integers,
% forming the truncation of our number. In other words,
% \[
% \left(
% \sum_{i=1}^{4} \meta{X_i} \cdot 10^{-4i} \cdot 10^{-\meta{shift}}
% \right)
% - \bigl( \meta{X'_1} \cdot 10^{-8} + \meta{X'_2} \cdot 10^{-16} \bigr)
% = 0.\meta{extra-digits} \cdot 10^{-16}
% \in [0,10^{-16}).
% \]
% To round properly later, we need to remember some information
% about the difference. The \meta{rounding} digit is $0$ if and
% only if the difference is exactly $0$, and $5$ if and only if
% the difference is exactly $0.5\cdot 10^{-16}$. Otherwise, it
% is the (non-$0$, non-$5$) digit closest to $10^{17}$ times the
% difference. In particular, if the shift is $17$ or more, all
% the digits are dropped, \meta{rounding} is $1$ (not $0$), and
% \meta{X'_1} and \meta{X'_2} are both zero.
%
% If the shift is $1$, the \meta{rounding} digit is simply the
% only digit that was pushed out of the brace groups (this is
% important for subtraction). It would be more natural for the
% \meta{rounding} digit to be placed after the \meta{X'_i},
% but the choice we make involves less reshuffling.
%
% Note that this function treats negative \meta{shift} as $0$.
% \begin{macrocode}
\cs_new:Npn \@@_decimate:nNnnnn #1
{
\cs:w
@@_decimate_
\if_int_compare:w \__int_eval:w #1 > \c_@@_prec_int
tiny
\else:
\__int_to_roman:w \__int_eval:w #1
\fi:
:Nnnnn
\cs_end:
}
% \end{macrocode}
% Each of the auxiliaries see the function \meta{f_1},
% followed by $4$ blocks of $4$ digits.
% \end{macro}
%
% \begin{macro}[EXP]{\@@_decimate_:Nnnnn, \@@_decimate_tiny:Nnnnn}
% If the \meta{shift} is zero, or too big, life is very easy.
% \begin{macrocode}
\cs_new:Npn \@@_decimate_:Nnnnn #1 #2#3#4#5
{ #1 0 {#2#3} {#4#5} ; }
\cs_new:Npn \@@_decimate_tiny:Nnnnn #1 #2#3#4#5
{ #1 1 { 0000 0000 } { 0000 0000 } 0 #2#3#4#5 ; }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
% {
% \@@_decimate_auxi:Nnnnn, \@@_decimate_auxii:Nnnnn,
% \@@_decimate_auxiii:Nnnnn, \@@_decimate_auxiv:Nnnnn,
% \@@_decimate_auxv:Nnnnn, \@@_decimate_auxvi:Nnnnn,
% \@@_decimate_auxvii:Nnnnn, \@@_decimate_auxviii:Nnnnn,
% \@@_decimate_auxix:Nnnnn, \@@_decimate_auxx:Nnnnn,
% \@@_decimate_auxxi:Nnnnn, \@@_decimate_auxxii:Nnnnn,
% \@@_decimate_auxxiii:Nnnnn, \@@_decimate_auxxiv:Nnnnn,
% \@@_decimate_auxxv:Nnnnn, \@@_decimate_auxxvi:Nnnnn
% }
% \begin{syntax}
% \cs{@@_decimate_auxi:Nnnnn} \meta{f_1} \Arg{X_1} \Arg{X_2} \Arg{X_3} \Arg{X_4}
% \end{syntax}
% Shifting happens in two steps: compute the \meta{rounding} digit,
% and repack digits into two blocks of $8$. The sixteen functions
% are very similar, and defined through \cs{@@_tmp:w}.
% The arguments are as follows: |#1| indicates which function is
% being defined; after one step of expansion, |#2| yields the
% \enquote{extra digits} which are then converted by
% \cs{@@_round_digit:Nw} to the \meta{rounding} digit (note the |+|
% separating blocks of digits to avoid overflowing \TeX{}'s integers).
% This triggers the \texttt{f}-expansion of
% \cs{@@_decimate_pack:nnnnnnnnnnw},\footnote{No, the argument
% spec is not a mistake: the function calls an auxiliary to
% do half of the job.} responsible for building two blocks of
% $8$ digits, and removing the rest. For this to work, |#3|
% alternates between braced and unbraced blocks of $4$ digits,
% in such a way that the $5$ first and $5$ next token groups
% yield the correct blocks of $8$ digits.
% \begin{macrocode}
\cs_new:Npn \@@_tmp:w #1 #2 #3
{
\cs_new:cpn { @@_decimate_ #1 :Nnnnn } ##1 ##2##3##4##5
{
\exp_after:wN ##1
\__int_value:w
\exp_after:wN \@@_round_digit:Nw #2 ;
\@@_decimate_pack:nnnnnnnnnnw #3 ;
}
}
\@@_tmp:w {i} {\use_none:nnn #50}{ 0{#2}#3{#4}#5 }
\@@_tmp:w {ii} {\use_none:nn #5 }{ 00{#2}#3{#4}#5 }
\@@_tmp:w {iii} {\use_none:n #5 }{ 000{#2}#3{#4}#5 }
\@@_tmp:w {iv} { #5 }{ {0000}#2{#3}#4 #5 }
\@@_tmp:w {v} {\use_none:nnn #4#5 }{ 0{0000}#2{#3}#4 #5 }
\@@_tmp:w {vi} {\use_none:nn #4#5 }{ 00{0000}#2{#3}#4 #5 }
\@@_tmp:w {vii} {\use_none:n #4#5 }{ 000{0000}#2{#3}#4 #5 }
\@@_tmp:w {viii}{ #4#5 }{ {0000}0000{#2}#3 #4 #5 }
\@@_tmp:w {ix} {\use_none:nnn #3#4+#5}{ 0{0000}0000{#2}#3 #4 #5 }
\@@_tmp:w {x} {\use_none:nn #3#4+#5}{ 00{0000}0000{#2}#3 #4 #5 }
\@@_tmp:w {xi} {\use_none:n #3#4+#5}{ 000{0000}0000{#2}#3 #4 #5 }
\@@_tmp:w {xii} { #3#4+#5}{ {0000}0000{0000}#2 #3 #4 #5 }
\@@_tmp:w {xiii}{\use_none:nnn#2#3+#4#5}{ 0{0000}0000{0000}#2 #3 #4 #5 }
\@@_tmp:w {xiv} {\use_none:nn #2#3+#4#5}{ 00{0000}0000{0000}#2 #3 #4 #5 }
\@@_tmp:w {xv} {\use_none:n #2#3+#4#5}{ 000{0000}0000{0000}#2 #3 #4 #5 }
\@@_tmp:w {xvi} { #2#3+#4#5}{{0000}0000{0000}0000 #2 #3 #4 #5}
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_decimate_pack:nnnnnnnnnnw}
% The computation of the \meta{rounding} digit leaves an unfinished
% \cs{__int_value:w}, which expands the following functions. This
% allows us to repack nicely the digits we keep. Those digits come
% as an alternation of unbraced and braced blocks of $4$ digits,
% such that the first $5$ groups of token consist in $4$ single digits,
% and one brace group (in some order), and the next $5$ have the same
% structure. This is followed by some digits and a semicolon.
% \begin{macrocode}
\cs_new:Npn \@@_decimate_pack:nnnnnnnnnnw #1#2#3#4#5
{ \@@_decimate_pack:nnnnnnw { #1#2#3#4#5 } }
\cs_new:Npn \@@_decimate_pack:nnnnnnw #1 #2#3#4#5#6
{ {#1} {#2#3#4#5#6} }
% \end{macrocode}
% \end{macro}
% ^^A end[todo]
%
% \subsection{Functions for use within primitive conditional branches}
%
% The functions described in this section are not pretty and can easily
% be misused. When correctly used, each of them removes one \cs{fi:} as
% part of its parameter text, and puts one back as part of its
% replacement text.
%
% Many computation functions in \pkg{l3fp} must perform tests on the
% type of floating points that they receive. This is often done in an
% \cs{if_case:w} statement or another conditional statement, and only a
% few cases lead to actual computations: most of the special cases are
% treated using a few standard functions which we define now. A typical
% use context for those functions would be
% \begin{syntax}
% |\if_case:w| \meta{integer} |\exp_stop_f:|
% | \@@_case_return_o:Nw| \meta{fp var}
% |\or: \@@_case_use:nw| \Arg{some computation}
% |\or: \@@_case_return_same_o:w|
% |\or: \@@_case_return:nw| \Arg{something}
% |\fi:|
% \meta{junk}
% \meta{floating point}
% \end{syntax}
% In this example, the case $0$ returns the floating point
% \meta{fp~var}, expanding once after that floating point. Case $1$
% does \meta{some computation} using the \meta{floating point}
% (presumably compute the operation requested by the user in that
% non-trivial case). Case $2$ returns the \meta{floating point}
% without modifying it, removing the \meta{junk} and expanding once
% after. Case $3$ closes the conditional, removes the \meta{junk}
% and the \meta{floating point}, and expands \meta{something} next. In
% other cases, the \enquote{\meta{junk}} is expanded, performing some
% other operation on the \meta{floating point}. We provide similar
% functions with two trailing \meta{floating points}.
%
% \begin{macro}[EXP]{\@@_case_use:nw}
% This function ends a \TeX{} conditional, removes junk until the next
% floating point, and places its first argument before that floating
% point, to perform some operation on the floating point.
% \begin{macrocode}
\cs_new:Npn \@@_case_use:nw #1#2 \fi: #3 \s_@@ { \fi: #1 \s_@@ }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_case_return:nw}
% This function ends a \TeX{} conditional, removes junk and a floating
% point, and places its first argument in the input stream. A quirk
% is that we don't define this function requiring a floating point to
% follow, simply anything ending in a semicolon. This, in turn, means
% that the \meta{junk} may not contain semicolons.
% \begin{macrocode}
\cs_new:Npn \@@_case_return:nw #1#2 \fi: #3 ; { \fi: #1 }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_case_return_o:Nw}
% This function ends a \TeX{} conditional, removes junk and a floating
% point, and returns its first argument (an \meta{fp~var}) then expands
% once after it.
% \begin{macrocode}
\cs_new:Npn \@@_case_return_o:Nw #1#2 \fi: #3 \s_@@ #4 ;
{ \fi: \exp_after:wN #1 }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_case_return_same_o:w}
% This function ends a \TeX{} conditional, removes junk, and returns
% the following floating point, expanding once after it.
% \begin{macrocode}
\cs_new:Npn \@@_case_return_same_o:w #1 \fi: #2 \s_@@
{ \fi: \@@_exp_after_o:w \s_@@ }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_case_return_o:Nww}
% Same as \cs{@@_case_return_o:Nw} but with two trailing floating
% points.
% \begin{macrocode}
\cs_new:Npn \@@_case_return_o:Nww #1#2 \fi: #3 \s_@@ #4 ; #5 ;
{ \fi: \exp_after:wN #1 }
% \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_case_return_i_o:ww, \@@_case_return_ii_o:ww}
% Similar to \cs{@@_case_return_same_o:w}, but this returns the first
% or second of two trailing floating point numbers, expanding once
% after the result.
% \begin{macrocode}
\cs_new:Npn \@@_case_return_i_o:ww #1 \fi: #2 \s_@@ #3 ; \s_@@ #4 ;
{ \fi: \@@_exp_after_o:w \s_@@ #3 ; }
\cs_new:Npn \@@_case_return_ii_o:ww #1 \fi: #2 \s_@@ #3 ;
{ \fi: \@@_exp_after_o:w }
% \end{macrocode}
% \end{macro}
%
% \subsection{Integer floating points}
%
% \begin{macro}[EXP, pTF]{\@@_int:w}
% Tests if the floating point argument is an integer. For normal
% floating point numbers, this holds if the rounding digit resulting
% from \cs{@@_decimate:nNnnnn} is~$0$.
% \begin{macrocode}
\prg_new_conditional:Npnn \@@_int:w \s_@@ \@@_chk:w #1 #2 #3 #4;
{ TF , T , F , p }
{
\if_case:w #1 \exp_stop_f:
\prg_return_true:
\or:
\if_charcode:w 0
\@@_decimate:nNnnnn { \c_@@_prec_int - #3 }
\@@_use_i_until_s:nw #4
\prg_return_true:
\else:
\prg_return_false:
\fi:
\else: \prg_return_false:
\fi:
}
% \end{macrocode}
% \end{macro}
%
% \subsection{Small integer floating points}
%
% \begin{macro}[EXP]{\@@_small_int:wTF}
% \begin{macro}[EXP]
% {
% \@@_small_int_true:wTF,
% \@@_small_int_normal:NnwTF,
% \@@_small_int_test:NnnwNTF
% }
% Tests if the floating point argument is an integer or $\pm\infty$.
% If so, it is converted to an integer in the range $[-10^{8},10^{8}]$
% and fed as a braced argument to the \meta{true code}.
% Otherwise, the \meta{false code} is performed.
%
% First filter special cases: zeros and infinities are integers,
% \texttt{nan} is not. For normal numbers, decimate. If the rounding
% digit is not $0$ run the \meta{false code}. If it is, then the
% integer is |#2| |#3|; use |#3| if |#2| vanishes and otherwise
% $10^{8}$.
% \begin{macrocode}
\cs_new:Npn \@@_small_int:wTF \s_@@ \@@_chk:w #1#2
{
\if_case:w #1 \exp_stop_f:
\@@_case_return:nw { \@@_small_int_true:wTF 0 ; }
\or: \exp_after:wN \@@_small_int_normal:NnwTF
\or:
\@@_case_return:nw
{
\exp_after:wN \@@_small_int_true:wTF \__int_value:w
\if_meaning:w 2 #2 - \fi: 1 0000 0000 ;
}
\else: \@@_case_return:nw \use_ii:nn
\fi:
#2
}
\cs_new:Npn \@@_small_int_true:wTF #1; #2#3 { #2 {#1} }
\cs_new:Npn \@@_small_int_normal:NnwTF #1#2#3;
{
\@@_decimate:nNnnnn { \c_@@_prec_int - #2 }
\@@_small_int_test:NnnwNw
#3 #1
}
\cs_new:Npn \@@_small_int_test:NnnwNw #1#2#3#4; #5
{
\if_meaning:w 0 #1
\exp_after:wN \@@_small_int_true:wTF
\__int_value:w \if_meaning:w 2 #5 - \fi:
\if_int_compare:w #2 > 0 \exp_stop_f: