summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/source/latex/l3kernel/l3fp-parse.dtx
blob: 28257218e7eb5d632d9a77db63be86d74a988d20 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
% \iffalse meta-comment
%
%% File: l3fp-parse.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{The \textsf{l3fp-parse} package\\
%   Floating point expression parsing}
% \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 2020-06-18}
%
% \maketitle
%
% \begin{documentation}
%
% \end{documentation}
%
% \begin{implementation}
%
% \section{\pkg{l3fp-parse} implementation}
%
%    \begin{macrocode}
%<*initex|package>
%    \end{macrocode}
%
%    \begin{macrocode}
%<@@=fp>
%    \end{macrocode}
%
% \subsection{Work plan}
%
% The task at hand is non-trivial, and some previous failed attempts
% show that the code leads to unreadable logs, so we had better get it
% (almost) right the first time.  Let us first describe our goal, then
% discuss the design precisely before writing any code.
%
% In this file at least, a \meta{floating point object} is a floating
% point number or tuple.  This can be extended to anything that starts
% with \cs{s_@@} or \cs{s_@@_\meta{type}} and ends with |;| with some
% internal structure that depends on the \meta{type}.
%
% \begin{macro}[EXP]{\@@_parse:n}
%   \begin{syntax}
%     \cs{@@_parse:n} \Arg{fpexpr}
%   \end{syntax}
%   Evaluates the \meta{floating point expression} and leaves the result
%   in the input stream as a floating point object.  This
%   function forms the basis of almost all public \pkg{l3fp} functions.
%   During evaluation, each token is fully \texttt{f}-expanded.
%
%   \cs{@@_parse_o:n} does the same but expands once after its result.
%   \begin{texnote}
%     Registers (integers, toks, etc.) are automatically unpacked,
%     without requiring a function such as \cs{int_use:N}.  Invalid
%     tokens remaining after \texttt{f}-expansion lead to
%     unrecoverable low-level \TeX{} errors.
%   \end{texnote}
% \end{macro}
%
% \begin{variable}
%   {
%     \c_@@_prec_func_int,
%     \c_@@_prec_hatii_int,
%     \c_@@_prec_hat_int,
%     \c_@@_prec_not_int,
%     \c_@@_prec_juxt_int,
%     \c_@@_prec_times_int,
%     \c_@@_prec_plus_int,
%     \c_@@_prec_comp_int,
%     \c_@@_prec_and_int,
%     \c_@@_prec_or_int,
%     \c_@@_prec_quest_int,
%     \c_@@_prec_colon_int,
%     \c_@@_prec_comma_int,
%     \c_@@_prec_tuple_int,
%     \c_@@_prec_end_int,
%   }
%   Floating point expressions are composed of numbers, given in various
%   forms, infix operators, such as |+|, |**|, or~|,| (which joins two
%   numbers into a list), and prefix operators, such as the unary~|-|,
%   functions, or opening parentheses.  Here is a list of precedences
%   which control the order of evaluation (some distinctions are
%   irrelevant for the order of evaluation, but serve as signals), from
%   the tightest binding to the loosest binding.
%   \begin{itemize}
%     \item[16] Function calls.
%     \item[13/14] Binary |**| and~|^| (right to left).
%     \item[12] Unary |+|, |-|, |!| (right to left).
%     \item[11] Juxtaposition (implicit~|*|) with no parenthesis.
%     \item[10] Binary |*| and~|/|.
%     \item[9] Binary |+| and~|-|.
%     \item[7] Comparisons.
%     \item[6] Logical \texttt{and}, denoted by~|&&|.
%     \item[5] Logical \texttt{or}, denoted by~\verb*+||+.
%     \item[4] Ternary operator |?:|, piece~|?|.
%     \item[3] Ternary operator |?:|, piece~|:|.
%     \item[2] Commas.
%     \item[1] Place where a comma is allowed and generates a tuple.
%     \item[0] Start and end of the expression.
%   \end{itemize}
%    \begin{macrocode}
\int_const:Nn \c_@@_prec_func_int   { 16 }
\int_const:Nn \c_@@_prec_hatii_int  { 14 }
\int_const:Nn \c_@@_prec_hat_int    { 13 }
\int_const:Nn \c_@@_prec_not_int    { 12 }
\int_const:Nn \c_@@_prec_juxt_int   { 11 }
\int_const:Nn \c_@@_prec_times_int  { 10 }
\int_const:Nn \c_@@_prec_plus_int   { 9 }
\int_const:Nn \c_@@_prec_comp_int   { 7 }
\int_const:Nn \c_@@_prec_and_int    { 6 }
\int_const:Nn \c_@@_prec_or_int     { 5 }
\int_const:Nn \c_@@_prec_quest_int  { 4 }
\int_const:Nn \c_@@_prec_colon_int  { 3 }
\int_const:Nn \c_@@_prec_comma_int  { 2 }
\int_const:Nn \c_@@_prec_tuple_int  { 1 }
\int_const:Nn \c_@@_prec_end_int    { 0 }
%    \end{macrocode}
% \end{variable}
%
% \subsubsection{Storing results}
%
% The main question in parsing expressions expandably is to decide where
% to put the intermediate results computed for various subexpressions.
%
% One option is to store the values at the start of the expression, and
% carry them together as the first argument of each macro.  However, we
% want to \texttt{f}-expand tokens one by one in the expression (as
% \cs{int_eval:n} does), and with this approach, expanding the next
% unread token forces us to jump with \cs{exp_after:wN} over every value
% computed earlier in the expression.  With this approach, the run-time
% grows at least quadratically in the length of the expression, if
% not as its cube (inserting the \cs{exp_after:wN} is tricky and slow).
%
% A second option is to place those values at the end of the expression.
% Then expanding the next unread token is straightforward, but this
% still hits a performance issue: for long expressions we would be
% reaching all the way to the end of the expression at every step of the
% calculation.  The run-time is again quadratic.
%
% A variation of the above attempts to place the intermediate results
% which appear when computing a parenthesized expression near the
% closing parenthesis.  This still lets us expand tokens as we go, and
% avoids performance problems as long as there are enough parentheses.
% However, it would be better to avoid requiring the closing
% parenthesis to be present as soon as the corresponding opening
% parenthesis is read: the closing parenthesis may still be hidden in a
% macro yet to be expanded.
%
% Hence, we need to go for some fine expansion control: the result is
% stored \emph{before} the start!
%
% Let us illustrate this idea in a simple model: adding positive
% integers which may be resulting from the expansion of macros, or may
% be values of registers.  Assume that one number, say, $12345$, has
% already been found, and that we want to parse the next number. The
% current status of the code may look as follows.
% \begin{syntax}
%   \cs{exp_after:wN} |\add:ww| \cs{int_value:w} 12345 \cs{exp_after:wN} ;
%   \cs{exp:w} |\operand:w| \meta{stuff}
% \end{syntax}
% One step of expansion expands \cs{exp_after:wN}, which triggers the
% primitive \cs{int_value:w}, which reads the five digits we have
% already found, |12345|.  This integer is unfinished, causing the
% second \cs{exp_after:wN} to expand, and to trigger the construction
% \cs{exp:w}, which expands |\operand:w|, defined to read
% what follows and make a number out of it, then leave \cs{exp_end:}, the
% number, and a semicolon in the input stream.  Once |\operand:w| is
% done expanding, we obtain essentially
% \begin{syntax}
%   \cs{exp_after:wN} |\add:ww| \cs{int_value:w} 12345 ;
%   \cs{exp:w} \cs{exp_end:} 333444 ;
% \end{syntax}
% where in fact \cs{exp_after:wN} has already been expanded,
% \cs{int_value:w} has already seen |12345|, and
% \cs{exp:w} is still looking for a number.  It finds
% \cs{exp_end:}, hence expands to nothing.  Now, \cs{int_value:w} sees
% the \texttt{;}, which cannot be part of a number.  The expansion
% stops, and we are left with
% \begin{syntax}
%   |\add:ww| 12345 ; 333444 ;
% \end{syntax}
% which can safely perform the addition by grabbing two arguments
% delimited by~|;|.
%
% If we were to continue parsing the expression, then the following
% number should also be cleaned up before the next use of a binary
% operation such as |\add:ww|.  Just like \cs{int_value:w} |12345|
% \cs{exp_after:wN}~|;| expanded what follows once, we need |\add:ww|
% to do the calculation, and in the process to expand the following
% once.  This is also true in our real application: all the functions of
% the form \cs[no-index]{@@_\ldots_o:ww} expand what follows once.  This comes at the
% cost of leaving tokens in the input stack, and we need to be
% careful not to waste this memory.  All of our discussion above is nice
% but simplistic, as operations should not simply be performed in the
% order they appear.
%
% \subsubsection{Precedence and infix operators}
%
% The various operators we will encounter have different precedences,
% which influence the order of calculations: $1+2\times 3 = 1+(2\times
% 3)$ because $\times$~has a higher precedence than~$+$.  The true
% analog of our macro |\operand:w| must thus take care of that.  When
% looking for an operand, it needs to perform calculations until
% reaching an operator which has lower precedence than the one which
% called |\operand:w|.  This means that |\operand:w| must know what the
% previous binary operator is, or rather, its precedence: we thus rename
% it |\operand:Nw|.  Let us describe as an example how we plan to do
% the calculation |41-2^3*4+5|.  More precisely we describe how to
% perform the first operation in this expression.  Here, we abuse
% notations: the first argument of |\operand:Nw| should be an integer
% constant (\cs{c_@@_prec_plus_int}, \ldots{}) equal to the precedence
% of the given operator, not directly the operator itself.
% \begin{itemize}
%   \item Clean up~|41| and find~|-|.  We call |\operand:Nw|~|-| to find
%     the second operand.
%   \item Clean up~|2| and find~|^|.
%   \item Compare the precedences of |-| and~|^|.  Since the latter is
%     higher, we need to compute the exponentiation.  For this, find the
%     second operand with a nested call to |\operand:Nw|~|^|.
%   \item Clean up~|3| and find~|*|.
%   \item Compare the precedences of |^| and~|*|.  Since the former is
%     higher, |\operand:Nw|~|^| has found the second operand of the
%     exponentiation, which is computed: $2^{3} = 8$.
%   \item We now have |41-8*4+5|, and |\operand:Nw|~|-| is still
%     looking for a second operand for the subtraction.  Is it~$8$?
%   \item Compare the precedences of |-| and~|*|.  Since the latter is
%     higher, we are not done with~$8$.  Call |\operand:Nw|~|*| to find
%     the second operand of the multiplication.
%   \item Clean up~|4|, and find~|+|.
%   \item Compare the precedences of |*| and~|+|.  Since the former is
%     higher, |\operand:Nw|~|*| has found the second operand of the
%     multiplication, which is computed: $8*4 = 32$.
%   \item We now have |41-32+5|, and |\operand:Nw|~|-| is still looking
%     for a second operand for the subtraction.  Is it~$32$?
%   \item Compare the precedences of |-| and~|+|.  Since they are equal,
%     |\operand:Nw|~|-| has found the second operand for the
%     subtraction, which is computed: $41-32=9$.
%   \item We now have |9+5|.
% \end{itemize}
% The procedure above stops short of performing all computations, but
% adding a surrounding call to |\operand:Nw| with a very low precedence
% ensures that all computations are performed before |\operand:Nw|
% is done.  Adding a trailing marker with the same very low precedence
% prevents the surrounding |\operand:Nw| from going beyond the marker.
%
% The pattern above to find an operand for a given operator, is to find
% one number and the next operator, then compare precedences to know if
% the next computation should be done.  If it should, then perform it
% after finding its second operand, and look at the next operator, then
% compare precedences to know if the next computation should be done.
% This continues until we find that the next computation should not be
% done.  Then, we stop.
%
% We are now ready to get a bit more technical and describe which of the
% \pkg{l3fp-parse} functions correspond to each step above.
%
% First, \cs{@@_parse_operand:Nw} is the |\operand:Nw| function above,
% with small modifications due to expansion issues discussed later.  We
% denote by \meta{precedence} the argument of \cs{@@_parse_operand:Nw},
% that is, the precedence of the binary operator whose operand we are
% trying to find.  The basic action is to read numbers from the input
% stream.  This is done by \cs{@@_parse_one:Nw}.  A first approximation
% of this function is that it reads one \meta{number}, performing no
% computation, and finds the following binary \meta{operator}.  Then it
% expands to
% \begin{quote}
%   \meta{number}\\
%   |  \__fp_parse_infix_|\meta{operator}|:N| \meta{precedence}
% \end{quote}
% expanding the \texttt{infix} auxiliary before leaving the above in the
% input stream.
%
% We now explain the \texttt{infix} auxiliaries.  We need some
% flexibility in how we treat the case of equal precedences: most often,
% the first operation encountered should be performed, such as |1-2-3|
% being computed as |(1-2)-3|, but |2^3^4| should be evaluated as
% |2^(3^4)| instead.  For this reason, and to support the equivalence
% between |**| and~|^| more easily, each binary operator is converted to
% a control sequence |\__fp_parse_infix_|\meta{operator}|:N| when it is
% encountered for the first time.  Instead of passing both precedences
% to a test function to do the comparison steps above, we pass the
% \meta{precedence} (of the earlier operator) to the \texttt{infix}
% auxiliary for the following \meta{operator}, to know whether to
% perform the computation of the \meta{operator}.  If it should not be
% performed, the \texttt{infix} auxiliary expands to
% \begin{syntax}
%   |@| \cs{use_none:n} |\__fp_parse_infix_|\meta{operator}|:N|
% \end{syntax}
% and otherwise it calls \cs{@@_parse_operand:Nw} with the precedence of
% the \meta{operator} to find its second operand \meta{number_2} and the
% next \meta{operator_2}, and expands to
% \begin{syntax}
%   |@| \cs{@@_parse_apply_binary:NwNwN}
%   ~~~~\meta{operator} \meta{number_2}
%   |@| |\__fp_parse_infix_|\meta{operator_2}|:N|
% \end{syntax}
% The \texttt{infix} function is responsible for comparing precedences,
% but cannot directly call the computation functions, because the first
% operand \meta{number} is before the \texttt{infix} function in the
% input stream.  This is why we stop the expansion here and give control
% to another function to close the loop.
%
% A definition of \cs{@@_parse_operand:Nw} \meta{precedence} with some
% of the expansion control removed is
% \begin{syntax}
%   \cs{exp_after:wN} \cs{@@_parse_continue:NwN}
%   \cs{exp_after:wN} \meta{precedence}
%   \cs{exp:w} \cs{exp_end_continue_f:w}
%   ~~\cs{@@_parse_one:Nw} \meta{precedence}
% \end{syntax}
% This expands \cs{@@_parse_one:Nw} \meta{precedence} completely, which
% finds a number, wraps the next \meta{operator} into an \texttt{infix}
% function, feeds this function the \meta{precedence}, and expands it,
% yielding either
% \begin{syntax}
%   \cs{@@_parse_continue:NwN} \meta{precedence}
%   \meta{number} |@|
%   \cs{use_none:n} |\__fp_parse_infix_|\meta{operator}|:N|
% \end{syntax}
% or
% \begin{syntax}
%   \cs{@@_parse_continue:NwN} \meta{precedence}
%   \meta{number} |@|
%   \cs{@@_parse_apply_binary:NwNwN}
%   ~~\meta{operator} \meta{number_2}
%   |@| |\__fp_parse_infix_|\meta{operator_2}|:N|
% \end{syntax}
% The definition of \cs{@@_parse_continue:NwN} is then very simple:
% \begin{syntax}
%   |\cs_new:Npn \__fp_parse_continue:NwN #1#2@#3 { #3 #1 #2 @ }|
% \end{syntax}
% In the first case, |#3|~is \cs{use_none:n}, yielding
% \begin{syntax}
%   \cs{use_none:n} \meta{precedence} \meta{number} |@|
%   |\__fp_parse_infix_|\meta{operator}|:N|
% \end{syntax}
% then \meta{number} |@| |\__fp_parse_infix_|\meta{operator}|:N|.  In
% the second case, |#3|~is \cs{@@_parse_apply_binary:NwNwN}, whose role
% is to compute \meta{number} \meta{operator} \meta{number_2} and to
% prepare for the next comparison of precedences: first we get
% \begin{syntax}
%   \cs{@@_parse_apply_binary:NwNwN}
%   ~~\meta{precedence} \meta{number} |@|
%   ~~\meta{operator} \meta{number_2}
%   |@| |\__fp_parse_infix_|\meta{operator_2}|:N|
% \end{syntax}
% then
% \begin{syntax}
%   \cs{exp_after:wN} \cs{@@_parse_continue:NwN}
%   \cs{exp_after:wN} \meta{precedence}
%   \cs{exp:w} \cs{exp_end_continue_f:w}
%   |\__fp_|\meta{operator}|_o:ww| \meta{number} \meta{number_2}
%   \cs{exp:w} \cs{exp_end_continue_f:w}
%   |\__fp_parse_infix_|\meta{operator_2}|:N| \meta{precedence}
% \end{syntax}
% where |\__fp_|\meta{operator}|_o:ww| computes \meta{number}
% \meta{operator} \meta{number_2} and expands after the result, thus
% triggers the comparison of the precedence of the \meta{operator_2} and
% the \meta{precedence}, continuing the loop.
%
% We have introduced the most important functions here, and the next few
% paragraphs we describe various subtleties.
%
% \subsubsection{Prefix operators, parentheses, and functions}
%
% Prefix operators (unary |-|, |+|,~|!|) and parentheses are taken care
% of by the same mechanism, and functions (\texttt{sin}, \texttt{exp},
% etc.) as well.  Finding the argument of the unary~|-|, for instance,
% is very similar to grabbing the second operand of a binary infix
% operator, with a subtle precedence explained below.  Once that operand
% is found, the operator can be applied to it (for the unary~|-|, this
% simply flips the sign).  A left parenthesis is just a prefix operator
% with a very low precedence equal to that of the closing parenthesis
% (which is treated as an infix operator, since it normally appears just
% after numbers), so that all computations are performed until the
% closing parenthesis.  The prefix operator associated to the left
% parenthesis does not alter its argument, but it removes the closing
% parenthesis (with some checks).
%
% Prefix operators are the reason why we only summarily described the
% function \cs{@@_parse_one:Nw} earlier.  This function is responsible
% for reading in the input stream the first possible \meta{number} and
% the next infix \meta{operator}.  If what follows \cs{@@_parse_one:Nw}
% \meta{precedence} is a prefix operator, then we must find the operand
% of this prefix operator through a nested call to
% \cs{@@_parse_operand:Nw} with the appropriate precedence, then apply
% the operator to the operand found to yield the result of
% \cs{@@_parse_one:Nw}.  So far, all is simple.
%
% The unary operators |+|, |-|,~|!| complicate things a little bit:
% |-3**2| should be $-(3^2)=-9$, and not $(-3)^2=9$.  This would easily
% be done by giving~|-| a lower precedence, equal to that of the infix
% |+| and~|-|.  Unfortunately, this fails in cases such as |3**-2*4|,
% yielding $3^{-2\times 4}$ instead of the correct $3^{-2}\times 4$.  A
% second attempt would be to call \cs{@@_parse_operand:Nw} with the
% \meta{precedence} of the previous operator, but |0>-2+3| is then
% parsed as |0>-(2+3)|: the addition is performed because it binds more
% tightly than the comparision which precedes~|-|.  The correct approach
% is for a unary~|-| to perform operations whose precedence is greater
% than both that of the previous operation, and that of the unary~|-|
% itself.  The unary~|-| is given a precedence higher than
% multiplication and division.  This does not lead to any surprising
% result, since $-(x/y) = (-x)/y$ and similarly for multiplication, and
% it reduces the number of nested calls to \cs{@@_parse_operand:Nw}.
%
% Functions are implemented as prefix operators with very high
% precedence, so that their argument is the first number that can
% possibly be built.
%
% Note that contrarily to the \texttt{infix} functions discussed
% earlier, the \texttt{prefix} functions do perform tests on the
% previous \meta{precedence} to decide whether to find an argument or
% not, since we know that we need a number, and must never stop there.
%
% \subsubsection{Numbers and reading tokens one by one}
%
% So far, we have glossed over one important point: what is a
% \enquote{number}?  A number is typically given in the form
% \meta{significand}|e|\meta{exponent}, where the \meta{significand} is
% any non-empty string composed of decimal digits and at most one
% decimal separator (a period), the exponent
% \enquote{\texttt{e}\meta{exponent}} is optional and is composed of an
% exponent mark~|e| followed by a possibly empty string of signs
% |+| or~|-| and a non-empty string of decimal digits.  The
% \meta{significand} can also be an integer, dimension, skip, or muskip
% variable, in which case dimensions are converted from points (or mu
% units) to floating points, and the \meta{exponent} can also be an
% integer variable.  Numbers can also be given as floating point
% variables, or as named constants such as |nan|, |inf| or~|pi|.  We may
% add more types in the future.
%
% When \cs{@@_parse_one:Nw} is looking for a \enquote{number}, here is
% what happens.
% \begin{itemize}
%   \item If the next token is a control sequence with the meaning of
%     \cs{scan_stop:}, it can be: \cs{s_@@}, in which case our job is
%     done, as what follows is an internal floating point number, or
%     \cs{s_@@_expr_mark}, in which case the expression has come to an early
%     end, as we are still looking for a number here, or something else,
%     in which case we consider the control sequence to be a bad
%     variable resulting from \texttt{c}-expansion.
%   \item If the next token is a control sequence with a different
%     meaning, we assume that it is a register, unpack it with
%     \cs{tex_the:D}, and use its value (in \texttt{pt} for dimensions
%     and skips, \texttt{mu} for muskips) as the \meta{significand} of a
%     number: we look for an exponent.
%   \item If the next token is a digit, we remove any leading zeros,
%     then read a significand larger than~$1$ if the next character is a
%     digit, read a significand smaller than~$1$ if the next character
%     is a period, or we have found a significand equal to~$0$
%     otherwise, and look for an exponent.
%   \item If the next token is a letter, we collect more letters until
%     the first non-letter: the resulting word may denote a function
%     such as |asin|, a constant such as |pi| or be unknown.  In the
%     first case, we call \cs{@@_parse_operand:Nw} to find the argument
%     of the function, then apply the function, before declaring that we
%     are done.  Otherwise, we are done, either with the value of the
%     constant, or with the value |nan| for unknown words.
%   \item If the next token is anything else, we check whether it is a
%     known prefix operator, in which case \cs{@@_parse_operand:Nw}
%     finds its operand.  If it is not known, then either a number is
%     missing (if the token is a known infix operator) or the token is
%     simply invalid in floating point expressions.
% \end{itemize}
% Once a number is found, \cs{@@_parse_one:Nw} also finds an infix
% operator.  This goes as follows.
% \begin{itemize}
%   \item If the next token is a control sequence, it could be the
%     special marker \cs{s_@@_expr_mark}, and
%     otherwise it is a case of juxtaposing numbers, such as
%     |2\c_zero_int|, with an implied multiplication.
%   \item If the next token is a letter, it is also a case of
%     juxtaposition, as letters cannot be proper infix operators.
%   \item Otherwise (including in the case of digits), if the token is a
%     known infix operator, the appropriate
%     |\__fp_infix_|\meta{operator}|:N| function is built, and if it
%     does not exist, we complain.  In particular, the juxtaposition
%     |\c_zero_int 2| is disallowed.
% \end{itemize}
%
% In the above, we need to test whether a character token~|#1| is a
% digit:
% \begin{verbatim}
% \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
%   is a digit
% \else:
%   not a digit
% \fi:
% \end{verbatim}
% To exclude |0|, replace |9| by |10|. The use of
% \cs{token_to_str:N} ensures that a digit with any catcode is detected.
% To test if a character token is a letter, we need to work with its
% character code, testing if |`#1| lies in $[65,90]$ (uppercase letters)
% or $[97,112]$ (lowercase letters)
% \begin{verbatim}
% \if_int_compare:w \__fp_int_eval:w
%     ( `#1 \if_int_compare:w `#1 > `Z - 32 \fi: ) / 26 = 3 \exp_stop_f:
%   is a letter
% \else:
%   not a letter
% \fi:
% \end{verbatim}
% At all steps, we try to accept all category codes: when |#1|~is kept
% to be used later, it is almost always converted to category code other
% through \cs{token_to_str:N}.  More precisely, catcodes $\{3, 6, 7, 8,
% 11, 12\}$ should work without trouble, but not $\{1, 2, 4, 10, 13\}$,
% and of course $\{0, 5, 9\}$ cannot become tokens.
%
% Floating point expressions should behave as much as possible like
% \eTeX{}-based integer expressions and dimension expressions.  In
% particular, \texttt{f}-expansion should be performed as the expression
% is read, token by token, forcing the expansion of protected macros,
% and ignoring spaces.  One advantage of expanding at every step is that
% restricted expandable functions can then be used in floating point
% expressions just as they can be in other kinds of expressions.
% Problematically, spaces stop \texttt{f}-expansion: for instance, the
% macro~|\X| below would not be expanded if we simply performed
% \texttt{f}-expansion.
% \begin{verbatim}
%   \DeclareDocumentCommand {\test} {m} { \fp_eval:n {#1} }
%   \ExplSyntaxOff
%   \test { 1 + \X }
% \end{verbatim}
% Of course, spaces typically do not appear in a code setting, but may very
% easily come in document-level input, from which some expressions may
% come.  To avoid this problem, at every step, we do essentially what
% \cs{use:f} would do: take an argument, put it back in the input
% stream, then \texttt{f}-expand it.  This is not a complete solution,
% since a macro's expansion could contain leading spaces which would stop
% the \texttt{f}-expansion before further macro calls are performed.
% However, in practice it should be enough: in particular, floating
% point numbers are correctly expanded to the underlying \cs{s_@@}
% \ldots{} structure.  The \texttt{f}-expansion is performed by
% \cs{@@_parse_expand:w}.
%
% ^^A begin[todo]
%
% \subsection{Main auxiliary functions}
%
% \begin{macro}[rEXP]{\@@_parse_operand:Nw}
%   \begin{syntax}
%     \cs{exp:w} \cs{@@_parse_operand:Nw} \meta{precedence} \cs{@@_parse_expand:w}
%   \end{syntax}
%   Reads the \enquote{\ttfamily\ldots{}}, performing every computation
%   with a precedence higher than \meta{precedence}, then expands to
%   \begin{syntax}
%     \meta{result} |@| |\__fp_parse_infix_|\meta{operation}|:N| \ldots{}
%   \end{syntax}
%   where the \meta{operation} is the first operation with a lower
%   precedence, possibly \texttt{end}, and the
%   \enquote{\ttfamily\ldots{}} start just after the \meta{operation}.
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_infix_+:N}
%   \begin{syntax}
%     \cs{@@_parse_infix_+:N} \meta{precedence} \ldots{}
%   \end{syntax}
%   If |+|~has a precedence higher than the \meta{precedence}, cleans up
%   a second \meta{operand} and finds the \meta{operation_2} which
%   follows, and expands to
%   \begin{syntax}
%     |@| \cs{@@_parse_apply_binary:NwNwN} |+| \meta{operand} |@| \cs{@@_parse_infix_\meta{operation_2}:N} \ldots{}
%   \end{syntax}
%   Otherwise expands to
%   \begin{syntax}
%     |@| \cs{use_none:n} \cs{@@_parse_infix_+:N} \ldots{}
%   \end{syntax}
%   A similar function exists for each infix operator.
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_one:Nw}
%   \begin{syntax}
%     \cs{@@_parse_one:Nw} \meta{precedence} \ldots{}
%   \end{syntax}
%   Cleans up one or two operands depending on how the precedence of the
%   next operation compares to the \meta{precedence}.  If the following
%   \meta{operation} has a precedence higher than \meta{precedence},
%   expands to
%   \begin{syntax}
%     \meta{operand_1} |@| \cs{@@_parse_apply_binary:NwNwN} \meta{operation} \meta{operand_2} |@| |\__fp_parse_infix_|\meta{operation_2}|:N| \ldots{}
%   \end{syntax}
%   and otherwise expands to
%   \begin{syntax}
%     \meta{operand} |@| \cs{use_none:n} |\__fp_parse_infix_|\meta{operation}|:N| \ldots{}
%   \end{syntax}
% \end{macro}
%
% ^^A end[todo]
%
% \subsection{Helpers}
%
% \begin{macro}[rEXP]{\@@_parse_expand:w}
%   \begin{syntax}
%     \cs{exp:w} \cs{@@_parse_expand:w} \meta{tokens}
%   \end{syntax}
%   This function must always come within a \cs{exp:w} expansion.
%   The \meta{tokens} should be the part of the expression that we have
%   not yet read.  This requires in particular closing all conditionals
%   properly before expanding.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_expand:w #1 { \exp_end_continue_f:w #1 }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_return_semicolon:w}
%   This very odd function swaps its position with the following
%   \cs{fi:} and removes \cs{@@_parse_expand:w} normally responsible for
%   expansion.  That turns out to be useful.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_return_semicolon:w
    #1 \fi: \@@_parse_expand:w { \fi: ; #1 }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]
%   {
%     \@@_parse_digits_vii:N  ,
%     \@@_parse_digits_vi:N   ,
%     \@@_parse_digits_v:N    ,
%     \@@_parse_digits_iv:N   ,
%     \@@_parse_digits_iii:N  ,
%     \@@_parse_digits_ii:N   ,
%     \@@_parse_digits_i:N    ,
%     \@@_parse_digits_:N
%   }
%   These functions must be called within an \cs{int_value:w} or
%   \cs{@@_int_eval:w} construction.  The first token which follows must
%   be \texttt{f}-expanded prior to calling those functions.  The
%   functions read tokens one by one, and output digits into the input
%   stream, until meeting a non-digit, or up to a number of digits equal
%   to their index.  The full expansion is
%   \begin{syntax}
%     \meta{digits} |;| \meta{filling 0} |;| \meta{length}
%   \end{syntax}
%   where \meta{filling 0} is a string of zeros such that \meta{digits}
%   \meta{filling 0} has the length given by the index of the function,
%   and \meta{length} is the number of zeros in the \meta{filling 0}
%   string.  Each function puts a digit into the input stream and calls
%   the next function, until we find a non-digit.  We are careful to
%   pass the tested tokens through \cs{token_to_str:N} to normalize
%   their category code.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1 #2 #3
  {
    \cs_new:cpn { @@_parse_digits_ #1 :N } ##1
      {
        \if_int_compare:w 9 < 1 \token_to_str:N ##1 \exp_stop_f:
          \token_to_str:N ##1 \exp_after:wN #2 \exp:w
        \else:
          \@@_parse_return_semicolon:w #3 ##1
        \fi:
        \@@_parse_expand:w
      }
  }
\@@_tmp:w {vii}  \@@_parse_digits_vi:N   { 0000000 ; 7 }
\@@_tmp:w {vi}   \@@_parse_digits_v:N    { 000000 ; 6 }
\@@_tmp:w {v}    \@@_parse_digits_iv:N   { 00000 ; 5 }
\@@_tmp:w {iv}   \@@_parse_digits_iii:N  { 0000 ; 4 }
\@@_tmp:w {iii}  \@@_parse_digits_ii:N   { 000 ; 3 }
\@@_tmp:w {ii}   \@@_parse_digits_i:N    { 00 ; 2 }
\@@_tmp:w {i}    \@@_parse_digits_:N     { 0 ; 1 }
\cs_new:Npn \@@_parse_digits_:N { ; ; 0 }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Parsing one number}
%
% \begin{macro}[EXP]{\@@_parse_one:Nw}
%   This function finds one number, and packs the symbol which follows
%   in an \cs[no-index]{@@_parse_infix_\ldots{}} csname.
%   |#1|~is the previous \meta{precedence},
%   and |#2|~the first token of the operand.  We distinguish four cases:
%   |#2|~is equal to \cs{scan_stop:} in meaning, |#2|~is a different
%   control sequence, |#2|~is a digit, and |#2|~is something else (this
%   last case is split further later).  Despite the earlier
%   \texttt{f}-expansion, |#2|~may still be expandable if it was
%   protected by \cs{exp_not:N}, as may happen with the \LaTeXe{} command
%   \tn{protect}.  Using a well placed \cs{reverse_if:N}, this case is
%   sent to \cs{@@_parse_one_fp:NN} which deals with it robustly.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one:Nw #1 #2
  {
    \if_catcode:w \scan_stop: \exp_not:N #2
      \exp_after:wN \if_meaning:w \exp_not:N #2 #2 \else:
        \exp_after:wN \reverse_if:N
      \fi:
      \if_meaning:w \scan_stop: #2
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_one_fp:NN
      \else:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_one_register:NN
      \fi:
    \else:
      \if_int_compare:w 9 < 1 \token_to_str:N #2 \exp_stop_f:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_one_digit:NN
      \else:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_one_other:NN
      \fi:
    \fi:
    #1 #2
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_one_fp:NN,
%     \@@_exp_after_expr_mark_f:nw,
%     \@@_exp_after_?_f:nw
%   }
%   This function receives a \meta{precedence} and a control sequence
%   equal to \cs{scan_stop:} in meaning.  There are three cases.
%   \begin{itemize}
%     \item \cs{s_@@} starts a floating point number, and we call
%       \cs{@@_exp_after_f:nw}, which |f|-expands after the floating
%       point.
%     \item \cs{s_@@_expr_mark} is a premature end, we call
%       \cs{@@_exp_after_expr_mark_f:nw}, which triggers an |fp-early-end|
%       error.
%     \item For a control sequence not containing \cs[no-index]{s_@@}, we call
%       \cs{@@_exp_after_?_f:nw}, causing a |bad-variable| error.
%   \end{itemize}
%   This scheme is extensible: additional types can be added by starting
%   the variables with a scan mark of the form \cs[no-index]{s_@@_\meta{type}} and
%   defining |\__fp_exp_after_|\meta{type}|_f:nw|.  In all cases, we
%   make sure that the second argument of \cs{@@_parse_infix:NN} is
%   correctly expanded.
%   A special case only enabled in \LaTeXe{} is that if \tn{protect} is
%   encountered then the error message mentions the control sequence
%   which follows it rather than \tn{protect} itself.  The test for
%   \LaTeXe{} uses \tn{@unexpandable@protect} rather than \tn{protect}
%   because \tn{protect} is often \cs{scan_stop:} hence \enquote{does
%   not exist}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one_fp:NN #1
  {
    \@@_exp_after_any_f:nw
      {
        \exp_after:wN \@@_parse_infix:NN
        \exp_after:wN #1 \exp:w \@@_parse_expand:w
      }
  }
\cs_new:Npn \@@_exp_after_expr_mark_f:nw #1
  {
    \int_case:nnF { \exp_after:wN \use_i:nnn \use_none:nnn #1 }
      {
        \c_@@_prec_comma_int { }
        \c_@@_prec_tuple_int { }
        \c_@@_prec_end_int
          {
            \exp_after:wN \c_@@_empty_tuple_fp
            \exp:w \exp_end_continue_f:w
          }
      }
      {
        \__kernel_msg_expandable_error:nn { kernel } { fp-early-end }
        \exp_after:wN \c_nan_fp \exp:w \exp_end_continue_f:w
      }
    #1
  }
\cs_new:cpn { @@_exp_after_?_f:nw } #1#2
  {
    \__kernel_msg_expandable_error:nnn { kernel } { bad-variable }
      {#2}
    \exp_after:wN \c_nan_fp \exp:w \exp_end_continue_f:w #1
  }
%<*package>
\cs_set_protected:Npn \@@_tmp:w #1
  {
    \cs_if_exist:NT #1
      {
        \cs_gset:cpn { @@_exp_after_?_f:nw } ##1##2
          {
            \exp_after:wN \c_nan_fp \exp:w \exp_end_continue_f:w ##1
            \str_if_eq:nnTF {##2} { \protect }
              {
                \cs_if_eq:NNTF ##2 #1 { \use_i:nn } { \use:n }
                {
                  \__kernel_msg_expandable_error:nnn { kernel }
                    { fp-robust-cmd }
                }
              }
              {
                \__kernel_msg_expandable_error:nnn { kernel }
                  { bad-variable } {##2}
              }
          }
      }
  }
\exp_args:Nc \@@_tmp:w { @unexpandable@protect }
%</package>
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_one_register:NN,
%     \@@_parse_one_register_aux:Nw,
%     \@@_parse_one_register_auxii:wwwNw,
%     \@@_parse_one_register_int:www,
%     \@@_parse_one_register_mu:www,
%     \@@_parse_one_register_dim:ww,
%   }
%   This is called whenever~|#2| is a control sequence other than
%   \cs{scan_stop:} in meaning.  We special-case \tn{wd}, \tn{ht}, \tn{dp}
%   (see later) and otherwise assume that it is a register, but
%   carefully unpack it with \cs{tex_the:D} within braces.  First, we
%   find the exponent following~|#2|.  Then we unpack~|#2| with
%   \cs{tex_the:D}, and the \texttt{auxii} auxiliary distinguishes
%   integer registers from dimensions/skips from muskips, according to
%   the presence of a period and/or of |pt|.  For integers, simply
%   convert \meta{value}|e|\meta{exponent} to a floating point number
%   with \cs{@@_parse:n} (this is somewhat wasteful).  For other
%   registers, the decimal rounding provided by \TeX{} does not
%   accurately represent the binary value that it manipulates, so we
%   extract this binary value as a number of scaled points with
%   \cs{int_value:w} \cs{dim_to_decimal_in_sp:n} |{| \meta{decimal value} |pt| |}|, and
%   use an auxiliary of \cs{dim_to_fp:n}, which performs the
%   multiplication by $2^{-16}$, correctly rounded.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one_register:NN #1#2
  {
    \exp_after:wN \@@_parse_infix_after_operand:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
      \@@_parse_one_register_special:N #2
      \exp_after:wN \@@_parse_one_register_aux:Nw
      \exp_after:wN #2
      \int_value:w
        \exp_after:wN \@@_parse_exponent:N
        \exp:w \@@_parse_expand:w
  }
\cs_new:Npx \@@_parse_one_register_aux:Nw #1
  {
    \exp_not:n
      {
        \exp_after:wN \use:nn
        \exp_after:wN \@@_parse_one_register_auxii:wwwNw
      }
    \exp_not:N \exp_after:wN { \exp_not:N \tex_the:D #1 }
      ; \exp_not:N \@@_parse_one_register_dim:ww
      \tl_to_str:n { pt } ; \exp_not:N \@@_parse_one_register_mu:www
      . \tl_to_str:n { pt } ; \exp_not:N \@@_parse_one_register_int:www
      \s_@@_stop
  }
\exp_args:Nno \use:nn
  { \cs_new:Npn \@@_parse_one_register_auxii:wwwNw #1 . #2 }
    { \tl_to_str:n { pt } #3 ; #4#5 \s_@@_stop }
    { #4 #1.#2; }
\exp_args:Nno \use:nn
  { \cs_new:Npn \@@_parse_one_register_mu:www #1 }
    { \tl_to_str:n { mu } ; #2 ; }
    { \@@_parse_one_register_dim:ww #1 ; }
\cs_new:Npn \@@_parse_one_register_int:www #1; #2.; #3;
  { \@@_parse:n { #1 e #3 } }
\cs_new:Npn \@@_parse_one_register_dim:ww #1; #2;
  {
    \exp_after:wN \@@_from_dim_test:ww
    \int_value:w #2 \exp_after:wN ,
    \int_value:w \dim_to_decimal_in_sp:n { #1 pt } ;
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}
%   {
%     \@@_parse_one_register_special:N,
%     \@@_parse_one_register_math:NNw,
%     \@@_parse_one_register_wd:w,
%     \@@_parse_one_register_wd:Nw
%   }
%   The \tn{wd}, \tn{dp}, \tn{ht} primitives expect an integer argument.
%   We abuse the exponent parser to find the integer argument: simply
%   include the exponent marker~|e|.  Once that \enquote{exponent} is
%   found, use \cs{tex_the:D} to find the box dimension and then copy
%   what we did for dimensions.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one_register_special:N #1
  {
    \if_meaning:w \box_wd:N #1 \@@_parse_one_register_wd:w \fi:
    \if_meaning:w \box_ht:N #1 \@@_parse_one_register_wd:w \fi:
    \if_meaning:w \box_dp:N #1 \@@_parse_one_register_wd:w \fi:
    \if_meaning:w \infty #1
      \@@_parse_one_register_math:NNw \infty #1
    \fi:
    \if_meaning:w \pi #1
      \@@_parse_one_register_math:NNw \pi #1
    \fi:
  }
\cs_new:Npn \@@_parse_one_register_math:NNw
    #1#2#3#4 \@@_parse_expand:w
  {
    #3
    \str_if_eq:nnTF {#1} {#2}
      {
        \__kernel_msg_expandable_error:nnn
          { kernel } { fp-infty-pi } {#1}
        \c_nan_fp
      }
      { #4 \@@_parse_expand:w }
  }
\cs_new:Npn \@@_parse_one_register_wd:w
    #1#2 \exp_after:wN #3#4 \@@_parse_expand:w
  {
    #1
    \exp_after:wN \@@_parse_one_register_wd:Nw
    #4 \@@_parse_expand:w e
  }
\cs_new:Npn \@@_parse_one_register_wd:Nw #1#2 ;
  {
    \exp_after:wN \@@_from_dim_test:ww
    \exp_after:wN 0 \exp_after:wN ,
    \int_value:w \dim_to_decimal_in_sp:n { #1 #2 } ;
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_one_digit:NN}
%   A digit marks the beginning of an explicit floating point number.
%   Once the number is found, we catch the case of overflow and
%   underflow with \cs{@@_sanitize:wN}, then
%   \cs{@@_parse_infix_after_operand:NwN} expands \cs{@@_parse_infix:NN}
%   after the number we find, to wrap the following infix operator as
%   required.  Finding the number itself begins by removing leading
%   zeros: further steps are described later.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one_digit:NN #1
  {
    \exp_after:wN \@@_parse_infix_after_operand:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
      \exp_after:wN \@@_sanitize:wN
      \int_value:w \@@_int_eval:w 0 \@@_parse_trim_zeros:N
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_one_other:NN}
%   For this function, |#2|~is a character token which is not a digit.
%   If it is an \textsc{ascii} letter, \cs{@@_parse_letters:N} beyond this one and give
%   the result to \cs{@@_parse_word:Nw}.  Otherwise, the character is
%   assumed to be a prefix operator, and we build
%   |\__fp_parse_prefix_|\meta{operator}|:Nw|.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_one_other:NN #1 #2
  {
    \if_int_compare:w
        \@@_int_eval:w
          ( `#2 \if_int_compare:w `#2 > `Z - 32 \fi: ) / 26
        = 3 \exp_stop_f:
      \exp_after:wN \@@_parse_word:Nw
      \exp_after:wN #1
      \exp_after:wN #2
      \exp:w \exp_after:wN \@@_parse_letters:N
      \exp:w
    \else:
      \exp_after:wN \@@_parse_prefix:NNN
      \exp_after:wN #1
      \exp_after:wN #2
      \cs:w
        @@_parse_prefix_ \token_to_str:N #2 :Nw
        \exp_after:wN
      \cs_end:
      \exp:w
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_word:Nw}
% \begin{macro}[rEXP]{\@@_parse_letters:N}
%   Finding letters is a simple recursion.  Once \cs{@@_parse_letters:N}
%   has done its job, we try to build a control sequence from the
%   word~|#2|.  If it is a known word, then the corresponding action is
%   taken, and otherwise, we complain about an unknown word, yield
%   \cs{c_nan_fp}, and look for the following infix operator.  Note that
%   the unknown word could be a mistyped function as well as a mistyped
%   constant, so there is no way to tell whether to look for arguments;
%   we do not.
%   The standard requires \enquote{inf} and \enquote{infinity} and
%   \enquote{nan} to be recognized regardless of case, but we probably
%   don't want to allow every \pkg{l3fp} word to have an arbitrary
%   mixture of lower and upper case, so we test and use a
%   differently-named control sequence.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_word:Nw #1#2;
  {
    \cs_if_exist_use:cF { @@_parse_word_#2:N }
      {
        \cs_if_exist_use:cF
          { @@_parse_caseless_ \str_foldcase:n {#2} :N }
          {
            \__kernel_msg_expandable_error:nnn
              { kernel } { unknown-fp-word } {#2}
            \exp_after:wN \c_nan_fp \exp:w \exp_end_continue_f:w
            \@@_parse_infix:NN
          }
      }
      #1
  }
\cs_new:Npn \@@_parse_letters:N #1
  {
    \exp_end_continue_f:w
    \if_int_compare:w
        \if_catcode:w \scan_stop: \exp_not:N #1
          0
        \else:
          \@@_int_eval:w
            ( `#1 \if_int_compare:w `#1 > `Z - 32 \fi: ) / 26
        \fi:
        = 3 \exp_stop_f:
      \exp_after:wN #1
      \exp:w \exp_after:wN \@@_parse_letters:N
      \exp:w
    \else:
      \@@_parse_return_semicolon:w #1
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]
%   {\@@_parse_prefix:NNN, \@@_parse_prefix_unknown:NNN}
%   For this function, |#1|~is the previous \meta{precedence}, |#2|~is
%   the operator just seen, and |#3|~is a control sequence which
%   implements the operator if it is a known operator.  If this control
%   sequence is \cs{scan_stop:}, then the operator is in fact unknown.
%   Either the expression is missing a number there (if the operator is
%   valid as an infix operator), and we put \texttt{nan}, wrapping the
%   infix operator in a csname as appropriate, or the character is
%   simply invalid in floating point expressions, and we continue
%   looking for a number, starting again from \cs{@@_parse_one:Nw}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_prefix:NNN #1#2#3
  {
    \if_meaning:w \scan_stop: #3
      \exp_after:wN \@@_parse_prefix_unknown:NNN
      \exp_after:wN #2
    \fi:
    #3 #1
  }
\cs_new:Npn \@@_parse_prefix_unknown:NNN #1#2#3
  {
    \cs_if_exist:cTF { @@_parse_infix_ \token_to_str:N #1 :N }
      {
        \__kernel_msg_expandable_error:nnn
          { kernel } { fp-missing-number } {#1}
        \exp_after:wN \c_nan_fp \exp:w \exp_end_continue_f:w
        \@@_parse_infix:NN #3 #1
      }
      {
        \__kernel_msg_expandable_error:nnn
          { kernel } { fp-unknown-symbol } {#1}
        \@@_parse_one:Nw #3
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Numbers: trimming leading zeros}
%
% Numbers are parsed as follows: first we trim leading zeros, then
% if the next character is a digit, start reading a significand $\geq 1$
% with the set of functions |\__fp_parse_large|\ldots{}; if it is a
% period, the significand is~$<1$; and otherwise it is zero.  In the
% second case, trim additional zeros after the period, counting them for
% an exponent shift $\meta{exp_1}<0$, then read the significand with the
% set of functions |\__fp_parse_small|\ldots{} Once the significand is
% read, read the exponent if |e|~is present.
%
% \begin{macro}[rEXP]{\@@_parse_trim_zeros:N, \@@_parse_trim_end:w}
%   This function expects an already expanded token.  It removes any
%   leading zero, then distinguishes three cases: if the first non-zero
%   token is a digit, then call \cs{@@_parse_large:N} (the significand
%   is $\geq 1$); if it is |.|, then continue trimming zeros with
%   \cs{@@_parse_strim_zeros:N}; otherwise, our number is exactly zero,
%   and we call \cs{@@_parse_zero:} to take care of that case.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_trim_zeros:N #1
  {
    \if:w 0 \exp_not:N #1
      \exp_after:wN \@@_parse_trim_zeros:N
      \exp:w
    \else:
      \if:w . \exp_not:N #1
        \exp_after:wN \@@_parse_strim_zeros:N
        \exp:w
      \else:
        \@@_parse_trim_end:w #1
      \fi:
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_trim_end:w #1 \fi: \fi: \@@_parse_expand:w
  {
      \fi:
    \fi:
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      \exp_after:wN \@@_parse_large:N
    \else:
      \exp_after:wN \@@_parse_zero:
    \fi:
    #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]
%   {\@@_parse_strim_zeros:N, \@@_parse_strim_end:w}
%   If we have removed all digits until a period (or if the body started
%   with a period), then enter the \enquote{\texttt{small_trim}} loop
%   which outputs $-1$ for each removed~$0$.  Those $-1$ are added to an
%   integer expression waiting for the exponent.  If the first non-zero
%   token is a digit, call \cs{@@_parse_small:N} (our significand is
%   smaller than~$1$), and otherwise, the number is an exact zero.  The
%   name \texttt{strim} stands for \enquote{small trim}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_strim_zeros:N #1
  {
    \if:w 0 \exp_not:N #1
      - 1
      \exp_after:wN \@@_parse_strim_zeros:N \exp:w
    \else:
      \@@_parse_strim_end:w #1
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_strim_end:w #1 \fi: \@@_parse_expand:w
  {
    \fi:
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      \exp_after:wN \@@_parse_small:N
    \else:
      \exp_after:wN \@@_parse_zero:
    \fi:
    #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_zero:}
%   After reading a significand of~$0$, find any exponent, then put a
%   sign of~|1| for \cs{@@_sanitize:wN}, which removes everything
%   and leaves an exact zero.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_zero:
  {
    \exp_after:wN ; \exp_after:wN 1
    \int_value:w \@@_parse_exponent:N
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Number: small significand}
%
% \begin{macro}[rEXP]{\@@_parse_small:N}
%   This function is called after we have passed the decimal separator
%   and removed all leading zeros from the significand.  It is followed
%   by a non-zero digit (with any catcode).  The goal is to read up to
%   $16$ digits.  But we can't do that all at once, because
%   \cs{int_value:w} (which allows us to collect digits and continue
%   expanding) can only go up to $9$ digits.  Hence we grab digits in
%   two steps of $8$ digits.  Since |#1| is a digit, read seven more
%   digits using \cs{@@_parse_digits_vii:N}.  The \texttt{small_leading}
%   auxiliary leaves those digits in the \cs{int_value:w}, and
%   grabs some more, or stops if there are no more digits.  Then the
%   \texttt{pack_leading} auxiliary puts the various parts in the
%   appropriate order for the processing further up.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_small:N #1
  {
    \exp_after:wN \@@_parse_pack_leading:NNNNNww
    \int_value:w \@@_int_eval:w 1 \token_to_str:N #1
      \exp_after:wN \@@_parse_small_leading:wwNN
      \int_value:w 1
        \exp_after:wN \@@_parse_digits_vii:N
        \exp:w \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_small_leading:wwNN}
%   \begin{syntax}
%     \cs{@@_parse_small_leading:wwNN} |1| \meta{digits} |;| \meta{zeros} |;| \meta{number~of~zeros}
%   \end{syntax}
%   We leave \meta{digits} \meta{zeros} in the input stream: the
%   functions used to grab digits are such that this constitutes digits
%   $1$ through~$8$ of the significand.  Then prepare to pack $8$~more
%   digits, with an exponent shift of zero (this shift is used in
%   the case of a large significand).  If |#4|~is a digit, leave it
%   behind for the packing function, and read $6$~more digits to reach a
%   total of $15$~digits: further digits are involved in the rounding.
%   Otherwise put $8$~zeros in to complete the significand, then look
%   for an exponent.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_small_leading:wwNN 1 #1 ; #2; #3 #4
  {
    #1 #2
    \exp_after:wN \@@_parse_pack_trailing:NNNNNNww
    \exp_after:wN 0
    \int_value:w \@@_int_eval:w 1
      \if_int_compare:w 9 < 1 \token_to_str:N #4 \exp_stop_f:
        \token_to_str:N #4
        \exp_after:wN \@@_parse_small_trailing:wwNN
        \int_value:w 1
          \exp_after:wN \@@_parse_digits_vi:N
          \exp:w
      \else:
        0000 0000 \@@_parse_exponent:Nw #4
      \fi:
      \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_small_trailing:wwNN}
%   \begin{syntax}
%     \cs{@@_parse_small_trailing:wwNN} |1| \meta{digits} |;| \meta{zeros} |;| \meta{number~of~zeros} \meta{next~token}
%   \end{syntax}
%   Leave digits $10$ to~$15$ (arguments |#1| and |#2|) in the input
%   stream.  If the \meta{next~token} is a digit, it is the $16$th
%   digit, we keep it, then the \texttt{small_round} auxiliary considers
%   this digit and all further digits to perform the rounding: the
%   function expands to nothing, to |+0| or to |+1|.
%   Otherwise, there is no $16$-th digit, so we put a~$0$, and look for
%   an exponent.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_small_trailing:wwNN 1 #1 ; #2; #3 #4
  {
    #1 #2
    \if_int_compare:w 9 < 1 \token_to_str:N #4 \exp_stop_f:
      \token_to_str:N #4
      \exp_after:wN \@@_parse_small_round:NN
      \exp_after:wN #4
      \exp:w
    \else:
      0 \@@_parse_exponent:Nw #4
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]
%   {
%     \@@_parse_pack_trailing:NNNNNNww ,
%     \@@_parse_pack_leading:NNNNNww   ,
%     \@@_parse_pack_carry:w
%   }
%   Those functions are expanded after all the digits are found, we took
%   care of the rounding, as well as the exponent.  The last argument is
%   the exponent.  The previous five arguments are $8$~digits which we
%   pack in groups of~$4$, and the argument before that is~$1$, except
%   in the rare case where rounding lead to a carry, in which case the
%   argument is~$2$.  The \texttt{trailing} function has an exponent
%   shift as its first argument, which we add to the exponent found in
%   the |e...| syntax.  If the trailing digits cause a carry, the
%   integer expression for the leading digits is incremented (|+1|
%   in the code below).  If the leading digits propagate this carry all
%   the way up, the function \cs{@@_parse_pack_carry:w} increments the
%   exponent, and changes the significand from |0000...| to |1000...|:
%   this is simple because such a carry can only occur to give rise to a
%   power of~$10$.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_pack_trailing:NNNNNNww #1 #2 #3#4#5#6 #7; #8 ;
  {
    \if_meaning:w 2 #2 + 1 \fi:
    ; #8 + #1 ; {#3#4#5#6} {#7};
  }
\cs_new:Npn \@@_parse_pack_leading:NNNNNww #1 #2#3#4#5 #6; #7;
  {
    + #7
    \if_meaning:w 2 #1 \@@_parse_pack_carry:w \fi:
    ; 0 {#2#3#4#5} {#6}
  }
\cs_new:Npn \@@_parse_pack_carry:w \fi: ; 0 #1
  { \fi: + 1 ; 0 {1000} }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Number: large significand}
%
% Parsing a significand larger than~$1$ is a little bit more difficult
% than parsing small significands.  We need to count the number of
% digits before the decimal separator, and add that to the final
% exponent.  We also need to test for the presence of a dot each time we
% run out of digits, and branch to the appropriate \texttt{parse_small}
% function in those cases.
%
% \begin{macro}[EXP]{\@@_parse_large:N}
%   This function is followed by the first non-zero digit of a
%   \enquote{large} significand ($\geq 1$).  It is called within an
%   integer expression for the exponent.  Grab up to $7$~more digits,
%   for a total of $8$~digits.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_large:N #1
  {
    \exp_after:wN \@@_parse_large_leading:wwNN
    \int_value:w 1 \token_to_str:N #1
      \exp_after:wN \@@_parse_digits_vii:N
      \exp:w \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_large_leading:wwNN}
%   \begin{syntax}
%     \cs{@@_parse_large_leading:wwNN} |1| \meta{digits} |;| \meta{zeros} |;| \meta{number~of~zeros} \meta{next~token}
%   \end{syntax}
%   We shift the exponent by the number of digits in~|#1|, namely the
%   target number, $8$, minus the \meta{number of zeros} (number of
%   digits missing).  Then prepare to pack the $8$~first digits.  If the
%   \meta{next token} is a digit, read up to $6$~more digits (digits
%   $10$ to~$15$).  If it is a period, try to grab the end of our
%   $8$~first digits, branching to the \texttt{small} functions since
%   the number of digit does not affect the exponent anymore.  Finally,
%   if this is the end of the significand, insert the \meta{zeros} to
%   complete the $8$~first digits, insert $8$~more, and look for an
%   exponent.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_large_leading:wwNN 1 #1 ; #2; #3 #4
  {
    + \c_@@_half_prec_int - #3
    \exp_after:wN \@@_parse_pack_leading:NNNNNww
    \int_value:w \@@_int_eval:w 1 #1
      \if_int_compare:w 9 < 1 \token_to_str:N #4 \exp_stop_f:
        \exp_after:wN \@@_parse_large_trailing:wwNN
        \int_value:w 1 \token_to_str:N #4
          \exp_after:wN \@@_parse_digits_vi:N
          \exp:w
      \else:
        \if:w . \exp_not:N #4
          \exp_after:wN \@@_parse_small_leading:wwNN
          \int_value:w 1
            \cs:w
              @@_parse_digits_
              \@@_int_to_roman:w #3
              :N \exp_after:wN
            \cs_end:
            \exp:w
        \else:
          #2
          \exp_after:wN \@@_parse_pack_trailing:NNNNNNww
          \exp_after:wN 0
          \int_value:w 1 0000 0000
          \@@_parse_exponent:Nw #4
        \fi:
      \fi:
      \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_large_trailing:wwNN}
%   \begin{syntax}
%     \cs{@@_parse_large_trailing:wwNN} |1| \meta{digits} |;| \meta{zeros} |;| \meta{number~of~zeros} \meta{next~token}
%   \end{syntax}
%   We have just read $15$~digits.  If the \meta{next token} is a digit,
%   then the exponent shift caused by this block of $8$~digits is~$8$,
%   first argument to the \texttt{pack_trailing} function.  We keep the
%   \meta{digits} and this $16$-th digit, and find how this should be
%   rounded using \cs{@@_parse_large_round:NN}.  Otherwise, the exponent
%   shift is the number of \meta{digits}, $7$~minus the \meta{number of
%     zeros}, and we test for a decimal point.  This case happens in
%   |123451234512345.67| with exactly $15$ digits before the decimal
%   separator.  Then branch to the appropriate \texttt{small} auxiliary,
%   grabbing a few more digits to complement the digits we already
%   grabbed.  Finally, if this is truly the end of the significand, look
%   for an exponent after using the \meta{zeros} and providing a $16$-th
%   digit of~$0$.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_large_trailing:wwNN 1 #1 ; #2; #3 #4
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #4 \exp_stop_f:
      \exp_after:wN \@@_parse_pack_trailing:NNNNNNww
      \exp_after:wN \c_@@_half_prec_int
      \int_value:w \@@_int_eval:w 1 #1 \token_to_str:N #4
        \exp_after:wN \@@_parse_large_round:NN
        \exp_after:wN #4
        \exp:w
    \else:
      \exp_after:wN \@@_parse_pack_trailing:NNNNNNww
      \int_value:w \@@_int_eval:w 7 - #3 \exp_stop_f:
      \int_value:w \@@_int_eval:w 1 #1
        \if:w . \exp_not:N #4
          \exp_after:wN \@@_parse_small_trailing:wwNN
          \int_value:w 1
            \cs:w
              @@_parse_digits_
              \@@_int_to_roman:w #3
              :N \exp_after:wN
            \cs_end:
            \exp:w
        \else:
          #2 0 \@@_parse_exponent:Nw #4
        \fi:
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Number: beyond 16 digits, rounding}
%
% \begin{macro}[rEXP]{\@@_parse_round_loop:N, \@@_parse_round_up:N}
%   This loop is called when rounding a number (whether the mantissa is
%   small or large).  It should appear in an integer expression.  This
%   function reads digits one by one, until reaching a non-digit, and
%   adds~$1$ to the integer expression for each digit.  If all digits
%   found are~$0$, the function ends the expression by |;0|,
%   otherwise by |;1|.  This is done by switching the loop to
%   |round_up| at the first non-zero digit, thus we avoid to test
%   whether digits are~$0$ or not once we see a first non-zero digit.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_round_loop:N #1
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      + 1
      \if:w 0 \token_to_str:N #1
        \exp_after:wN \@@_parse_round_loop:N
        \exp:w
      \else:
        \exp_after:wN \@@_parse_round_up:N
        \exp:w
      \fi:
    \else:
      \@@_parse_return_semicolon:w 0 #1
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_round_up:N #1
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      + 1
      \exp_after:wN \@@_parse_round_up:N
      \exp:w
    \else:
      \@@_parse_return_semicolon:w 1 #1
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_round_after:wN}
%   After the loop \cs{@@_parse_round_loop:N}, this function fetches an
%   exponent with \cs{@@_parse_exponent:N}, and combines it with the
%   number of digits counted by \cs{@@_parse_round_loop:N}.  At the same
%   time, the result |0| or |1| is added to the
%   surrounding integer expression.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_round_after:wN #1; #2
  {
    + #2 \exp_after:wN ;
    \int_value:w \@@_int_eval:w #1 + \@@_parse_exponent:N
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]
%   {\@@_parse_small_round:NN, \@@_parse_round_after:wN}
%   Here, |#1|~is the digit that we are currently rounding (we only care
%   whether it is even or odd).  If |#2|~is not a digit, then fetch an
%   exponent and expand to |;|\meta{exponent} only.  Otherwise, we
%   expand to |+0| or |+1|, then |;|\meta{exponent}.  To
%   decide which, call \cs{@@_round_s:NNNw} to know whether to round up,
%   giving it as arguments a sign~$0$ (all explicit numbers are
%   positive), the digit |#1|~to round, the first following digit~|#2|,
%   and either |+0| or |+1| depending on whether the
%   following digits are all zero or not.  This last argument is
%   obtained by \cs{@@_parse_round_loop:N}, whose number of digits we
%   discard by multiplying it by~$0$.  The exponent which follows the
%   number is also fetched by \cs{@@_parse_round_after:wN}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_small_round:NN #1#2
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #2 \exp_stop_f:
      +
      \exp_after:wN \@@_round_s:NNNw
      \exp_after:wN 0
      \exp_after:wN #1
      \exp_after:wN #2
      \int_value:w \@@_int_eval:w
        \exp_after:wN \@@_parse_round_after:wN
        \int_value:w \@@_int_eval:w 0 * \@@_int_eval:w 0
          \exp_after:wN \@@_parse_round_loop:N
          \exp:w
    \else:
      \@@_parse_exponent:Nw #2
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
%
% \begin{macro}[rEXP]
%   {
%     \@@_parse_large_round:NN,
%     \@@_parse_large_round_test:NN,
%     \@@_parse_large_round_aux:wNN,
%   }
%   Large numbers are harder to round, as there may be a period in the
%   way.  Again, |#1|~is the digit that we are currently rounding (we
%   only care whether it is even or odd).  If there are no more digits
%   (|#2|~is not a digit), then we must test for a period: if there is
%   one, then switch to the rounding function for small significands,
%   otherwise fetch an exponent.  If there are more digits (|#2|~is a
%   digit), then round, checking with \cs{@@_parse_round_loop:N} if all
%   further digits vanish, or some are non-zero.  This loop is not
%   enough, as it is stopped by a period.  After the loop, the
%   \texttt{aux} function tests for a period: if it is present, then we
%   must continue looking for digits, this time discarding the number of
%   digits we find.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_large_round:NN #1#2
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #2 \exp_stop_f:
      +
      \exp_after:wN \@@_round_s:NNNw
      \exp_after:wN 0
      \exp_after:wN #1
      \exp_after:wN #2
      \int_value:w \@@_int_eval:w
        \exp_after:wN \@@_parse_large_round_aux:wNN
        \int_value:w \@@_int_eval:w 1
          \exp_after:wN \@@_parse_round_loop:N
    \else: %^^A could be dot, or e, or other
      \exp_after:wN \@@_parse_large_round_test:NN
      \exp_after:wN #1
      \exp_after:wN #2
    \fi:
  }
\cs_new:Npn \@@_parse_large_round_test:NN #1#2
  {
    \if:w . \exp_not:N #2
      \exp_after:wN \@@_parse_small_round:NN
      \exp_after:wN #1
      \exp:w
    \else:
      \@@_parse_exponent:Nw #2
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_large_round_aux:wNN #1 ; #2 #3
  {
    + #2
    \exp_after:wN \@@_parse_round_after:wN
    \int_value:w \@@_int_eval:w #1
      \if:w . \exp_not:N #3
        + 0 * \@@_int_eval:w 0
          \exp_after:wN \@@_parse_round_loop:N
          \exp:w \exp_after:wN \@@_parse_expand:w
      \else:
        \exp_after:wN ;
        \exp_after:wN 0
        \exp_after:wN #3
      \fi:
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Number: finding the exponent}
%
% Expansion is a little bit tricky here, in part because we accept input
% where multiplication is implicit.
% \begin{syntax}
%   \cs{@@_parse:n} |{ 3.2 erf(0.1) }|
%   \cs{@@_parse:n} |{ 3.2 e\l_my_int }|
%   \cs{@@_parse:n} |{ 3.2 \c_pi_fp }|
% \end{syntax}
% The first case indicates that just looking one character ahead for an
% \enquote{\texttt{e}} is not enough, since we would mistake the
% function \texttt{erf} for an exponent of \enquote{\texttt{rf}}.  An
% alternative would be to look two tokens ahead and check if what
% follows is a sign or a digit, considering in that case that we must be
% finding an exponent.  But taking care of the second case requires that
% we unpack registers after \texttt{e}.  However, blindly expanding the
% two tokens ahead completely would break the third example (unpacking
% is even worse).  Indeed, in the course of reading $3.2$, \cs{c_pi_fp}
% is expanded to \cs{s_@@} \cs{@@_chk:w} |1| |0| |{-1}| |{3141}|
% $\cdots$ |;| and \cs{s_@@} stops the expansion.  Expanding two tokens
% ahead would then force the expansion of \cs{@@_chk:w} (despite it
% being protected), and that function tries to produce an error.
%
% What can we do?  Really, the reason why this last case breaks is that
% just as \TeX{} does, we should read ahead as little as possible.
% Here, the only case where there may be an exponent is if the first
% token ahead is |e|.  Then we expand (and possibly unpack) the second
% token.
%
% \begin{macro}[rEXP]{\@@_parse_exponent:Nw}
%   This auxiliary is convenient to smuggle some material through
%   \cs{fi:} ending conditional processing.  We place those \cs{fi:}
%   (argument~|#2|) at a very odd place because this allows us to insert
%   \cs{@@_int_eval:w} \ldots{} there if needed.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_exponent:Nw #1 #2 \@@_parse_expand:w
  {
    \exp_after:wN ;
    \int_value:w #2 \@@_parse_exponent:N #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]
%   {\@@_parse_exponent:N, \@@_parse_exponent_aux:N}
%   This function should be called within an \cs{int_value:w}
%   expansion (or within an integer expression).  It leaves digits of the
%   exponent behind it in the input stream, and terminates the expansion
%   with a semicolon.  If there is no~|e| (or~|E|), leave an exponent of~$0$.  If
%   there is an~|e| or~|E|, expand the next token to run some tests on it.  The
%   first rough test is that if the character code of~|#1| is greater
%   than that of~|9| (largest code valid for an exponent, less than any
%   code valid for an identifier), there was in fact no exponent;
%   otherwise, we search for the sign of the exponent.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_exponent:N #1
  {
    \if:w e \if:w E \exp_not:N #1 e \else: \exp_not:N #1 \fi:
      \exp_after:wN \@@_parse_exponent_aux:N
      \exp:w
    \else:
      0 \@@_parse_return_semicolon:w #1
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_exponent_aux:N #1
  {
    \if_int_compare:w \if_catcode:w \scan_stop: \exp_not:N #1
                0 \else: `#1 \fi: > `9 \exp_stop_f:
      0 \exp_after:wN ; \exp_after:wN e
    \else:
      \exp_after:wN \@@_parse_exponent_sign:N
    \fi:
    #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_exponent_sign:N}
%   Read signs one by one (if there is any).
%    \begin{macrocode}
\cs_new:Npn \@@_parse_exponent_sign:N #1
  {
    \if:w + \if:w - \exp_not:N #1 + \fi: \token_to_str:N #1
      \exp_after:wN \@@_parse_exponent_sign:N
      \exp:w \exp_after:wN \@@_parse_expand:w
    \else:
      \exp_after:wN \@@_parse_exponent_body:N
      \exp_after:wN #1
    \fi:
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_exponent_body:N}
%   An exponent can be an explicit integer (most common case), or
%   various other things (most of which are invalid).
%    \begin{macrocode}
\cs_new:Npn \@@_parse_exponent_body:N #1
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      \token_to_str:N #1
      \exp_after:wN \@@_parse_exponent_digits:N
      \exp:w
    \else:
      \@@_parse_exponent_keep:NTF #1
        { \@@_parse_return_semicolon:w #1 }
        {
          \exp_after:wN ;
          \exp:w
        }
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_exponent_digits:N}
%   Read digits one by one, and leave them behind in the input stream.
%   When finding a non-digit, stop, and insert a semicolon.  Note that
%   we do not check for overflow of the exponent, hence there can be a
%   \TeX{} error.  It is mostly harmless, except when parsing
%   |0e9876543210|, which should be a valid representation of $0$, but
%   is not.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_exponent_digits:N #1
  {
    \if_int_compare:w 9 < 1 \token_to_str:N #1 \exp_stop_f:
      \token_to_str:N #1
      \exp_after:wN \@@_parse_exponent_digits:N
      \exp:w
    \else:
      \@@_parse_return_semicolon:w #1
    \fi:
    \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[rEXP]{\@@_parse_exponent_keep:NTF}
%   This is the last building block for parsing exponents.  The
%   argument~|#1| is already fully expanded, and neither |+| nor~|-| nor
%   a digit.  It can be:
%   \begin{itemize}
%   \item \cs{s_@@}, marking the start of an internal floating point,
%     invalid here;
%   \item another control sequence equal to \tn{relax}, probably a bad
%     variable;
%   \item a register: in this case we make sure that it is an integer
%     register, not a dimension;
%   \item a character other than |+|, |-| or digits, again, an error.
%   \end{itemize}
%    \begin{macrocode}
\prg_new_conditional:Npnn \@@_parse_exponent_keep:N #1 { TF }
  {
    \if_catcode:w \scan_stop: \exp_not:N #1
      \if_meaning:w \scan_stop: #1
        \if_int_compare:w
            \@@_str_if_eq:nn { \s_@@ } { \exp_not:N #1 }
            = 0 \exp_stop_f:
          0
          \__kernel_msg_expandable_error:nnn
            { kernel } { fp-after-e } { floating~point~ }
          \prg_return_true:
        \else:
          0
          \__kernel_msg_expandable_error:nnn
            { kernel } { bad-variable } {#1}
          \prg_return_false:
        \fi:
      \else:
        \if_int_compare:w
            \@@_str_if_eq:nn { \int_value:w #1 } { \tex_the:D #1 }
            = 0 \exp_stop_f:
          \int_value:w #1
        \else:
          0
          \__kernel_msg_expandable_error:nnn
            { kernel } { fp-after-e } { dimension~#1 }
        \fi:
        \prg_return_false:
      \fi:
    \else:
      0
      \__kernel_msg_expandable_error:nnn
        { kernel } { fp-missing } { exponent }
      \prg_return_true:
    \fi:
  }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Constants, functions and prefix operators}
%
% \subsubsection{Prefix operators}
%
% \begin{macro}[EXP]{\@@_parse_prefix_+:Nw}
%   A unary~|+| does nothing: we should continue looking for a number.
%    \begin{macrocode}
\cs_new_eq:cN { @@_parse_prefix_+:Nw } \@@_parse_one:Nw
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_apply_function:NNNwN}
%   Here, |#1| is a precedence, |#2| is some extra data used by some
%   functions, |#3| is \emph{e.g.}, \cs{@@_sin_o:w}, and expands once
%   after the calculation, |#4| is the operand, and |#5| is a
%   \cs[no-index]{@@_parse_infix_\ldots{}:N} function.  We feed the data~|#2|, and the
%   argument~|#4|, to the function~|#3|, which expands
%   \cs{exp:w} thus the \texttt{infix} function~|#5|.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_apply_function:NNNwN #1#2#3#4@#5
  {
    #3 #2 #4 @
    \exp:w \exp_end_continue_f:w #5 #1
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_apply_unary:NNNwN}
% \begin{macro}[EXP]{\@@_parse_apply_unary_chk:NwNw, \@@_parse_apply_unary_chk:nNNNw}
% \begin{macro}[EXP]{\@@_parse_apply_unary_type:NNN, \@@_parse_apply_unary_error:NNw}
%   In contrast to \cs{@@_parse_apply_function:NNNwN}, this checks that
%   the operand |#4| is a single argument (namely there is a single
%   |;|).  We use the fact that any floating point starts with a
%   \enquote{safe} token like \cs{s_@@}.  If there is no argument
%   produce the |fp-no-arg| error; if there are at least two produce
%   |fp-multi-arg|.  For the error message extract the mathematical
%   function name (such as |sin|) from the \pkg{expl3} function that
%   computes it, such as \cs{@@_sin_o:w}.
%
%   In addition, since there is a single argument we can dispatch on
%   type and check that the resulting function exists.  This catches
%   things like |sin((1,2))| where it does not make sense to take the
%   sine of a tuple.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_apply_unary:NNNwN #1#2#3#4@#5
  {
    \@@_parse_apply_unary_chk:NwNw #4 @ ; . \s_@@_stop
    \@@_parse_apply_unary_type:NNN
    #3 #2 #4 @
    \exp:w \exp_end_continue_f:w #5 #1
  }
\cs_new:Npn \@@_parse_apply_unary_chk:NwNw #1#2 ; #3#4 \s_@@_stop
  {
    \if_meaning:w @ #3 \else:
      \token_if_eq_meaning:NNTF . #3
        { \@@_parse_apply_unary_chk:nNNNNw { no } }
        { \@@_parse_apply_unary_chk:nNNNNw { multi } }
    \fi:
  }
\cs_new:Npn \@@_parse_apply_unary_chk:nNNNNw #1#2#3#4#5#6 @
  {
    #2
    \@@_error:nffn { fp-#1-arg } { \@@_func_to_name:N #4 } { } { }
    \exp_after:wN #4 \exp_after:wN #5 \c_nan_fp @
  }
\cs_new:Npn \@@_parse_apply_unary_type:NNN #1#2#3
  {
    \@@_change_func_type:NNN #3 #1 \@@_parse_apply_unary_error:NNw
    #2 #3
  }
\cs_new:Npn \@@_parse_apply_unary_error:NNw #1#2#3 @
  { \@@_invalid_operation_o:fw { \@@_func_to_name:N #1 } #3 }
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_prefix_-:Nw, \@@_parse_prefix_!:Nw}
%   The unary~|-| and boolean not are harder: we parse the operand using
%   a precedence equal to the maximum of the previous precedence~|##1|
%   and the precedence \cs{c_@@_prec_not_int} of the unary operator, then call
%   the appropriate |\__fp_|\meta{operation}|_o:w| function,
%   where the \meta{operation} is |set_sign| or |not|.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1#2#3#4
  {
    \cs_new:cpn { @@_parse_prefix_ #1 :Nw } ##1
      {
        \exp_after:wN \@@_parse_apply_unary:NNNwN
        \exp_after:wN ##1
        \exp_after:wN #4
        \exp_after:wN #3
        \exp:w
        \if_int_compare:w #2 < ##1
          \@@_parse_operand:Nw ##1
        \else:
          \@@_parse_operand:Nw #2
        \fi:
        \@@_parse_expand:w
      }
  }
\@@_tmp:w - \c_@@_prec_not_int \@@_set_sign_o:w 2
\@@_tmp:w ! \c_@@_prec_not_int \@@_not_o:w ?
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_prefix_.:Nw}
%   Numbers which start with a decimal separator (a~period) end up here.
%   Of course, we do not look for an operand, but for the rest of the
%   number.  This function is very similar to \cs{@@_parse_one_digit:NN}
%   but calls \cs{@@_parse_strim_zeros:N} to trim zeros after the
%   decimal point, rather than the \texttt{trim_zeros} function for
%   zeros before the decimal point.
%    \begin{macrocode}
\cs_new:cpn { @@_parse_prefix_.:Nw } #1
  {
    \exp_after:wN \@@_parse_infix_after_operand:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
      \exp_after:wN \@@_sanitize:wN
      \int_value:w \@@_int_eval:w 0 \@@_parse_strim_zeros:N
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {\@@_parse_prefix_(:Nw, \@@_parse_lparen_after:NwN}
%   The left parenthesis is treated as a unary prefix operator because
%   it appears in exactly the same settings.  If the previous precedence
%   is \cs{c_@@_prec_func_int} we are parsing arguments of a function
%   and commas should not build tuples; otherwise commas should build
%   tuples.  We distinguish these cases by precedence:
%   \cs{c_@@_prec_comma_int} for the case of arguments,
%   \cs{c_@@_prec_tuple_int} for the case of tuples.
%   Once the operand is found, the \texttt{lparen_after} auxiliary makes
%   sure that there was a closing parenthesis (otherwise it complains),
%   and leaves in the input stream an operand,
%   fetching the following infix operator.
%    \begin{macrocode}
\cs_new:cpn { @@_parse_prefix_(:Nw } #1
  {
    \exp_after:wN \@@_parse_lparen_after:NwN
    \exp_after:wN #1
    \exp:w
    \if_int_compare:w #1 = \c_@@_prec_func_int
      \@@_parse_operand:Nw \c_@@_prec_comma_int
    \else:
      \@@_parse_operand:Nw \c_@@_prec_tuple_int
    \fi:
    \@@_parse_expand:w
  }
\cs_new:Npx \@@_parse_lparen_after:NwN #1#2 @ #3
  {
    \exp_not:N \token_if_eq_meaning:NNTF #3
      \exp_not:c { @@_parse_infix_):N }
      {
        \exp_not:N \@@_exp_after_array_f:w #2 \s_@@_expr_stop
        \exp_not:N \exp_after:wN
        \exp_not:N \@@_parse_infix_after_paren:NN
        \exp_not:N \exp_after:wN #1
        \exp_not:N \exp:w
        \exp_not:N \@@_parse_expand:w
      }
      {
        \exp_not:N \__kernel_msg_expandable_error:nnn
          { kernel } { fp-missing } { ) }
        \exp_not:N \tl_if_empty:nT {#2} \exp_not:N \c_@@_empty_tuple_fp
        #2 @
        \exp_not:N \use_none:n #3
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_prefix_):Nw}
%   The right parenthesis can appear as a prefix in two similar cases:
%   in an empty tuple or tuple ending with a comma, or in an empty
%   argument list or argument list ending with a comma, such as in
%   |max(1,2,)| or in |rand()|.
%    \begin{macrocode}
\cs_new:cpn { @@_parse_prefix_):Nw } #1
  {
    \if_int_compare:w #1 = \c_@@_prec_comma_int
    \else:
      \if_int_compare:w #1 = \c_@@_prec_tuple_int
        \exp_after:wN \c_@@_empty_tuple_fp \exp:w
      \else:
        \__kernel_msg_expandable_error:nnn
          { kernel } { fp-missing-number } { ) }
        \exp_after:wN \c_nan_fp \exp:w
      \fi:
      \exp_end_continue_f:w
    \fi:
    \@@_parse_infix_after_paren:NN #1 )
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Constants}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_word_inf:N  , \@@_parse_word_nan:N   ,
%     \@@_parse_word_pi:N   , \@@_parse_word_deg:N   ,
%     \@@_parse_word_true:N , \@@_parse_word_false:N ,
%   }
%   Some words correspond to constant floating points.  The floating
%   point constant is left as a result of \cs{@@_parse_one:Nw} after
%   expanding \cs{@@_parse_infix:NN}.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1 #2
  {
    \cs_new:cpn { @@_parse_word_#1:N }
      { \exp_after:wN #2 \exp:w \exp_end_continue_f:w \@@_parse_infix:NN }
  }
\@@_tmp:w { inf } \c_inf_fp
\@@_tmp:w { nan } \c_nan_fp
\@@_tmp:w { pi  } \c_pi_fp
\@@_tmp:w { deg } \c_one_degree_fp
\@@_tmp:w { true } \c_one_fp
\@@_tmp:w { false } \c_zero_fp
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_caseless_inf:N,
%     \@@_parse_caseless_infinity:N,
%     \@@_parse_caseless_nan:N
%   }
%   Copies of \cs[no-index]{@@_parse_word_\ldots{}:N} commands, to allow
%   arbitrary case as mandated by the standard.
%    \begin{macrocode}
\cs_new_eq:NN \@@_parse_caseless_inf:N \@@_parse_word_inf:N
\cs_new_eq:NN \@@_parse_caseless_infinity:N \@@_parse_word_inf:N
\cs_new_eq:NN \@@_parse_caseless_nan:N \@@_parse_word_nan:N
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_word_pt:N , \@@_parse_word_in:N ,
%     \@@_parse_word_pc:N , \@@_parse_word_cm:N , \@@_parse_word_mm:N ,
%     \@@_parse_word_dd:N , \@@_parse_word_cc:N , \@@_parse_word_nd:N ,
%     \@@_parse_word_nc:N , \@@_parse_word_bp:N , \@@_parse_word_sp:N ,
%   }
%   Dimension units are also floating point constants but their value is
%   not stored as a floating point constant.  We give the values
%   explicitly here.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1 #2
  {
    \cs_new:cpn { @@_parse_word_#1:N }
      {
        \@@_exp_after_f:nw { \@@_parse_infix:NN }
        \s_@@ \@@_chk:w 10 #2 ;
      }
  }
\@@_tmp:w {pt} { {1} {1000} {0000} {0000} {0000} }
\@@_tmp:w {in} { {2} {7227} {0000} {0000} {0000} }
\@@_tmp:w {pc} { {2} {1200} {0000} {0000} {0000} }
\@@_tmp:w {cm} { {2} {2845} {2755} {9055} {1181} }
\@@_tmp:w {mm} { {1} {2845} {2755} {9055} {1181} }
\@@_tmp:w {dd} { {1} {1070} {0085} {6496} {0630} }
\@@_tmp:w {cc} { {2} {1284} {0102} {7795} {2756} }
\@@_tmp:w {nd} { {1} {1066} {9783} {4645} {6693} }
\@@_tmp:w {nc} { {2} {1280} {3740} {1574} {8031} }
\@@_tmp:w {bp} { {1} {1003} {7500} {0000} {0000} }
\@@_tmp:w {sp} { {-4} {1525} {8789} {0625} {0000} }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_word_em:N, \@@_parse_word_ex:N}
%   The font-dependent units |em| and |ex| must be evaluated on the fly.
%   We reuse an auxiliary of \cs{dim_to_fp:n}.
%    \begin{macrocode}
\tl_map_inline:nn { {em} {ex} }
  {
    \cs_new:cpn { @@_parse_word_#1:N }
      {
        \exp_after:wN \@@_from_dim_test:ww
        \exp_after:wN 0 \exp_after:wN ,
        \int_value:w \dim_to_decimal_in_sp:n { 1 #1 } \exp_after:wN ;
        \exp:w \exp_end_continue_f:w \@@_parse_infix:NN
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Functions}
%
% ^^A begin[todo]
%
% \begin{macro}[EXP]
%   {\@@_parse_unary_function:NNN, \@@_parse_function:NNN}
%    \begin{macrocode}
\cs_new:Npn \@@_parse_unary_function:NNN #1#2#3
  {
    \exp_after:wN \@@_parse_apply_unary:NNNwN
    \exp_after:wN #3
    \exp_after:wN #2
    \exp_after:wN #1
    \exp:w
    \@@_parse_operand:Nw \c_@@_prec_func_int \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_function:NNN #1#2#3
  {
    \exp_after:wN \@@_parse_apply_function:NNNwN
    \exp_after:wN #3
    \exp_after:wN #2
    \exp_after:wN #1
    \exp:w
    \@@_parse_operand:Nw \c_@@_prec_func_int \@@_parse_expand:w
  }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Main functions}
%
% \begin{macro}[EXP]{\@@_parse:n, \@@_parse_o:n}
% \begin{macro}[EXP]{\@@_parse_after:ww}
%   Start an \cs{exp:w} expansion so that \cs{@@_parse:n} expands
%   in two steps.  The \cs{@@_parse_operand:Nw} function performs
%   computations until reaching an operation with precedence
%   \cs{c_@@_prec_end_int} or less, namely, the end of the expression.  The
%   marker \cs{s_@@_expr_mark} indicates that the next token is an already
%   parsed version of an infix operator, and \cs{@@_parse_infix_end:N}
%   has infinitely negative precedence.  Finally, clean up a
%   (well-defined) set of extra tokens and stop the initial expansion
%   with \cs{exp_end:}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse:n #1
  {
    \exp:w
      \exp_after:wN \@@_parse_after:ww
      \exp:w
        \@@_parse_operand:Nw \c_@@_prec_end_int
        \@@_parse_expand:w #1
        \s_@@_expr_mark \@@_parse_infix_end:N
      \s_@@_expr_stop
    \exp_end:
  }
\cs_new:Npn \@@_parse_after:ww
    #1@ \@@_parse_infix_end:N \s_@@_expr_stop #2 { #2 #1 }
\cs_new:Npn \@@_parse_o:n #1
  {
    \exp:w
      \exp_after:wN \@@_parse_after:ww
      \exp:w
        \@@_parse_operand:Nw \c_@@_prec_end_int
        \@@_parse_expand:w #1
        \s_@@_expr_mark \@@_parse_infix_end:N
      \s_@@_expr_stop
    {
      \exp_end_continue_f:w
      \@@_exp_after_any_f:nw { \exp_after:wN \exp_stop_f: }
    }
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_operand:Nw}
% \begin{macro}[EXP]{\@@_parse_continue:NwN}
%   This is just a shorthand which sets up both \cs{@@_parse_continue:NwN}
%   and \cs{@@_parse_one:Nw} with the same precedence. Note the
%   trailing \cs{exp:w}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_operand:Nw #1
  {
    \exp_end_continue_f:w
    \exp_after:wN \@@_parse_continue:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
    \exp_after:wN \@@_parse_one:Nw
    \exp_after:wN #1
    \exp:w
  }
\cs_new:Npn \@@_parse_continue:NwN #1 #2 @ #3 { #3 #1 #2 @ }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_apply_binary:NwNwN}
% \begin{macro}[EXP]
%   {\@@_parse_apply_binary_chk:NN, \@@_parse_apply_binary_error:NNN}
%   Receives \meta{precedence} \meta{operand_1} |@| \meta{operation}
%   \meta{operand_2} |@| \meta{infix command}.  Builds the appropriate
%   call to the \meta{operation}~|#3|, dispatching on both types.
%   If the resulting control sequence does not exist, the operation is
%   not allowed.
%
%   This is redefined in \pkg{l3fp-extras}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_apply_binary:NwNwN #1 #2#3@ #4 #5#6@ #7
  {
    \exp_after:wN \@@_parse_continue:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
      \exp_after:wN \@@_parse_apply_binary_chk:NN
        \cs:w
          @@
          \@@_type_from_scan:N #2
          _#4
          \@@_type_from_scan:N #5
          _o:ww
        \cs_end:
        #4
      #2#3 #5#6
    \exp:w \exp_end_continue_f:w #7 #1
  }
\cs_new:Npn \@@_parse_apply_binary_chk:NN #1#2
  {
    \if_meaning:w \scan_stop: #1
      \@@_parse_apply_binary_error:NNN #2
    \fi:
    #1
  }
\cs_new:Npn \@@_parse_apply_binary_error:NNN #1#2#3
  {
    #2
    \@@_invalid_operation_o:Nww #1
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_binary_type_o:Nww, \@@_binary_rev_type_o:Nww}
%   Applies the operator |#1| to its two arguments, dispatching
%   according to their types, and expands once after the result.
%   The |rev| version swaps its arguments before doing this.
%    \begin{macrocode}
\cs_new:Npn \@@_binary_type_o:Nww #1 #2#3 ; #4
  {
    \exp_after:wN \@@_parse_apply_binary_chk:NN
      \cs:w
        @@
        \@@_type_from_scan:N #2
        _ #1
        \@@_type_from_scan:N #4
        _o:ww
      \cs_end:
      #1
    #2 #3 ; #4
  }
\cs_new:Npn \@@_binary_rev_type_o:Nww #1 #2#3 ; #4#5 ;
  {
    \exp_after:wN \@@_parse_apply_binary_chk:NN
      \cs:w
        @@
        \@@_type_from_scan:N #4
        _ #1
        \@@_type_from_scan:N #2
        _o:ww
      \cs_end:
      #1
    #4 #5 ; #2 #3 ;
  }
%    \end{macrocode}
% \end{macro}
%
% \subsection{Infix operators}
%
% \begin{macro}[EXP]{\@@_parse_infix_after_operand:NwN}
%    \begin{macrocode}
\cs_new:Npn \@@_parse_infix_after_operand:NwN #1 #2;
  {
    \@@_exp_after_f:nw { \@@_parse_infix:NN #1 }
    #2;
  }
\cs_new:Npn \@@_parse_infix:NN #1 #2
  {
    \if_catcode:w \scan_stop: \exp_not:N #2
      \if_int_compare:w
          \@@_str_if_eq:nn { \s_@@_expr_mark } { \exp_not:N #2 }
          = 0 \exp_stop_f:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_mark:NNN
      \else:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_juxt:N
      \fi:
    \else:
      \if_int_compare:w
          \@@_int_eval:w
            ( `#2 \if_int_compare:w `#2 > `Z - 32 \fi: ) / 26
          = 3 \exp_stop_f:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_juxt:N
      \else:
        \exp_after:wN \@@_parse_infix_check:NNN
        \cs:w
          @@_parse_infix_ \token_to_str:N #2 :N
          \exp_after:wN \exp_after:wN \exp_after:wN
        \cs_end:
      \fi:
    \fi:
    #1
    #2
  }
\cs_new:Npn \@@_parse_infix_check:NNN #1#2#3
  {
    \if_meaning:w \scan_stop: #1
      \__kernel_msg_expandable_error:nnn
        { kernel } { fp-missing } { * }
      \exp_after:wN \@@_parse_infix_mul:N
      \exp_after:wN #2
      \exp_after:wN #3
    \else:
      \exp_after:wN #1
      \exp_after:wN #2
      \exp:w \exp_after:wN \@@_parse_expand:w
    \fi:
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\@@_parse_infix_after_paren:NN}
%   Variant of \cs{@@_parse_infix:NN} for use after a closing
%   parenthesis.  The only difference is that \cs{@@_parse_infix_juxt:N}
%   is replaced by \cs{@@_parse_infix_mul:N}.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_infix_after_paren:NN #1 #2
  {
    \if_catcode:w \scan_stop: \exp_not:N #2
      \if_int_compare:w
          \@@_str_if_eq:nn { \s_@@_expr_mark } { \exp_not:N #2 }
          = 0 \exp_stop_f:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_mark:NNN
      \else:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_mul:N
      \fi:
    \else:
      \if_int_compare:w
          \@@_int_eval:w
            ( `#2 \if_int_compare:w `#2 > `Z - 32 \fi: ) / 26
          = 3 \exp_stop_f:
        \exp_after:wN \exp_after:wN
        \exp_after:wN \@@_parse_infix_mul:N
      \else:
        \exp_after:wN \@@_parse_infix_check:NNN
        \cs:w
          @@_parse_infix_ \token_to_str:N #2 :N
          \exp_after:wN \exp_after:wN \exp_after:wN
        \cs_end:
      \fi:
    \fi:
    #1
    #2
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Closing parentheses and commas}
%
% \begin{macro}[EXP]{\@@_parse_infix_mark:NNN}
%   As an infix operator, \cs{s_@@_expr_mark} means that the next
%   token~(|#3|) has already gone through \cs{@@_parse_infix:NN} and
%   should be provided the precedence~|#1|.  The scan mark~|#2| is
%   discarded.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_infix_mark:NNN #1#2#3 { #3 #1 }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_infix_end:N}
%   This one is a little bit odd: force every previous operator to end,
%   regardless of the precedence.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_infix_end:N #1
  { @ \use_none:n \@@_parse_infix_end:N }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]+\@@_parse_infix_):N+
%   This is very similar to \cs{@@_parse_infix_end:N}, complaining about
%   an extra closing parenthesis if the previous operator was the
%   beginning of the expression, with precedence \cs{c_@@_prec_end_int}.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1
  {
    \cs_new:Npn #1 ##1
      {
        \if_int_compare:w ##1 > \c_@@_prec_end_int
          \exp_after:wN @
          \exp_after:wN \use_none:n
          \exp_after:wN #1
        \else:
          \__kernel_msg_expandable_error:nnn { kernel } { fp-extra } { ) }
          \exp_after:wN \@@_parse_infix:NN
          \exp_after:wN ##1
          \exp:w \exp_after:wN \@@_parse_expand:w
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_):N }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[verb, EXP]{\__fp_parse_infix_,:N}
% \begin{macro}[EXP]{\@@_parse_infix_comma:w, \@@_parse_apply_comma:NwNwN}
%   As for other infix operations, if the previous operations has higher
%   precedence the comma waits.  Otherwise we call
%   \cs{@@_parse_operand:Nw} to read more comma-delimited arguments that
%   \cs{@@_parse_infix_comma:w} simply concatenates into a |@|-delimited
%   array.  The first comma in a tuple that is not a function argument
%   is distinguished: in that case call \cs{@@_parse_apply_comma:NwNwN}
%   whose job is to convert the first item of the tuple and an array of
%   the remaining items into a tuple.  In contrast to
%   \cs{@@_parse_apply_binary:NwNwN} this function's operands are not
%   single-object arrays.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1
  {
    \cs_new:Npn #1 ##1
      {
        \if_int_compare:w ##1 > \c_@@_prec_comma_int
          \exp_after:wN @
          \exp_after:wN \use_none:n
          \exp_after:wN #1
        \else:
          \if_int_compare:w ##1 < \c_@@_prec_comma_int
            \exp_after:wN @
            \exp_after:wN \@@_parse_apply_comma:NwNwN
            \exp_after:wN ,
            \exp:w
          \else:
            \exp_after:wN \@@_parse_infix_comma:w
            \exp:w
          \fi:
          \@@_parse_operand:Nw \c_@@_prec_comma_int
          \exp_after:wN \@@_parse_expand:w
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_,:N }
\cs_new:Npn \@@_parse_infix_comma:w #1 @
  { #1 @ \use_none:n }
\cs_new:Npn \@@_parse_apply_comma:NwNwN #1 #2@ #3 #4@ #5
  {
    \exp_after:wN \@@_parse_continue:NwN
    \exp_after:wN #1
    \exp:w \exp_end_continue_f:w
    \@@_exp_after_tuple_f:nw { }
      \s_@@_tuple \@@_tuple_chk:w { #2 #4 } ;
    #5 #1
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsubsection{Usual infix operators}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_infix_+:N, \@@_parse_infix_-:N,
%     \@@_parse_infix_juxt:N,
%     \@@_parse_infix_/:N, \@@_parse_infix_mul:N,
%     \@@_parse_infix_and:N, \@@_parse_infix_or:N,
%   }
% \begin{macro}[EXP]+\@@_parse_infix_^:N+
%   As described in the \enquote{work plan}, each infix operator has an
%   associated |\..._infix_...| function, a computing function, and
%   precedence, given as arguments to \cs{@@_tmp:w}.  Using the general
%   mechanism for arithmetic operations.  The power operation must be
%   associative in the opposite order from all others.  For this, we use
%   two distinct precedences.
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1#2#3#4
  {
    \cs_new:Npn #1 ##1
      {
        \if_int_compare:w ##1 < #3
          \exp_after:wN @
          \exp_after:wN \@@_parse_apply_binary:NwNwN
          \exp_after:wN #2
          \exp:w
          \@@_parse_operand:Nw #4
          \exp_after:wN \@@_parse_expand:w
        \else:
          \exp_after:wN @
          \exp_after:wN \use_none:n
          \exp_after:wN #1
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_^:N }   ^
  \c_@@_prec_hatii_int \c_@@_prec_hat_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_juxt:N } *
  \c_@@_prec_juxt_int \c_@@_prec_juxt_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_/:N }   /
  \c_@@_prec_times_int \c_@@_prec_times_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_mul:N } *
  \c_@@_prec_times_int \c_@@_prec_times_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_-:N }   -
  \c_@@_prec_plus_int  \c_@@_prec_plus_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_+:N }   +
  \c_@@_prec_plus_int  \c_@@_prec_plus_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_and:N } &
  \c_@@_prec_and_int   \c_@@_prec_and_int
\exp_args:Nc \@@_tmp:w { @@_parse_infix_or:N }  |
  \c_@@_prec_or_int    \c_@@_prec_or_int
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsubsection{Juxtaposition}
%
% \begin{macro}[EXP]+\@@_parse_infix_(:N+
%   When an opening parenthesis appears where we expect an infix
%   operator, we compute the product of the previous operand and the
%   contents of the parentheses using \cs{@@_parse_infix_mul:N}.
%    \begin{macrocode}
\cs_new:cpn { @@_parse_infix_(:N } #1
  { \@@_parse_infix_mul:N #1 ( }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Multi-character cases}
%
% \begin{macro}[EXP]{\@@_parse_infix_*:N}
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1
  {
    \cs_new:cpn { @@_parse_infix_*:N } ##1##2
      {
        \if:w * \exp_not:N ##2
          \exp_after:wN #1
          \exp_after:wN ##1
        \else:
          \exp_after:wN \@@_parse_infix_mul:N
          \exp_after:wN ##1
          \exp_after:wN ##2
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_^:N }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]+\@@_parse_infix_|:Nw+
% \begin{macro}[EXP]+\@@_parse_infix_&:Nw+
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1#2#3
  {
    \cs_new:Npn #1 ##1##2
      {
        \if:w #2 \exp_not:N ##2
          \exp_after:wN #1
          \exp_after:wN ##1
          \exp:w \exp_after:wN \@@_parse_expand:w
        \else:
          \exp_after:wN #3
          \exp_after:wN ##1
          \exp_after:wN ##2
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_|:N } | \@@_parse_infix_or:N
\exp_args:Nc \@@_tmp:w { @@_parse_infix_&:N } & \@@_parse_infix_and:N
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsubsection{Ternary operator}
%
% \begin{macro}[EXP]{\@@_parse_infix_?:N, \@@_parse_infix_::N}
%    \begin{macrocode}
\cs_set_protected:Npn \@@_tmp:w #1#2#3#4
  {
    \cs_new:Npn #1 ##1
      {
        \if_int_compare:w ##1 < \c_@@_prec_quest_int
          #4
          \exp_after:wN @
          \exp_after:wN #2
          \exp:w
          \@@_parse_operand:Nw #3
          \exp_after:wN \@@_parse_expand:w
        \else:
          \exp_after:wN @
          \exp_after:wN \use_none:n
          \exp_after:wN #1
        \fi:
      }
  }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_?:N }
  \@@_ternary:NwwN \c_@@_prec_quest_int { }
\exp_args:Nc \@@_tmp:w { @@_parse_infix_::N }
  \@@_ternary_auxii:NwwN \c_@@_prec_colon_int
  {
    \__kernel_msg_expandable_error:nnnn
      { kernel } { fp-missing } { ? } { ~for~?: }
  }
%    \end{macrocode}
% \end{macro}
%
% \subsubsection{Comparisons}
%
% \begin{macro}[EXP]
%   {
%     \@@_parse_infix_<:N, \@@_parse_infix_=:N,
%     \@@_parse_infix_>:N, \@@_parse_infix_!:N
%   }
% \begin{macro}[EXP]
%   {
%     \@@_parse_excl_error:,
%     \@@_parse_compare:NNNNNNN,
%     \@@_parse_compare_auxi:NNNNNNN,
%     \@@_parse_compare_auxii:NNNNN,
%     \@@_parse_compare_end:NNNNw,
%     \@@_compare:wNNNNw,
%   }
%    \begin{macrocode}
\cs_new:cpn { @@_parse_infix_<:N } #1
  { \@@_parse_compare:NNNNNNN #1 1 0 0 0 0 < }
\cs_new:cpn { @@_parse_infix_=:N } #1
  { \@@_parse_compare:NNNNNNN #1 1 0 0 0 0 = }
\cs_new:cpn { @@_parse_infix_>:N } #1
  { \@@_parse_compare:NNNNNNN #1 1 0 0 0 0 > }
\cs_new:cpn { @@_parse_infix_!:N } #1
  {
    \exp_after:wN \@@_parse_compare:NNNNNNN
    \exp_after:wN #1
    \exp_after:wN 0
    \exp_after:wN 1
    \exp_after:wN 1
    \exp_after:wN 1
    \exp_after:wN 1
  }
\cs_new:Npn \@@_parse_excl_error:
  {
    \__kernel_msg_expandable_error:nnnn
      { kernel } { fp-missing } { = } { ~after~!. }
  }
\cs_new:Npn \@@_parse_compare:NNNNNNN #1
  {
    \if_int_compare:w #1 < \c_@@_prec_comp_int
      \exp_after:wN \@@_parse_compare_auxi:NNNNNNN
      \exp_after:wN \@@_parse_excl_error:
    \else:
      \exp_after:wN @
      \exp_after:wN \use_none:n
      \exp_after:wN \@@_parse_compare:NNNNNNN
    \fi:
  }
\cs_new:Npn \@@_parse_compare_auxi:NNNNNNN #1#2#3#4#5#6#7
  {
    \if_case:w
      \@@_int_eval:w \exp_after:wN ` \token_to_str:N #7 - `<
        \@@_int_eval_end:
         \@@_parse_compare_auxii:NNNNN #2#2#4#5#6
    \or: \@@_parse_compare_auxii:NNNNN #2#3#2#5#6
    \or: \@@_parse_compare_auxii:NNNNN #2#3#4#2#6
    \or: \@@_parse_compare_auxii:NNNNN #2#3#4#5#2
    \else: #1 \@@_parse_compare_end:NNNNw #3#4#5#6#7
    \fi:
  }
\cs_new:Npn \@@_parse_compare_auxii:NNNNN #1#2#3#4#5
  {
    \exp_after:wN \@@_parse_compare_auxi:NNNNNNN
    \exp_after:wN \prg_do_nothing:
    \exp_after:wN #1
    \exp_after:wN #2
    \exp_after:wN #3
    \exp_after:wN #4
    \exp_after:wN #5
    \exp:w \exp_after:wN \@@_parse_expand:w
  }
\cs_new:Npn \@@_parse_compare_end:NNNNw #1#2#3#4#5 \fi:
  {
    \fi:
    \exp_after:wN @
    \exp_after:wN \@@_parse_apply_compare:NwNNNNNwN
    \exp_after:wN \c_one_fp
    \exp_after:wN #1
    \exp_after:wN #2
    \exp_after:wN #3
    \exp_after:wN #4
    \exp:w
    \@@_parse_operand:Nw \c_@@_prec_comp_int \@@_parse_expand:w #5
  }
\cs_new:Npn \@@_parse_apply_compare:NwNNNNNwN
    #1 #2@ #3 #4#5#6#7 #8@ #9
  {
    \if_int_odd:w
        \if_meaning:w \c_zero_fp #3
          0
        \else:
          \if_case:w \@@_compare_back_any:ww #8 #2 \exp_stop_f:
            #5 \or: #6 \or: #7 \else: #4
          \fi:
        \fi:
        \exp_stop_f:
      \exp_after:wN \@@_parse_apply_compare_aux:NNwN
      \exp_after:wN \c_one_fp
    \else:
      \exp_after:wN \@@_parse_apply_compare_aux:NNwN
      \exp_after:wN \c_zero_fp
    \fi:
    #1 #8 #9
  }
\cs_new:Npn \@@_parse_apply_compare_aux:NNwN #1 #2 #3; #4
  {
    \if_meaning:w \@@_parse_compare:NNNNNNN #4
      \exp_after:wN \@@_parse_continue_compare:NNwNN
      \exp_after:wN #1
      \exp_after:wN #2
      \exp:w \exp_end_continue_f:w
      \@@_exp_after_o:w #3;
      \exp:w \exp_end_continue_f:w
    \else:
      \exp_after:wN \@@_parse_continue:NwN
      \exp_after:wN #2
      \exp:w \exp_end_continue_f:w
      \exp_after:wN #1
      \exp:w \exp_end_continue_f:w
    \fi:
    #4 #2
  }
\cs_new:Npn \@@_parse_continue_compare:NNwNN #1#2 #3@ #4#5
  { #4 #2 #3@ #1 }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \subsection{Tools for functions}
%
% \begin{macro}[EXP]{\@@_parse_function_all_fp_o:fnw}
%   Followed by \Arg{function name} \Arg{code} \meta{float array} |@|
%   this checks all floats are floating point numbers (no tuples).
%    \begin{macrocode}
\cs_new:Npn \@@_parse_function_all_fp_o:fnw #1#2#3 @
  {
    \@@_array_if_all_fp:nTF {#3}
      { #2 #3 @ }
      {
        \@@_error:nffn { fp-bad-args }
          {#1}
          { \fp_to_tl:n { \s_@@_tuple \@@_tuple_chk:w {#3} ; } }
          { }
        \exp_after:wN \c_nan_fp
      }
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_parse_function_one_two:nnw}
% \begin{macro}[EXP]
%   {
%     \@@_parse_function_one_two_error_o:w,
%     \@@_parse_function_one_two_aux:nnw,
%     \@@_parse_function_one_two_auxii:nnw
%   }
%   This is followed by \Arg{function name} \Arg{code} \meta{float
%   array} |@|.  It checks that the \meta{float array} consists of one
%   or two floating point numbers (not tuples), then leaves the
%   \meta{code} (if there is one float) or its tail (if there are two
%   floats) followed by the \meta{float array}.  The \meta{code} should
%   start with a single token such as \cs{@@_atan_default:w} that deals
%   with the single-float case.
%
%   The first \cs{@@_if_type_fp:NTwFw} test catches the case of no
%   argument and the case of a tuple argument.  The next one
%   distinguishes the case of a single argument (no error, just add
%   \cs{c_one_fp}) from a tuple second argument.  Finally check there is
%   no further argument.
%    \begin{macrocode}
\cs_new:Npn \@@_parse_function_one_two:nnw #1#2#3
  {
    \@@_if_type_fp:NTwFw
      #3 { } \s_@@ \@@_parse_function_one_two_error_o:w \s_@@_stop
    \@@_parse_function_one_two_aux:nnw {#1} {#2} #3
  }
\cs_new:Npn \@@_parse_function_one_two_error_o:w #1#2#3#4 @
  {
    \@@_error:nffn { fp-bad-args }
      {#2}
      { \fp_to_tl:n { \s_@@_tuple \@@_tuple_chk:w {#4} ; } }
      { }
    \exp_after:wN \c_nan_fp
  }
\cs_new:Npn \@@_parse_function_one_two_aux:nnw #1#2 #3; #4
  {
    \@@_if_type_fp:NTwFw
      #4 { }
      \s_@@
      {
        \if_meaning:w @ #4
          \exp_after:wN \use_iv:nnnn
        \fi:
        \@@_parse_function_one_two_error_o:w
      }
      \s_@@_stop
    \@@_parse_function_one_two_auxii:nnw {#1} {#2} #3; #4
  }
\cs_new:Npn \@@_parse_function_one_two_auxii:nnw #1#2#3; #4; #5
  {
    \if_meaning:w @ #5 \else:
      \exp_after:wN \@@_parse_function_one_two_error_o:w
    \fi:
    \use_ii:nn {#1} { \use_none:n #2 } #3; #4; #5
  }
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_tuple_map_o:nw, \@@_tuple_map_loop_o:nw}
%   Apply |#1| to all items in the following tuple and expand once
%   afterwards.  The code |#1| should itself expand once after its
%   result.
%    \begin{macrocode}
\cs_new:Npn \@@_tuple_map_o:nw #1 \s_@@_tuple \@@_tuple_chk:w #2 ;
  {
    \exp_after:wN \s_@@_tuple
    \exp_after:wN \@@_tuple_chk:w
    \exp_after:wN {
      \exp:w \exp_end_continue_f:w
      \@@_tuple_map_loop_o:nw {#1} #2
        { \s_@@ \prg_break: } ;
      \prg_break_point:
    \exp_after:wN } \exp_after:wN ;
  }
\cs_new:Npn \@@_tuple_map_loop_o:nw #1#2#3 ;
  {
    \use_none:n #2
    #1 #2 #3 ;
    \exp:w \exp_end_continue_f:w
    \@@_tuple_map_loop_o:nw {#1}
  }
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}[EXP]{\@@_tuple_mapthread_o:nww, \@@_tuple_mapthread_loop_o:nw}
%   Apply |#1| to pairs of items in the two following tuples and expand once
%   afterwards.
%    \begin{macrocode}
\cs_new:Npn \@@_tuple_mapthread_o:nww #1
    \s_@@_tuple \@@_tuple_chk:w #2 ;
    \s_@@_tuple \@@_tuple_chk:w #3 ;
  {
    \exp_after:wN \s_@@_tuple
    \exp_after:wN \@@_tuple_chk:w
    \exp_after:wN {
      \exp:w \exp_end_continue_f:w
      \@@_tuple_mapthread_loop_o:nw {#1}
        #2 { \s_@@ \prg_break: } ; @
        #3 { \s_@@ \prg_break: } ;
      \prg_break_point:
    \exp_after:wN } \exp_after:wN ;
  }
\cs_new:Npn \@@_tuple_mapthread_loop_o:nw #1#2#3 ; #4 @ #5#6 ;
  {
    \use_none:n #2
    \use_none:n #5
    #1 #2 #3 ; #5 #6 ;
    \exp:w \exp_end_continue_f:w
    \@@_tuple_mapthread_loop_o:nw {#1} #4 @
  }
%    \end{macrocode}
% \end{macro}
%
% ^^A end[todo]
%
% \subsection{Messages}
%
%    \begin{macrocode}
\__kernel_msg_new:nnn { kernel } { fp-deprecated }
  { '#1'~deprecated;~use~'#2' }
\__kernel_msg_new:nnn { kernel } { unknown-fp-word }
  { Unknown~fp~word~#1. }
\__kernel_msg_new:nnn { kernel } { fp-missing }
  { Missing~#1~inserted #2. }
\__kernel_msg_new:nnn { kernel } { fp-extra }
  { Extra~#1~ignored. }
\__kernel_msg_new:nnn { kernel } { fp-early-end }
  { Premature~end~in~fp~expression. }
\__kernel_msg_new:nnn { kernel } { fp-after-e }
  { Cannot~use~#1 after~'e'. }
\__kernel_msg_new:nnn { kernel } { fp-missing-number }
  { Missing~number~before~'#1'. }
\__kernel_msg_new:nnn { kernel } { fp-unknown-symbol }
  { Unknown~symbol~#1~ignored. }
\__kernel_msg_new:nnn { kernel } { fp-extra-comma }
  { Unexpected~comma~turned~to~nan~result. }
\__kernel_msg_new:nnn { kernel } { fp-no-arg }
  { #1~got~no~argument;~used~nan. }
\__kernel_msg_new:nnn { kernel } { fp-multi-arg }
  { #1~got~more~than~one~argument;~used~nan. }
\__kernel_msg_new:nnn { kernel } { fp-num-args }
  { #1~expects~between~#2~and~#3~arguments. }
\__kernel_msg_new:nnn { kernel } { fp-bad-args }
  { Arguments~in~#1#2~are~invalid. }
\__kernel_msg_new:nnn { kernel } { fp-infty-pi }
  { Math~command~#1 is~not~an~fp }
%<*package>
\cs_if_exist:cT { @unexpandable@protect }
  {
    \__kernel_msg_new:nnn { kernel } { fp-robust-cmd }
      { Robust~command~#1 invalid~in~fp~expression! }
  }
%</package>
%    \end{macrocode}
%
%    \begin{macrocode}
%</initex|package>
%    \end{macrocode}
%
% \end{implementation}
%
% \PrintChanges
%
% \PrintIndex