summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/l3kernel/l3fp-round.dtx
blob: 5063fa4105afa5447bc442ddca7ff6cfa9f895fa (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
% \iffalse meta-comment
%
%% File: l3fp-round.dtx
%
% Copyright (C) 2011-2020 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-round} package\\ Rounding 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 2020-03-06}
%
% \maketitle
%
% \begin{documentation}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{l3fp-round} implementation}
%
%    \begin{macrocode}
%<*initex|package>
%    \end{macrocode}
%
%    \begin{macrocode}
%<@@=fp>
%    \end{macrocode}
%
% ^^A todo: provide an interface for rounding modes.
% ^^A todo: provide a \l_@@_rounding_mode_int giving the current mode.
% ^^A todo: make transcendental function obey the correct rounding mode.
% ^^A todo: optimize all rounding functions for various rounding modes.
% ^^A todo: reduce the number of almost identical functions.
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_word_trunc:N,
%     \@@_parse_word_floor:N,
%     \@@_parse_word_ceil:N
%   }
%    \begin{macrocode}
\cs_new:Npn \@@_parse_word_trunc:N
  { \@@_parse_function:NNN \@@_round_o:Nw \@@_round_to_zero:NNN }
\cs_new:Npn \@@_parse_word_floor:N
  { \@@_parse_function:NNN \@@_round_o:Nw \@@_round_to_ninf:NNN }
\cs_new:Npn \@@_parse_word_ceil:N
  { \@@_parse_function:NNN \@@_round_o:Nw \@@_round_to_pinf:NNN }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_word_round:N, \@@_parse_round:Nw,
%   }
%    \begin{macrocode}
\cs_new:Npn \@@_parse_word_round:N #1#2
  {
    \@@_parse_function:NNN
      \@@_round_o:Nw \@@_round_to_nearest:NNN #1
    #2
  }
\cs_new:Npn \@@_parse_round:Nw #1 #2 \@@_round_to_nearest:NNN #3#4
  { #2 #1 #3 }

%    \end{macrocode}
% \end{macro}
%
% \subsection{Rounding tools}
%
% \begin{variable}{\c_@@_five_int}
%   This is used as the half-point for which numbers are rounded
%   up/down.
%    \begin{macrocode}
\int_const:Nn \c_@@_five_int { 5 }
%    \end{macrocode}
% \end{variable}
%
% Floating point operations often yield a result that cannot be exactly
% represented in a significand with $16$ digits.  In that case, we need to
% round the exact result to a representable number.  The \textsc{ieee}
% standard defines four rounding modes:
% \begin{itemize}
%   \item Round to nearest: round to the representable floating point
%     number whose absolute difference with the exact result is the
%     smallest.  If the exact result lies exactly at the mid-point
%     between two consecutive representable floating point numbers,
%     round to the floating point number whose last digit is even.
%   \item Round towards negative infinity: round to the greatest
%     floating point number not larger than the exact result.
%   \item Round towards zero: round to a floating point number with the
%     same sign as the exact result, with the largest absolute value not
%     larger than the absolute value of the exact result.
%   \item Round towards positive infinity: round to the least floating
%     point number not smaller than the exact result.
% \end{itemize}
% This is not fully implemented in \pkg{l3fp} yet, and transcendental
% functions fall back on the \enquote{round to nearest} mode.  All
% rounding for basic algebra is done through the functions defined in
% this module, which can be redefined to change their rounding behaviour
% (but there is not interface for that yet).
%
% The rounding tools available in this module are many variations on a
% base function \cs{@@_round:NNN}, which expands to |0\exp_stop_f:| or
% |1\exp_stop_f:| depending on whether the final result should be rounded up
% or down.
% \begin{itemize}
%   \item \cs{@@_round:NNN} \meta{sign} \meta{digit_1} \meta{digit_2}
%     can expand to |0\exp_stop_f:| or |1\exp_stop_f:|.
%   \item \cs{@@_round_s:NNNw} \meta{sign} \meta{digit_1} \meta{digit_2}
%     \meta{more digits}|;| can expand to |0\exp_stop_f:;| or |1\exp_stop_f:;|.
%   \item \cs{@@_round_neg:NNN} \meta{sign} \meta{digit_1} \meta{digit_2}
%     can expand to |0\exp_stop_f:| or |1\exp_stop_f:|.
% \end{itemize}
% See implementation comments for details on the syntax.
%
% \begin{macro}[rEXP]{\@@_round:NNN}
% \begin{macro}[rEXP]
%   {
%     \@@_round_to_nearest:NNN,
%     \@@_round_to_nearest_ninf:NNN,
%     \@@_round_to_nearest_zero:NNN,
%     \@@_round_to_nearest_pinf:NNN,
%     \@@_round_to_ninf:NNN,
%     \@@_round_to_zero:NNN,
%     \@@_round_to_pinf:NNN
%   }
%   \begin{syntax}
%     \cs{@@_round:NNN} \meta{final sign} \meta{digit_1} \meta{digit_2}
%   \end{syntax}
%   If rounding the number $\meta{final sign}
%   \meta{digit_1}.\meta{digit_2}$ to an integer rounds it towards zero
%   (truncates it), this function expands to |0\exp_stop_f:|, and otherwise
%   to |1\exp_stop_f:|.  Typically used within the scope of an
%   \cs{@@_int_eval:w}, to add~$1$ if needed, and thereby round
%   correctly.  The result depends on the rounding mode.
%
%   It is very important that \meta{final sign} be the final sign of the
%   result. Otherwise, the result would be incorrect in the case of
%   rounding towards~$-\infty$ or towards~$+\infty$. Also recall that
%   \meta{final sign} is~$0$ for positive, and~$2$ for negative.
%
%   By default, the functions below return |0\exp_stop_f:|, but this is
%   superseded by \cs{@@_round_return_one:}, which instead returns
%   |1\exp_stop_f:|, expanding everything and removing |0\exp_stop_f:| in the
%   process.  In the case of rounding towards~$\pm\infty$ or
%   towards~$0$, this is not really useful, but it prepares us for the
%   \enquote{round to nearest, ties to even} mode.
%
%   The \enquote{round to nearest} mode is the default.  If the
%   \meta{digit_2} is larger than~$5$, then round up.  If it is less
%   than~$5$, round down.  If it is exactly $5$, then round such that
%   \meta{digit_1} plus the result is even.  In other words, round up if
%   \meta{digit_1} is odd.
%
%   The \enquote{round to nearest} mode has three variants, which differ
%   in how ties are rounded: down towards $-\infty$, truncated towards $0$,
%   or up towards $+\infty$.
%    \begin{macrocode}
\cs_new:Npn \@@_round_return_one:
  { \exp_after:wN 1 \exp_after:wN \exp_stop_f: \exp:w }
\cs_new:Npn \@@_round_to_ninf:NNN #1 #2 #3
  {
    \if_meaning:w 2 #1
      \if_int_compare:w #3 > 0 \exp_stop_f:
        \@@_round_return_one:
      \fi:
    \fi:
    0 \exp_stop_f:
  }
\cs_new:Npn \@@_round_to_zero:NNN #1 #2 #3 { 0 \exp_stop_f: }
\cs_new:Npn \@@_round_to_pinf:NNN #1 #2 #3
  {
    \if_meaning:w 0 #1
      \if_int_compare:w #3 > 0 \exp_stop_f:
        \@@_round_return_one:
      \fi:
    \fi:
    0 \exp_stop_f:
  }
\cs_new:Npn \@@_round_to_nearest:NNN #1 #2 #3
  {
    \if_int_compare:w #3 > \c_@@_five_int
      \@@_round_return_one:
    \else:
      \if_meaning:w 5 #3
        \if_int_odd:w #2 \exp_stop_f:
          \@@_round_return_one:
        \fi:
      \fi:
    \fi:
    0 \exp_stop_f:
  }
\cs_new:Npn \@@_round_to_nearest_ninf:NNN #1 #2 #3
  {
    \if_int_compare:w #3 > \c_@@_five_int
      \@@_round_return_one:
    \else:
      \if_meaning:w 5 #3
        \if_meaning:w 2 #1
            \@@_round_return_one:
        \fi:
      \fi:
    \fi:
    0 \exp_stop_f:
  }
\cs_new:Npn \@@_round_to_nearest_zero:NNN #1 #2 #3
  {
    \if_int_compare:w #3 > \c_@@_five_int
      \@@_round_return_one:
    \fi:
    0 \exp_stop_f:
  }
\cs_new:Npn \@@_round_to_nearest_pinf:NNN #1 #2 #3
  {
    \if_int_compare:w #3 > \c_@@_five_int
      \@@_round_return_one:
    \else:
      \if_meaning:w 5 #3
        \if_meaning:w 0 #1
            \@@_round_return_one:
        \fi:
      \fi:
    \fi:
    0 \exp_stop_f:
  }
\cs_new_eq:NN \@@_round:NNN \@@_round_to_nearest:NNN
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round_s:NNNw}
%   \begin{syntax}
%     \cs{@@_round_s:NNNw} \meta{final sign} \meta{digit} \meta{more digits} |;|
%   \end{syntax}
%   Similar to \cs{@@_round:NNN}, but with an extra semicolon, this
%   function expands to |0\exp_stop_f:;| if rounding $\meta{final sign}
%   \meta{digit}.\meta{more digits}$ to an integer truncates, and to
%   |1\exp_stop_f:;| otherwise.  The \meta{more digits} part must be a digit,
%   followed by something that does not overflow a \cs{int_use:N}
%   \cs{@@_int_eval:w} construction.  The only relevant information about
%   this piece is whether it is zero or not.
%    \begin{macrocode}
\cs_new:Npn \@@_round_s:NNNw #1 #2 #3 #4;
  {
    \exp_after:wN \@@_round:NNN
    \exp_after:wN #1
    \exp_after:wN #2
    \int_value:w \@@_int_eval:w
      \if_int_odd:w 0 \if_meaning:w 0 #3 1 \fi:
                      \if_meaning:w 5 #3 1 \fi:
                \exp_stop_f:
        \if_int_compare:w \@@_int_eval:w #4 > 0 \exp_stop_f:
          1 +
        \fi:
      \fi:
      #3
    ;
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round_digit:Nw}
%   \begin{syntax}
%     \cs{int_value:w} \cs{@@_round_digit:Nw} \meta{digit} \meta{intexpr} |;|
%   \end{syntax}
%   This function should always be called within an \cs{int_value:w}
%   or \cs{@@_int_eval:w} expansion; it may add an extra
%   \cs{@@_int_eval:w}, which means that the integer or integer
%   expression should not be ended with a synonym of \tn{relax}, but
%   with a semi-colon for instance.
%    \begin{macrocode}
\cs_new:Npn \@@_round_digit:Nw #1 #2;
  {
    \if_int_odd:w \if_meaning:w 0 #1 1 \else:
                  \if_meaning:w 5 #1 1 \else:
                  0 \fi: \fi: \exp_stop_f:
      \if_int_compare:w \@@_int_eval:w #2 > 0 \exp_stop_f:
        \@@_int_eval:w 1 +
      \fi:
    \fi:
    #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round_neg:NNN}
% \begin{macro}[EXP]
%   {
%     \@@_round_to_nearest_neg:NNN,
%     \@@_round_to_nearest_ninf_neg:NNN,
%     \@@_round_to_nearest_zero_neg:NNN,
%     \@@_round_to_nearest_pinf_neg:NNN,
%     \@@_round_to_ninf_neg:NNN,
%     \@@_round_to_zero_neg:NNN,
%     \@@_round_to_pinf_neg:NNN
%   }
%   \begin{syntax}
%     \cs{@@_round_neg:NNN} \meta{final sign} \meta{digit_1} \meta{digit_2}
%   \end{syntax}
%   This expands to |0\exp_stop_f:| or |1\exp_stop_f:| after doing the following
%   test. Starting from a number of
%   the form $ \meta{final sign}0.\meta{15 digits}\meta{digit_1} $ with exactly
%   $15$ (non-all-zero) digits before \meta{digit_1}, subtract from it
%   $\meta{final sign}0.0\ldots{}0\meta{digit_2}$, where there are $16$~zeros.
%   If in the current rounding mode the result should be rounded down,
%   then this function returns |1\exp_stop_f:|. Otherwise, \emph{i.e.},
%   if the result is rounded back to the first operand, then this function
%   returns |0\exp_stop_f:|.
%
%   It turns out that this negative \enquote{round to nearest}
%   is identical to the positive one. And this is the default mode.
%    \begin{macrocode}
\cs_new_eq:NN \@@_round_to_ninf_neg:NNN \@@_round_to_pinf:NNN
\cs_new:Npn \@@_round_to_zero_neg:NNN #1 #2 #3
  {
    \if_int_compare:w #3 > 0 \exp_stop_f:
      \@@_round_return_one:
    \fi:
    0 \exp_stop_f:
  }
\cs_new_eq:NN \@@_round_to_pinf_neg:NNN \@@_round_to_ninf:NNN
\cs_new_eq:NN \@@_round_to_nearest_neg:NNN \@@_round_to_nearest:NNN
\cs_new_eq:NN \@@_round_to_nearest_ninf_neg:NNN
  \@@_round_to_nearest_pinf:NNN
\cs_new:Npn \@@_round_to_nearest_zero_neg:NNN #1 #2 #3
  {
    \if_int_compare:w #3 < \c_@@_five_int \else:
      \@@_round_return_one:
    \fi:
    0 \exp_stop_f:
  }
\cs_new_eq:NN \@@_round_to_nearest_pinf_neg:NNN
  \@@_round_to_nearest_ninf:NNN
\cs_new_eq:NN \@@_round_neg:NNN \@@_round_to_nearest_neg:NNN
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{The \texttt{round} function}
%
% ^^A todo: This macro is intermingled with l3fp-parse.
% ^^A todo: Add explanations.
% \begin{macro}[EXP]{\@@_round_o:Nw, \@@_round_aux_o:Nw}
%   First check that all arguments are floating point numbers.
%   The |trunc|, |ceil| and |floor| functions expect one or two
%   arguments (the second is $0$ by default), and the |round| function
%   also accepts a third argument (\texttt{nan} by default), which
%   changes |#1| from \cs{@@_round_to_nearest:NNN} to one of its
%   analogues.
%    \begin{macrocode}
\cs_new:Npn \@@_round_o:Nw #1
  {
    \@@_parse_function_all_fp_o:fnw
      { \@@_round_name_from_cs:N #1 }
      { \@@_round_aux_o:Nw #1 }
  }
\cs_new:Npn \@@_round_aux_o:Nw #1#2 @
  {
    \if_case:w
      \@@_int_eval:w \@@_array_count:n {#2} \@@_int_eval_end:
         \@@_round_no_arg_o:Nw #1 \exp:w
    \or: \@@_round:Nwn #1 #2 {0} \exp:w
    \or: \@@_round:Nww #1 #2 \exp:w
    \else: \@@_round:Nwww #1 #2 @ \exp:w
    \fi:
    \exp_after:wN \exp_end:
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round_no_arg_o:Nw}
%    \begin{macrocode}
\cs_new:Npn \@@_round_no_arg_o:Nw #1
  {
    \cs_if_eq:NNTF #1 \@@_round_to_nearest:NNN
      { \@@_error:nnnn { fp-num-args } { round () } { 1 } { 3 } }
      {
        \@@_error:nffn { fp-num-args }
          { \@@_round_name_from_cs:N #1 () } { 1 } { 2 }
      }
    \exp_after:wN \c_nan_fp
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round:Nwww}
%   Having three arguments is only allowed for |round|, not |trunc|,
%   |ceil|, |floor|, so check for that case.  If all is well, construct
%   one of \cs{@@_round_to_nearest:NNN}, \cs{@@_round_to_nearest_zero:NNN},
%   \cs{@@_round_to_nearest_ninf:NNN}, \cs{@@_round_to_nearest_pinf:NNN}
%   and act accordingly.
%    \begin{macrocode}
\cs_new:Npn \@@_round:Nwww #1#2 ; #3 ; \s_@@ \@@_chk:w #4#5#6 ; #7 @
  {
    \cs_if_eq:NNTF #1 \@@_round_to_nearest:NNN
      {
        \tl_if_empty:nTF {#7}
          {
            \exp_args:Nc \@@_round:Nww
              {
                @@_round_to_nearest
                \if_meaning:w 0 #4 _zero \else:
                \if_case:w #5 \exp_stop_f: _pinf \or: \else: _ninf \fi: \fi:
                :NNN
              }
            #2 ; #3 ;
          }
          {
            \@@_error:nnnn { fp-num-args } { round () } { 1 } { 3 }
            \exp_after:wN \c_nan_fp
          }
      }
      {
        \@@_error:nffn { fp-num-args }
          { \@@_round_name_from_cs:N #1 () } { 1 } { 2 }
        \exp_after:wN \c_nan_fp
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round_name_from_cs:N}
%    \begin{macrocode}
\cs_new:Npn \@@_round_name_from_cs:N #1
  {
    \cs_if_eq:NNTF #1 \@@_round_to_zero:NNN { trunc }
      {
        \cs_if_eq:NNTF #1 \@@_round_to_ninf:NNN { floor }
          {
            \cs_if_eq:NNTF #1 \@@_round_to_pinf:NNN { ceil }
              { round }
          }
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_round:Nww, \@@_round:Nwn}
% \begin{macro}[EXP]
%   {
%     \@@_round_normal:NwNNnw ,
%     \@@_round_normal:NnnwNNnn ,
%     \@@_round_pack:Nw ,
%     \@@_round_normal:NNwNnn ,
%     \@@_round_normal_end:wwNnn ,
%     \@@_round_special:NwwNnn ,
%     \@@_round_special_aux:Nw
%   }
%   If the number of digits to round to is an integer or infinity all is
%   good; if it is \texttt{nan} then just produce a \texttt{nan};
%   otherwise invalid as we have something like |round(1,3.14)| where
%   the number of digits is not an integer.
%    \begin{macrocode}
\cs_new:Npn \@@_round:Nww #1#2 ; #3 ;
  {
    \@@_small_int:wTF #3; { \@@_round:Nwn #1#2; }
      {
        \if:w 3 \@@_kind:w #3 ;
          \exp_after:wN \use_i:nn
        \else:
          \exp_after:wN \use_ii:nn
        \fi:
        { \exp_after:wN \c_nan_fp }
        {
          \@@_invalid_operation_tl_o:ff
            { \@@_round_name_from_cs:N #1 }
            { \@@_array_to_clist:n { #2; #3; } }
        }
      }
  }
\cs_new:Npn \@@_round:Nwn #1 \s_@@ \@@_chk:w #2#3#4; #5
  {
    \if_meaning:w 1 #2
      \exp_after:wN \@@_round_normal:NwNNnw
      \exp_after:wN #1
      \int_value:w #5
    \else:
      \exp_after:wN \@@_exp_after_o:w
    \fi:
    \s_@@ \@@_chk:w #2#3#4;
  }
\cs_new:Npn \@@_round_normal:NwNNnw #1#2 \s_@@ \@@_chk:w 1#3#4#5;
  {
    \@@_decimate:nNnnnn { \c_@@_prec_int - #4 - #2 }
      \@@_round_normal:NnnwNNnn #5 #1 #3 {#4} {#2}
  }
\cs_new:Npn \@@_round_normal:NnnwNNnn #1#2#3#4; #5#6
  {
    \exp_after:wN \@@_round_normal:NNwNnn
    \int_value:w \@@_int_eval:w
      \if_int_compare:w #2 > 0 \exp_stop_f:
        1 \int_value:w #2
        \exp_after:wN \@@_round_pack:Nw
        \int_value:w \@@_int_eval:w 1#3 +
      \else:
        \if_int_compare:w #3 > 0 \exp_stop_f:
          1 \int_value:w #3 +
        \fi:
      \fi:
      \exp_after:wN #5
      \exp_after:wN #6
      \use_none:nnnnnnn #3
      #1
      \@@_int_eval_end:
      0000 0000 0000 0000 ; #6
  }
\cs_new:Npn \@@_round_pack:Nw #1
  { \if_meaning:w 2 #1 + 1 \fi: \@@_int_eval_end: }
\cs_new:Npn \@@_round_normal:NNwNnn #1 #2
  {
    \if_meaning:w 0 #2
      \exp_after:wN \@@_round_special:NwwNnn
      \exp_after:wN #1
    \fi:
    \@@_pack_twice_four:wNNNNNNNN
    \@@_pack_twice_four:wNNNNNNNN
    \@@_round_normal_end:wwNnn
    ; #2
  }
\cs_new:Npn \@@_round_normal_end:wwNnn #1;#2;#3#4#5
  {
    \exp_after:wN \@@_exp_after_o:w \exp:w \exp_end_continue_f:w
    \@@_sanitize:Nw #3 #4 ; #1 ;
  }
\cs_new:Npn \@@_round_special:NwwNnn #1#2;#3;#4#5#6
  {
    \if_meaning:w 0 #1
      \@@_case_return:nw
        { \exp_after:wN \@@_zero_fp:N \exp_after:wN #4 }
    \else:
      \exp_after:wN \@@_round_special_aux:Nw
      \exp_after:wN #4
      \int_value:w \@@_int_eval:w 1
        \if_meaning:w 1 #1 -#6 \else: +#5 \fi:
    \fi:
    ;
  }
\cs_new:Npn \@@_round_special_aux:Nw #1#2;
  {
    \exp_after:wN \@@_exp_after_o:w \exp:w \exp_end_continue_f:w
    \@@_sanitize:Nw #1#2; {1000}{0000}{0000}{0000};
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
%    \begin{macrocode}
%</initex|package>
%    \end{macrocode}
%
% \end{implementation}
%
% \PrintChanges
%
% \PrintIndex