summaryrefslogtreecommitdiff
path: root/macros/generic/minifp/minifp.dtx
blob: 2e88dd1e47f90f21c3a403e518968a09ea6b9ab6 (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
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
3793
3794
3795
3796
3797
3798
3799
3800
3801
3802
3803
3804
3805
3806
3807
3808
3809
3810
3811
3812
3813
3814
3815
3816
3817
3818
3819
3820
3821
3822
3823
3824
3825
3826
3827
3828
3829
3830
3831
3832
3833
3834
3835
3836
3837
3838
3839
3840
3841
3842
3843
3844
3845
3846
3847
3848
3849
3850
3851
3852
3853
3854
3855
3856
3857
3858
3859
3860
3861
3862
3863
3864
3865
3866
3867
3868
3869
3870
3871
3872
3873
3874
3875
3876
3877
3878
3879
3880
3881
3882
3883
3884
3885
3886
3887
3888
3889
3890
3891
3892
3893
3894
3895
3896
3897
3898
3899
3900
3901
3902
3903
3904
3905
3906
3907
3908
3909
3910
3911
3912
3913
3914
3915
3916
3917
3918
3919
3920
3921
3922
3923
3924
3925
3926
3927
3928
3929
3930
3931
3932
3933
3934
3935
3936
3937
3938
3939
3940
3941
3942
3943
3944
3945
3946
3947
3948
3949
3950
3951
3952
3953
3954
3955
3956
3957
3958
3959
3960
3961
3962
3963
3964
3965
3966
3967
3968
3969
3970
3971
3972
3973
3974
3975
3976
3977
3978
3979
3980
3981
3982
3983
3984
3985
3986
3987
3988
3989
3990
3991
3992
3993
3994
3995
3996
3997
3998
3999
4000
4001
4002
4003
4004
4005
4006
4007
4008
4009
4010
4011
4012
4013
4014
4015
4016
4017
4018
4019
4020
4021
4022
4023
4024
4025
4026
4027
4028
4029
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
4051
4052
4053
4054
4055
4056
4057
4058
4059
4060
4061
4062
4063
4064
4065
4066
4067
4068
4069
4070
4071
4072
4073
4074
4075
4076
4077
4078
4079
4080
4081
4082
4083
4084
4085
4086
4087
4088
4089
4090
4091
4092
4093
4094
4095
4096
4097
4098
4099
4100
4101
4102
4103
4104
4105
4106
4107
4108
4109
4110
4111
4112
4113
4114
4115
4116
4117
4118
4119
4120
4121
4122
4123
4124
4125
4126
4127
4128
4129
4130
4131
4132
4133
4134
4135
4136
% \iffalse
% -------------------------------------------------------------------
%
% Copyright 2008--2013, Daniel H. Luecking
%
% minifp may be distributed and/or modified under the conditions of the
% LaTeX Project Public License, either version 1.3b of this license or (at
% your option) any later version. The latest version of this license is in
%    <http://www.latex-project.org/lppl.txt>
% and version 1.3c or later is part of all distributions of LaTeX version
% 2008/12/01 or later.
%
% minifp has maintenance status "author-maintained". The Current Maintainer
% is Daniel H. Luecking. The Base Interpreter is TeX (plain TeX or LaTeX).
%<*driver|sty>
\def\MFPfiledate{2013/12/30}%
\def\MFPfileversion{0.96}%
%</driver|sty>
%
%<*driver>
\ProvidesFile{minifp.dtx}
  [\MFPfiledate\space v\MFPfileversion. Macros for real number operations and a
    stack-based programing language.]%
\documentclass[draft]{ltxdoc}

\addtolength{\textwidth}{1pt}

\usepackage[morefloats=5]{morefloats}
\usepackage{amssymb}
% This avoids messages about nonexistent font variants (e.g., in \section):
\def\mytt{\upshape\mdseries\ttfamily}
% I use it instead of \texttt:
\renewcommand\marg[1]{{\mytt\{#1\}}}
\renewcommand\oarg[1]{{\mytt [#1]}}
\renewcommand\parg[1]{{\mytt (#1)}}
\renewcommand \arg[1]{{\mytt \##1}}
\renewcommand\#{\char`\#\relax}
\DeclareRobustCommand\cs[1]{{\mytt\char`\\#1}}
% sometimes I want a <meta> without enclosing braces:
\renewcommand{\meta}[1]{\mbox{$\langle$\rmfamily\itshape#1\/$\rangle$}}
% and sometimes I want the braces:
\newcommand\mmarg[1]{\marg{\meta{#1}}}

\def\prog#1{{\mdseries\scshape #1}}
\def\mfp{\prog{minifp}}
\def\Mfp{\prog{Minifp}}
\def\file#1{{\mytt #1}}
\let\dim\file
\let\env\file
\def\sgn{\mathop{\mathrm{sgn}}\nolimits}
% \op is for abstract operations (e.g., \op{add}) as opposed to
% the macro that performs it (e.g., \cs{Radd}). And \reg is for
% a "register" (e.g., the 3 macros \MFP@x@Sgn, \MFP@x@Int and \MFP@x@Frc)
% conceived of as a single entity.
\let\op\textit
\def\reg#1{$#1$}
% The occasional bare \tt braces
\renewcommand\{{\char`\{}
\renewcommand\}{\char`\}}
% this gives the alternative symbol in BNF productions, i.e., the bar
% in: { this | that }
\renewcommand\|{${}\mathrel{|}{}$}

\makeatletter
\newcommand\bsl{{\mytt\@backslashchar}}
% better lists
\def\@listi{\leftmargin\leftmargini
  \parsep \z@ \@plus\p@ \@minus\z@
  \topsep 4\p@ \@plus\p@ \@minus2\p@
  \itemsep\parsep}
\let\@listI\@listi \@listi
\renewcommand\labelitemi{\normalfont\bfseries \textendash}
\renewcommand\labelitemii{\textasteriskcentered}
\renewcommand\labelitemiii{\textperiodcentered}
\leftmargini\parindent
% better index
\def\usage#1{\textrm{#1}}
\def\index@prologue{\section*{Index}\markboth{Index}{Index}%
  Numbers refer to the page(s) where the corresponding entry is described.}
\def\IndexParms{%
  \parindent \z@ \columnsep 15pt
  \parskip 0pt plus 1pt
  \rightskip 5pt plus2em \mathsurround \z@
  \parfillskip-5pt \small
  % less hanging:
  \def\@idxitem{\par\hangindent 20pt}%
  \def\subitem{\@idxitem\hspace*{15pt}}%
  \def\subsubitem{\@idxitem\hspace*{25pt}}%
  \def\indexspace{\par\vspace{10pt plus 2pt minus 3pt}}}
\makeatother

\title{The \mfp{} package\thanks{This file has version number
        \fileversion, last revised \filedate. The code described here
        was developed by Dan Luecking.}}
\author{Dan Luecking}
\date{\filedate}
\DisableCrossrefs
\CodelineIndex
\AlsoImplementation

\begin{document}
  \DeleteShortVerb{\|}
  \DocInput{minifp.dtx}
\end{document}
%</driver>
%\fi
% \CheckSum{3541}
% \CharacterTable
%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%   Digits        \0\1\2\3\4\5\6\7\8\9
%   Exclamation   \!     Double quote  \"     Hash (number) \#
%   Dollar        \$     Percent       \%     Ampersand     \&
%   Acute accent  \'     Left paren    \(     Right paren   \)
%   Asterisk      \*     Plus          \+     Comma         \,
%   Minus         \-     Point         \.     Solidus       \/
%   Colon         \:     Semicolon     \;     Less than     \<
%   Equals        \=     Greater than  \>     Question mark \?
%   Commercial at \@     Left bracket  \[     Backslash     \\
%   Right bracket \]     Circumflex    \^     Underscore    \_
%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%   Right brace   \}     Tilde         \~}
%
% \GetFileInfo{minifp.dtx}
% \maketitle
%
% \begin{abstract}
% This package provides minimal fixed point exact decimal arithmetic
% operations. `Minimal' means numbers are limited to eight digits on
% either side of the decimal point. `Exact' means that when a number
% \emph{can} be represented exactly within those limits, it will be.
% \end{abstract}
%
% \StopEventually{\PrintIndex}
% \tableofcontents
%
% \section{Introduction}
% In working on an application that needed to be able to automatically
% generate numeric labels on the axes of a graph, I needed to be able
% to make simple calculations with real numbers. What \TeX{} provides is
% far too limited. In fact, its only native user-level support for real
% numbers is as factors for dimensions. For example one can ``multiply''
% $3.1\times 0.2$ by the following: \verb$\dimen0=0.2pt \dimen0=3.1\dimen0$.
%
% Unfortunately \TeX{} stores dimensions as integer multiples of the
% ``scaled points'' (\dim{sp}) with \dim{sp}${}=2^{-16}$\dim{pt}, and
% therefore \dim{.2pt} is approximated by $\frac{13107}{65536}$, which is
% not exact. Then mutiplying by $3.1$ produces $\frac{40631}{65536}$. If
% we ask \TeX{} to display this, it produces $0.61998$\dim{pt} and not the
% exact value $0.62$. This is sufficiently accurate for positioning
% elements on a page, but not for displaying automatically computed axis
% labels if five digit accuracy is needed.
%
% The \mfp{} package was written to provide the necessary calculations
% with the necessary accuracy for this application. The implementation
% would have been an order of magnitude smaller and faster if only four digit
% accuracy were provided (and I may eventually do that for the application
% under consideration), but I have decided to clean up what I have
% produced and release it as is. The full \mfp{} package provides nearly
% the same operations as a subset of the \prog{fp} package, but the latter
% carries calculations to 18 decimal places, which is far more than
% necessary for my purposes. I want something small and fast to embed in
% the \prog{mfpic} drawing package.
%
% I decided on eight digits on both sides of the decimal point essentially
% because I wanted at least five digits and the design I chose made multiples
% of four the easiest to work with.
%
% \Mfp{} also provides a simple stack-based language for writing assembly
% language-like programs. Originally, this was to be the native
% calculation method, but it turned out to be too unwieldy for ordinary
% use. I left it in because it adds only about 10\% overhead to the code.
%
% But why \emph{only} eight digits? \TeX{} only works with integers, and
% since the maximum integer allowed is about $2\,000\,000\,000$, the
% largest numbers that can be added are limited to about $999\,999\,999$.
% It is very little trouble to add numbers by adding their fractional
% parts and integer parts separately as 9-digit integers. So it would seem
% multiples of $9$ digits would be easy to implement.
%
% However, something we have to do repeatedly in \emph{division} is
% multiply the integer and fractional parts of a number by a one-digit
% number. For that purpose, nine digits would be too much, but eight
% digits is just right. For nine digits, we would have to inconveniently
% break the number into more than two parts. Limiting our numbers to
% eight-digit parts drastically simplifies division.
%
% Another simplification: multiplication has to be done by breaking the
% number into parts. \TeX{} can multiply any two 4-digit integers without
% overflow, but it cannot multiply most pairs of 5-digit integers. Two
% 8-digit numbers conveniently break into four 4-digit parts. To get even
% nine digits of accuracy would require six parts (five, if we don't
% insist on a separation occuring at the decimal point). The complexity of
% the multiplication process goes up as the square of the number of parts,
% so six parts would more than double the complexity.
%
% A final simplification: \TeX{} places a limit of nine on the number of
% arguments a macro can have. Quite often the last argument is needed to
% clear out unused text to be discarded. Thus, a string of eight digits
% can quite often be processed with one execution of one nine-argument
% macro.
%
% Addition and subtraction can be exact, multiplication and division can
% extend numbers past the 8-digit limit so they might be rounded.
% However, when the exact answer fits in the 8-digit limit, our code
% should produce it. Overflow (in the sense that the integer part can
% exceed the allowed eight digits) is always possible, but is much more
% likely with multiplication and division.
%
% Multiplication is carried out internally to an exact answer, with 16
% digits on each side of the decimal point. The underflow digits (places 9
% through 16 after the decimal point) are used to round to an 8-digit
% result. Overflow digits (those to the left of the lowest 8 in the
% integer part) are discarded, usually without warning. Division is
% internally carried to nine digits after the decimal, which is then also
% rounded to an 8-digit result. Overflow digits are ignored for division
% also.
%
% We supply two kinds of operations in this package. There are stack-based
% operations, in which the operands are \op{popped} from a stack and the
% results \op{pushed} onto it, and argument-based, in which the operands (and a
% macro to hold the result^^A
%         \footnote{Unlike most other packages for decimal
%         arithmetic, \mfp{} puts the macro to hold the result
%         last. This allows the calculation to be performed before the
%         macro is even read, and this makes it somewhat easier for the
%         stack- and argument-based versions to share code.}^^A
% ) are arguments of a macro. Both types load the arguments into internal
% macros (think of them as ``registers''), then call internal commands
% (think ``microcode'') which return the results in internal macros.
% These results are then \op{pushed} onto the stack (stack-based
% operations) or stored in a supplied macro argument (think ``variable'').
% The difference lies entirely in where the operands come from (arguments
% or stack) and where they go (macro or stack).
%
% The stack is implemented as an internal macro which is redefined with
% each command. The binary operations act on the last two \op{pushed} objects
% in the order they were \op{pushed}. For example, the sequence ``\op{push} 5,
% \op{push} 3, \op{subtract}'' performs $5-3$ by popping $3$ and $5$ into
% registers (thereby removing them from the stack), subtracting them
% and then pushing the result ($2$) onto the stack.
%
% Our implementation of the \op{push} operation first prepares the number
% in a standard form. Thus, stack-based operations always obtain numbers
% in this form. The argument based operations will prepare the arguments
% in the same way. The internal commands will thus have a standard form to
% operate on. All results are returned in standard form.
%
% The standard form referred to above is an integer part (one to eight digits
% with no unnecessary leading zeros nor unnecessary sign) followed by the
% decimal point (always a dot, which is ASCII \number`\.), followed by exactly
% eight digits, all of this preceded by a minus sign if the number is
% negative. Thus, $-{-0.25}$ would be processed and stored as
% ``\texttt{0.25000000}'' and $-.333333$ as ``\texttt{-0.33333300}''.
%
%
% \section{User macros}
%
% \Mfp{} provides (so far) six binary operations (that act on a pair of
% numbers): addition, subtraction, multiplication, division, maximum and
% minimum, as well as fourteen unary operations (that act on one number):
% negation, absolute value, doubling, halving, integer part, fractional
% part, floor, ceiling, signum, squaring, increment, decrement and
% inversion. With the ``\texttt{extra}'' option, the unary operations
% sine, cosine, logarithm, powers, square root and random number are
% available, and the binary operation angle. See section~\ref{extras}.
%
% These extra operations are made available using the \texttt{extra}
% option in \LaTeX{}:
% \begin{verbatim}
%   \usepackage[extra]{minifp} \end{verbatim}
% In plain \TeX{}, they will be loaded if you give the macro
% \cs{MFPextra} a definition (any definition) before inputting
% \file{minifp.sty}:
% \begin{verbatim}
%   \def\MFPextra{} \input minifp.sty \end{verbatim}
% The extras can also be loaded by means of the command
% \cs{MFPloadextra}, issued after \file{minifp.sty} is loaded.
% As of version 0.95 \file{mfpextra} can be directly \cs{input}.
% It will detect whether \file{minifp.sty} has been loaded and input it
% if not. This will work only in plain \TeX{}.
%
% If the extra operations are not needed, some memory and time might be
% saved by using \file{minifp.sty} alone. I have not seriously tried to
% keep \file{mfpextra.tex} as small or fast as possible, but I do try
% to improve the accuracy when I can.
%
% As previously mentioned, each of these operations come in two versions:
% a version that acts on operands and stores the result in a macro, and a
% version that acts on the stack. The former all have names that begin
% \cs{MFP} and the latter begin with \cs{R}. The former can be used
% anywhere, while the latter can only be used in a ``program''.
% A program is started with \cs{startMFPprogram} and terminated with
% \cs{stopMFPprogram}. The \texttt{R} in the names is for `real'. This is
% because it is possible that stacks of other types will be implemented in
% the future.
%
% For example, \verb$\MFPadd{1.2}{3.4}\X$ will add $1.20000000$ to
% $3.40000000$ and then define \cs{X} to be the resulting
% \texttt{4.60000000}. These operand forms do not alter or even address
% the stack in any way. The stack-based version of the same operation
% would look like the following:
% \begin{verbatim}
%   \Rpush{1.2}
%   \Rpush{3.4}
%   \Radd
%   \Rpop\X \end{verbatim}
% which would \op{push} first \texttt{1.20000000} then \texttt{3.40000000} onto
% the stack, then replace them with \texttt{4.60000000}, then remove that
% and store it in \verb$\X$. Clearly the stack is intended for
% calculations that produce a lot of intermediate values and only the
% final result needs to be stored.
%
% \SpecialUsageIndex{\startMFPprogram}
% The command \cs{startMFPprogram} starts a group. That group should be
% ended by \cs{stopMFPprogram}.
% \SpecialUsageIndex{\stopMFPprogram}
% Changes to the stack and defined macros are local to that group. Thus
% the macro \cs{X} in the example above might seem to be useful only as a
% temporary storage for later calculations in the same program group.
% However, there are commands provided to force such a macro to survive
% the group, and even to force the contents of the stack to survive the
% group (see the end of subsection~\ref{stack}). Do not try to turn a
% \mfp{} program into a \LaTeX{} environment. The extra grouping added by
% environments would defeat the effects of these commands.
%
% \subsection{Nonstack-based operations}
%
% In the following tables, an argument designated \meta{num} can be any
% decimal real number with at most 8 digits on each side of the decimal
% point, or it can be a macro that contains such a number. If the decimal
% dot is absent, the fractional part will be taken to be zero, if the
% integer part or the fractional part is absent, it will be taken to be
% zero. (One consequence of these rules is that all the following
% arguments produce the same internal representation of zero: \marg{0.0},
% \marg{0.}, \marg{.0}, \marg{0}, \marg{.}, and \marg{}\,.) Spaces may
% appear anywhere in the \meta{num} arguments and are stripped out before
% the number is used. For example, \marg{3 . 1415 9265} is a valid
% argument. Commas are not permitted. The decimal point \emph{must} be
% ASCII 46 (variously called a dot, period, or fullstop) with category 12
% (`other'). If an input encoding is used that allows more than one `dot',
% the user must be sure to enter this one. If some babel language
% definitions make it a shorthand, it must be inactivated before use.
%
% The \cs{macro} argument is any legal macro. The result of using one of
% these commands is that the macro is defined (or redefined, there is no
% checking done) to contain the standard form of the result. If the
% \meta{num} is a macro, the braces surrounding it are optional.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3in}}
% \textit{Binary Operations}&\\[3pt]
% \hline \hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\MFPadd}^^A
%   \cs{MFPadd}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the result of \meta{num$_1$}${}+{}$\meta{num$_2$} in \cs{macro}\\
%   \SpecialUsageIndex{\MFPsub}^^A
%   \cs{MFPsub}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the result of \meta{num$_1$}${}-{}$\meta{num$_2$} in \cs{macro}\\
%   \SpecialUsageIndex{\MFPmul}^^A
%   \cs{MFPmul}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the result of \meta{num$_1$}${}\times{}$\meta{num$_2$},
%     rounded to 8 places after the decimal point, in \cs{macro}\\
%   \SpecialUsageIndex{\MFPmpy}^^A
%   \cs{MFPmpy}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Same as \cs{MFPmul}\\
%   \SpecialUsageIndex{\MFPdiv}^^A
%   \cs{MFPdiv}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the result of \meta{num$_1$}${}/{}$\meta{num$_2$},
%     rounded to 8 places after the decimal point, in \cs{macro}\\
%   \SpecialUsageIndex{\MFPmin}^^A
%   \cs{MFPmin}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the smaller of \meta{num$_1$} and \meta{num$_2$} in \cs{macro}\\
%   \SpecialUsageIndex{\MFPmax}^^A
%   \cs{MFPmax}\mmarg{num$_1$}\mmarg{num$_2$}\cs{macro}&
%     Stores the larger of \meta{num$_1$} and \meta{num$_2$} in \cs{macro}
% \end{tabular}}
%
%\bigskip
%
% \centerline{%
% \begin{tabular}{lp{3.4in}}
% \textit{Unary Operations}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\MFPchs}^^A
%   \cs{MFPchs}\mmarg{num}\cs{macro}&
%     Stores $-{}$\meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPabs}^^A
%   \cs{MFPabs}\mmarg{num}\cs{macro}&
%     Stores $|$\meta{num}$|$ in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPdbl}^^A
%   \cs{MFPdbl}\mmarg{num}\cs{macro}&
%     Stores $2\times{}$\meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPhalve}^^A
%   \cs{MFPhalve}\mmarg{num}\cs{macro}&
%     Stores \meta{num}/2, rounded to 8 places after the decimal point, in
%     \cs{macro}.\\
%   \SpecialUsageIndex{\MFPint}^^A
%   \cs{MFPint}\mmarg{num}\cs{macro}&
%     Replaces the part of \meta{num} after the decimal point with zeros
%     (keeps the sign unless the result is zero) and stores the result in
%     \cs{macro}.\\
%   \SpecialUsageIndex{\MFPfrac}^^A
%   \cs{MFPfrac}\mmarg{num}\cs{macro}&
%     Replaces the part of \meta{num} before the decimal point with zero
%     (keeps the sign unless the result is zero) and stores the result in
%     \cs{macro}.\\
%   \SpecialUsageIndex{\MFPfloor}^^A
%   \cs{MFPfloor}\mmarg{num}\cs{macro}&
%     Stores the largest integer not more than \meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPceil}^^A
%   \cs{MFPceil}\mmarg{num}\cs{macro}&
%     Stores the smallest integer not less than \meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPsgn}^^A
%   \cs{MFPsgn}\mmarg{num}\cs{macro}&
%     Stores $-1$, $0$ or $1$ (in standard form) in \cs{macro} according
%     to whether \meta{num} is negative, zero, or positive.\\
%   \SpecialUsageIndex{\MFPsq}^^A
%   \cs{MFPsq}\mmarg{num}\cs{macro}&
%     Stores the square of \meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPinv}^^A
%   \cs{MFPinv}\mmarg{num}\cs{macro}&
%     Stores 1/\meta{num}, rounded to 8 places after the decimal point, in
%     \cs{macro}.\\
%   \SpecialUsageIndex{\MFPincr}^^A
%   \cs{MFPincr}\mmarg{num}\cs{macro}&
%     Stores \meta{num}${}+1$ in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPdecr}^^A
%   \cs{MFPdecr}\mmarg{num}\cs{macro}&
%     Stores \meta{num}${}-1$ in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPzero}^^A
%   \cs{MFPzero}\mmarg{num}\cs{macro}&
%     Ignores \meta{num} and stores {0.00000000} in the \cs{macro}.\\
%   \SpecialUsageIndex{\MFPstore}^^A
%   \cs{MFPstore}\mmarg{num}\cs{macro}&
%     Stores the \meta{num}, converted to standard form, in \cs{macro}
% \end{tabular}}
%
%\bigskip
%
% The command \cs{MFPzero} is useful for ``macro programs''. If you want
% to do something to a number depending on the outcome of a test, you may
% occasionally want to simply absorbed the number and output a default
% result. This is more efficient than multiplying by zero (but less
% efficient than simply defining the \cs{macro} to be zero.)
%
% Note that one could easily double, halve, square, increment,
% decrement or invert a \meta{num} using the binary versions of
% \cs{MFPadd}, \cs{MFPsub}, \cs{MFPmul} or \cs{MFPdiv}. The commands
% \cs{MFPdbl}, \cs{MFPhalve}, \cs{MFPsq}, \cs{MFPincr}, \cs{MFPdecr} and
% \cs{MFPinv} are designed to be more efficient versions, since they are
% used repeatedly in internal code.
%
% Also, multiplication is far more efficient than division, so even if you
% use the two argument versions, \cs{MFPmul}\mmarg{num}\marg{.5} is faster than
% \cs{MFPdiv}\mmarg{num}\marg{2}.
%
% There is one command that takes no argument and returns no value:
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3.4in}}
% \textit{Do Nothing}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\MFPnoop}\cs{MFPnoop}& Does nothing.
% \end{tabular}}
%
% \bigskip
% The following are not commands at all, but macros that contain
% convenient constants.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3.9in}}
% \textit{Constants}&\\[3pt]
% \hline\hline
%   \textbf{Constant}&\textbf{value}\\
% \hline
%   \SpecialUsageIndex{\MFPpi}^^A
%   \cs{MFPpi}& \texttt{3.14159265}, the eight-digit approximation to
%     $\pi$.\\
%   \SpecialUsageIndex{\MFPe}^^A
%   \cs{MFPe}& \texttt{2.71828183}, the eight-digit approximation to
%     $e$.\\
%   \SpecialUsageIndex{\MFPphi}^^A
%   \cs{MFPphi}& \texttt{1.61803399}, the eight-digit approximation to
%     the golden ratio $\phi.$
% \end{tabular}}
%
% \bigskip
% There also exist commands to check the sign of a number and the
% relative size of two numbers:
%
% \medskip
% \indent \SpecialUsageIndex{\MFPchk}\cs{MFPchk}\mmarg{num}\\
% \indent \SpecialUsageIndex{\MFPcmp}\cs{MFPcmp}\mmarg{num$_1$}\mmarg{num$_2$}
%
% \medskip
% \noindent These influence the behavior of six commands:
%
% \medskip
% \indent \SpecialUsageIndex{\IFneg}\cs{IFneg}\mmarg{true text}\mmarg{false text}\\
% \indent \SpecialUsageIndex{\IFzero}\cs{IFzero}\mmarg{true text}\mmarg{false text}\\
% \indent \SpecialUsageIndex{\IFpos}\cs{IFpos}\mmarg{true text}\mmarg{false text}\\
% \indent \SpecialUsageIndex{\IFlt}\cs{IFlt}\mmarg{true text}\mmarg{false text}\\
% \indent \SpecialUsageIndex{\IFeq}\cs{IFeq}\mmarg{true text}\mmarg{false text}\\
% \indent \SpecialUsageIndex{\IFgt}\cs{IFgt}\mmarg{true text}\mmarg{false text}
%
% \medskip
% Issuing \verb$\MFPchk{\X}$ will check the sign of the number stored in
% the macro \cs{X}. Then \verb$\IFneg{A}{B}$ will produce `\verb$A$' if it
% is negative and `\verb$B$' if it is zero or positive. Similarly,
% \verb$\MFPcmp{\X}{1}$ will compare the number stored in \cs{X} to $1$.
% Afterward, \verb$\IFlt{A}{B}$ will produce `\verb$A$' if \cs{X} is less
% than $1$ and `\verb$B$' if \cs{X} is equal to or greater than $1$.
%
% If users finds it tiresome to type two separate commands, they can
% easily define a single command that both checks a value and runs
% \cs{IF...}. For example\\
% \indent\verb$\def\IFisneg#1{\MFPchk{#1}\IFneg}$\\
% Used like\\
% \indent\verb$\IFisneg{\X}{A}{B}$\\
% this will check the value of \cs{X} and run \cs{IFneg} on the pair of
% alternatives that follow.
%
% The user might never need to use \cs{MFPchk} because every one of the
% operators provided by the \mfp{} package runs an internal version of
% \cs{MFPchk} on the result of the operation before storing it in the
% \cs{macro}. For example, after \cs{MFPzero} the command \cs{IFzero} will
% always return the first argument. For this reason one should not insert
% any \mfp{} operations between a check/compare and the \cs{IF...} command
% that depends on it.
%
% Note: the behavior of all six \cs{IF...} commands is influenced by
% \emph{both} \cs{MFPchk} and \cs{MFPcmp}. This is because internally
% \verb$\MFPchk{\X}$ (for example) and \verb$\MFPcmp{\X}{0}$ do
% essentially the same thing. In fact there are only three internal
% booleans that govern the behavior of the six \cs{IF...} commands. The
% different names are for clarity: \cs{IFgt} after a compare is less
% confusing than the entirely equivalent \cs{IFpos}.
%
% It should probably be pointed out that the settings for the \cs{IF...}
% macros are local to any \TeX{} groups they are contained in.
%
%
% \subsection{Commands to process numbers for printing}
%
% After \verb$\MFPadd{1}{2}\X$ one can use \cs{X} anywhere and get
% $3.00000000$. One might may well prefer $3.0$, and so commands are
% provided to truncate a number or round it to some number of decimal
% places. Note: these are provided for printing and they will not invoke
% the above \cs{MFPchk}. They do not have any stack-based versions.
% The commands are\\
% \indent\SpecialUsageIndex{\MFPtruncate}\cs{MFPtruncate}\mmarg{int}\mmarg{num}\cs{macro}\\
% \indent\SpecialUsageIndex{\MFPround}\cs{MFPround}\mmarg{int}\mmarg{num}\cs{macro}\\
% \indent\SpecialUsageIndex{\MFPstrip}\cs{MFPstrip}\mmarg{num}\cs{macro}\\
% where \meta{int} is a whole number between $-8$ and $8$ (inclusive). The
% other two arguments are as before.
%
% These commands merely process \meta{num} and define \cs{macro} to
% produce a truncated or rounded version, or one stripped of trailing
% zeros, or one with added trailing zeros. Note that truncating or
% rounding a number to a number of digits greater than it already has will
% actually lengthen it with added zeros. For example,
%   \verb$\MFPround{4}{3.14159}\X$
% will cause \cs{X} to be defined to contain \texttt{3.1416}, while
%   \verb$\MFPround{6}{3.14159}\X$
% will cause \cs{X} to contain \texttt{3.141590}.
% If \cs{Y} contains \texttt{3.14159}, then
%   \verb$\MFPtruncate{4}\Y\Y$
% will redefine \cs{Y} to contain \texttt{3.1415}. Also,
%   \verb$\MFPstrip{1.20000000}\Z$
% will cause \cs{Z} to contain \texttt{1.2}. All these commands first
% normalize the \meta{num}, so any spaces are removed and redundant signs
% are discarded.
%
% If \meta{int} is negative, places are counted to the left of the decimal
% point and $0$\,s are substituted for lower order digits. That is,
%   \verb$\MFPtruncate{-2}{1864.3}\X$
% will give \cs{X} the value \texttt{1800} and
%   \verb$\MFPround{-2}{1864}\X$
% will give \cs{X} the value \texttt{1900}.
%
% If the first argument of \cs{MFPround} or \cs{MFPtruncate} is zero or
% negative then the dot is also omitted from the result. If \cs{MFPstrip} is
% applied to a number with all zeros after the dot, then one 0 is
% retained. There is a star form where the dot and the zero are dropped.
%
% For these three commands, the sign of the number is irrelevant. That
% is, the results for negative numbers are the negatives of the results
% for the absolute values. The processing will remove redundant signs
% along with redundant leading zeros: \verb$\MFPtruncate{-3}{-+123.456}$
% will produce \texttt{0}. The rounding rule is as follows: round up if
% the digit to the right of the rounding point is $5$ or more, round down if
% the digit is $4$ or less.
%
%
% \subsection{Stack-based macros}\label{stack}
%
% The stack-based macros can only be used in a \mfp{} program group.
% This group is started by the command \cs{startMFPprogram} and ended by
% \cs{stopMFPprogram}. None of the stack-based macros takes an argument,
% but merely operate on values on the stack, replacing them with the
% results. There are also commands to manipulate the stack and save a
% value on the stack into a macro. Finally, since all changes to the stack
% (and to macros) are local and therefore lost after \cs{stopMFPprogram},
% there are commands to selectively cause them to be retained.
%
% To place numbers on the stack we have \cs{Rpush} and to get them
% off we have \cs{Rpop}. The syntax is\\
% \indent \SpecialUsageIndex{\Rpush}\cs{Rpush}\marg{\meta{num}}\\
% \indent \SpecialUsageIndex{\Rpop}\cs{Rpop}\cs{macro}\\
% The first will preprocess the \meta{num} as previously discussed and
% put it on the stack, the second will remove the last number from the stack
% and define the given macro to have that number as its definition.
%
% All the binary operations remove the last two numbers from the stack,
% operate on them in the order they were put on the stack, and \op{push} the
% result on the stack. Thus the program
% \begin{verbatim}
%   \Rpush{1.2}
%   \Rpush{3.4}
%   \Rsub \end{verbatim}
% will first put \texttt{1.20000000} and \texttt{3.40000000} on the stack
% and then replace them with \texttt{-2.20000000}. Note the order: $1.2-3.4$.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{4.0in}}
%   \textit{Binary Operations}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Radd}\cs{Radd}&
%     Adds the last two numbers on the stack.\\
%   \SpecialUsageIndex{\Rsub}\cs{Rsub}&
%     Subtracts the last two numbers on the stack.\\
%   \SpecialUsageIndex{\Rmul}\cs{Rmul}&
%     Multiplies the last two numbers on the stack, rounding to 8 decimal
%     places.\\
%   \SpecialUsageIndex{\Rmpy}\cs{Rmpy}&
%     Same as \cs{Rmul}.\\
%   \SpecialUsageIndex{\Rdiv}\cs{Rdiv}&
%     Divides the last two numbers on the stack, rounding to 8 decimal
%     places.\\
%   \SpecialUsageIndex{\Rmin}\cs{Rmin}&
%     Replaces the last two elements on the stack with the smaller one.\\
%   \SpecialUsageIndex{\Rmax}\cs{Rmax}&
%     Replaces the last two elements on the stack with the larger one.
% \end{tabular}}
%
%\bigskip
%
% The unary operations replace the last number on the stack with the
% result of the operation performed on it.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{4.0in}}
%   \textit{Unary Operations}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Rchs}\cs{Rchs}&
%     Changes the sign.\\
%   \SpecialUsageIndex{\Rabs}\cs{Rabs}&
%     Obtains the absolute value.\\
%   \SpecialUsageIndex{\Rdbl}\cs{Rdbl}&
%     Doubles the value.\\
%   \SpecialUsageIndex{\Rhalve}\cs{Rhalve}&
%     Halves the value, rounding to 8 places.\\
%   \SpecialUsageIndex{\Rint}\cs{Rint}&
%     Replaces the fractional part with zeros. If the result equals $0.0$, any
%     negative sign will be dropped.\\
%   \SpecialUsageIndex{\Rfrac}\cs{Rfrac}&
%     Replaces the integer part with \texttt{0}. If the result equals
%     $0.0$, any negative sign will be dropped.\\
%   \SpecialUsageIndex{\Rfloor}\cs{Rfloor}&
%     Obtains the largest integer not greater than the number.\\
%   \SpecialUsageIndex{\Rceil}\cs{Rceil}&
%     Obtains the smallest integer not less than the number.\\
%   \SpecialUsageIndex{\Rsgn}\cs{Rsgn}&
%     Obtains $-1$, $0$ or $1$ according to whether the number
%     is negative, zero, or positive. These numbers are pushed onto the
%     stack with the usual decimal point followed by 8 zeros.\\
%   \SpecialUsageIndex{\Rsq}\cs{Rsq}&
%     Obtains the square. Slightly more efficient than the equivalent
%     \cs{Rdup}\cs{Rmul}. See below for \cs{Rdup}.\\
%   \SpecialUsageIndex{\Rinv}\cs{Rinv}&
%     Obtains the reciprocal. Slightly more efficient than the equivalent
%     division.\\
%   \SpecialUsageIndex{\Rincr}\cs{Rincr}&
%     Increases by $1$. Slightly more efficient than the equivalent
%     addition.\\
%   \SpecialUsageIndex{\Rdecr}\cs{Rdecr}&
%     Decreases by $1$. Slightly more efficient than the equivalent
%     subtraction.\\
%   \SpecialUsageIndex{\Rzero}\cs{Rzero}&
%     Replaces the number with zero. Slightly more convenient than the
%     equivalent \cs{Rpop}\cs{X} followed by a \cs{Rpush}\marg{0}.\\
% \end{tabular}}
%
%\bigskip
%
%
% There is one operation, which does not read the stack nor change it
% (nor do anything else).
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3.8in}}
%   \textit{Do Nothing}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Rnoop}\cs{Rnoop}&
%     Does nothing.
% \end{tabular}}
%
% \bigskip
% There also exist commands to check the sign of the last number, and the
% relative size of the last two numbers on the stack:\\
% \indent \SpecialUsageIndex{\Rchk}\cs{Rchk}\\
% \indent \SpecialUsageIndex{\Rcmp}\cs{Rcmp}\\
% They do not remove anything from the stack.
% Just like the nonstack counterparts, they influence the behavior of
% six commands: \cs{IFneg}, \cs{IFzero}, \cs{IFpos}, \cs{IFlt},
% \cs{IFeq} and \cs{IFgt}. Issuing \verb$\Rchk$ will check the sign of the
% last number on the stack, while \verb$\Rcmp$ will compare the last two
% in the order they were pushed. For example, in the following
% \begin{verbatim}
%   \Rpush{1.3}
%   \Rpush{-2.3}
%   \Rcmp
%   \IFgt{\Radd}{\Rsub}
%   \Rpush\X
%   \Rchk
%   \IFneg{\Radd}{\Rsub} \end{verbatim}
% \verb$\Rcmp$ will compare $1.3$ to $-2.3$. Since the first is greater
% than the second, \verb$\IFgt$ will be true and they will be added,
% producing $-1.0$. Following this the contents of the macro \cs{X} are
% pushed, it is examined by \verb$\Rchk$ and then either added to or
% subtracted from $-1.0$.
%
% The user might never need to use \cs{Rchk} because every operator that
% puts something on the stack also runs \cs{Rchk}. In the above program,
% in fact, \verb$\Rchk$ is redundant since \verb$\Rpush$ will already have
% run it on the contents of \cs{X}.
%
% There exist stack manipulation commands that allow the contents of the
% stack to be changed without performing any operations. These are really
% just conveniences, as there effects could be obtained with appropriate
% combinations of \verb$\Rpop$ and \verb$\Rpush$. These commands, however, do
% not run \verb$\Rchk$ as \cs{Rpush} would.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3.8in}}
%   \textit{Stack Manipulations}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Rdup}\cs{Rdup}&
%     Puts another copy of the last element of the stack onto the stack.\\
%   \SpecialUsageIndex{\Rexch}\cs{Rexch}&
%     Exchanges the last two elements on the stack.
% \end{tabular}}
%
% \bigskip
%
% After \cs{stopMFPprogram}, any changes to macros or to the stack are
% lost, unless arrangements have been made to save them. There are four
% commands provided. Two act on a macro which is the only argument, the
% other two have no arguments and act on the stack. The macro must
% simply contain a value, it cannot be more complicated and certainly
% cannot take an argument.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3.8in}}
%   \textit{Exporting changed values}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Export}\cs{Export}\cs{macro}&
%     \raggedright
%     Causes the definition of \cs{macro} to survive the
%     program group.\tabularnewline
%   \SpecialUsageIndex{\Global}\cs{Global}\cs{macro}&
%     Causes the definition of \cs{macro} to be global.\\
%   \SpecialUsageIndex{\ExportStack}\cs{ExportStack}&
%     \raggedright
%     Causes the contents of the stack to survive the program
%     group.\tabularnewline
%   \SpecialUsageIndex{\GlobalStack}\cs{GlobalStack}&
%     Causes the contents of the stack to be global.\\
% \end{tabular}}
%
% \bigskip
% The difference between \cs{Export} and \cs{Global} is solely in how
% \emph{other} grouping is handled. If the program group is contained in
% another group (for example, inside an environment), then the result of
% \cs{Global}\cs{X} is that the definition of \cs{X} survives that group
% (and all containing groups) as well. On the other hand, after
% \cs{Export}\cs{X}, then the definition survives the program group, but
% not other containing groups.
%
% If \TeX{} grouping is used \emph{inside} a program group, then using
% \cs{Export} inside that group has no effect at all, while \cs{Global}
% works as before.
%
% The stack versions are implemented by running \cs{Export} or
% \cs{Global} on the internal macro that defines the stack, so they
% have the same behavior.
%
% \subsection{Errors}
%
% If one tries to \op{pop} from an empty stack, an error message will be
% issued. Ignoring the error causes the macro to have the value stored
% in the macro \SpecialUsageIndex{\EndofStack}\verb$\EndofStack$.
% Its default is \texttt{0.00000000}.
%
% If one tries to divide by zero, an error message will be issued.
% Ignoring the error causes the result to be one of the following:
% \begin{itemize}
% \item Dividing $0$ by $0$ gives a result whose integer part is stored
%       in \verb$\ZeroOverZeroInt$\SpecialUsageIndex{\ZeroOverZeroInt}
%       and whose fractional part is stored in
%       \SpecialUsageIndex{\ZeroOverZeroFrac}\verb$\ZeroOverZeroFrac$.
%       The default is \texttt{0.00000000}
% \item Dividing a nonzero $x$ by $0$ gives a result whose integer part is
%       stored in \SpecialUsageIndex{\xOverZeroInt}\verb$\xOverZeroInt$
%       and whose fractional part is stored in
%       \SpecialUsageIndex{\xOverZeroFrac}\verb$\xOverZeroFrac$. The
%       defaults are both equal to \texttt{99999999}. The sign of the
%       result will be the sign of $x$.
% \end{itemize}
%
% You can change any of these macros, but make sure they produce a
% result which is a number in standard form (as described earlier).
% These macros are copied directly into the result without checking.
% Then further processing steps may require the result to be a number in
% standard form.
%
% Error messages may result from trying to process numbers given in
% incorrect format. However, there are so many ways for numbers to be
% incorrect that this package does not even try to detect them. Thus, they
% will only be caught if some \TeX{} operation encounters something it
% cannot handle. (The \LaTeX{} manual calls these ``weird errors'' because
% the messages tend to be uninformative.) Incorrectly formed numbers may even
% pass unnoticed, but leave unexpected printed characters on the paper, or odd
% spacing.
%
% \section{Implementation}
%
% \subsection{Utility macros}
%
% We announce ourself, and our purpose. We save the catcode of
% \texttt{@} and change it to letter. Several other catcodes are saved
% and set to other in this file. We also make provisions to load the
% extra definitions, either directly with \cs{MFPloadextra} or through a
% declared option in \LaTeX{}.
%    \begin{macrocode}
%<*sty>
\expandafter
\ifx \csname MFP@finish\endcsname\relax
\else \expandafter\endinput \fi
\expandafter\edef\csname MFP@finish\endcsname{%
  \catcode64=\the\catcode64 \space
  \catcode46=\the\catcode46 \space
  \catcode60=\the\catcode60 \space
  \catcode62=\the\catcode62 \space}%
\ifx\ProvidesPackage\UndEfInEd
  \newlinechar`\^^J%
  \message{%
      Package minifp: \MFPfiledate\space v\MFPfileversion. %
      Macros for real number operations %
      ^^Jand a stack-based programing language.^^J}%
\else
  \ProvidesPackage{minifp}[\MFPfiledate\space v\MFPfileversion. %
      Macros for real number operations %
      and a stack-based programing language.]%
  \DeclareOption{extra}{\def\MFPextra{}}%
  \ProcessOptions\relax
\fi
\catcode64=11
\ifx\MFPextra\UndEfInEd
  \def\MFP@loadextra{}%
\else
  \def\MFP@loadextra{\input mfpextra\relax}%
\fi
\def\MFPloadextra{\input mfpextra\relax}%
\catcode46=12
\catcode60=12
\catcode62=12
%    \end{macrocode}
%
% We check for \LaTeX{} (ignoring \LaTeX209); \cs{MFP@ifnoLaTeX}\dots\cs{MFP@end}
% is skipped in \LaTeX{} and executed otherwise.
%    \begin{macrocode}
\long\def\gobbleto@MFP@end#1\MFP@end{}%
\def\MFP@end{\@empty}%
\ifx\documentclass\UndEfInEd
  \def\MFP@ifnoLaTeX{}%
\else
  \let\MFP@ifnoLaTeX\gobbleto@MFP@end
\fi
%    \end{macrocode}
%
% We have \LaTeX{}'s \cs{zap@space}. It pretty much \emph{must} be used
% inside \cs{edef} or other purely expansion context. The rest of these
% are standard \LaTeX{} internals. Note that the token list that
% \cs{zap@space} is applied to should probably never contain braces or
% expandable tokens.\\
%  \indent Usage: \verb*$\edef\X{\zap@space<tokens> \@empty}$\\
% The space is necessary in case none exist; the \cs{@empty} terminates
% the loop.
%    \begin{macrocode}
\let\@xp\expandafter
\def\@XP{\@xp\@xp\@xp}%
\MFP@ifnoLaTeX
  \def\@empty{}%
  \long\def\@gobble#1{}%
  \def\zap@space#1 #2{#1\ifx#2\@empty\else\@xp\zap@space\fi#2}%
  \long\def\@ifnextchar#1#2#3{%
    \let\reserved@d#1%
    \def\reserved@a{#2}%
    \def\reserved@b{#3}%
    \futurelet\@let@token\@ifnch}%
  \def\@ifnch{%
    \ifx\@let@token\@sptoken
      \let\reserved@c\@xifnch
    \else
      \ifx\@let@token\reserved@d
        \let\reserved@c\reserved@a
      \else
        \let\reserved@c\reserved@b
      \fi
    \fi
    \reserved@c}%
  {%
    \def\:{\global\let\@sptoken= }\: %
    \def\:{\@xifnch}\@xp\gdef\: {\futurelet\@let@token\@ifnch}%
  }%
  \def\@ifstar#1{\@ifnextchar*{\@firstoftwo{#1}}}%
  \long\def\@firstofone #1{#1}%
  \long\def\@firstoftwo #1#2{#1}%
  \long\def\@secondoftwo#1#2{#2}%
\MFP@end
%    \end{macrocode}
%
% We need to divide by both $10^4$ and $10^8$ several times. I could
% have allocated two count registers, but have taken the approach of
% reserving those for intermediate calculations.
%    \begin{macrocode}
\def\MFP@tttfour {10000}% ttt = Ten To The
\def\MFP@ttteight{100000000}%
%    \end{macrocode}
%
% These are for manipulating digits. The \verb$\...ofmany$ commands
% require a sequence of arguments (brace groups or tokens) followed by
% \verb$\MFP@end$. The minimum number of required parameters is surely
% obvious. For example, \cs{MFP@ninthofmany} must be used like\\
% \indent\cs{MFP@ninthofmany}\meta{9 or more arguments}\cs{MFP@end}\\
% All these are fully expandable.
%    \begin{macrocode}
\def\MFP@oneofmany#1#2\MFP@end{#1}%
\def\MFP@fifthofmany#1#2#3#4#5#6\MFP@end{#5}%
\def\MFP@ninthofmany#1#2#3#4#5#6#7#8{\MFP@oneofmany}%
\def\MFP@eightofmany#1#2#3#4#5#6#7#8#9\MFP@end{#1#2#3#4#5#6#7#8}%
%    \end{macrocode}
%
% \subsection{Processing numbers and the stack}
%
% Our stack stores elements in groups, like \\
% \indent \verb${-1.234567890}{0.00001234}\MFP@eos$\\
% with an end marker. The purpose of the marker is to prevent certain
% parameter manipulations from stripping off braces. This means we can't
% use \cs{@empty} to test for an empty stack. At the moment, only
% \cs{Rpop} actually checks, but all other stack commands (so far) use
% \cs{Rpop} to get their arguments.
%    \begin{macrocode}
\let\MFP@eos\relax
\def\MFP@EOS{\MFP@eos}%
\def\MFP@initRstack{\def\MFP@Rstack{\MFP@eos}}%
\MFP@initRstack
%    \end{macrocode}
%
% Define some scratch registers for arithmetic operations. We don't care
% that these might be already in use, as we only use them inside a group.
% However, we need one counter that will not be messed with by any of
% our operations. I must be sure not to use commands that change
% \cs{MFP@loopctr} in code that depends on it.
%    \begin{macrocode}
\countdef \MFP@tempa 0
\countdef \MFP@tempb 2
\countdef \MFP@tempc 4
\countdef \MFP@tempd 6
\countdef \MFP@tempe 8
\countdef \MFP@tempf 10
\newcount \MFP@loopctr
%    \end{macrocode}
%
% The following can only be used where unrestricted expansion is robust.
% It will allow results obtained inside a group to survive the group,
% but not be unrestrictedly global.
% Example: the code\\
% \indent \verb$\MFP@endgroup@after{\def\noexpand\MFP@z@Val{\MFP@z@Val}}$\\
% becomes\\
% \indent \verb$\edef\x{\endgroup\def\noexpand\MFP@z@Val{\MFP@z@Val}}\x$\\
% which gives, upon expansion of \verb$\x$,\\
% \indent
%   \cs{endgroup}\cs{def}\cs{MFP@z@Val}\marg{\meta{expansion-of-\cs{MFP@z@Val}}}\\
% which defines \cs{MFP@z@Val} outside the current group to equal its expansion
% within the current group (provided the group was started with
% \cs{begingroup}).
%
% We define a \cs{MFP@returned@values} to make all the conceivable produced
% values survive the group. The \cs{MFPcurr@Sgn} part is to permit testing
% the sign of the result and allow conditional code based on it.
%
% I have been lax at making sure \cs{MFP@z@Ovr} is properly initiallized
% and properly checked whenever it could be relevant, and properly
% passed on. I think every internal command \cs{MFP@R}\textit{xxx}
% should ensure it starts being zero and ends with a numerical value. At
% one time division could leave it undefined.
%
% \cs{MFP@subroutine} executes its argument (typically a single command) with
% a wrapper that initializes all the macros that might need initializing,
% and returns the necessary results.
%    \begin{macrocode}
\def\MFP@endgroup@after#1{\edef\x{\endgroup#1}\x}%
\def\MFP@afterdef{\def\noexpand}%
\def\MFP@returned@values{%
  \MFP@afterdef\MFP@z@Val{\MFP@z@Sign\MFP@z@Int.\MFP@z@Frc}%
  \MFP@afterdef\MFP@z@Ovr{\MFP@z@Ovr}%
  \MFP@afterdef\MFP@z@Und{\MFP@z@Und}%
  \MFP@afterdef\MFPcurr@Sgn{\MFP@z@Sgn}}%
\def\MFP@subroutine#1{%
  \begingroup
    \MFP@Rzero
    \def\MFP@z@Ovr{0}%
    \def\MFP@z@Und{0}%
    #1%
  \MFP@endgroup@after\MFP@returned@values}%
\def\MFP@Rzero{%
  \def\MFP@z@Sgn{0}%
  \def\MFP@z@Int{0}%
  \def\MFP@z@Frc{00000000}}%
%    \end{macrocode}
%
% \DescribeMacro{\EndofStack}
% We define here the error messages: popping from an empty stack and
% dividing by zero. In addition to the error messages, we provide some
% default values that hopefully allow some operations to continue.
%
% We also have a warning or two.
%    \begin{macrocode}
\def\MFP@errmsg#1#2{%
\begingroup
  \newlinechar`\^^J\let~\space
  \def\MFP@msgbreak{^^J~~~~~~~~~~~~~~}%
  \edef\reserved@a{\errhelp{#2}}\reserved@a
  \errmessage{MiniFP error: #1}%
\endgroup}%
\def\MFP@popempty@err{%
    \MFP@errmsg{cannot POP from an empty stack}%
      {There were no items on the stack for the POP operation. %
       If you continue, ^^Jthe macro will contain the %
       value \EndofStack.}}%
\def\EndofStack{0.00000000}%
\def\MFP@dividebyzero@err{%
  \MFP@errmsg{division by zero}%
    {You tried to divide by zero. What were you thinking? %
     If you continue, ^^Jthe value assigned will be either %
     \ZeroOverZeroInt.\ZeroOverZeroFrac~(numerator=0) or %
     ^^J+/-\xOverZeroInt.\xOverZeroFrac~(numerator<>0).}}%
\def\MFP@warn#1{%
\begingroup
  \newlinechar`\^^J\let~\space
  \def\MFP@msgbreak{^^J~~~~~~~~~~~~~~~~}%
  \immediate\write16{^^JMiniFP warning: #1.^^J}%
\endgroup}%
%    \end{macrocode}
%
% \DescribeMacro{\MaxRealInt}These are the largest possible integer and
% fractional parts of a real
% \DescribeMacro{\MaxRealFrac}number. They are returned for division by
% zero, for logarithm of zero, and when overflow is detected in the
% exponential function.
%    \begin{macrocode}
\def\MaxRealInt      {99999999}%
\def\MaxRealFrac     {99999999}%
%    \end{macrocode}
%
% \SpecialUsageIndex{\MaxRealInt}
% \SpecialUsageIndex{\MaxRealFrac}
% These are the results returned when trying to divide by zero. Two are
% \DescribeMacro{\xOverZeroInt}
% \DescribeMacro{\xOverZeroFrac}
% used when dividing a nonzero number by zero and and two when trying to
% divide zero by zero.
% \DescribeMacro{\ZeroOverZeroInt}
% \DescribeMacro{\ZeroOverZeroFrac}
%    \begin{macrocode}
\def\xOverZeroInt    {\MaxRealInt}%
\def\xOverZeroFrac   {\MaxRealFrac}%
\def\ZeroOverZeroInt {0}%
\def\ZeroOverZeroFrac{00000000}%
%    \end{macrocode}
%
% These macros strip the spaces, process a number into sign, integer and
% fractional parts, and pad the fractional part out to eight decimals. They
% are used in \op{push} so that the stack will only contains reals in a
% normalized form. Some of them are also used to preprocess the reals in
% the operand versions of commands
%
% The \cs{MFP@*@Int} and \cs{MFP@*@Frc} parts are always positive, the sign being
% stored in \cs{MFP@*@Sgn} as $-1$, $0$ or $1$.
%
% We strip the spaces and pad the fractional parts separately because
% they are unnecessary when processing \op{pop}ped reals (though they wouldn't
% hurt).
%
% The number to be parsed is \arg4 and the macros to contain the parts
% are the first three arguments. Since we normally call \cs{MFPparse@real}
% with one of two sets of macros, we have two shortcuts for those cases.
%    \begin{macrocode}
\def\MFPparse@real#1#2#3#4{%
  \MFPnospace@def\MFPtemp@Val{#4}%
  \MFPprocess@into@parts\MFPtemp@Val#1#2#3%
  \MFP@padtoeight#3}%
\def\MFPparse@x{\MFPparse@real\MFP@x@Sgn\MFP@x@Int\MFP@x@Frc}%
\def\MFPparse@y{\MFPparse@real\MFP@y@Sgn\MFP@y@Int\MFP@y@Frc}%
%    \end{macrocode}
%
% This macro strips all spaces out of the number (not just before and
% after). It takes a macro that will hold the result, followed by the
% number (as a macro or a group of actual digits).
%    \begin{macrocode}
\def\MFPnospace@def#1#2{%
  \edef#1{#2\space}\edef#1{\@xp\zap@space#1\@empty}}%
%    \end{macrocode}
%
% This is the process that splits a number into parts. The biggest
% difficulty is obtaining the sign. All four arguments are macros, with
% the first one holding the number. Following that are the macros to hold
% the sign, integer and fractional parts.
%    \begin{macrocode}
\def\MFPprocess@into@parts#1#2#3#4{%
  \@xp\MFPsplit@dot#1..\MFP@end #3#4%
%    \end{macrocode}
%
% At this point \arg3 holds the part before the dot (or the whole thing
% if there was no dot) and \arg4 holds the part after the dot, (or
% nothing). Now is the first place where having at most eight digits
% simplifies things. Note that \arg3 could contain any number of
% consecutive signs followed by up to eight digits. It could be zero or
% empty, so to avoid losing the sign we append a \texttt{1} (for up to
% nine digits). We temporarily define the sign based on the result, but
% may need to drop it if both the integer and fractional parts are zero.
%
% Prepending a zero to the fractional part pemits it to be empty.
% In the final \cs{edef}, \arg3 is made positive.
%    \begin{macrocode}
  \ifnum#31<0 \def#2{-1}%
  \else \def#2{1}%
  \fi
  \ifnum #30=0
    \def#3{0}%
    \ifnum 0#4=0 \def#2{0}\fi
  \fi
  \edef#3{\number \ifnum #2<0 -\fi#3}}%
%    \end{macrocode}
%
% This only copies the parts before and after the dot, \arg1 and \arg2,
% into macros \arg4 and \arg5.
%    \begin{macrocode}
\def\MFPsplit@dot#1.#2.#3\MFP@end#4#5{\edef#4{#1}\edef#5{#2}}%
%    \end{macrocode}
%
% This is used to pad the fractional part to eight places with zeros. If
% a number with more than eight digits survives to this point, it gets
% truncated.
%    \begin{macrocode}
\def\MFP@padtoeight#1{%
  \edef#1{\@xp\MFP@eightofmany#100000000\MFP@end}}%
%    \end{macrocode}
%
% These take operands off the stack. We know already that there are no
% spaces and that the fractional part has eight digits.
%    \begin{macrocode}
\def\MFPgetoperand@x{\Rpop\MFP@x@Val
  \MFPprocess@into@parts\MFP@x@Val\MFP@x@Sgn\MFP@x@Int\MFP@x@Frc}%
\def\MFPgetoperand@y{\Rpop\MFP@y@Val
  \MFPprocess@into@parts\MFP@y@Val\MFP@y@Sgn\MFP@y@Int\MFP@y@Frc}%
%    \end{macrocode}
%
% Concatenate an argument (or two) to the front of stack. The material
% must already be in correct format. Note: `front' is where they go
% visually (i.e., leftmost) but it can be useful to imagine the stack
% growin rightward (or sometimes even downward).
%
% Note that the result of \verb$\MFP@cattwo{#1}{#2}$ is the same as
% \verb$\MFP@cat{#2}$ followed by \verb$\MFP@cat{#1}$. It seemed that
% reversing the arguments in \cs{MFP@Rcattwo} confused me more than this
% fact.
%    \begin{macrocode}
\def\MFP@Rcat#1{\edef\MFP@Rstack{{#1}\MFP@Rstack}}%
\def\MFP@Rcattwo#1#2{\edef\MFP@Rstack{{#1}{#2}\MFP@Rstack}}%
%    \end{macrocode}
%
% Convert from a signum (a number) to a sign ($-$ or nothing):
%    \begin{macrocode}
\def\MFP@Sign#1{\ifnum#1<0 -\fi}%
\def\MFP@x@Sign{\MFP@Sign\MFP@x@Sgn}%
\def\MFP@y@Sign{\MFP@Sign\MFP@y@Sgn}%
\def\MFP@z@Sign{\MFP@Sign\MFP@z@Sgn}%
%    \end{macrocode}
%
% Sometimes only parts of the number needs changing (used in CHS, ABS).
% This copies the integer and fractional parts of $x$ into $z$.
%    \begin{macrocode}
\def\copyMFP@x{\edef\MFP@z@Int{\MFP@x@Int}\edef\MFP@z@Frc{\MFP@x@Frc}}%
%    \end{macrocode}
%
% We use \cs{MFPpush@result} to put the result of internal operations onto
% the stack. For convenience, we also have it set the sign flags.
%    \begin{macrocode}
\def\MFPpush@result{\MFP@Rchk\MFPcurr@Sgn\MFP@Rcat\MFP@z@Val}%
%    \end{macrocode}
%
% When \op{pop} encounters an empty stack it gobbles the code that would
% perform the \op{pop} (\arg1) and defines the macro (\arg2) to contain
% \cs{EndofStack}. The default meaning for this macro is $0$.
%    \begin{macrocode}
\def\if@EndofStack{%
  \ifx\MFP@EOS\MFP@Rstack
    \@xp\@firstoftwo
  \else
    \@xp\@secondoftwo
  \fi}%
%    \end{macrocode}
%
% The macro \cs{Rpop} calls \cs{MFP@popit} followed by the contents of the
% stack, the token \cs{MFP@end} and the macro to \op{pop} into. If the stack is
% not empty, \cs{doMFP@popit} will read the first group \arg1 into that macro
% \arg3, and then redefine the stack to be the rest of the argument \arg2.
% If the stack is empty, \cs{doMFP@EOS} will equate the macro to
% \cs{EndofStack} (initialized to {\tt0.00000000}) after issuing an error
% message.
%    \begin{macrocode}
\def\MFP@popit{\if@EndofStack\doMFP@EOS\doMFP@popit}%
\def\doMFP@EOS#1\MFP@end#2{\MFP@popempty@err\let#2\EndofStack}%
\def\doMFP@popit#1#2\MFP@end#3{\edef\MFP@Rstack{#2}\edef#3{#1}}%
%    \end{macrocode}
%
% \subsection{The user-level operations}
%
% All operations that can be done on arguments as well as the stack will
% have a common format: The stack version pops the requisite numbers and
% splits them into internal macros (\cs{MFPgetoperand@*}), runs an internal
% command that operates on these internal macros, then ``pushes'' the result
% returned. The internal commands take care to return the result in proper
% form so we don't actually run \cs{Rpush}, but only \cs{MFPpush@result}.
%
% The operand version processes the operands into normalized form (as if
% pushed, using \cs{MFPparse@*}), then proceeds as in the stack version, but
% copies the result into the named macro instead of to the stack
% (\cs{MFPstore@result}).
%
% For unary operations we process one argument or stack element. We call
% it $x$ and use the \texttt{x} version of all macros. All internal
% commands (\arg1) return the results in \texttt{z} versions.
%
% \DescribeMacro{\MFPchk}
% The \cs{MFPchk} command examines its argument and sets a flag according to
% its sign.
%    \begin{macrocode}
\def\MFPchk#1{%
  \MFPparse@x{#1}%
  \MFP@Rchk\MFP@x@Sgn}%
%    \end{macrocode}
%
% We make \cs{MFP@Rchk} a little more general than is strictly needed here,
% by giving it an argument (instead of only examining \cs{MFP@x@Sgn}). This is
% so we can apply it to the results of operations (which would be in
% \cs{MFPcurr@Sgn}).
%    \begin{macrocode}
\def\MFP@Rchk#1{%
  \MFPclear@flags
  \ifnum#1>0  \MFP@postrue
  \else\ifnum#1<0  \MFP@negtrue
  \else  \MFP@zerotrue
  \fi\fi}%
\def\MFPclear@flags{\MFP@zerofalse \MFP@negfalse \MFP@posfalse}%
%    \end{macrocode}
%
% \DescribeMacro{\IFzero}
% \DescribeMacro{\IFneg}
% \DescribeMacro{\IFpos}
% These are the user interface to the internal \cs{ifMFP@zero},
% \cs{ifMFP@neg}, \cs{ifMFP@pos}
%    \begin{macrocode}
\def\IFzero{\ifMFP@zero\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
\def\IFneg {\ifMFP@neg \@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
\def\IFpos {\ifMFP@pos \@xp\@firstoftwo\else\@xp\@secondoftwo\fi}%
\newif\ifMFP@zero \newif\ifMFP@neg \newif\ifMFP@pos
%    \end{macrocode}
%
% Our comparison commands parallel the check-sign commands. They even
% \DescribeMacro{\MFPcmp}
% reuse the same internal booleans. The differences: the internal
% \DescribeMacro{\IFeq}
% \cs{MFP@Rcmp} doesn't take arguments and the comparison test is a little
% \DescribeMacro{\IFlt}
% more involved. We could simply subtract, which automatically sets the
% \DescribeMacro{\IFgt}
% internal booleans, but it is way more efficient to compare sizes
% directly.
%    \begin{macrocode}
\newif\ifMFPdebug
\def\MFPcmp#1#2{\MFPparse@x{#1}\MFPparse@y{#2}\MFP@Rcmp}%
\def\MFP@Rcmp{\MFPclear@flags
  \ifnum \MFP@x@Sign\MFP@x@Int>\MFP@y@Sign\MFP@y@Int\relax
    \MFP@postrue
  \else\ifnum \MFP@x@Sign\MFP@x@Int<\MFP@y@Sign\MFP@y@Int\relax
    \MFP@negtrue
  \else\ifnum \MFP@x@Sign\MFP@x@Frc>\MFP@y@Sign\MFP@y@Frc\relax
    \MFP@postrue
  \else\ifnum \MFP@x@Sign\MFP@x@Frc<\MFP@y@Sign\MFP@y@Frc\relax
    \MFP@negtrue
  \else
    \MFP@zerotrue
  \fi\fi\fi\fi}%
\let\IFeq\IFzero\let\IFlt\IFneg \let\IFgt\IFpos
%    \end{macrocode}
%
% Given an operation (\op{pop}, \op{chs}, or whatever), the stack version will
% have the same name with ``\texttt{R}'' (for ``real'') prepended. The operand
% versions will have the same name with ``\texttt{MFP}'' prepended. The
% internal version has the same name as the stack version, with an
% ``\texttt{MFP@}'' prepended.
%
% The unary operations are:
% \begin{description}
%   \item[chs]   change sign of $x$.
%   \item[abs]   absolute value of $x$.
%   \item[dbl]   double $x$.
%   \item[halve] halve $x$.
%   \item[sgn]   $+1$, $-1$ or $0$ depending on the sign of $x$.
%   \item[sq]    square $x$.
%   \item[int]   zero out the fractional part of $x$.
%   \item[frac]  zero out the integer part of $x$.
%   \item[floor] largest integer not exceeding $x$.
%   \item[ceil]  smallest integer not less than $x$.
% \end{description}
%
% The binary operations are ($x$ represents the first and $y$ second):
% \begin{description}
%   \item[add]   add $x$ and $y$.
%   \item[sub]   subtract $y$ from $x$.
%   \item[mul]   multiply $x$ and $y$.
%   \item[div]   divide $x$ by $y$.
% \end{description}
%
% There are also some operations that do not actually change any
% values, but may change the stack or the state of some boolean:
% \begin{description}
%   \item[cmp] compare $x$ and $y$ (stack version does not change stack).
%   \item[chk] examine the sign of $x$ (stack version does not change stack).
%   \item[dup] stack only, duplicate the top element of the stack.
%   \item[push] stack only, put a value onto the top of the stack.
%   \item[pop] stack only, remove the top element of the stack,
%              store it in a variable.
%   \item[exch] stack only, exchange top two elements of the stack.
% \end{description}
%
% \DescribeMacro{\startMFPprogram}
% The purpose of \cs{startMFPprogram} is to start the group, inside of
% which all the stack operations can be used. The ensuing
% \DescribeMacro{\stopMFPprogram}
% \cs{stopMFPprogram} closes the group.
%    \begin{macrocode}
\def\startMFPprogram{%
\begingroup
%    \end{macrocode}
%
% \DescribeMacro{\Rchs}
% \DescribeMacro{\Rabs}
% \DescribeMacro{\Rdbl}
% \DescribeMacro{\Rhalve}
% \DescribeMacro{\Rsgn}
% Then  we give definitions to all the stack-based macros.
% These definitions are all lost after the group ends.
%
% \DescribeMacro{\Rsq}
% \DescribeMacro{\Rinv}
% \DescribeMacro{\Rint}
% \DescribeMacro{\Rfrac}
% \DescribeMacro{\Rfloor}
% \DescribeMacro{\Rceil}
% \DescribeMacro{\Rincr}
% \DescribeMacro{\Rdecr}
% \DescribeMacro{\Rzero}
% We start with the unary operations. Note that all they do is call a
% wrapper macro \cs{MFP@stack@Unary} with an argument which is the internal
% version of the command.
%    \begin{macrocode}
  \def\Rchs  {\MFP@stack@Unary\MFP@Rchs}%
  \def\Rabs  {\MFP@stack@Unary\MFP@Rabs}%
  \def\Rdbl  {\MFP@stack@Unary\MFP@Rdbl}%
  \def\Rhalve{\MFP@stack@Unary\MFP@Rhalve}%
  \def\Rsgn  {\MFP@stack@Unary\MFP@Rsgn}%
  \def\Rsq   {\MFP@stack@Unary\MFP@Rsq}%
  \def\Rinv  {\MFP@stack@Unary\MFP@Rinv}%
  \def\Rint  {\MFP@stack@Unary\MFP@Rint}%
  \def\Rfrac {\MFP@stack@Unary\MFP@Rfrac}%
  \def\Rfloor{\MFP@stack@Unary\MFP@Rfloor}%
  \def\Rceil {\MFP@stack@Unary\MFP@Rceil}%
  \def\Rincr {\MFP@stack@Unary\MFP@Rincr}%
  \def\Rdecr {\MFP@stack@Unary\MFP@Rdecr}%
  \def\Rzero {\MFP@stack@Unary\MFP@Rzero}%
%    \end{macrocode}
%
% \DescribeMacro{\Radd}
% \DescribeMacro{\Rsub}
% \DescribeMacro{\Rmul}
% \DescribeMacro{\Rmpy}
% \DescribeMacro{\Rdiv}
% \DescribeMacro{\Rmin}
% \DescribeMacro{\Rmax}
% Then the binary operations, which again call a wrapper around
% the internal version.
%    \begin{macrocode}
  \def\Radd  {\MFP@stack@Binary\MFP@Radd}%
  \def\Rmul  {\MFP@stack@Binary\MFP@Rmul}%
  \let\Rmpy\Rmul
  \def\Rsub  {\MFP@stack@Binary\MFP@Rsub}%
  \def\Rdiv  {\MFP@stack@Binary\MFP@Rdiv}%
  \def\Rmin  {\MFP@stack@Binary\MFP@Rmin}%
  \def\Rmax  {\MFP@stack@Binary\MFP@Rmax}%
%    \end{macrocode}
%
% \DescribeMacro{\Rnoop}
% \DescribeMacro{\Rcmp}
% \DescribeMacro{\Rchk}
% \DescribeMacro{\Rpush}
% \DescribeMacro{\Rpop}
% \DescribeMacro{\Rexch}
% \DescribeMacro{\Rdup}
% And finally some special commands. There is a no-op and commands for
% comparing, checking, and manipulation of the stack. Note that
% \cs{Rcmp} parses the last two elements on the stack, then puts them back
% before calling the internal command that operates on the parsed parts.
% The same is true of \cs{Rchk}, but only the last stack element is
% examined.
%    \begin{macrocode}
  \let\Rnoop\relax
  \def\Rcmp{%
    \MFPgetoperand@y\MFPgetoperand@x
    \MFP@Rcattwo\MFP@y@Val\MFP@x@Val
    \MFP@Rcmp}%
  \def\Rchk{%
    \MFPgetoperand@x
    \MFP@Rcat\MFP@x@Val
    \MFP@Rchk\MFP@x@Sgn}%
  \def\Rpush##1{%
    \MFPparse@x{##1}%
    \edef\MFP@z@Val{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}%
    \edef\MFPcurr@Sgn{\MFP@x@Sgn}%
    \MFPpush@result}%
  \def\Rpop{\@xp\MFP@popit\MFP@Rstack\MFP@end}%
  \def\Rexch{%
    \Rpop\MFP@y@Val\Rpop\MFP@x@Val
    \MFP@Rcattwo\MFP@x@Val\MFP@y@Val}%
  \def\Rdup{%
    \Rpop\MFP@x@Val
    \MFP@Rcattwo\MFP@x@Val\MFP@x@Val}%
%    \end{macrocode}
%
% If \file{mfpextra.tex} is input, then \cs{MFP@Rextra} makes the
% additional commands in that file available to an \mfp{} program.
%
% \DescribeMacro{\Global}
% \DescribeMacro{\GlobalStack}
% \DescribeMacro{\Export}
% \DescribeMacro{\ExportStack}
% The last four commands allow computed values to be made available
% outside the program group
%    \begin{macrocode}
  \MFP@Rextra
  \let\Global\MFP@Global
  \let\GlobalStack\MFP@GlobalStack
  \let\Export\MFP@Export
  \let\ExportStack\MFP@ExportStack}%
\def\stopMFPprogram{\@xp\endgroup\MFPprogram@returns}%
\let\MFP@Rextra\@empty
\let\MFPprogram@returns\@empty
%    \end{macrocode}
%
% \DescribeMacro{\MFPchs}
% \DescribeMacro{\MFPabs}
% \DescribeMacro{\MFPdbl}
% \DescribeMacro{\MFPhalve}
% \DescribeMacro{\MFPsgn}
% \DescribeMacro{\MFPsq}
% \DescribeMacro{\MFPinv}
% Now we define the operand versions. These also are defined via a
% wrapper command that executes the very same internal commands as the
% stack versions.
%
% \DescribeMacro{\MFPint}
% \DescribeMacro{\MFPfrac}
% \DescribeMacro{\MFPfloor}
% \DescribeMacro{\MFPceil}
% \DescribeMacro{\MFPincr}
% \DescribeMacro{\MFPdecr}
% \DescribeMacro{\MFPzero}
% \DescribeMacro{\MFPstore}
% First the unary operations.
%    \begin{macrocode}
\def\MFPchs  {\MFP@op@Unary\MFP@Rchs}%
\def\MFPabs  {\MFP@op@Unary\MFP@Rabs}%
\def\MFPdbl  {\MFP@op@Unary\MFP@Rdbl}%
\def\MFPhalve{\MFP@op@Unary\MFP@Rhalve}%
\def\MFPsgn  {\MFP@op@Unary\MFP@Rsgn}%
\def\MFPsq   {\MFP@op@Unary\MFP@Rsq}%
\def\MFPinv  {\MFP@op@Unary\MFP@Rinv}%
\def\MFPint  {\MFP@op@Unary\MFP@Rint}%
\def\MFPfrac {\MFP@op@Unary\MFP@Rfrac}%
\def\MFPfloor{\MFP@op@Unary\MFP@Rfloor}%
\def\MFPceil {\MFP@op@Unary\MFP@Rceil}%
\def\MFPincr {\MFP@op@Unary\MFP@Rincr}%
\def\MFPdecr {\MFP@op@Unary\MFP@Rdecr}%
\def\MFPzero {\MFP@op@Unary\MFP@Rzero}%
\def\MFPstore{\MFP@op@Unary\MFP@Rstore}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPadd}
% \DescribeMacro{\MFPsub}
% \DescribeMacro{\MFPmul}
% \DescribeMacro{\MFPmpy}
% \DescribeMacro{\MFPdiv}
% \DescribeMacro{\MFPmin}
% \DescribeMacro{\MFPmax}
% Then the binary operations.
%    \begin{macrocode}
\def\MFPadd{\MFP@op@Binary\MFP@Radd}%
\def\MFPmul{\MFP@op@Binary\MFP@Rmul}%
\let\MFPmpy\MFPmul
\def\MFPsub{\MFP@op@Binary\MFP@Rsub}%
\def\MFPdiv{\MFP@op@Binary\MFP@Rdiv}%
\def\MFPmin{\MFP@op@Binary\MFP@Rmin}%
\def\MFPmax{\MFP@op@Binary\MFP@Rmax}%
%    \end{macrocode}
%
% A \emph{nullary} operation is one that produces a result with no
% operand. Thus, it could return a fixed constant, or it could perform
% calculations that obtain input from the system (e.g., current time). At
% the moment we don't define any.
%    \begin{macrocode}
\def\MFP@stack@Nullary#1{%
  \MFP@subroutine{#1}\MFPpush@result}%
\def\MFP@op@Nullary#1{%
  \MFP@subroutine{#1}\MFPstore@result}%
%    \end{macrocode}
%
% These are the wrappers for unary operations. The operand versions have a
% second argument, the macro that stores the result. But this will be the
% argument of \cs{MFPstore@result}.
%    \begin{macrocode}
\def\MFP@stack@Unary#1{%
  \MFPgetoperand@x
  \MFP@subroutine{#1}\MFPpush@result}%
\def\MFP@op@Unary#1#2{%
  \MFPparse@x{#2}%
  \MFP@subroutine{#1}\MFPstore@result}%
\def\MFPstore@result#1{\MFP@Rchk\MFPcurr@Sgn\edef#1{\MFP@z@Val}}%
%    \end{macrocode}
%
% These are the wrappers for binary operations. The top level definitions
% are almost identical to those of the unary operations. The only difference
% is they \op{pop} or parse two operands.
%    \begin{macrocode}
\def\MFP@stack@Binary#1{%
  \MFPgetoperand@y \MFPgetoperand@x
  \MFP@subroutine{#1}\MFPpush@result}%
\def\MFP@op@Binary#1#2#3{%
  \MFPparse@x{#2}\MFPparse@y{#3}%
  \MFP@subroutine{#1}\MFPstore@result}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPnoop}
% We end with a traditional, but generally useless command, the no-op,
% which does nothing. It doesn't even have a wrapper.
%    \begin{macrocode}
\let\MFPnoop\relax
%    \end{macrocode}
%
% \subsection{The internal computations}
%
% To change the sign or get the absolute value, we just need to set the
% value of \cs{MFP@x@Sgn}.
%    \begin{macrocode}
\def\MFP@Rabs{%
  \copyMFP@x \edef\MFP@z@Sgn{\ifnum\MFP@x@Sgn=0 0\else1\fi}}%
\def\MFP@Rchs{\copyMFP@x \edef\MFP@z@Sgn{\number-\MFP@x@Sgn}}%
%    \end{macrocode}
%
% The doubling and halving operations are more efficient ways to
% multiply or divide a number by $2$. For doubling, copy $x$ to $y$
% and add. For halving, we use basic \TeX{} integer division, more
% efficient than multiplying by $0.5$ and far more than using
% \cs{MFP@Rdiv}.
%
% In \cs{MFP@Rhalve}. we add $1$ to the fractional part for rounding
% purposes, and we move any odd 1 from the end of the integer part to the
% start of the fractional part.
%    \begin{macrocode}
\def\MFP@Rdbl{\MFP@Rcopy xy\MFP@Radd}%
\def\MFP@Rhalve{%
  \MFP@tempa\MFP@x@Int
  \MFP@tempb\MFP@x@Frc\relax
  \ifodd\MFP@tempb
    \def\MFP@z@Und{5}%
    \advance\MFP@tempb 1
    \ifnum\MFP@ttteight=\MFP@tempb
       \MFP@tempb0 \advance\MFP@tempa1
    \fi
  \fi
  \ifodd \MFP@tempa
    \advance\MFP@tempb \MFP@ttteight\relax
  \fi
  \divide\MFP@tempa 2
  \divide\MFP@tempb 2
  \MFP@Rloadz\MFP@x@Sgn\MFP@tempa\MFP@tempb}%
%    \end{macrocode}
%
% The signum is $0.0$, $1.0$ or $-1.0$ to match the sign of $x$.
%    \begin{macrocode}
\def\MFP@Rsgn{\MFP@Rloadz \MFP@x@Sgn{\ifnum\MFP@x@Sgn=0 0\else1\fi}0}%
%    \end{macrocode}
%
% The squaring operation just calls \cs{MFP@Rmul} after copying $x$ to
% $y$. Its gain in efficiency over a multiplication is that it can skip
% preprocessing of the second (identical) operand.
%    \begin{macrocode}
\def\MFP@Rsq{\MFP@Rcopy xy\MFP@Rmul}%
%    \end{macrocode}
%
% The inversion operation just calls \cs{MFP@Rdiv} after copying $x$ to
% $y$ and $1$ to $x$. Its advantage over a divide is it skips the
% preprocessing of $1$ as an operand.
%    \begin{macrocode}
\def\MFP@Rinv{\MFP@Rcopy xy\MFP@Rload x110\MFP@Rdiv}%
%    \end{macrocode}
%
% Integer part: replace fractional part with zeros.
%    \begin{macrocode}
\def\MFP@Rint{%
  \MFP@Rloadz {\ifnum\MFP@x@Int=0 0\else\MFP@x@Sgn\fi}\MFP@x@Int 0}%
%    \end{macrocode}
%
% Fractional part: replace integer part with a zero.
%    \begin{macrocode}
\def\MFP@Rfrac{%
  \MFP@Rloadz {\ifnum\MFP@x@Frc=0 0\else\MFP@x@Sgn\fi}0\MFP@x@Frc}%
%    \end{macrocode}
%
% To increment and decrement by $1$, except in border cases, we need only
% address the integer part of a number. This doesn't seem so simple
% written out but, even so, it is more efficient than full-blown addition.
% It would be very slightly more efficient if \cs{MFP@Rdecr} did not call
% \cs{MFP@Rincr}, but instead was similarly coded.
%    \begin{macrocode}
\def\MFP@Rincr{%
 \ifnum\MFP@x@Sgn<0
   \ifcase\MFP@x@Int
     \MFP@tempa\MFP@ttteight
     \advance\MFP@tempa -\MFP@x@Frc\relax
     \MFP@Rloadz 10\MFP@tempa
   \or
      \MFP@Rloadz{\ifnum\MFP@x@Frc=0 0\else -1\fi}0\MFP@x@Frc
   \else
     \MFP@tempa\MFP@x@Int
     \advance\MFP@tempa -1
     \MFP@Rloadz{-1}\MFP@tempa\MFP@x@Frc
   \fi
 \else
   \MFP@tempa\MFP@x@Int
   \advance\MFP@tempa 1
   \MFP@Rloadz 1\MFP@tempa\MFP@x@Frc
 \fi}%
\def\MFP@Rdecr{%
  \edef\MFP@x@Sgn{\number -\MFP@x@Sgn}\MFP@Rincr
  \edef\MFP@z@Sgn{\number -\MFP@z@Sgn}}%
\def\MFP@Rstore{\MFP@Rcopy xz}%
%    \end{macrocode}
%
% The floor of a real number $x$ is the largest integer not larger than
% $x$. The ceiling is the smallest integer not less than $x$. For
% positive $x$, floor is the same as integer part. Not true for negative
% $x$. Example: $\mathop{\mathrm{int}}(-1.5) = -1$ but
% $\mathop{\mathrm{floor}}=-2$
%
% We use the same code to get floor or ceiling, the
% appropriate inequality character being its argument.
%    \begin{macrocode}
\def\MFP@Rfloorceil#1{%
  \MFP@tempa\MFP@x@Int\relax
  \ifnum \MFP@x@Sgn #10
    \ifnum\MFP@x@Frc=0
    \else
      \advance\MFP@tempa1
    \fi
  \fi
  \MFP@Rloadz{\ifnum\MFP@x@Int=0 0\else\MFP@x@Sgn\fi}\MFP@tempa0}%
\def\MFP@Rfloor{\MFP@Rfloorceil<}%
\def\MFP@Rceil {\MFP@Rfloorceil>}%
%    \end{macrocode}
%
% For multiplication, after the usual break into integer and fractional
% parts, we further split these parts into $4$-digit pieces with
% \cs{MFP@split}. The first argument (\arg1) holds the eight digit number,
% then \arg2 is a macro that will hold the top four digits and \arg3 will
% hold the bottom four.
%    \begin{macrocode}
\def\MFP@split#1#2#3{%
  \begingroup
    \MFP@tempa#1\relax
    \MFP@tempb\MFP@tempa
    \divide\MFP@tempb by\MFP@tttfour
    \edef#2{\number\MFP@tempb}%
    \multiply\MFP@tempb by\MFP@tttfour
    \advance\MFP@tempa-\MFP@tempb
  \MFP@endgroup@after{%
    \MFP@afterdef#2{#2}%
    \MFP@afterdef#3{\number\MFP@tempa}%
  }}%
%
\def\MFP@@split{%
  \MFP@split\MFP@x@Int\MFP@x@Int@ii\MFP@x@Int@i
  \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii
  \MFP@split\MFP@y@Int\MFP@y@Int@ii\MFP@y@Int@i
  \MFP@split\MFP@y@Frc\MFP@y@Frc@i\MFP@y@Frc@ii}%
%    \end{macrocode}
%
% We will store the intermediate and final products in \cs{MFP@z@*}. Each one
% is ultimately reduced to four digits, like the parts of $x$ and $y$. As each
% base-$10000$ digit of $y$ is multiplied by a digit of $x$, we add the
% result to the appropriate digit of the partial result $z$.
%
% The underflow ends up in \cs{MFP@z@Frc@iv} and \cs{MFP@z@Frc@iii}.
% Overflow will be in \cs{MFP@z@Int@iii}. Unlike the rest, it can be up to
% eight digits because we do not need to carry results out of it.
%
% This command prepends zeros so a number fills four slots. Here \arg1 is
% a macro holding the value and it is redefined to contain the result. A
% macro that calls this should ensure that \arg1 is not empty and is less
% than 10,000.
%    \begin{macrocode}
\def\makeMFP@fourdigits#1{%
  \edef#1{\@xp\MFP@fifthofmany\number#1{}{0}{00}{000}\MFP@end\number#1}}%
%    \end{macrocode}
%
% This is the same, but produces eight digits. Similarly \arg1 should be
% nonempty and less than 100,000,000.
%    \begin{macrocode}
\def\makeMFP@eightdigits#1{%
  \edef#1{\@xp\MFP@ninthofmany\number#1%
    {}{0}{00}{000}{0000}{00000}{000000}{0000000}\MFP@end\number#1}}%
%    \end{macrocode}
%
% The following macros implement carrying. The macros \cs{MFP@carrya} and
% \cs{MFP@carrym} should be followed by two macros that hold numbers. The
% first number can have too many digits. These macros remove extra digits
% from the front and add their value to the number in the second macro
% (the ``carry''). Both act by calling \cs{MFP@carry}, which is told the
% number of digits to keep via \arg1 (10,000 for four digits,
% 100,000,000 for eight). The ``\texttt{a}'' in \cs{MFP@carrya} is for
% addition and ``\texttt{m}'' is for multiplication, which indicates where
% these will mainly be used.
%    \begin{macrocode}
\def\MFP@carrya{\MFP@carry\MFP@ttteight}%
\def\MFP@carrym{\MFP@carry\MFP@tttfour}%
\def\MFP@carry#1#2#3{%
  \begingroup
    \MFP@carryi{#1}#2#3%
  \MFP@endgroup@after{%
    \MFP@afterdef#3{\number\MFP@tempa}%
    \MFP@afterdef#2{\number\MFP@tempb}%
  }}%
%    \end{macrocode}
%
% This is the ``internal'' carry. \arg1, \arg2, and \arg3 are as in
% \cs{MFP@carry}. Its advantage is that it can be used used where \arg2
% and \arg3 are not macros, leaving the result in \cs{MFP@tempa} and
% \cs{MFP@tempb} with \cs{MFP@tempb} in the correct range,
% $[0,\mbox{\arg1})$. Its disadvantage is it does not protect temporary
% registers. Warning: do not use it with \arg2=\cs{MFP@tempa} and do not
% use it without grouping if you want to preserve the values in these
% temporary count registers.
%    \begin{macrocode}
\def\MFP@carryi#1#2#3{%
    \MFP@tempa=#3\relax
    \MFP@tempb=#2\relax
    \MFP@tempc=\MFP@tempb
    \divide  \MFP@tempc #1\relax
    \advance \MFP@tempa \MFP@tempc
    \multiply\MFP@tempc #1\relax
    \advance \MFP@tempb -\MFP@tempc}%
%    \end{macrocode}
%
% This adds \arg1 to \arg2, the result goes into macro \arg3. This does no
% checking. It is basicly used to add with macros instead of registers.
%    \begin{macrocode}
\def\MFP@addone#1#2#3{%
  \begingroup
    \MFP@tempa#1%
    \advance\MFP@tempa#2\relax
  \MFP@endgroup@after{%
    \MFP@afterdef#3{\number\MFP@tempa}%
  }}%
%    \end{macrocode}
%
% Multiply \arg1 by \cs{MFP@tempb} and add to \arg2. \cs{MFP@tempb} is one digit
% (base=10000) of $y$ in multiplying $x\times y$, \arg1 (usually a macro)
% holds one digit of $x$. \arg2 is a macro that will hold one digit of the
% final product $z$. The product is added to it (overflow is taken care of
% later by the carry routines).
%    \begin{macrocode}
\def\MFP@multiplyone#1#2{%
  \MFP@tempa#1%
  \multiply\MFP@tempa\MFP@tempb
  \advance\MFP@tempa#2%
  \edef#2{\number\MFP@tempa}}%
%    \end{macrocode}
%
% This does the above multiplication-addition for all four ``digits'' of
% $x$. This is where \cs{MFP@tempb} is initialized for \cs{MFP@multiplyone}. The
% first argument represents a digit of $y$, the remaining four arguments
% are macros representing the digits of $z$ that are involved in
% multiplying the digits of $x$ by \arg1.
%    \begin{macrocode}
\def\MFP@multiplyfour#1#2#3#4#5{%
  \MFP@tempb #1\relax
  \MFP@multiplyone\MFP@x@Int@ii #2%
  \MFP@multiplyone\MFP@x@Int@i  #3%
  \MFP@multiplyone\MFP@x@Frc@i  #4%
  \MFP@multiplyone\MFP@x@Frc@ii #5}%
%    \end{macrocode}
%
% Now we begin the internal implementations of the binary operations. All
% four expect macros \cs{MFP@x@Sgn}, \cs{MFP@x@Int}, \cs{MFP@x@Frc}, \cs{MFP@y@Sgn},
% \cs{MFP@y@Int} and \cs{MFP@y@Frc} to be the normalized parts of two real numbers
% $x$ and $y$.
%
% \cs{MFP@Rsub} just changes the sign of $y$ and then calls \cs{MFP@Radd}.
%
% \cs{MFP@Radd} checks whether $x$ and $y$ have same or different signs. In
% the first case we need only add absolute values and the sign of the
% result will match that of the operands. In the second case, finding the
% sign of the result is a little more involve (and ``borrowing'' may be
% needed).
%    \begin{macrocode}
\def\MFP@Rsub{\edef\MFP@y@Sgn{\number-\MFP@y@Sgn}\MFP@Radd}%
\def\MFP@Radd{%
  \MFP@tempa\MFP@x@Sgn
  \multiply\MFP@tempa\MFP@y@Sgn\relax
  \ifcase\MFP@tempa
    \ifnum \MFP@x@Sgn=0
      \MFP@Rcopy yz%
    \else
      \MFP@Rcopy xz%
    \fi
  \or
    \@xp\MFP@Radd@same
  \else
    \@xp\MFP@Radd@diff
  \fi}%
%    \end{macrocode}
%
% \cs{MFP@Radd@same} adds two numbers which have the same sign. The sign
% of the result is the common sign. The fractional and integer parts are
% added separately, then a carry is invoked. The overflow (\cs{MFP@z@Ovr})
% could be only a single digit 0 or 1.
%    \begin{macrocode}
\def\MFP@Radd@same{%
  \MFP@addone\MFP@x@Frc\MFP@y@Frc\MFP@z@Frc
  \MFP@addone\MFP@x@Int\MFP@y@Int\MFP@z@Int
  \MFP@carrya\MFP@z@Frc\MFP@z@Int
  \MFP@carrya\MFP@z@Int\MFP@z@Ovr
  \makeMFP@eightdigits\MFP@z@Frc
  \edef\MFP@z@Sgn{\MFP@x@Sgn}}%
%    \end{macrocode}
%
% We are now adding two numbers with opposite sign. Since $x\ne 0$ this
% is the same as $\sgn(x)(|x| - |y|)$ . So we subtract absolute values,
% save the result in \cs{MFP@z@Sgn}, \cs{MFP@z@Int} and \cs{MFP@z@Frc}
% (with the last two nonnegative, as usual), then change the sign of
% \cs{MFP@z@Sgn} if \cs{MFP@x@Sgn} is negative. Since the difference
% between numbers in $[0,10^8)$ has absolute value in that range, there is
% no carrying. However, there may be borrowing.
%    \begin{macrocode}
\def\MFP@Radd@diff{%
  \MFP@addone\MFP@x@Frc{-\MFP@y@Frc}\MFP@z@Frc
  \MFP@addone\MFP@x@Int{-\MFP@y@Int}\MFP@z@Int
%    \end{macrocode}
%
% Now we need to establish the sign and arrange the borrow. The sign of
% the result is the sign of \cs{MFP@z@Int} unless it is 0; in that case
% it, is the sign of \cs{MFP@z@Frc}. There must be a simpler coding,
% though.
%    \begin{macrocode}
  \MFP@tempa=\MFP@z@Int
  \MFP@tempb=\MFP@z@Frc\relax
  \ifnum\MFP@tempa=0 \else \MFP@tempa=\MFP@Sign\MFP@tempa 1 \fi
  \ifnum\MFP@tempb=0 \else \MFP@tempb=\MFP@Sign\MFP@tempb 1 \fi
  \ifnum\MFP@tempa=0 \MFP@tempa=\MFP@tempb \fi
%    \end{macrocode}
%
% Now we have the sign of $|x| - |y|$ in \cs{MFP@tempa}, and we multiply
% that sign by the sign of $x$ to get \cs{MFP@z@Sgn}. Then we multiply the
% current value of $z$ by that sign to get the absolute value, stored in
% \cs{MFP@tempa} and \cs{MFP@tempb}.
%    \begin{macrocode}
  \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@tempa}%
  \MFP@tempb\MFP@tempa
  \multiply\MFP@tempa \MFP@z@Int
  \multiply\MFP@tempb \MFP@z@Frc\relax
%    \end{macrocode}
%
% What we should have now is a positive number which might still be
% represented with a negative fractional part. A human being performing
% the subtraction would have borrowed first. Being a computer, we do it
% last, and we're done.
%    \begin{macrocode}
  \ifnum\MFP@tempb<0
    \advance\MFP@tempb\MFP@ttteight
    \advance\MFP@tempa-1
  \fi
  \edef\MFP@z@Int{\number\MFP@tempa}%
  \edef\MFP@z@Frc{\number\MFP@tempb}%
  \makeMFP@eightdigits\MFP@z@Frc}%
%    \end{macrocode}
%
% \cs{MFP@Rmul} first computes the (theoretical) sign of the product: if
% it is zero, return zero, otherwise provisionally set the sign of the product
% and call \cs{MFP@@Rmul}.
%    \begin{macrocode}
\def\MFP@Rmul{%
  \ifnum\MFP@x@Sgn=0 \MFP@Rzero
  \else\ifnum\MFP@y@Sgn=0 \MFP@Rzero
  \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}%
    \@XP\MFP@@Rmul
  \fi\fi}%
%    \end{macrocode}
%
% \cs{MFP@@Rmul} first initializes the macros that will hold the
% base-10000 digits of $z$. Then it splits the four expected macros into
% eight macros that hold the base-10000 digits for each of $x$ and $y$.
% Then each digit of $y$ is used to multiply the four digits of $x$ and the
% results are added to corresponding digits of $z$.
%    \begin{macrocode}
\def\MFP@@Rmul{%
  \def\MFP@z@Frc@iv {0}\def\MFP@z@Frc@iii{0}%
  \def\MFP@z@Frc@ii {0}\def\MFP@z@Frc@i  {0}%
  \def\MFP@z@Int@i  {0}\def\MFP@z@Int@ii {0}%
  \def\MFP@z@Int@iii{0}%
  \MFP@@split
  \MFP@multiplyfour \MFP@y@Frc@ii \MFP@z@Frc@i
    \MFP@z@Frc@ii \MFP@z@Frc@iii \MFP@z@Frc@iv
  \MFP@multiplyfour \MFP@y@Frc@i  \MFP@z@Int@i
    \MFP@z@Frc@i  \MFP@z@Frc@ii  \MFP@z@Frc@iii
  \MFP@multiplyfour \MFP@y@Int@i  \MFP@z@Int@ii
    \MFP@z@Int@i  \MFP@z@Frc@i   \MFP@z@Frc@ii
  \MFP@multiplyfour \MFP@y@Int@ii \MFP@z@Int@iii
    \MFP@z@Int@ii \MFP@z@Int@i   \MFP@z@Frc@i
%    \end{macrocode}
% Now apply the carry routines on the underflow digits\dots
%    \begin{macrocode}
  \MFP@carrym\MFP@z@Frc@iv\MFP@z@Frc@iii
  \MFP@carrym\MFP@z@Frc@iii\MFP@z@Frc@ii
%    \end{macrocode}
% \dots and pause to round the lowest digit that will be kept\dots
%    \begin{macrocode}
  \ifnum\MFP@z@Frc@iii<5000 \else
    \MFP@tempb\MFP@z@Frc@ii
    \advance\MFP@tempb1
    \edef\MFP@z@Frc@ii{\number\MFP@tempb}%
  \fi
%    \end{macrocode}
% \dots and continue carrying.
%    \begin{macrocode}
  \MFP@carrym\MFP@z@Frc@ii\MFP@z@Frc@i
  \MFP@carrym\MFP@z@Frc@i \MFP@z@Int@i
  \MFP@carrym\MFP@z@Int@i \MFP@z@Int@ii
  \MFP@carrym\MFP@z@Int@ii\MFP@z@Int@iii
%    \end{macrocode}
% To end, we arrange for all macros to hold four digits (except
% \cs{MFP@z@Int@ii} and \cs{MFP@z@Int@iii} which don't need leading 0s)
% and load them into the appropriate 8-digit macros. The underflow digits
% are stored in \cs{MFP@z@Und} in case we ever need to examine them (we
% now do: in our unit conversion routine \cs{MFP@DPmul}), and the overflow
% in \cs{MFP@z@Ovr} in case we ever want to implement an overflow error.
% Theoretically $z \ne 0$, but it is possible that $z=0$ after reducing to
% eight places. If so, we must reset \cs{MFP@z@Sgn}.
%    \begin{macrocode}
  \makeMFP@fourdigits\MFP@z@Frc@iv
  \makeMFP@fourdigits\MFP@z@Frc@iii
  \makeMFP@fourdigits\MFP@z@Frc@ii
  \makeMFP@fourdigits\MFP@z@Frc@i
  \makeMFP@fourdigits\MFP@z@Int@i
  \edef\MFP@z@Int{\number\MFP@z@Int@ii\MFP@z@Int@i}%
  \edef\MFP@z@Frc{\MFP@z@Frc@i\MFP@z@Frc@ii}%
  \edef\MFP@z@Ovr{\number\MFP@z@Int@iii}%
  \edef\MFP@z@Und{\MFP@z@Frc@iii\MFP@z@Frc@iv}%
  \ifnum\MFP@z@Int>0
  \else\ifnum\MFP@z@Frc>0
  \else \def\MFP@z@Sgn{0}%
  \fi\fi}%
%    \end{macrocode}
%
% For division, we will obtain the result one digit at a time until the
% $9$th digit after the decimal is found. That $9$th will be used to round
% to eight digits (and stored as underflow). We normalize the denominator
% by shifting left until the integer part is eight digits. We do the same for
% the numerator. The integer quotient of the integer parts will be one digit
% (possibly a 0). If the denominator is shifted $d$ digits left and the
% numerator $n$ digits left, the quotient will have to be shifted $n-d$
% places right or $d-n$ places left. Since the result is supposed to have
% $9$ digits after the dot, our quotient needs $9+d-n+1$ total digits.
% Since $d$ can be as high as $15$ and $n$ as low as $0$, we could need
% $25$ repetitions. However, that extreme would put $15$ or $16$ digits in
% the integer part, a $7$ or $8$ digit overflow. (It can be argued that
% only $16$ significant digits should be retained in any case.) If $d$ is
% $0$ and $n$ is $15$ we would need $-5$ digits. That means the first
% nonzero digit is in the 15th or 16th place after the dot and the
% quotient is effectively zero.
%
% Here I explain why we normalize the parts in this way. If a numerator
% has the form $n_1.n_2$ and the denominator has the form $d_1.d_2$ then
% \TeX{} can easily obtain the integer part of $n_1/d_1$, because these
% are within its range for integers. The resulting quotient (let's call it
% $q_1$) is the largest integer satisfying $q_1d_1 \le n_1$. What we seek,
% however is the largest integer $q$ such that $q(d_1.d_2) \le n_1.n_2$.
% It can easily be shown that $q \le q_1$. It is true, but not so easily
% shown, that $q \ge q_1 - 1$. This is only true if $d_1$ is large enough,
% in our case it has to be at least five digits. Thus we only have to do one
% simple division and decide if we need to reduce the quotient by one. If
% we arrange for $d_1$ to have eight digits, then $q_1$ will be one digit and
% the test for whether we need to reduce it becomes easier.
%
% This test is done as follows. The first trial quotient, $q_1$, will work
% if
% \[
%     q_1 d_1 (10)^8 + q_1 d_2 \le n_1 (10)^8 + n_2
% \]
% This means
% \begin{equation}\label{crucial}
%     0 \le (n_1 - q_1 d_1) (10)^8 + n_2 - q_1 d_2 .
% \end{equation}
% Since $d_2$ is no more than eight digits, $q_1 d_2$ is less than $9
% (10)^8$. Inequality (\ref{crucial}) is therefore satisfied if $n_1 - q_1
% d_1 \ge 9$. If that is not the case then the right side of
% (\ref{crucial}) is computable within \TeX's integer ranges and we can
% easily test the inequality. If the inequality holds, then $q = q_1$,
% otherwise $q = q_1 - 1$.
%
% Note also that when $q = q_1$, then both terms in (\ref{crucial})
% (ignoring the $10^8$ factor) will be needed to calculate the remainder.
% If $q = q_1 - 1$, we simply add $d_1$ and $d_2$ to the respective parts.
% Thus we will save these values for that use.
%
% Now I need to get it organized. \cs{MFP@Rdiv} will have \cs{MFP@x@*} and
% \cs{MFP@y@*} available. One step (could be first or last). Is to calculate
% the sign. Let's do it first (because we need to check for zero anyway).
%
% We invoke an error message upon division by zero, but nevertheless return
% a value. By default it is $0$ for $0/0$ and the maximum possible real
% for $x/0$ when $x$ is not zero. If the numerator is zero and the
% denominator not, we could do nothing as $z$ was initialized to be zero.
% However, we play it safe by explicitly setting $z$ to zero.
%
% If neither is zero, we calculate the sign of the result and call
% \cs{MFP@@Rdiv} to divide the absolute values.
%    \begin{macrocode}
\def\MFP@Rdiv{%
  \ifnum\MFP@y@Sgn=0 \MFP@dividebyzero@err
    \ifnum\MFP@x@Sgn=0
      \edef\MFP@z@Int{\ZeroOverZeroInt}%
      \edef\MFP@z@Frc{\ZeroOverZeroFrac}%
    \else
      \edef\MFP@z@Int{\xOverZeroInt}%
      \edef\MFP@z@Frc{\xOverZeroFrac}%
    \fi
    \edef\MFP@z@Sgn{\MFP@x@Sgn}%
  \else\ifnum\MFP@x@Sgn=0 \MFP@Rzero
  \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}\MFP@@Rdiv
  \fi\fi}%
%    \end{macrocode}
%
% Now we have two positive values to divide. Our first step is to shift
% the denominator ($y$) left and keep track of how many places. We store
% the shift in \cs{MFP@tempa}. This actually changes the value of $y$,
% but knowing the shift will give us the correct quotient in the end.
%
% We first arrange that \cs{MFP@y@Int} is nonzero by making it \cs{MFP@y@Frc} if
% it is zero (a shift of eight digits). Then the macro
% \cs{MFP@numdigits@toshift} computes $8$ minus the number of digits in
% \cs{MFP@y@Int}, which is how many positions left $y$ will be shifted.
% We then call \cs{MFP@doshift@y} on the concatenation of the digits in
% the integer and fractional parts (padded with zeros to ensure there are
% at least 16). All this macro does is read the first eight digits into
% \cs{MFP@y@Int} and the next eight into \cs{MFP@y@Frc}.
%    \begin{macrocode}
\def\MFP@@Rdiv{%
  \ifnum\MFP@y@Int=0
    \edef\MFP@y@Int{\number\MFP@y@Frc}%
    \def\MFP@y@Frc{00000000}%
    \MFP@tempa=8
  \else
    \MFP@tempa=0
  \fi
  \advance\MFP@tempa\MFP@numdigits@toshift\MFP@y@Int\relax
  \@XP\MFP@doshift@y\@xp\MFP@y@Int\MFP@y@Frc0000000\MFP@end
%    \end{macrocode}
%
% We repeat all that on the numerator $x$, except shifting its digits
% left means the final outcome will need a corresponding \emph{right}
% shift. We record that fact by reducing \cs{MFP@tempa}, which ends up
% holding the net shift necesary.
%
% This has the advantage that we know the result will be in the range
% $[0.1, 10)$. It also means we can reduce the number of places we will
% need to shift left as well as reduce the number of iterations of the
% loop that calculates the digits.
%    \begin{macrocode}
  \ifnum\MFP@x@Int=0
    \edef\MFP@x@Int{\number\MFP@x@Frc}%
    \def\MFP@x@Frc{00000000}%
    \advance\MFP@tempa -8
  \fi
  \advance\MFP@tempa-\MFP@numdigits@toshift\MFP@x@Int\relax
  \@XP\MFP@doshift@x\@xp\MFP@x@Int\MFP@x@Frc0000000\MFP@end
%    \end{macrocode}
%
% Since our result will have at most one digit in the integer part, a
% rightward shift of $10$ places will make every digit $0$ including the
% rounding digit, so we return $0$.
%    \begin{macrocode}
  \ifnum\MFP@tempa<-9
    \MFP@Rzero
  \else
%    \end{macrocode}
%
% Now we perform the division, which is a loop repeated $10 +
% {}$\cs{MFP@tempa} times. Therefore, we add 10 to \cs{MFP@tempa} in
% \cs{MFP@tempf}, our loop counter. We also initialize the macro that
% will store the digits and then, after the division, shift and split it
% into parts.
%    \begin{macrocode}
    \MFP@tempf\MFP@tempa
    \advance\MFP@tempf 10
    \def\MFP@z@digits{}%
    \MFP@Rdivloop
    \MFPshiftandsplit@z@digits
%    \end{macrocode}
%
% The last remaining step is to round and carry and get the fractional
% part in the appropriate 8-digit form..
%    \begin{macrocode}
    \ifnum\MFP@z@Und>4
      \MFP@addone\MFP@z@Frc1\MFP@z@Frc
      \MFP@carrya\MFP@z@Frc\MFP@z@Int
      \MFP@carrya\MFP@z@Int\MFP@z@Ovr
      \makeMFP@eightdigits\MFP@z@Frc
    \fi
  \fi}%
%    \end{macrocode}
%
% If \arg1 of \cs{MFP@numdigits@toshift}, has $n$ digits then
% \cs{MFP@numdigits@toshift} picks out the value $8-n$. \cs{MFP@doshift@x}
% reads the first eight digits into \cs{MFP@x@Int} and then pulls out eight more
% from the rest (\arg9) inside \cs{MFP@x@Frc}. The same with
% \cs{MFP@doshift@y}.
%    \begin{macrocode}
\def\MFP@numdigits@toshift#1{\@xp\MFP@ninthofmany#101234567\MFP@end}%
\def\MFP@doshift@x#1#2#3#4#5#6#7#8#9\MFP@end{%
  \def\MFP@x@Int{#1#2#3#4#5#6#7#8}%
  \edef\MFP@x@Frc{\MFP@eightofmany#9\MFP@end}}%
\def\MFP@doshift@y#1#2#3#4#5#6#7#8#9\MFP@end{%
  \def \MFP@y@Int{#1#2#3#4#5#6#7#8}%
  \edef\MFP@y@Frc{\MFP@eightofmany#9\MFP@end}}%
%    \end{macrocode}
%
% The loop counter is \cs{MFP@tempf}, \cs{MFP@tempa} is reserved for the
% shift required later, the quotient digit will be \cs{MFP@tempb}. The
% remainder will be calculated in \cs{MFP@tempc} and \cs{MFP@tempd}.
% \cs{MFP@tempe} will hold the value whose size determines whether the
% quotient needs to be reduced.
%    \begin{macrocode}
\def\MFP@Rdivloop{%
  \MFP@tempb\MFP@x@Int                    % \MFP@tempb = n_1
  \MFP@tempc\MFP@y@Int                    % \MFP@tempc = d_1
  \divide\MFP@tempb \MFP@tempc        % \MFP@tempb = n_1/d_1 = q_1
  \multiply \MFP@tempc \MFP@tempb     % \MFP@tempc = q_1 d_1
  \MFP@tempd \MFP@y@Frc                   % \MFP@tempd = d_2
  \multiply \MFP@tempd \MFP@tempb     % \MFP@tempd = q_1 d_2
  \MFP@tempe \MFP@tempc
  \advance \MFP@tempe -\MFP@x@Int\relax   % \MFP@tempe = -n_1 + q_1 d_1
  \ifnum \MFP@tempe > -9              % n_1 - q_1 d_1 < 9
    \multiply \MFP@tempe\MFP@ttteight    % -(n_1 - q_1 d_1)(10)^8
    \advance \MFP@tempe \MFP@tempd       % add q_1 d_2
    \advance \MFP@tempe -\MFP@x@Frc\relax % add -n_2
    \ifnum \MFP@tempe>0               % Crucial inequality fails
      \advance\MFP@tempb -1           % new q = q_1 - 1
      \advance\MFP@tempc -\MFP@y@Int      % q_1 d_1 - d_1 = q d_1
      \advance\MFP@tempd -\MFP@y@Frc\relax% q_1 d_2 - d_2 = q d_2
    \fi
  \fi
  \edef\MFP@z@digits{\MFP@z@digits\number\MFP@tempb}%
%    \end{macrocode}
%
% It remains to:
% \begin{itemize}
%   \item Do the carry from \cs{MFP@tempd} to \cs{MFP@tempc}. Then
%         \cs{MFP@tempc.}\cs{MFP@tempd} will represent $q\cdot y$.
%   \item Subtract them from \cs{MFP@x@Int} and \cs{MFP@x@Frc} (i.e. remainder =
%         $x - qy$).
%   \item Borrow, if needed, and we will have the remainder in
%         \cs{MFP@x@Int.}\cs{MFP@x@Frc}.
% \end{itemize}
% Then we decrement the loop counter, and decide whether to repeat this
% loop. If so, we need to shift the remainder right one digit (multiply
% by 10). We don't use \cs{MFP@carrya} since it requires macros; its
% internal code, \cs{MFP@carryi} just leaves the results in
% \cs{MFP@tempa.}\cs{MFP@tempb}.
%    \begin{macrocode}
  \begingroup
    \MFP@carryi\MFP@ttteight\MFP@tempd\MFP@tempc
  \MFP@endgroup@after{%
    \MFP@tempc=\number\MFP@tempa
    \MFP@tempd=\number\MFP@tempb\relax
  }%
% subtract
  \MFP@addone\MFP@x@Int{-\MFP@tempc}\MFP@x@Int
  \MFP@addone\MFP@x@Frc{-\MFP@tempd}\MFP@x@Frc
% borrow
  \ifnum\MFP@x@Frc<0
    \MFP@addone\MFP@x@Frc\MFP@ttteight\MFP@x@Frc
    \MFP@addone\MFP@x@Int{-1}\MFP@x@Int
  \fi
  \advance\MFP@tempf -1
  \ifnum\MFP@tempf>0
    \edef\MFP@x@Int{\MFP@x@Int0}%
    \edef\MFP@x@Frc{\MFP@x@Frc0}%
    \MFP@carrya\MFP@x@Frc\MFP@x@Int
    \@xp\MFP@Rdivloop
  \fi}%
%    \end{macrocode}
%
% Now \cs{MFPshiftandsplit@z@digits}. At this point, the digits of the
% quotient are stored in \cs{MFP@z@digits}. We need to shift the decimal
% \cs{MFP@tempa} places left, and perform the rounding. There are
% \cs{MFP@tempa}${}+10$ digits. This could be as little as $1$ or as great
% as $25$. In the first case \cs{MFP@tempa} is $-9$, and this (rightward)
% shift produces $0$ plus a rounding digit. In the latter case \cs{MFP@tempa}
% is $15$, and the shift produces $8$ digits overflow, an $8$-digit
% integer part, an $8$-digit fractional part and a rounding digit. In the
% example $0123456$, \cs{MFP@tempa}${}+10$ is $7$, so \cs{MFP@tempa} is $-3$.
% The shift produces $0.0001\,2345\,6$. The rounding digit ($6$) makes the
% answer $0.0001\,2346$.
%
% We take two cases:
% \begin{itemize}
%   \item \cs{MFP@tempa}${}\le 7$, prepend $7-{}$\cs{MFP@tempa} zeros. The first
%         $8$ digits will become the integer part, and there should be
%         exactly $9$ more digits.
%   \item \cs{MFP@tempa}${} > 7$, pluck \cs{MFP@tempa}${}-7$ digits for
%         overflow, the next $8$ for integer part, leaving $9$ more digits
% \end{itemize}
% In either case, the $9$ last digits will be processed into a fractional
% part (with possible carry if the rounding increases it to $10^8$).
%
% After this, we will return to \cs{MFP@Rdiv} so overwriting \cs{MFP@temp*}
% won't cause any problems.
%    \begin{macrocode}
\def\MFPshiftandsplit@z@digits{%
    \advance \MFP@tempa -7
    \ifnum\MFP@tempa>0
      \def\MFP@z@Ovr{}%
      \@xp\MFPget@Ovrdigits\MFP@z@digits\MFP@end
    \else
      \ifnum\MFP@tempa<-7
        \edef\MFP@z@digits{00000000\MFP@z@digits}%
        \advance\MFP@tempa8
      \fi
      \ifnum\MFP@tempa<-3
        \edef\MFP@z@digits{0000\MFP@z@digits}%
        \advance\MFP@tempa4
      \fi
      \edef\MFP@z@digits{%
        \ifcase-\MFP@tempa\or
          0\or
          00\or
          000\or
          0000\else
          00000%
        \fi \MFP@z@digits}%
      \@xp\MFPget@Intdigits\MFP@z@digits\MFP@end
    \fi}%
%    \end{macrocode}
%
% The macro \cs{MFPget@Ovrdigits} is a loop that loads the first \cs{MFP@tempa}
% digits of what follows into \cs{MFP@z@Ovr}. It does this one digit (\arg1)
% at a time. Once the counter reaches $0$, we call the macro that
% processes the integer part digits.
%    \begin{macrocode}
\def\MFPget@Ovrdigits#1{%
  \edef\MFP@z@Ovr{\MFP@z@Ovr#1}%
  \advance\MFP@tempa -1
  \ifnum\MFP@tempa>0
    \@xp\MFPget@Ovrdigits
  \else
    \@xp\MFPget@Intdigits
  \fi}%
%    \end{macrocode}
%
% The macro \cs{MFPget@Intdigits} should have exactly 17 digits following it.
% It puts eight of them in \cs{MFP@z@Int}, then calls \cs{MFPget@Frcdigits} to
% read the fractional part. That requires exactly nine digits follow it,
% putting eight in \cs{MFP@z@Frc} and the last in \cs{MFP@z@Und}. Still, to
% allow a graceful exit should there be more, we gobble the rest of the
% digits.
%    \begin{macrocode}
\def\MFPget@Intdigits#1#2#3#4#5#6#7#8{%
  \def\MFP@z@Int{\number#1#2#3#4#5#6#7#8}%
  \MFPget@Frcdigits}%
\def\MFPget@Frcdigits#1#2#3#4#5#6#7#8#9{%
  \def\MFP@z@Frc{#1#2#3#4#5#6#7#8}%
  \def\MFP@z@Und{#9}\gobbleto@MFP@end}%
%    \end{macrocode}
%
% The max amd min operations simply run the compare operation and use
% and use the resultant booleans to copy $x$ or $y$ to $z$.
%    \begin{macrocode}
\def\MFP@Rmax{%
  \MFP@Rcmp \ifMFP@neg \MFP@Rcopy yz\else\MFP@Rcopy xz\fi}%
\def\MFP@Rmin{%
  \MFP@Rcmp \ifMFP@pos \MFP@Rcopy yz\else\MFP@Rcopy xz\fi}%
%    \end{macrocode}
%
% \subsection{Commands to format for printing}
%
% \DescribeMacro{\MFPtruncate}
% This first runs the parsing command so the fractional part has exactly
% eight digits. These become the arguments of \cs{MFP@@Rtrunc}, which just
% keeps the right number. For negative truncations we prepend zeros to the
% integer part so it too is exactly eight digits. These become the
% arguments of \cs{MFP@@iRtrunc}, which substitutes 0 for the last
% \texttt{-}\cs{MFP@tempa} of them.
%
% The macro to store the result in follows \arg2. It is read and
% defined by either \cs{MFP@Rtrunc} or \cs{MFP@iRtrunc}.
%    \begin{macrocode}
\def\MFPtruncate#1#2{%
  \begingroup
    \MFP@tempa#1\relax
    \MFPparse@x{#2}%
    \ifnum\MFP@tempa<1
      \@xp\MFP@iRtrunc
    \else
      \@xp\MFP@Rtrunc
    \fi}%
\def\MFP@Rtrunc#1{%
    \edef\MFP@x@Frc{\@xp\MFP@@Rtrunc\MFP@x@Frc\MFP@end}%
    \ifnum\MFP@x@Int=0
      \ifnum\MFP@x@Frc=0
        \def\MFP@x@Sgn{0}%
      \fi
    \fi
  \MFP@endgroup@after{%
    \MFP@afterdef#1{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}}}%
\def\MFP@@Rtrunc#1#2#3#4#5#6#7#8#9\MFP@end{%
  \ifcase\MFP@tempa\or
    #1\or
    #1#2\or
    #1#2#3\or
    #1#2#3#4\or
    #1#2#3#4#5\or
    #1#2#3#4#5#6\or
    #1#2#3#4#5#6#7\else
    #1#2#3#4#5#6#7#8\fi}%
\def\MFP@iRtrunc#1{%
  \makeMFP@eightdigits\MFP@x@Int
  \edef\MFP@x@Val{\number\MFP@x@Sign\@xp\MFP@@iRtrunc\MFP@x@Int\MFP@end}%
  \MFP@endgroup@after{\MFP@afterdef#1{\MFP@x@Val}}}%
\def\MFP@@iRtrunc#1#2#3#4#5#6#7#8#9\MFP@end{%
  \ifcase-\MFP@tempa
    #1#2#3#4#5#6#7#8\or
    #1#2#3#4#5#6#70\or
    #1#2#3#4#5#600\or
    #1#2#3#4#5000\or
    #1#2#3#40000\or
    #1#2#300000\or
    #1#2000000\or
    #10000000\else
    00000000\fi}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPround}
% For rounding we simply add the appropriate fraction and truncate.
% The macro in which to store the result will follow \arg2, and be
% picked up by the \cs{MFPtruncate} command.
%    \begin{macrocode}
\def\MFPround#1#2{%
  \begingroup
    \MFP@tempa#1\relax
    \ifnum 0>\MFP@tempa
      \edef\MFP@y@Tmp{%
        \ifcase-\MFP@tempa\or
          5\or
          50\or
          500\or
          5000\or
          50000\or
          500000\or
          5000000\else
          50000000\fi
      }%
    \else
      \edef\MFP@y@Tmp{%
        \ifcase\MFP@tempa
          .5\or
          .05\or
          .005\or
          .0005\or
          .00005\or
          .000005\or
          .0000005\or
          .00000005\else
        0\fi
      }%
    \fi
    \MFPchk{#2}\ifMFP@neg\edef\MFP@y@Tmp{-\MFP@y@Tmp}\fi
    \MFPadd{#2}\MFP@y@Tmp\MFP@z@Tmp
  \MFP@endgroup@after{\MFP@afterdef\MFP@z@Tmp{\MFP@z@Tmp}}%
  \MFPtruncate{#1}\MFP@z@Tmp}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPstrip}
% Stripping zeros from the right end of the fractional part. The star form
% differs only in the handling of a zero fractional part. So we check
% whether it is zero and when it is, we either append `\texttt{.0}' or
% nothing. The rest of the code grabs a digit at a time and stops when the
% rest are zero.
%    \begin{macrocode}
\def\MFPstrip{%
  \@ifstar{\MFP@strip{}}{\MFP@strip{.0}}}%
\def\MFP@strip#1#2#3{%
  \MFPparse@x{#2}%
  \ifnum \MFP@x@Frc=0
    \edef#3{\MFP@x@Sign\MFP@x@Int#1}%
  \else
    \edef#3{\MFP@x@Sign\MFP@x@Int.\@xp\MFP@@strip\MFP@x@Frc\MFP@end}%
  \fi}%
\def\MFP@@strip#1#2\MFP@end{%
  #1%
  \ifnum 0#2>0
    \@xp\MFP@@strip
  \else
    \@xp\gobbleto@MFP@end
  \fi#2\MFP@end}%
%    \end{macrocode}
%
% \subsection{Miscellaneous}
%
% Here is the code that allows definitions to survive after
% \cs{stopMFPprogram}. The \cs{Global} variants are easiest.
%    \begin{macrocode}
\def\MFP@Global#1{\toks@\@xp{#1}\xdef#1{\the\toks@}}%
\def\MFP@GlobalStack{\MFP@Global\MFP@Rstack}%
%    \end{macrocode}
%
% The \cs{Export} command adds the command and its definition to a macro
% that is executed after the closing group of the program.
%    \begin{macrocode}
\def\MFP@Export#1{%
  \begingroup
    \toks@\@xp{\MFPprogram@returns}%
  \MFP@endgroup@after{%
    \MFP@afterdef\MFPprogram@returns{\the\toks@ \MFP@afterdef#1{#1}}%
  }}%
\def\MFP@ExportStack{\MFP@Export\MFP@Rstack}%
%    \end{macrocode}
%
% The various operations \cs{MFP@R...} together make up a ``microcode'' in
% terms of which the stack language and the operand language are both
% defined. As a language in its own right, it lacks only convenient ways
% to move numbers around, as well as a few extra registers for saving
% intermediate results. In this language, numbers are represented by a
% three part data structure, consisting of a signum, an integer part and a
% fractional part.
%
% Here we define extra commands to remedy this lack, starting with a way
% to load a number (or rather, a three part data structure representing a
% number) directly into a register. Here \arg1 is a register name (we
% always us a single letter) and the remaining arguments are the signum,
% the integer part and the fractional part (automatically normalized to 8
% digits). The ``register'' is just a set of three macros created from the
% name given.
%
% We make loading a number into a register a little more general than
% strictly needed, allowing the parts to be specified as anything \TeX{}
% recognizes as a number and allowing any register name. This generality
% might reduce efficiency but it simplifies code. Because register
% \reg{z} is by far the most common one to load, we make more efficient
% version of it.
%    \begin{macrocode}
\def\MFP@Rload #1#2#3#4{%
  \@xp\edef\csname MFP@#1@Sgn\endcsname{\number#2}%
  \@xp\edef\csname MFP@#1@Int\endcsname{\number#3}%
  \@xp\edef\csname MFP@#1@Frc\endcsname{\number#4}%
  \@xp\makeMFP@eightdigits\csname MFP@#1@Frc\endcsname}%
\def\MFP@Rcopy#1#2{%
  \MFP@Rload #2{\csname MFP@#1@Sgn\endcsname}%
               {\csname MFP@#1@Int\endcsname}%
               {\csname MFP@#1@Frc\endcsname}}%
\def\MFP@Rloadz#1#2#3{%
  \edef\MFP@z@Sgn{\number#1}%
  \edef\MFP@z@Int{\number#2}%
  \edef\MFP@z@Frc{\number#3}%
  \makeMFP@eightdigits\MFP@z@Frc}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPpi}
% These are some miscellaneous constants. The 8-digit approximation to
% $\pi$, is \cs{MFPpi} and the constant mathematicians call $e$ is
% \DescribeMacro{\MFPe}
% \cs{MFPe}. Finally, the golden ratio (often called $\phi$) is obtained
% by
% \DescribeMacro{\MFPphi}
% \cs{MFPphi}.
%    \begin{macrocode}
\def\MFPpi{3.14159265}%
\def\MFPe{2.71828183}%
\def\MFPphi{1.61803399}%
%    \end{macrocode}
% Load (conditionally) \file{mfpextra.tex}.
%    \begin{macrocode}
\MFP@loadextra
\MFP@finish
%</sty>
%    \end{macrocode}
%
% \section{Extras}\label{extras}
%
% The extras consist so far of sine, cosine, angle, logarithm, powers,
% square root, and random number. For completeness, here is the table of
% user-level commands available.
%
% \medskip
% \centerline{%
% \begin{tabular}{lp{3in}}
% \textit{Operand versions}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\MFPsin}^^A
%   \cs{MFPsin}\mmarg{num}\cs{macro}&
%     Stores $\sin(\meta{num})$ in \cs{macro}, where \meta{num} is an
%       angle in degrees.\\
%   \SpecialUsageIndex{\MFPcos}^^A
%   \cs{MFPcos}\mmarg{num}\cs{macro}&
%     Stores $\cos(\meta{num})$ in \cs{macro}, where \meta{num} is an
%       angle in degrees.\\
%   \SpecialUsageIndex{\MFPangle}^^A
%   \cs{MFPangle}\mmarg{$x$}\mmarg{$y$}\cs{macro}&
%     Stores in \cs{macro} the polar angle coordinate $\theta$ of the point
%       $(x,y)$, where $-180<\theta\le 180$.\\
%   \SpecialUsageIndex{\MFPrad}^^A
%   \cs{MFPrad}\mmarg{num}\cs{macro}&
%     The angle \meta{num} in degrees is converted to radians,
%       and result is stored in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPdeg}^^A
%   \cs{MFPdeg}\mmarg{num}\cs{macro}&
%     The angle \meta{num} in radians is converted to degrees,
%       and result is stored in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPlog}^^A
%   \cs{MFPlog}\mmarg{num}\cs{macro}&
%     Stores $\log(\meta{num})$ in \cs{macro} (base 10 logarithm).\\
%   \SpecialUsageIndex{\MFPln}^^A
%   \cs{MFPln}\mmarg{num}\cs{macro}&
%     Stores $\ln(\meta{num})$ in \cs{macro} (natural logarithm).\\
%   \SpecialUsageIndex{\MFPexp}^^A
%   \cs{MFPexp}\mmarg{num}\cs{macro}&
%     Stores $\exp(\meta{num})$ (i.e., $e^x$) in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPsqrt}^^A
%   \cs{MFPsqrt}\mmarg{num}\cs{macro}&
%     Stores the square root of \meta{num} in \cs{macro}.\\
%   \SpecialUsageIndex{\MFPrand}^^A
%   \cs{MFPrand}\mmarg{num}\cs{macro}&
%     Stores a random real number between $0$ amd \meta{num} in
%     \cs{macro}. If \meta{num} is negative, so is the result.\\
%   \SpecialUsageIndex{\MFPpow}^^A
%   \cs{MFPpow}\mmarg{num}\mmarg{int}\cs{macro}&
%     Stores the \meta{int} power of \meta{num} in \cs{macro}. The
%     second operand must be an integer (positive or negative).
% \end{tabular}}
%
% In addition, there is \SpecialUsageIndex{\MFPsetseed}\cs{MFPsetseed} for
% setting the internal random number seed. It takes one argument, the seed
% value, which must be an integer greater than or equal to $1$ and less
% than or equal to $2^{31}-2 = 2\,147\,483\,646$. If the seed is set to
% zero or a negative number then the first use of the random number
% generator will replace it with a seed value based on the current time
% and date. The randum number seed is a global value.
%
% There are actually three random number generators and they can be
% selected with the commands
% \SpecialUsageIndex{\MFPrandgenA}\cs{MFPrandgenA},
% \SpecialUsageIndex{\MFPrandgenB}\cs{MFPrandgenB}, or
% \SpecialUsageIndex{\MFPrandgenC}\cs{MFPrandgenC}. The first uses the
% code and multiplier value from the well-known macro file
% \file{random.tex}. It is the default. The other two use different
% multipliers which are alleged to have better statistical behavior. If
% any of these commands is used inside a group, that generator is in force
% during that group only.
%
% \bigskip
% \centerline{%
% \begin{tabular}{lp{3.9in}}
%   \textit{Stack versions}&\\[3pt]
% \hline\hline
%   \textbf{Command}&\textbf{operation}\\
% \hline
%   \SpecialUsageIndex{\Rsin}\cs{Rsin}&
%     The number is interpreted as degrees, and its sine is computed.\\
%   \SpecialUsageIndex{\Rcos}\cs{Rcos}&
%     The number is interpreted as degrees, and its cosine is computed.\\
%   \SpecialUsageIndex{\Rangle}\cs{Rangle}&
%     The top two numbers are interpreted as coordinates of a point $P$
%     in the order they were pushed. The polar angle coordinate $\theta$
%       of $P$, with $-180 < \theta \le 180$ is computed.\\
%   \SpecialUsageIndex{\Rrad}\cs{Rrad}&
%     The number of degrees is converted to radians.\\
%   \SpecialUsageIndex{\Rdeg}\cs{Rdeg}&
%     The number of radians is converted to degrees.\\
%   \SpecialUsageIndex{\Rlog}\cs{Rlog}&
%     Computes the base-10 logarithm.\\
%   \SpecialUsageIndex{\Rln}\cs{Rln}&
%     Computes the natural logarithm.\\
%   \SpecialUsageIndex{\Rexp}\cs{Rexp}&
%     Computes the exponential of the number (i.e., $e^x$).\\
%   \SpecialUsageIndex{\Rsqrt}\cs{Rsqrt}&
%     Computes the square root of the number.\\
%   \SpecialUsageIndex{\Rrand}\cs{Rrand}&
%     Returns a random real number between $0$ and the number, keeping the
%     sign.\\
%   \SpecialUsageIndex{\Rpow}\cs{Rpow}&
%     Computes $x^y$. The last number pushed ($y$) must be an
%     integer.
% \end{tabular}}
%
% \bigskip
% The user could easily convert between radians and degrees using
% multiplication and/or division. One could similarly convert between
% natural logarithms and base ten logarithms.  The commands \cs{Rdeg},
% \cs{Rrad}, \cs{Rlog} and \cs{Rln} (and their \cs{MFP...} counterparts)
% aim for more accurate results.
%
% \subsection{Loading the extras}
%
% \DescribeMacro{\Rsin}\DescribeMacro{\Rcos}
% \DescribeMacro{\Rangle}
% \DescribeMacro{\Rrad}\DescribeMacro{\Rdeg}
% \DescribeMacro{\Rlog}\DescribeMacro{\Rln}
% \DescribeMacro{\Rexp}\DescribeMacro{\Rsqrt}
% \DescribeMacro{\Rrand}\DescribeMacro{\Rpow}
% We start \file{mfpextra} with the hook \cs{MFP@Rextra} that
% \cs{startMFPprogram} will call to make available the extra operations
% defined here. If \file{minifp.sty} has been loaded, this macro is
% \cs{@empty}, otherwise it should be undefined. If it is undefined we
% load \file{minifp.sty}. If it is then not \cs{@empty} we assume
% \file{mfpextra.tex} was previously loaded and end input here.
%    \begin{macrocode}
%<*extra>
% check if mfpextra already loaded:
\expandafter\ifx\csname MFP@xfinish\endcsname\relax
\else \expandafter\endinput\fi
\expandafter\edef\csname MFP@xfinish\endcsname{%
  \catcode64=\the\catcode64 \space
  \catcode46=\the\catcode46 \space
  \catcode60=\the\catcode60 \space
  \catcode62=\the\catcode62 \space}%
\catcode64=11 % @
\catcode46=12 % . (period)
\catcode60=12 % <
\catcode62=12 % >
\ifx\MFP@Rextra\UndEfInEd \input minifp.sty \fi
\ifx\MFP@Rextra\@empty
\else
  \immediate\write16{mfpextra.tex: already loaded.^^J}%
  \MFP@xfinish
  \expandafter\endinput
\fi
\immediate\write16{%
    mfpextra.tex: extra operations for the MiniFP package.^^J}%
\def\MFP@Rextra{%
  \def\Rcos  {\MFP@stack@Unary\MFP@Rcos }%
  \def\Rsin  {\MFP@stack@Unary\MFP@Rsin }%
  \def\Rangle{\MFP@stack@Binary\MFP@Rangle}%
  \def\Rrad  {\MFP@stack@Unary\MFP@Rrad }%
  \def\Rdeg  {\MFP@stack@Unary\MFP@Rdeg }%
  \def\Rlog  {\MFP@stack@Unary\MFP@Rlog }%
  \def\Rln   {\MFP@stack@Unary\MFP@Rln  }%
  \def\Rexp  {\MFP@stack@Unary\MFP@Rexp }%
  \def\Rsqrt {\MFP@stack@Unary\MFP@Rsqrt}%
  \def\Rrand {\MFP@stack@Unary\MFP@Rrand}%
  \def\Rpow  {\MFP@stack@Binary\MFP@Rpow}}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPsin}\DescribeMacro{\MFPcos}
% \DescribeMacro{\MFPrad}\DescribeMacro{\MFPdeg}
% \DescribeMacro{\MFPlog}\DescribeMacro{\MFPln}
% \DescribeMacro{\MFPexp}\DescribeMacro{\MFPsqrt}
% \DescribeMacro{\MFPrand}\DescribeMacro{\MFPpow}
% Then the wrappers for the operand versions.
%    \begin{macrocode}
\def\MFPcos   {\MFP@op@Unary\MFP@Rcos }%
\def\MFPsin   {\MFP@op@Unary\MFP@Rsin }%
\def\MFPangle {\MFP@op@Binary\MFP@Rangle}%
\def\MFPrad   {\MFP@op@Unary\MFP@Rrad }%
\def\MFPdeg   {\MFP@op@Unary\MFP@Rdeg }%
\def\MFPlog   {\MFP@op@Unary\MFP@Rlog }%
\def\MFPln    {\MFP@op@Unary\MFP@Rln  }%
\def\MFPexp   {\MFP@op@Unary\MFP@Rexp }%
\def\MFPsqrt  {\MFP@op@Unary\MFP@Rsqrt}%
\def\MFPrand  {\MFP@op@Unary\MFP@Rrand}%
\def\MFPpow   {\MFP@op@Binary\MFP@Rpow}%
%    \end{macrocode}
%
% \subsection{Error messages}
%
% These extra commands come with a few possible new warnings and errors.
%
% \DescribeMacro{\LogOfZeroInt}
% \DescribeMacro{\LogOfZeroFrac}
% Trying to take the logarithm of zero will result in an error message.
% If one allows \TeX{} to continue, the returned value will be negative,
% with an integer part whose absolute value is equal to the contents of
% \cs{LogOfZeroInt} and a fractional part equal to the contents of
% \cs{LogOfZeroFrac}. The defaults are both $99999999$.
%
% Trying to take the logarithm of a negative number will produce the
% warning
% \begin{verbatim}
%   MFP warning: Log of a negative number is complex.
%                Only the real part will be computed. \end{verbatim}
% The log of the absolute value is returned.
%
% Trying to take the square root of a negative number has similar
% behavior. It produces a warning and returns $0$.
%
% \SpecialUsageIndex{\MaxRealInt}
% \SpecialUsageIndex{\MaxRealFrac}
% Trying to take the exponential of a number larger than about $18.42$
% will cause an error and the number returned has integer part
% $99999999$ and fractional part $99999999$.
%
% Trying to take a negative power of $0$ produces an error and returns
% the same value as trying to divide $1$ by $0$.
%
% Messages for errors related to impossible powers and logarithms.
%    \begin{macrocode}
\def\MFP@logofzero@err{%
  \MFP@errmsg{logarithm of zero}%
    {You tried to take the logarithm of zero. What were you %
     thinking? If you ^^Jcontinue, the value %
     assigned will be -\LogOfZeroInt.\LogOfZeroFrac.}}%
\def\LogOfZeroInt {\MaxRealInt}%
\def\LogOfZeroFrac{\MaxRealFrac}%
\def\MFP@expoverflow@err{%
  \MFP@errmsg{Power too large}%
    {The power you tried to calculate is too large for %
     8 digits. If you continue, ^^Jthe value assigned will be %
     \MaxRealInt.\MaxRealFrac.}}%
\def\MFP@badpower@err{%
  \MFP@errmsg{negative power of zero}%
    {You tried to take a negative power of zero. What were you
     thinking? If you ^^Jcontinue, the value assigned will be %
     \xOverZeroInt.\xOverZeroFrac.}}%
%    \end{macrocode}
%
% A debugging utility, \cs{MFPshowreg} displays the contents of a
% register.
%    \begin{macrocode}
\def\MFPshowreg #1{%
\ifMFPdebug
\begingroup
  \edef\theregister{%
    #1 =  \expandafter \MFP@Sign
          \csname MFP@#1@Sgn\endcsname %
          \csname MFP@#1@Int\endcsname.%
          \csname MFP@#1@Frc\endcsname}%
  \show\theregister
\endgroup
\fi}%
%    \end{macrocode}
%
% \subsection{Sine and Cosine}
%
% For iterated code, the most common register to copy is $z$ and
% the most common place to copy it is to $x$ or $y$ so we
% make single commands to do those.
%    \begin{macrocode}
\def\MFP@Rcopyz#1{\MFP@Rload {#1}\MFP@z@Sgn\MFP@z@Int\MFP@z@Frc}%
\def\MFP@Rcopyzx{\MFP@Rcopyz x}%
\def\MFP@Rcopyzy{\MFP@Rcopyz y}%
%    \end{macrocode}
%
% Our code assumes the number $x$ is an angle in degrees. To get sine and
% cosine of numbers as radians, simply convert your radians to degrees
% using \cs{MFPdeg} or \cs{Rdeg}. Then find the sine or cosine of the
% result. For example, if \cs{X} holds the angle in in radians and you
% want the result to be stored in \cs{S}:
% \begin{verbatim}
%   \MFPdeg\X\Y \MFPsin\Y\S \end{verbatim}
%
% For unit conversions such as radian to degree we try to be more accurate
% than a multiplication by an eight-digit conversion factor allows.
% If $x$ is large and the factor is off by $0.5\times 10^{-8}$, then the
% result can be significantly off. But if we are able to give the
% conversion factor 16 digits precision, then only the imprecision of $x$
% will significantly affect the result.
%
% We express the conversion factor as an integer part and two eight-digit
% fractional parts. We multiply $x$ by the integer and first fractional
% part (\arg1 and \arg2) with a normal \cs{MFP@Rmul}, but we save the
% underflow digits and undo the rounding that occured at the 8th digit.
% Together these give us an essentially exact result. Then we multiply $x$
% by the second fractional part (\arg3) and add the saved underflow to the
% result. Finally, we round and add the result to the first product.
% Argument \arg3, as well as the underflow digits, represent numbers less
% than $10^{-8}$, so we effectively scale them up by $10^8$, round the
% result to an integer and scale that back down.
%
% The registers $w$ and $v$ are used to save intermediate results.
% The ``\texttt{DP}'' in \cs{MFP@DPmul} refers to the fact that we are
% multiplying by a ``double precision'' real. The conversion factors are
% required to be positive.
%    \begin{macrocode}
\def\MFP@DPmul#1#2#3{%
  \ifnum\MFP@x@Sgn=0
    \MFP@Rzero
  \else
    \MFP@Rcopy xv%
    \MFP@Rload y1{#1}{#2}\MFP@Rmul
    \edef\MFP@w@Und{\MFP@z@Und}%
    \ifnum\MFP@z@Frc@iii>4999
      \MFP@tempa\MFP@z@Frc \advance\MFP@tempa-1
      \edef\MFP@z@Frc{\number\MFP@tempa}%
      \makeMFP@eightdigits\MFP@z@Frc
    \fi
    \MFP@Rcopyz w%
    \MFP@Rcopy vx\MFP@Rload y10{#3}\MFP@Rmul
    \MFP@Rcopyzx\MFP@Rload y\MFP@v@Sgn 0{\MFP@w@Und}\MFP@Radd
    \MFP@tempa\MFP@z@Int\relax
    \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi
    \ifnum\MFP@tempa<\MFP@ttteight\relax
      \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa
    \else
      \MFP@Rload x\MFP@z@Sgn10%
    \fi
    \MFP@Rcopy wy\MFP@Radd
  \fi}%
%    \end{macrocode}
%
% Conversion factors:
% \begin{itemize}
%   \item radians to degrees: $57.2957795130823209$
%   \item degrees to radians: $0.0174532925199433$
%   \item natural log to common log: $0.4342944819032518$
%   \item common log to natural log: $2.3025850929940457$
% \end{itemize}
%
% Note that the comparatively large size of the first number means that
% the $\pm0.5\cdot10^{-8}$ imprecision that $x$ implicitly carries will
% be multiplied to approximately $\pm29.6\cdot 10^{-8}$ in the result.
% The only way around this would be to operate with higher precision
% internally. We do that in the code for computing angles.
%    \begin{macrocode}
\def\MFP@Rdeg{\MFP@DPmul{57}{29577951}{30823209}}%
\def\MFP@Rrad{\MFP@DPmul{0}{01745329}{25199433}}%
\def\MFP@RbaseX{\MFP@DPmul{0}{43429448}{19032518}}%
\def\MFP@RbaseE{\MFP@DPmul{2}{30258509}{29940457}}%
%    \end{macrocode}
%
% There are very few angles that are expressible in eight digits whose sine
% or cosine can be expressed exactly in eight digits. For these, we do obtain
% an exact result. Other values produce inexact results. It would be nice
% if we could at least obtain these correctly rounded to eight decimals, but
% unfortunately our methods will often produce a result off by $1$ in the
% eighth decimal from the correctly rounded value. Anything that
% involves the addition of two or more rounded results can have this
% problem. The only way to get correctly rounded results is to carry out
% all operations internally to additional places. Even then, there will be
% the occasional $.4999\dots$ that should round to $0$ but rounds to $1$
% instead.
%
% For the cosine, just compute $\sin(90-x)$.
%    \begin{macrocode}
\def\MFP@Rcos{%
  \MFP@Rcopy xy\MFP@Rload x1{90}0\MFP@Rsub
  \MFP@Rcopyzx\MFP@Rsin}%
%    \end{macrocode}
%
% Reduce $|x|$ by subtracting $180$ from the integer part until it is less
% than $180$. Of course, $\sin x = \sgn(x)\sin(|x|)$ so we only need to
% compute $\sin(|x|)$. The sign will be that of $x$; each reduction by
% $180$ changes the sign, but the reduction code keeps track of that. If
% $x$ is 0 after the reduction, return zero.
%    \begin{macrocode}
\def\MFP@Rsin{%
  \MFP@tempa\MFP@x@Int
  \MFP@tempb\MFP@x@Frc
  \MFP@tempc\MFP@x@Sgn\relax
  \MFP@reduce@angle
  \ifnum\MFP@tempa>0  \MFP@@Rsin
  \else\ifnum\MFP@tempb>0  \MFP@@Rsin
  \else  \MFP@Rzero
  \fi\fi}%
%    \end{macrocode}
%
% This following reduces $|x|$ to the case $0 \le |x| < 180$. It assumes
% the integer part is in count register \cs{MFP@tempa}, the sign in
% \cs{MFP@tempc}.
%    \begin{macrocode}
\def\MFP@reduce@angle{%
  \ifnum\MFP@tempa<180
  \else
    \advance\MFP@tempa-180
    \MFP@tempc-\MFP@tempc
    \@xp\MFP@reduce@angle
  \fi}%
%    \end{macrocode}
%
% At this point, $|x|$ is represented by \cs{MFP@tempa} (integer part) and
% \cs{MFP@tempb} (fractional part). Also, we already know the sign stored
% in \cs{MFP@tempc}. Moreover $0 < {}$\cs{MFP@tempa}${} < 180$. We now
% reduce to $0 < |x| \le 90$ using $\sin(x) = \sin(180-|x|)$, and return
% $1$ if equal to $90$.
%
% The calculation of $180-x$ is optimized, taking advantage of the fact
% that both $x$ and the result are known to be positive. If the fractional
% part is positive, we borrow $1$ by reducing $180$ to $179$.
%    \begin{macrocode}
\def\MFP@@Rsin{%
  \ifnum\MFP@tempa<90
  \else
    \MFP@tempa -\MFP@tempa
    \ifnum\MFP@tempb>0
      \MFP@tempb -\MFP@tempb
      \advance\MFP@tempb \MFP@ttteight\relax
      \advance\MFP@tempa 179
    \else  \advance\MFP@tempa 180
    \fi
  \fi
  \ifnum\MFP@tempa=90
    \MFP@Rloadz \MFP@tempc10%
  \else
%    \end{macrocode}
%
% We would need to convert $x$ to radians (multiply by $\pi/180$) to use
% the standard power series, but instead we will incorporate the
% conversion factor into the power series coefficients.
%
% We will, however, try to increase accuracy by reducing the size of $x$
% and correspondingly increasing the appropriate factors. Since the
% number of significant figures of a product is limited by the least
% number of significant figures of the two factors, the bottleneck on
% accuracy is that of the smaller term: all our numbers have eight digits
% so if a number is small, the number of nonzero digits is small.
%
% Dividing by 100 seems a good choice (so our units are
% ``hectodegrees''). This makes $0 < x < .9$ and the integer part
% (\cs{MFP@tempa}) will be henceforth ignored.
%
% The addition of 50 is for rounding purposes. After that, our
% computations amount to concatenating the top six digits of
% \cs{MFP@tempb} to the digits of \cs{MFP@tempa}. This will produce the
% integer form of the fractional part of $x/100$ (the integer part of
% $x/100$ is zero).
%
% Division by $100$ can turn a number into $0$. This is one place we can
% lose accuracy (up to $\pm1$ in the last digit of the result). In
% compensation, the rest of the calculations become very much more
% accurate.
%    \begin{macrocode}
    \advance\MFP@tempb 50 \divide\MFP@tempb 100
    \multiply\MFP@tempa 1000000 \advance\MFP@tempb\MFP@tempa
    \ifnum\MFP@tempb=0
      \MFP@Rzero
    \else
%    \end{macrocode}
%
% We save some multiplications by working with $t=x^2$. As we don't need
% the original $x$ anymore, we simply replace it with the newly reduced
% value. We also save this reduced $x$ in another register, $s$, as
% we will need it again at the end, and our intermediate calculations do
% not preserve the $x$ register. Then we square $x$ and, if that
% square is $0$ we can skip all the power series and simply return $x$
% converted to radians. If $x^2$ is not zero, we save it in temporary
% register $t$ and call our power series. When this program is
% finished, all that remains is the final multiplication by a conversion
% factor (\cs{MFP@DPmul}).
%    \begin{macrocode}
      \MFP@Rload s\MFP@tempc0\MFP@tempb
      \MFP@Rcopy sx%
      \MFP@Rsq
      \ifnum \MFP@z@Frc>0
        \MFP@Rcopyz t\MFP@Rsin@prog
      \else
        \MFP@Rcopy sx%
      \fi
      \MFP@DPmul 1{74532925}{19943296}%
    \fi
  \fi}%
%    \end{macrocode}
%
% \cs{MFP@Rsin@prog} is the power series computation. The power series
% need only go to the $x^{13}$ term as the next is less than $10^{-9}$ and
% in our 8-place computations is indistingushable from $0$. Our series is:
% $$
% rx(1 - r^2t/3! + r^4t^2/5! - r^6t^3/7! + r^8t^4/9! - r^{10}t^5/11! +
%  r^{12}t^6/13!)
% $$
% where $r$ is the factor that converts $x$ to radian measure
% (hectodegrees to radians). When $x$ is so small as to produce $t = 0$ we
% have skipped all this.
%
% We minimize any multiplications of tiny numbers by computing this as
% $$
%   rx(1 - ft(1 - et(1 - dt(1 - ct(1 - bt(1 - at)))))).
% $$
% In this format, additional terms might actually make a difference,
% because $at$ is not particularly small. However, the more computations
% we have, the more errors accumulate. Therefore we take the fewest that
% produce acceptable accuracy.
%
% Now $r = 1.7453292519943296$ and $a$, $b$, etc., have formulas:
% $$
% \vcenter{\centering
%  $\displaystyle a = r^2/13/12,\ b = r^2/11/10,\ c = r^2/9/8$,\\
%  $\displaystyle d = r^2/7/6,\ e = r^2/5/4,\ f = r^2/3/2$.\par
% }
% $$
% An alternative method would be to accumulate a sum, computing each term
% from the previous one (e.g., if $u = t^3/7!$ is the fourth term, the next
% one is $u*t*(1/(8*9))$). This is a bit more complicated to code and requires
% moving values around more. It would have the advantage that we can stop
% whenever a term evaluates to zero, making computation faster for small
% values of $x$. I have not determined whether it would compromise
% accuracy.
%
% We avoid divisions by precomputing the coefficients $a$, $b$, $c$, etc.
% Note that without the reduction in $x$, the value of $a$ for example
% would  be $0.00000195$, with only three significant figures of accuracy.
% Now we can have seven, and the accuracy is more-or-less determined by that
% of the reduced x.
% $$
% \vcenter{\centering
%   $\displaystyle a = 0.01952676,\ b = 0.02769249,\ c = 0.04230797,$,\\
%   $\displaystyle d = 0.07252796,\ e = 0.15230871,\ f = 0.50769570$.\par
% }
% $$
% It is important to note that the following operations step all over
% the \cs{MFP@temp}\textit{x} \cs{count} registers, so we have made sure
% that we no longer need them.
%
% The \cs{MFP@flipz} computes $1-z$, where $z$ is the result of the
% previous operation. Instead of simply subtracting, we optimize based
% on the fact that $z$ is known to be nonnegative and not larger than $1$.
%
% The macro \cs{MFP@com@iter} `flipz' the previous result then multiplies
% by $t$ and the indicated coefficient. (The name of this macro stands for
% ``common iterated'' code; it is reused for some other power series.)
%
% For extra efficiency, the power series uses a ``small'' version of
% multiplication \cs{MFP@Rsmul}, used only when the factors are sure to
% lie in $[0,1]$. This does not take into account the sign of $x$,
% whence the ending \cs{edef}.
%    \begin{macrocode}
\def\MFP@Rsin@prog{%
  \MFP@Rcopy tx\MFP@Rload y10{01952676}\MFP@Rsmul%
  \MFP@com@iter{02769249}\MFP@com@iter{04230797}\MFP@com@iter{07252796}%
  \MFP@com@iter{15230871}\MFP@com@iter{50769570}\MFP@flipz \MFP@Rcopyzx
  \MFP@Rcopy sy\MFP@Rsmul\MFP@Rcopyzx\edef\MFP@x@Sgn{\MFP@s@Sgn}}%
\def\MFP@flipz{%
  \ifnum\MFP@z@Sgn=0
    \MFP@Rloadz 110%
  \else
      \MFP@tempa\MFP@ttteight
      \advance\MFP@tempa-\MFP@z@Frc\relax
      \MFP@Rloadz{\ifcase\MFP@tempa 0\else1\fi}0\MFP@tempa
  \fi}%
\def\MFP@com@iter#1{\MFP@flipz
  \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul
  \MFP@Rcopyzx\MFP@Rload y10{#1}\MFP@Rsmul}%
%    \end{macrocode}
%
% As to the accuracy of these computations, we can certainly lose accuracy
% at each step. In principle, if $x$ is known to 10 significant figures
% ($x \ge 10$~degrees), then even though we lose two figures with division
% by 100, the accuracy bottleneck is the fact that our coefficients have
% only seven figures.  Now we have 17 multiplications, and while products
% are said to have the same number of significant figures as the factors,
% in the worse case we can accumulate inaccuracy of about $.5\times
% 10^{-8}$ per multiplication. So we are not guaranteed an accuracy of
% more than about $\pm 10^{-7}$. Numerical tests, however, show that it
% isn't that bad, probably because the direction of inaccuracies usually
% varies randomly, and inaccuracies in one direction compensate for those
% going the other way. I have not seen a case where the result is off by
% more than $1$ in the last decimal place (i.e., $\pm 1.5\times 10^{-8}$).
% In the case where we can know the result exactly, $x=30$, we get an
% exact answer, even though we don't single it out (as we do $0$, $90$ and
% $180$).
%
% The following is the ``small'' version of \cs{MFP@Rmul}. Limited to
% non-negative numbers less than or equal to $1$. Theoretically all the
% numbers are strictly between $0$ and $1$, but in practice a
% multiplication could round to $0$ and then, after subtraction, a $1$
% could occur. We handle those easy cases separately, so that in
% \cs{MFP@@Rsmul} we don't have to worry about the integer parts at all.
%
% Also, since these are completely internal, we don't even define the
% overflow and underflow macros.
%    \begin{macrocode}
\def\MFP@Rsmul{%
  \ifnum \MFP@x@Sgn=0 \MFP@Rzero
  \else\ifnum \MFP@y@Sgn=0 \MFP@Rzero
  \else\ifnum\MFP@x@Int>0 \MFP@Rcopy yz%
  \else\ifnum\MFP@y@Int>0 \MFP@Rcopy xz%
  \else \MFP@@Rsmul
  \fi\fi\fi\fi}%
\def\MFP@@Rsmul{%
  \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii
  \MFP@split\MFP@y@Frc\MFP@y@Frc@i\MFP@y@Frc@ii
  \def\MFP@z@Frc@i  {0}\def\MFP@z@Frc@ii {0}%
  \def\MFP@z@Frc@iii{0}\def\MFP@z@Frc@iv {0}%
  \MFP@tempb\MFP@y@Frc@ii\relax
  \MFP@multiplyone\MFP@x@Frc@ii\MFP@z@Frc@iv
  \MFP@multiplyone\MFP@x@Frc@i\MFP@z@Frc@iii
  \MFP@tempb\MFP@y@Frc@i\relax
  \MFP@multiplyone\MFP@x@Frc@ii\MFP@z@Frc@iii
  \MFP@multiplyone\MFP@x@Frc@i\MFP@z@Frc@ii
  \MFP@carrym\MFP@z@Frc@iv\MFP@z@Frc@iii
  \MFP@carrym\MFP@z@Frc@iii\MFP@z@Frc@ii
  \ifnum\MFP@z@Frc@iii<5000 \else
    \MFP@tempb\MFP@z@Frc@ii
    \advance\MFP@tempb1
    \edef\MFP@z@Frc@ii{\number\MFP@tempb}\fi
  \MFP@carrym\MFP@z@Frc@ii\MFP@z@Frc@i
  \makeMFP@fourdigits\MFP@z@Frc@ii
  \makeMFP@fourdigits\MFP@z@Frc@i
  \def\MFP@z@Int{0}%
  \edef\MFP@z@Frc{\MFP@z@Frc@i\MFP@z@Frc@ii}%
  \edef\MFP@z@Sgn{\ifnum\MFP@z@Frc=0 0\else 1\fi}}%
%    \end{macrocode}
%
% \subsection{Polar angle}
%
% Instead of supplying the arcsine and arccosine functions, we supply the
% more general angle function. This is a binary operation that accepts
% the two coordinates of a point and computes its angle in polar
% coordinates. One then has, for example, $\arctan x =
% \mathop{\mathrm{angle}}(1,x)$ and $\arccos x = \mathop{\mathrm{angle}}
% (x, \sqrt{1-x^2})$.
%
% We start, as usual, with a few reductions. When the $y$-part is $0$, we
% immediately return $0$ or $180$. If the $y$-part is negative, we compute
% the angle for $(x,|y|)$ and negate it. If the $x$-part is negative, we
% compute the angle for $|x|$ and subtract it from $180$. Finally,
% reduced to both coordinates positive, if $y>x$ we compute the angle of
% $(y,x)$ and subtract that from $90$. Ultimately, we apply a power
% series formula for $\mathop{\mathrm{angle}}(1,y/x)$ and get convergence
% when the argument is less than $1$, but convergence is poor unless the
% argument is less than $1/2$. When that is not the case, conceptually, we
% rotate the picture clockwise by the arctangent of $1/2$, compute the
% angle of the new point and then add a precomputed value of
% $\arctan(1/2)$.
%    \begin{macrocode}
\def\MFP@Rangle{%
  \ifcase\MFP@y@Sgn\relax
    \ifcase\MFP@x@Sgn\relax
      \MFP@warn{Point (0,0) has no angle. Returning 0 anyway}%
      \MFP@Rzero
    \or
      \MFP@Rzero
    \else
      \MFP@Rloadz 1{180}0%
    \fi
    \@xp\@gobble
  \or
    \def\MFP@angle@Sgn{1}\@xp\@firstofone
  \else
    \def\MFP@y@Sgn{1}%
    \def\MFP@angle@Sgn{-1}\@xp\@firstofone
  \fi
 {\ifcase\MFP@x@Sgn\relax
    \MFP@Rloadz1{90}0%
  \or  \MFP@@Rangle
  \else
    \def\MFP@x@Sgn{1}\MFP@@Rangle
    \MFP@Rcopyzy\MFP@Rload x1{180}0\MFP@Rsub
  \fi
  \let\MFP@z@Sgn\MFP@angle@Sgn}}%
\def\MFP@@Rangle{%
  \MFP@Rcmp
  \ifMFP@neg
    \MFP@Rcopy xw\MFP@Rcopy yx\MFP@Rcopy wy%
    \MFP@@@Rangle
    \MFP@Rload x1{90}0\MFP@Rcopyzy\MFP@Rsub
  \else
    \MFP@@@Rangle
  \fi}%
%    \end{macrocode}
%
% Precisely what we do when we are finally in the case $0<y<x$ is perform
% a couple of reductions. Ultimately we want to compute the arctan of
% $z = y/x$. We once again use a power series but, for fast convergence,
% we require $z$ to be considerably less than $1$. For reasons we discuss
% later, we won't be able to use the more efficient \cs{MFP@Rsmul} so we
% want to keep the number of iterations of our power series calculations
% low.
%
% So we start with two iterations of the algorithm used by Knuth: if $y/x
% > 1/2$ we transform the pair $(x,y)$ to a new one whose angle has been
% reduced by $\arctan (1/2)$. The new pair is $(x',y') = (2x+y, 2y-x)$.
% If we still have $y/x > 1/4$, we perform $(x'',y'') = (4x + y, 4y - x)$,
% which then satisfies $y''/x'' \le 1/4$. When either of these
% transformations is performed, we add the corresponding angle to the
% ``angle-so-far'' in register $a$.
%
% We could continue this iteration 32 times to get (theoretically) the
% angle in degrees to $\pm 10^{-8}$. That seems a bit long, plus the
% accumulation of errors over $32$ iterations could (in the worst case)
% produce less than $\pm10^{-7}$ accuracy.
%
% To get the accuracy we need, we work in ``scaled reals''. That is, we
% get 10 effective decimal places of accuracy by letting an $x$ in the
% range $0< x < 100$ stand for $0< x/100 < 1$.
%
% Our initial reductions can increase $x$ by a factor of about 13.
% Moreover, we ultimately need to scale y by 100 when we convert to
% scaled computations. Thus, if we make sure $x$ is less than
% $1\,000\,000$, we will prevent overflow in both cases.
%    \begin{macrocode}
\def\MFP@Rquad{\MFP@Rdbl\MFP@Rcopyzx\MFP@Rdbl}%
\def\MFP@@@Rangle{%
  \MFP@Rcopy xs\MFP@Rcopy yt%
  \ifnum\MFP@x@Int<1000000
  \else
    \MFP@RdivC \MFP@Rcopyz s%
    \MFP@Rcopy tx\MFP@RdivC \MFP@Rcopyz t%
  \fi
  \ifnum\MFP@t@Sgn=0 \MFP@Rzero
  \else
    \MFP@Rcopy tx\MFP@Rdbl\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp
    \ifMFP@pos
      \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rdbl
      \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd
      \MFP@Rcopyz s\MFP@Rcopy ut%
      \MFP@Rload a1{2656}{50511771}%
    \else
      \MFP@Rload a000%
    \fi
    \MFP@Rcopy tx\MFP@Rquad\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp
    \ifMFP@pos
      \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rquad
      \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd
      \MFP@Rcopyz s\MFP@Rcopy ut%
      \MFP@Rcopy ax\MFP@Rload y1{1403}{62434679}%
      \MFP@Radd\MFP@Rcopy za%
    \fi
    \MFP@Rcopy tx\MFP@RmulC
    \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rdiv
    \MFP@Rcopyzx\MFP@Ratanc
    \MFP@Rcopyzx\MFP@Rdeg
    \MFP@Rcopyzx\MFP@Rcopy ay\MFP@Radd
    \MFP@Rcopyzx\MFP@RdivC
  \fi}%
%    \end{macrocode}
%
% Here are fast multiplication and division by 100. We need these because
% we are going to compute the arctangent in radians to ten decimal places.
% We do this by computing with scaled reals in which, for example, $0.5$
% is represented by $50.0$. When we do this, multiplication requires a
% division by 100: $.5\times.5 = .25$ would be computed as $(50\times50) /
% 100 = 25$.
%    \begin{macrocode}
\def\MFP@twoofmany#1#2#3\MFP@end{#1#2}%
\def\MFP@gobbletwo#1#2{}%
\def\MFP@RmulC{%
  \edef\MFP@z@Int{\MFP@x@Int\@xp\MFP@twoofmany\MFP@x@Frc\MFP@end}%
  \edef\MFP@z@Frc{\@xp\MFP@gobbletwo\MFP@x@Frc00}%
  \edef\MFP@z@Sgn{\MFP@x@Sgn}}%
\def\MFP@RdivC{%
  \makeMFP@eightdigits\MFP@x@Int
  \makeMFP@eightdigits\MFP@x@Frc
  \@XP\MFP@@RdivC\@xp\MFP@x@Int\MFP@x@Frc\MFP@end}%
\def\MFP@@RdivC#1#2#3#4#5#6{%
  \edef\MFP@z@Int{\number#1#2#3#4#5#6}%
  \MFP@@@RdivC}%
\def\MFP@@@RdivC#1#2#3#4#5#6#7#8#9\MFP@end{%
  \MFP@tempa#1#2#3#4#5#6#7#8\relax
  \ifnum#9>49 \advance\MFP@tempa1 \fi
  \edef\MFP@z@Frc{\number\MFP@tempa}%
  \makeMFP@eightdigits\MFP@z@Frc
  \edef\MFP@z@Sgn{\MFP@x@Sgn}%
  \ifnum\MFP@tempa=0
    \ifnum\MFP@z@Int=0 \def\MFP@z@Sgn{0}\fi
  \fi}%
%    \end{macrocode}
%
% Finally, we compute the arctan of a scaled real producing a result
% as a scaled number (i..e., as ``centiradians''---$100$ times the number
% of radians) using a power series. Since that number could be
% around $0.25$ (represented by $25.0$), we have to sum to at least its
% $15$th power ($4^{-15}/15 \approx .6\times 10^{-10}$ and the next term
% in the series is effectively $0$). Fortunately, the power series has
% only odd terms, so there are only eight terms we actually need to calculate.
% The calculation proceeds much like the one for the sine, starting with
% the sum
% $$
%   x\left(1 - \frac{u}{3} + \frac{u^2}{5} - \frac{u^3}{7} + \cdots
%   - \frac{u^7}{15}\right),
% $$
% where $u = x^2$.
%
% We start with the common iterated code. It assumes a scaled value in $x$
% to be multiplied by the saved (scaled) value of $x^2$ (in register $u$)
% and by a coefficient (supplied in separate integer and fractional
% parts). It ends with the new value in $x$.
%    \begin{macrocode}
\def\MFP@scaledmul{\MFP@Rmul\MFP@Rcopyzx\MFP@RdivC}%
\def\MFP@atan@iter#1#2{%
  \MFP@Rcopy uy\MFP@scaledmul
  \MFP@Rcopyzx\MFP@Rload y1{#1}{#2}\MFP@scaledmul
  \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}%
  \MFP@Rsub\MFP@Rcopyzx}%
\def\MFP@Ratanc{%
  \MFP@Rcopy xs\MFP@Rcopy xy\MFP@scaledmul
  \ifnum \MFP@z@Sgn=0
    \MFP@Rcopy sz%
  \else
    \MFP@Rcopyz u\MFP@Rcopyzx
    \MFP@Rload y1{86}{66666667}\MFP@scaledmul
    \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}\MFP@Rsub\MFP@Rcopyzx
    \MFP@atan@iter{84}{61538462}\MFP@atan@iter{81}{81818182}%
    \MFP@atan@iter{77}{77777778}\MFP@atan@iter{71}{42857143}%
    \MFP@atan@iter{60}{00000000}\MFP@atan@iter{33}{33333333}%
    \MFP@Rcopy sy\MFP@scaledmul
  \fi}%
%    \end{macrocode}
%
% \subsection{Logarithms}
%
% Now for logarithms. We are going to compute both common logarithms
% (base $10$) and natural logarithms (base $e$). The first step of the
% calculation is be essentially trivial and works with base 10: to
% get the integer part of the log for numbers with positive integer part,
% count the digits in the integer part and subtract $1$. For numbers less
% than one, count the number of zeros at the beginning of the fractional
% part and add $1$ (subtract this from the result of the second part). This
% reduces the problem to numbers $1 \le x < 10$. A few divisions (when
% necessary) reduce to the case where $x = 1 + u$ with $u$ small enough
% that the power series for $\log (1 + u)$ can be computed accurately in
% an acceptable number of of terms. Then we proceed as in the code for
% sine.
%
% The power series produces a logarithm in base $e$ so we ultimately get
% the answer in two parts, with the parts calculated for different bases.
% The last step for the common log is to multiply the second part by a
% conversion factor and add the first to it. For natural log, convert the
% first and add the second. Which one is to be returned is passed as a
% boolean.
%
% We keep the value-so-far in register $s$ and the modified
% $x$-value in register $t$.
%    \begin{macrocode}
\newif\ifMFP@natural
\def\MFP@Rlog{\MFP@naturalfalse\MFP@Rlog@}%
\def\MFP@Rln{\MFP@naturaltrue\MFP@Rlog@}%
\def\MFP@Rlog@{%
  \ifnum\MFP@x@Sgn=0
    \MFP@logofzero@err
    \MFP@Rloadz{-1}\LogOfZeroInt\LogOfZeroFrac
  \else
    \ifnum \MFP@x@Sgn<0
      \MFP@warn{The logarithm of a negative number is complex.
      \MFP@msgbreak Only the real part will be computed}%
      \def\MFP@x@Sgn{1}%
    \fi
    \MFP@Rload s000%
%    \end{macrocode}
%
% If the integer part is $0$, the fractional part is not. Save the
% number of places that will be shifted in \cs{MFP@tempa}. We use
% \cs{number} to strip the leading zeros and (essentially) we count
% the number of digits that remain. Then we shift left, putting the first
% digit into the integer part of \reg{s} and the rest into the
% fractional part.
%    \begin{macrocode}
    \ifnum \MFP@x@Int=0
      \edef\MFP@x@Tmp{\number\MFP@x@Frc}%
      \MFP@tempa=\MFP@numshiftL\MFP@x@Tmp\relax
      \def\MFP@s@Sgn{-1}%
      \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}%
      \edef\MFP@t@Frc{\@xp\@gobble\MFP@x@Tmp0}%
      \MFP@padtoeight\MFP@t@Frc
    \else
%    \end{macrocode}
% When the integer part is not $0$, we get the number of digits to
% shift again in \cs{MFP@tempa}. It will be one less than the number of
% integer digits.
%    \begin{macrocode}
      \MFP@tempa\MFP@numshiftR\MFP@x@Int
      \edef\MFP@x@Tmp{\MFP@x@Int\MFP@x@Frc}%
      \ifnum\MFP@tempa>0 \def\MFP@s@Sgn{1}\fi
      \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}%
      \edef\MFP@x@Tmp{\@xp\@gobble\MFP@x@Tmp}%
      \edef\MFP@t@Frc{\@xp\MFP@eightofmany\MFP@x@Tmp\MFP@end}%
    \fi
%    \end{macrocode}
%
% Now the integer part of $\log_{10} x$ is known. We save it in $s$.
% Also set the sign of the reduced argument (positive). Then call
% \cs{MFP@Rlog@reduce}, which reduces $x$ to less than $1.161\,$ while
% possibly increasing $s$. For the natural log, we convert the value in
% $s$.
%
% If the reduced $x$ is $1$, return the value in $s$, otherwise call the
% power series program (discarding the integer part of $t$, which should
% be a $1$). Finally, convert the returned result if necessary and add
% register $s$ to it.
%    \begin{macrocode}
    \edef\MFP@s@Int{\number\MFP@tempa}%
    \def\MFP@t@Sgn{1}%
    \MFP@Rlog@reduce
    \ifMFP@natural \MFP@Rcopy sx\MFP@RbaseE \MFP@Rcopy zs\fi
    \ifnum\MFP@t@Frc=0
      \MFP@Rcopy sz%
    \else
      \def\MFP@t@Int{0}\MFP@Rlog@prog
      \ifMFP@natural\else \MFP@Rcopyzx \MFP@RbaseX \fi
      \MFP@Rcopy sy\MFP@Rcopyzx\MFP@Radd
    \fi
  \fi}%
%    \end{macrocode}
%
% We determine the size of a right shift by lining up the digits in
% the integer part, followed by the possible numbers, and picking out the
% ninth argument. Similarly, to get a left shift we line up the digits
% of the fractional part (minus the leading zeros) followed by the
% possible numbers, and again picking the ninth.
%    \begin{macrocode}
\def\MFP@numshiftR#1{\@xp\MFP@ninthofmany#176543210\MFP@end}%
\def\MFP@numshiftL#1{\@xp\MFP@ninthofmany#112345678\MFP@end}%
%    \end{macrocode}
%
% In \cs{MFP@Rlog@reduce} we divide by the square root of 10 if the number
% is significantly larger than that (adding $.5$ to value-so-far). We
% repeat with the 4th, 8th and 16th roots. It seems that this could be
% where errors can accumulate, so the divisions are done with double
% precision multiplication and $x$ is scaled by 100. Our check whether
% $x > \sqrt{10}$ is rather rough: comparing the first three digits only,
% but even in the worst case, the final $x$ is less than $1.1605$, so at
% most $0.161$ is fed to the power series.
%    \begin{macrocode}
\def\MFP@Rlog@reduce{%
  \MFP@Rcopy tx\MFP@RmulC\MFP@Rcopyz t%
  \MFP@reduceonce {316}{31622776}{60168379}{50000000}%
  \MFP@reduceonce {177}{56234132}{51903491}{25000000}%
  \MFP@reduceonce {133}{74989420}{93324558}{12500000}%
  \MFP@reduceonce {115}{86596432}{33600654}{06250000}%
  \MFP@Rcopy tx\MFP@RdivC\MFP@Rcopyz t}%
\def\MFP@reduceonce#1#2#3#4{%
  \ifnum\MFP@t@Int>#1\relax
    \MFP@Rcopy tx%
    \MFP@DPmul 0{#2}{#3}\MFP@Rcopyz t%
    \MFP@Rcopy sx\MFP@Rload y10{#4}\MFP@Radd
    \MFP@Rcopyz s%
  \fi}%
%    \end{macrocode}
%
% Now we have a value for $t$ of the form $1 + u$ with $0\le u < 0.161$.
% We will use the formula
% $$
%    \ln (1 + u) = \sum_{n=1}^\infty (-1)^{n-1} \frac{u^n}{n}.
% $$
% We only need to carry it far enough to assure that the remaining terms
% would be zero in our finite resolution arithmetic, that is
% $(.161)^k/k < .5\times 10^{-8}$. This is satisfied by $k=10$.
% So we carry the sum to 9 places.
%
% Again, we compute this by
% $$
%    u(1-au(1-bu(1-cu(1-du(1-eu(1-fu(1-gu(1-hu))))))))
% $$
% where $a= 1/2$, $b = 2/3$,\dots, $g=7/8$, and $h=8/9$
% This arrangement allows us to reuse \cs{MFP@com@iter}.
%    \begin{macrocode}
\def\MFP@Rlog@prog{%
  \MFP@Rcopy tx\MFP@Rload y10{88888889}\MFP@Rsmul
  \MFP@com@iter{87500000}\MFP@com@iter{85714286}\MFP@com@iter{83333333}%
  \MFP@com@iter{80000000}\MFP@com@iter{75000000}\MFP@com@iter{66666667}%
  \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul}%
%    \end{macrocode}
%
% \subsection{Powers}
%
% With the exponential function we immediately return $1$ if $x=0$. We
% call two separate handlers for positive and negative $x$. This is
% because the issues are different between positive and negative
% exponents.
%    \begin{macrocode}
\def\MFP@Rexp{%
  \ifcase\MFP@x@Sgn\relax
    \MFP@Rloadz 110%
  \or
    \MFP@Rexp@pos
  \else
    \def\MFP@x@Sgn{1}%
    \MFP@Rexp@neg
  \fi}%
%    \end{macrocode}
%
% One issue for positive exponents is overflow, so we issue an error
% message for that case. The largest mumber that will not produce
% overflow is $18.42068074$ so we first compare to that; if larger,
% issue the error message and return $99999999.99999999$.
%
% We compute the integer power first, using an \cs{ifcase}. Because there
% are only 19 cases to consider a table lookup is faster than
% multiplications.
%
% Then, we examine the first digit $d$ after the decimal and compute
% $e^{0.d}$, again by cases. This is multiplied by the integer power
% previously found. What remains is the rest of the fractional part of
% $x$, which is strictly less than $0.1$. The exponential of this is
% computed using the first several terms of the power series for $e^x$.
%    \begin{macrocode}
\def\MFP@Rexp@pos{%
  \MFP@Rload y1{18}{42068074}\MFP@Rcmp
  \ifMFP@pos
    \MFP@expoverflow@err
    \MFP@Rloadz 1\MaxRealInt\MaxRealFrac
  \else
    \MFP@tempa\MFP@x@Int
    \edef\MFP@powerof@e{%
     1\ifcase\MFP@tempa
      10\or
      2{71828183}\or
      7{38905610}\or
      {20}{08553692}\or
      {54}{59815003}\or
      {148}{41315910}\or
      {403}{42879349}\or
      {1096}{63315843}\or
      {2980}{95798704}\or
      {8103}{08392758}\or
      {22026}{46579481}\or
      {59874}{14171520}\or
      {162754}{79141900}\or
      {442413}{39200892}\or
      {1202604}{28416478}\or
      {3269017}{37247211}\or
      {8886110}{52050787}\or
      {24154952}{75357530}\or
      {65659969}{13733051}\else
      {\MaxRealInt}{\MaxRealFrac}\fi}%
    \@xp\MFP@Rloadz\MFP@powerof@e
    \ifnum\MFP@x@Frc=0
    \else
      \MFP@Rcopyz s%
      \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end
      \edef\MFP@powerof@e{%
      y1\ifcase\MFP@tempa
        10\or
        1{10517092}\or
        1{22140276}\or
        1{34985881}\or
        1{49182470}\or
        1{64872127}\or
        1{82211880}\or
        2{01375271}\or
        2{22554093}\or
        2{45960311}\else
        10\fi}%
      \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}%
      \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul
      \ifnum\MFP@t@Frc=0
      \else
        \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc
        \MFP@Rexp@pos@prog
        \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Rmul
      \fi
    \fi
  \fi}%
%    \end{macrocode}
%
% Since the $x$ value is now less than $0.1$, we can get eight places of
% accuracy with only six terms of the power series. We can also arrange to
% use the more efficient \cs{MFP@Rsmul} for multiplication.
%
% We organize the computation thusly
% $$
%   1 + (x + x/2(x + x/3(x + x/4(x + x/5(x + x/6)))))
% $$
% We start by loading $x$ (now in register $t$) into register
% $z$, then repeatedly run \cs{MFP@Rexp@iter} feeding it the
% successive values of $1/n$. This iterator first multiplies the most
% recent result (the $z$ register) by $1/n$, then that by $x$ and
% then adds $x$ to that. The final step is to add $1$.
%    \begin{macrocode}
\def\MFP@Rexp@pos@prog{%
  \MFP@Rcopy tz\MFP@Rexp@iter{14285714}\MFP@Rexp@iter{16666667}%
  \MFP@Rexp@iter{20000000}\MFP@Rexp@iter{25000000}%
  \MFP@Rexp@iter{33333333}\MFP@Rexp@iter{50000000}\MFP@Rcopyzx
  \MFP@Rincr}%
\def\MFP@Rexp@iter#1{%
  \MFP@Rcopyzx\MFP@Rload y10{#1}\MFP@Rsmul
  \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul
  \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd}%
%    \end{macrocode}
% It is impossible to get accuracy to the last digit when $e^x$ is large.
% This is because an absolute error in $x$ converts to a relative error
% in $e^x$, That is, knowing $x$ only to $10^{-8}$ means $e^x$ is off by
% (about) $e^x\cdot 10^{-8}$. Roughly speaking, this means only about $8$
% places of $e^x$ are accurate, so if (for example) the integer part of
% $e^x$ has six places then only two places after the decimal are
% significant.
%
% \bigskip
% The first issue with negative exponents is that it doesn't take much to
% produce a value of $e^{-x}$ that rounds to $0$. Any $x > 19.11382792$. So
% we start by comparing to that value and simply return zero if $x$ is
% larger.
%
% We perform exactly the same reductions as for positive exponents,
% handling the integer part and the first decimal separately. Then we call
% the power series program (not the same).
%    \begin{macrocode}
\def\MFP@Rexp@neg{%
  \MFP@Rload y1{19}{11382792}%
  \MFP@Rcmp
  \ifMFP@pos
    \MFP@Rloadz 000%
  \else
    \MFP@tempa\MFP@x@Int
    \edef\MFP@powerof@e{%
      \ifcase\MFP@tempa
      11{0}\or
      10{36787944}\or
      10{13533528}\or
      10{04978707}\or
      10{01831564}\or
      10{00673795}\or
      10{00247875}\or
      10{00091188}\or
      10{00033546}\or
      10{00012341}\or
      10{00004540}\or
      10{00001670}\or
      10{00000614}\or
      10{00000226}\or
      10{00000083}\or
      10{00000031}\or
      10{00000011}\or
      10{00000004}\or
      10{00000002}\or
      10{00000001}\else
      000\fi}%
    \@xp\MFP@Rloadz\MFP@powerof@e
    \ifnum\MFP@x@Frc=0
    \else
      \MFP@Rcopyz s%
      \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end
      \edef\MFP@powerof@e{%
      y1\ifcase\MFP@tempa
        10\or
        0{90483742}\or
        0{81873075}\or
        0{74081822}\or
        0{67032005}\or
        0{60653066}\or
        0{54881164}\or
        0{49658530}\or
        0{44932896}\or
        0{40656966}\else
        10\fi}%
      \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}%
      \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul
      \ifnum\MFP@t@Frc=0
      \else
        \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc
        \MFP@Rexp@neg@prog
        \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rmul
      \fi
    \fi
  \fi}%
%    \end{macrocode}
%
% Since $x$ is now positive we calculate $e^{-x}$. Again we need only up
% to the 6th power, organized as follows
% $$
% 1 - x(1 - x/2(1 - x/3(1 - x/4(1 - x/5(1 - x/6)))))
% $$
% Since this has exactly the same form as the the power series calculation
% for $\log$ and $\sin$, we can reuse the code in \cs{MFP@com@iter}. We
% end with the final multiplication by $x$ and the subtraction from 1
% rather than call \cs{MFP@com@iter} with a useless multiplication by $1$.
%    \begin{macrocode}
\def\MFP@Rexp@neg@prog{%
  \MFP@Rcopy tx\MFP@Rload y10{14285712}\MFP@Rsmul
  \MFP@com@iter{16666667}\MFP@com@iter{20000000}%
  \MFP@com@iter{25000000}\MFP@com@iter{33333333}%
  \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx
  \MFP@Rcopy ty\MFP@Rsmul\MFP@flipz}%
%    \end{macrocode}
%
% The most efficient way to take an integer power of a number $x$ is to
% scan the binary code for the exponent. Each digit $1$ in this code
% corresponds to a $2^k$ power of $x$, which can be computed by repeatedly
% squaring $x$. These \emph{dyadic} powers are mutiplied together. We can
% convert this idea to a simple loop illustrated by this example of
% finding $x^{13}$ ($13 = 1101$ in base $2$). Here $p$ holds the current
% product and $q$ holds the current dyadic power of $x$, initialized with
% $p=1$ and $q=x$:
% \begin{enumerate}
% \item Rightmost digit 1: update $p\leftarrow pq = x$ and $q\leftarrow
%   q^2 = x^2$.
% \item Next digit 0: Just update $q\leftarrow q^2 = x^4$.
% \item Next digit 1: update $p \leftarrow pq = x^5$ and $q\leftarrow
%   q^2 = x^8$.
% \item Next digit 1: update $p \leftarrow pq = x^{13}$, detect that we
%   are at the end and skip the update of $q$. Return $p$.
% \end{enumerate}
% Of course, this requires the binary digits of the exponent $n$. But the
% rightmost digit of $n$ is $1$ if and only if $n$ is odd, and we can
% examine each digit in turn if we divide $n$ by $2$ (discarding the
% remainder) at each stage. We detect the end when $n$ is reduced to $1$.
%
% Accuracy is partly a function of the number of multiplications.
% The above scheme requires at most  $\lfloor\log_2 n\rfloor$ squarings
% and at most $\lceil \log_2 n \rceil$ multiplications for $x^n$, while
% directly multiplying $x\cdot x \cdots x$ would require $n-1$
% multiplications.
%
% I have tested with an exponents around $8000$, which has 13 binary
% digits. Each squaring could double the relative error. For that
% large a power, the base has to be near 1 to avoid overflow or underflow.
% So the relative error is about $.5(10)^{-8}$. Doubling that 12 times
% would increase it to about $.00004$, and the result could have as little
% as 4 or 5 significant figures. In these tests, the results were actually
% accurate to 5 or 6 significant figures, starting with 8 figures. Raising
% to this power takes only about $25$ times as long as a single
% multiplication (rather than $7999$ times).
%
% For negative powers we can either find the positive power of $x$ and
% take its reciprocal or take the reciprocal of $x$ and find its positive
% power. We do the second so that overflow can be detected in
% \cs{MFP@@Rpow}.
%    \begin{macrocode}
\def\MFP@Rpow{%
  \ifnum\MFP@y@Frc>0
    \MFP@warn{The "pow" function requires an integer power.
      \MFP@msgbreak The fractional part will be ignored}%
  \fi
  \MFP@loopctr=\MFP@y@Int\relax
  \ifnum\MFP@loopctr=0
    \MFP@Rloadz 110%
  \else
    \ifnum\MFP@x@Sgn=0
      \ifnum\MFP@y@Sgn>0
        \MFP@Rloadz 000%
      \else
        \MFP@badpower@err
        \MFP@Rloadz 1\xOverZeroInt\xOverZeroFrac
      \fi
    \else
      \ifnum\MFP@x@Sgn>0
        \def\MFP@power@Sgn{1}%
      \else
        \edef\MFP@power@Sgn{\ifodd\MFP@loopctr -\fi 1}%
      \fi
      \ifnum\MFP@y@Sgn<0 \MFP@Rinv \MFP@Rcopyzx\fi
      \ifnum\MFP@loopctr=1
        \MFP@Rloadz \MFP@power@Sgn\MFP@x@Int\MFP@x@Frc
      \else
        \MFP@@Rpow
      \fi
    \fi
  \fi}%
%    \end{macrocode}
%
% This implements the algorithm discussed above. We save $x$ in register
% $q$, initialize the starting value of $1$ in \reg{p} and then
% run the loop. If the binary digit just read is a 1 (i.e., \cs{ifodd} is
% true), it multiplies $p$ and $q$. It also saves the last product (copies
% \reg{z} to \reg{p}). This need not be done on the last iteration,
% but must not be moved out of the \cs{ifodd} conditional because
% intervening computations modify $z$. If there are more iterations to do
% (i.e., the \cs{ifnum} is true), this squares $q$ and reduces the
% counter. Note that the exponents $0$ and $1$ do not occur since we have
% handled them separately.
%
% In case of overflow (either the multiplication or the squaring) we
% break the loop and return $\pm\infty$.
%    \begin{macrocode}
\def\MFP@@Rpow{%
  \MFP@Rcopy xq%
  \MFP@Rload p110%
  \MFP@Rpow@loop}%
\def\MFP@Rpow@loop{%
  \ifodd\MFP@loopctr
    \MFP@Rcopy px\MFP@Rcopy qy\MFP@Rmul
    \ifnum \MFP@z@Ovr>0 \MFP@handle@expoverflow
    \else
      \ifnum\MFP@loopctr>1 \MFP@Rcopyz p\fi
    \fi
  \fi
  \ifnum\MFP@loopctr>1
    \MFP@Rcopy qx\MFP@Rsq
    \ifnum \MFP@z@Ovr>0 \MFP@handle@expoverflow
    \else
      \MFP@Rcopyz q%
      \divide\MFP@loopctr 2
      \@XP\MFP@Rpow@loop
    \fi
  \fi}%
\def\MFP@handle@expoverflow{%
  \MFP@expoverflow@err
  \MFP@loopctr=0
  \MFP@Rloadz\MFP@power@Sgn\MaxRealInt\MaxRealFrac}%
%    \end{macrocode}
%
% \subsection{The square root}
%
% One can combine logarithms and exponentials to get any power: to get
% $x^y$, compute $e^{y\ln x}$. This has the disadvantage that it doesn't
% work if $x$ is negative. Most powers of negative numbers are not
% defined, but certainly integer powers are. Thus we have defined
% \cs{MFPpow} and \cs{Rpow} for that case.
%
% If we enforce a positive $x$, then $y$ can have any value. However,
% the computation of $e^{.5\ln x}$ cannot give a result as good as one can
% get from a special purpose algorithm for the square root. For example,
% the inaccuracies in computing $\ln x$ will make $e^{.5\ln 9}$ inexact,
% while the square root function we implement below will produce exactly
% $\sqrt{9} = 3$. In fact, if a square root can be expressed exactly
% within our 8-digit precision, our code will find it.
%
% For the square root we return zero if $x$ is not positive. If the integer
% part of $x$ is $0$, we copy the fractional part to the integer part
% (that is, we multiply by $10^{8}$, remembering to multiply by $10^{-4}$
% later). This makes the square root of such numbers rather more
% accurate. (To get around some other rare but annoying inaccuracies, we
% go through a similar process when the integer part of $x$ is at most $4$
% digits, multiplying by $10^4$ before and by $10^{-2}$ after.)
%
% We then compute the square root using an algorithm that will
% be exact whenever possible. We perform one additional processing step.
% To explain it, note that our algorithm actually produces the largest
% number $s$ with four digits right of the decimal place that satisfies $s^2
% \le x$. That is
% $$
%    s^2 \le x < \left( s + 10^{-4} \right)^2
% $$
% From this it follows that $x = (s+\epsilon)^2 = s^2 + 2s\epsilon +
% \epsilon^2$ with $\epsilon < 10^{-4}$ (and so $\epsilon^2 < 10^{-8}$).
% We estimate this $\epsilon$ and add that estimate to $s$. The estimate
% we use is obtained by discarding the very small $\epsilon^2$ and solving
% for the remaining $\epsilon$ get
% $$
%     \epsilon \approx \bar\epsilon = \frac{x-s^2}{2s}
% $$
% With this value, $s + \bar\epsilon$ misses the exact square root by at
% most $\epsilon^2/(2s) < .5\cdot 10^{-8}$,  because $s \ge 1$.
% The final result $s + \bar\epsilon$ is equivalent to computing the
% average of $s$ and $x/s$. This, possibly divided by $10^4$ or $10^2$ is the
% returned value.
%
% By tests, with rare exceptions, our computations produces a result
% correct in all eight decimal places. In the rare exception, the last
% place is within $1$ of the correct value.
%    \begin{macrocode}
\def\MFP@Rsqrt{%
  \ifcase\MFP@x@Sgn\relax
    \MFP@Rzero
  \or
    \ifnum\MFP@x@Int=0
      \def\MFP@sqrt@reduce{2}%
      \edef\MFP@x@Int{\number\MFP@x@Frc}%
      \edef\MFP@x@Frc{00000000}%
    \else\ifnum\MFP@x@Int<10000
      \def\MFP@sqrt@reduce{1}%
      \edef\MFP@x@Int{\MFP@x@Int\@xp\MFP@fourofmany\MFP@x@Frc\MFP@end}%
      \edef\MFP@x@Frc{\@xp\MFP@gobblefour\MFP@x@Frc0000}%
    \else
      \def\MFP@sqrt@reduce{0}%
    \fi\fi
    \MFP@Rcopy xt%
    \MFP@Isqrt
    \MFP@Rcopyz s\MFP@Rcopyzy
    \MFP@Rcopy tx\MFP@Rdiv
    \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Radd
    \MFP@Rcopyzx\MFP@Rhalve
    \ifcase \MFP@sqrt@reduce\relax
    \or
      \MFP@Rcopyzx\MFP@Rload y10{01000000}\MFP@Rmul
    \or
      \MFP@Rcopyzx\MFP@Rload y10{00010000}\MFP@Rmul
    \fi
  \else
    \MFP@warn{Square root of a negative number. Zero will be returned.}%
    \MFP@Rzero
  \fi}%
\def\MFP@fourofmany#1#2#3#4#5\MFP@end{#1#2#3#4}%
\def\MFP@gobblefour#1#2#3#4{}%
%    \end{macrocode}
%
% There is a rather straightforward pencil and paper algorithm that
% provides the square root digit by digit, and it produces an exact answer
% when that is possible. Unfortunately, the decimal version is not easy to
% code. Fortunately the same algorithm works in any number base and it is
% rather simple to code the binary version (because we only need to decide
% at each stage whether the ``next digit'' is $0$ or $1$. This produces a
% square root in binary digits, from which it is easy to compute the
% number itself. The result is exact if the answer would be a finite
% number of binary digits. We apply it to the integer $10^8 x$. While this
% number is too large for \TeX{} to handle as an integer, it is not that
% hard to convert it to a string of binary digits stored in a macro.
%
% The algorithm simplifies somewhat if we proces a base 4 integer,
% producing a base 2 result. Also, instead of producing the square root
% encoded in a string of binary digits, we simply build the numerical
% result as we discover the binary digits (multiply previous value by two
% and add the new digit.) Fortunately, the square root of $10^8 x$ (and
% the temporary scratch registers used in the code) will never exceed
% \TeX{}'s limit for integers.
%
% The macro \cs{MFP@ItoQ} implements the conversion to base-4 digits.
% The two arguments are the integer and fractional part of $x$. The
% result is stored in \cs{MFP@ItoQ@Tmp}, which is so far only used by the
% square root code.
%
% The test \cs{ifodd}\cs{MFP@tempb} is used to get the binary digits.
% Combining two of them yields the quadrenary digits. The
% \cs{ifodd}\cs{MFP@tempa} tests are there to check whether there
% will be a remainder after division by $2$, which should then be
% inserted at the front of \cs{MFP@tempb} before division by $2$. Two
% divisions by $2$ each iteration amounts to division by $4$. This is slightly
% more efficient than dividing by $4$ and determining the remainder.
%    \begin{macrocode}
\def\MFP@ItoQ#1#2{%
  \MFP@tempa#1\relax\MFP@tempb#2\relax
  \def\MFP@ItoQ@Tmp{}\MFP@ItoQ@loop}%
\def\MFP@ItoQ@loop{%
  \ifodd\MFP@tempb
    \ifodd\MFP@tempa \advance\MFP@tempb \MFP@ttteight\relax\fi
    \divide\MFP@tempa2 \divide\MFP@tempb2
    \edef\MFP@ItoQ@Tmp{\ifodd\MFP@tempb 3\else 1\fi\MFP@ItoQ@Tmp}%
  \else
    \ifodd\MFP@tempa \advance\MFP@tempb \MFP@ttteight\relax\fi
    \divide\MFP@tempa2 \divide\MFP@tempb2
    \edef\MFP@ItoQ@Tmp{\ifodd\MFP@tempb 2\else 0\fi\MFP@ItoQ@Tmp}%
  \fi
  \ifodd\MFP@tempa \advance\MFP@tempb \MFP@ttteight\relax\fi
  \divide\MFP@tempa 2 \divide\MFP@tempb 2
  \ifnum\MFP@tempa>0
    \@xp\MFP@ItoQ@loop
  \else\ifnum\MFP@tempb>0
    \@XP\MFP@ItoQ@loop
  \fi\fi}%
%    \end{macrocode}
%
% This integer square root $n$ is $10^4$ times the largest number $y$
% satisfying $y^2 \le x$ and having at most four decimal places. The rest of
% the code after the \cs{MFP@Isqrt@loop} is intended to divide $n$
% (returned in \cs{MFP@tempc}) by $10^4$ in order to get the number $y$
% itself.
%    \begin{macrocode}
\def\MFP@Isqrt{%
  \MFP@ItoQ\MFP@x@Int\MFP@x@Frc
  \MFP@tempa=0 \MFP@tempb=0 \MFP@tempc=0
  \expandafter\MFP@Isqrt@loop\MFP@ItoQ@Tmp\MFP@end
  \MFP@tempa=\MFP@tempc
  \divide\MFP@tempc\MFP@tttfour
  \edef\MFP@z@Int{\number\MFP@tempc}%
  \multiply\MFP@tempc \MFP@tttfour
  \advance\MFP@tempa -\MFP@tempc
  \edef\MFP@z@Frc{\number\MFP@tempa}%
  \makeMFP@fourdigits\MFP@z@Frc
  \edef\MFP@z@Frc{\MFP@z@Frc0000}%
  \def\MFP@z@Sgn{1}}%
%    \end{macrocode}
%
% The following is a loop that essentially performs a base-2 version of
% the base-10 algorithm that I learned at age 12 from my father
% (apparently it was taught in eighth or ninth grade in his day, but not
% in mine). Seeing it written out, I am surprise at how concise and
% elegant it is!
%    \begin{macrocode}
\def\MFP@Isqrt@loop#1{%
  \ifx\MFP@end #1%
  \else
    \multiply\MFP@tempa 2 \multiply\MFP@tempb 4 \multiply\MFP@tempc 2
    \advance \MFP@tempb#1\relax
    \ifnum\MFP@tempa<\MFP@tempb
       \advance\MFP@tempc 1 \advance\MFP@tempa 1
       \advance\MFP@tempb -\MFP@tempa
       \advance\MFP@tempa 1
    \fi
    \expandafter\MFP@Isqrt@loop
  \fi}%
%    \end{macrocode}
%
%^^A For my own benefit: the above code finds the next binary digit and
%^^A updates the square root (in \cs{MFP@tempc}) by appending that digit. The
%^^A new digit is also appended to the end of \cs{MFP@tempa}. This is
%^^A subtracted from \cs{MFP@tempb}, but only if the last digit is a 1. Then
%^^A the next quadrenary digit is appended to \cs{MFP@tempb}. Finally, the
%^^A last binary digit found is added (not appended) to \cs{MFP@tempa}. The
%^^A ``appending'' of a digit means a multiplication by $2$ (or $4$) and the
%^^A addition of the digit. We perform such additions only if the digit is a
%^^A 1, and we determine if the digit is 1 or 0 by the \cs{ifnum} test.
%
% \subsection{Random numbers}
%
% We borrow the code of \file{random.tex} to generate a random integer in
% the range $1$ to $2^{31}-2$, inclusive. Mathematically, this works
% because the modulus $m = 2^{31}-1$ is a prime number, and the
% multiplicative group of nonzero elements of $\mathbb{Z}_m$ is cyclic.
% The multiplier chosen (in our cases $16\,807$, $48\,271$, or $69\,621$)
% has to be a generator of that group.
%
% The first step is the code for \cs{nextrandom} from \file{random.tex}.
% We could omit this if it is already defined, or we could even input
% \file{random.tex} but, for better control, we define it ourselves with
% an internal name. This code leaves the next random number in
% \cs{MFP@randseed}. The initial seed is calculated from the time and
% date if it was not positive
%    \begin{macrocode}
\newcount\MFP@randseed % the random number (and starting seed)
\def\MFP@nextrand{\begingroup
  \ifnum\MFP@randseed<1
    \global\MFP@randseed\time
    \global\multiply\MFP@randseed388 \global\advance\MFP@randseed\year
    \global\multiply\MFP@randseed31 \global\advance\MFP@randseed\day
    \global\multiply\MFP@randseed97 \global\advance\MFP@randseed\month
    \MFP@nextrand \MFP@nextrand \MFP@nextrand
  \fi
  \MFP@tempa\MFP@randseed
  \divide\MFP@tempa \MFP@rand@q % modulus = m*q + r
  \MFP@tempb\MFP@tempa
  \multiply\MFP@tempa \MFP@rand@q
  \global\advance\MFP@randseed-\MFP@tempa % seed mod q
  \global\multiply\MFP@randseed \MFP@rand@m
  \multiply\MFP@tempb \MFP@rand@r
  \global\advance\MFP@randseed-\MFP@tempb
  \ifnum\MFP@randseed<\z@ \global\advance\MFP@randseed "7FFFFFFF\relax\fi
  \endgroup}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPrandgenA}\DescribeMacro{\MFPrandgenB}
% \DescribeMacro{\MFPrandgenC}
% We have paametrized \cs{MFP@nextrand} so that any suitable multiplier
% can be used. The following commands each select one of the three
% multipliers that we provide, plus precomputed values for the quotient
% and remainder. We default to generator ``A''.
%    \begin{macrocode}
\def\MFPrandgenA{\def\MFP@rand@m{16807 }\def\MFP@rand@q{127773 }%
  \def\MFP@rand@r{2836 }}%
\def\MFPrandgenB{\def\MFP@rand@m{48271 }\def\MFP@rand@q{44488 }%
  \def\MFP@rand@r{3399 }}%
\def\MFPrandgenC{\def\MFP@rand@m{69621 }\def\MFP@rand@q{30845 }%
  \def\MFP@rand@r{23902 }}%
\MFPrandgenA
%    \end{macrocode}
%
% The command \verb$\MFPranr{$\meta{x}\verb$}\X$ will take a parameter $x$
% and define \cs{X} to contain a (pseudo)random real number in the
% interval $[0,x]$. Theoretically, the number should lie in $[0,x)$, but
% rounding will make $x$ itself a possible value. Similarly, \cs{Rrand}
% will replace the $x$ on top of the stack with this random value. To get
% the result, we call \cs{MFP@getrand} twice to produce two random
% integers in the range $[0,99999999]$ and assemble them into a double
% precision multiplier less than $1$. Then we multiply $x$ by that with
% our \cs{MDP@DPmul}.
%
% The test at the end of \cs{MFP@getrand} fails only about 1 time in 50,
% so the odds are vanishingly small that more than a few tries are needed.
%    \begin{macrocode}
\def\MFP@getrand{% leaves result in \MFP@tempa
  \MFP@nextrand
  \MFP@tempa\MFP@randseed
  \advance\MFP@tempa-1
  \divide\MFP@tempa 21 % (2^31-3)= 100000000*21 + r
  \ifnum \MFP@ttteight> \MFP@tempa
  \else \@xp\MFP@getrand\fi}%
\def\MFP@Rrand{%
  \MFP@getrand \edef\MFP@a@Tmp{\number\MFP@tempa}%
  \MFP@getrand \edef\MFP@b@Tmp{\number\MFP@tempa}%
  \MFP@DPmul0\MFP@a@Tmp\MFP@b@Tmp}%
%    \end{macrocode}
%
% \DescribeMacro{\MFPsetseed}
% Finally, a user-level command to set the seed value.
%    \begin{macrocode}
\def\MFPsetseed#1{\global\MFP@randseed #1\relax}%
\MFP@xfinish
%</extra>
%    \end{macrocode}
%\Finale
%