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
|
% $Id: mpmathdecimal.w 1915 2013-06-13 10:17:31Z taco $
%
% This file is part of MetaPost;
% the MetaPost program is in the public domain.
% See the <Show version...> code in mpost.w for more info.
% Here is TeX material that gets inserted after \input webmac
\font\tenlogo=logo10 % font used for the METAFONT logo
\font\logos=logosl10
\def\MF{{\tenlogo META}\-{\tenlogo FONT}}
\def\MP{{\tenlogo META}\-{\tenlogo POST}}
\def\title{Math support functions for decNumber based math}
\pdfoutput=1
@ Introduction.
@c
#include <w2c/config.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "mpmathdecimal.h" /* internal header */
#define ROUND(a) floor((a)+0.5)
@h
@ @c
@<Declarations@>;
@ @(mpmathdecimal.h@>=
#ifndef MPMATHDECIMAL_H
#define MPMATHDECIMAL_H 1
#include "mplib.h"
#include "mpmp.h" /* internal header */
#define DECNUMDIGITS 1000
#include "decNumber.h"
@<Internal library declarations@>;
#endif
@* Math initialization.
First, here are some very important constants.
@d E_STRING "2.7182818284590452353602874713526624977572470936999595749669676277240766303535"
@d PI_STRING "3.1415926535897932384626433832795028841971693993751058209749445923078164062862"
@d fraction_multiplier 4096
@d angle_multiplier 16
@ Here are the functions that are static as they are not used elsewhere
@<Declarations@>=
#define DEBUG 0
static void mp_decimal_scan_fractional_token (MP mp, int n);
static void mp_decimal_scan_numeric_token (MP mp, int n);
static void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);
static void mp_decimal_crossing_point (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c);
static void mp_decimal_number_modulo (mp_number *a, mp_number b);
static void mp_decimal_print_number (MP mp, mp_number n);
static char * mp_decimal_number_tostring (MP mp, mp_number n);
static void mp_decimal_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig);
static void mp_decimal_square_rt (MP mp, mp_number *ret, mp_number x_orig);
static void mp_decimal_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin);
static void mp_init_randoms (MP mp, int seed);
static void mp_number_angle_to_scaled (mp_number *A);
static void mp_number_fraction_to_scaled (mp_number *A);
static void mp_number_scaled_to_fraction (mp_number *A);
static void mp_number_scaled_to_angle (mp_number *A);
static void mp_decimal_m_exp (MP mp, mp_number *ret, mp_number x_orig);
static void mp_decimal_m_log (MP mp, mp_number *ret, mp_number x_orig);
static void mp_decimal_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b);
static void mp_decimal_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b);
static void mp_decimal_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y);
static void mp_decimal_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf, mp_number cf, mp_number t);
static void mp_set_decimal_from_int(mp_number *A, int B);
static void mp_set_decimal_from_boolean(mp_number *A, int B);
static void mp_set_decimal_from_scaled(mp_number *A, int B);
static void mp_set_decimal_from_addition(mp_number *A, mp_number B, mp_number C);
static void mp_set_decimal_from_substraction (mp_number *A, mp_number B, mp_number C);
static void mp_set_decimal_from_div(mp_number *A, mp_number B, mp_number C);
static void mp_set_decimal_from_mul(mp_number *A, mp_number B, mp_number C);
static void mp_set_decimal_from_int_div(mp_number *A, mp_number B, int C);
static void mp_set_decimal_from_int_mul(mp_number *A, mp_number B, int C);
static void mp_set_decimal_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C);
static void mp_number_negate(mp_number *A);
static void mp_number_add(mp_number *A, mp_number B);
static void mp_number_substract(mp_number *A, mp_number B);
static void mp_number_half(mp_number *A);
static void mp_number_halfp(mp_number *A);
static void mp_number_double(mp_number *A);
static void mp_number_add_scaled(mp_number *A, int B); /* also for negative B */
static void mp_number_multiply_int(mp_number *A, int B);
static void mp_number_divide_int(mp_number *A, int B);
static void mp_decimal_abs(mp_number *A);
static void mp_number_clone(mp_number *A, mp_number B);
static void mp_number_swap(mp_number *A, mp_number *B);
static int mp_round_unscaled(mp_number x_orig);
static int mp_number_to_int(mp_number A);
static int mp_number_to_scaled(mp_number A);
static int mp_number_to_boolean(mp_number A);
static double mp_number_to_double(mp_number A);
static int mp_number_odd(mp_number A);
static int mp_number_equal(mp_number A, mp_number B);
static int mp_number_greater(mp_number A, mp_number B);
static int mp_number_less(mp_number A, mp_number B);
static int mp_number_nonequalabs(mp_number A, mp_number B);
static void mp_number_floor (mp_number *i);
static void mp_decimal_fraction_to_round_scaled (mp_number *x);
static void mp_decimal_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_decimal_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_decimal_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_decimal_number_take_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_new_number (MP mp, mp_number *n, mp_number_type t) ;
static void mp_free_number (MP mp, mp_number *n) ;
static void mp_set_decimal_from_double(mp_number *A, double B);
static void mp_free_decimal_math (MP mp);
static void mp_decimal_set_precision (MP mp);
static void mp_check_decNumber (MP mp, decNumber *dec, decContext *context);
static int decNumber_check (decNumber *dec, decContext *context);
static char * mp_decnumber_tostring (decNumber *n);
@ We do not want special numbers as return values for functions, so:
@c
int decNumber_check (decNumber *dec, decContext *context)
{
int test = false;
if (context->status & DEC_Overflow) {
test = true;
context->status &= ~DEC_Overflow;
}
if (context->status & DEC_Underflow) {
test = true;
context->status &= ~DEC_Underflow;
}
if (context->status & DEC_Errors) {
// fprintf(stdout, "DEC_ERROR %x (%s)\n", context->status, decContextStatusToString(context));
test = true;
decNumberZero(dec);
}
context->status = 0;
if (decNumberIsSpecial(dec)) {
test = true;
if (decNumberIsInfinite(dec)) {
if (decNumberIsNegative(dec)) {
decNumberCopyNegate(dec, &EL_GORDO_decNumber);
} else {
decNumberCopy(dec, &EL_GORDO_decNumber);
}
} else { // Nan
decNumberZero(dec);
}
}
if (decNumberIsZero(dec) && decNumberIsNegative(dec)) {
decNumberZero(dec);
}
return test;
}
void mp_check_decNumber (MP mp, decNumber *dec, decContext *context)
{
mp->arith_error = decNumber_check (dec, context);
}
@ There are a few short decNumber functions that do not exist, but
make life easier for us:
@d decNumberIsPositive(A) !(decNumberIsZero(A) || decNumberIsNegative(A))
@c
static decContext set;
static decContext limitedset;
static void checkZero (decNumber *ret) {
if (decNumberIsZero(ret) && decNumberIsNegative(ret))
decNumberZero(ret);
}
static int decNumberLess(decNumber *a, decNumber *b) {
decNumber comp;
decNumberCompare(&comp, a, b, &set);
return decNumberIsNegative(&comp);
}
static int decNumberGreater(decNumber *a, decNumber *b) {
decNumber comp;
decNumberCompare(&comp, a, b, &set);
return decNumberIsPositive(&comp);
}
static void decNumberFromDouble(decNumber *A, double B) {
char buf[1000];
char *c;
snprintf(buf,1000,"%-650.325lf",B);
c = buf;
while (*c++) {
if (*c == ' ') {
*c = '\0';
break;
}
}
decNumberFromString(A, buf, &set);
}
static double decNumberToDouble(decNumber *A) {
char *buffer = malloc(A->digits + 14);
double res = 0.0;
assert (buffer);
decNumberToString(A, buffer);
if (sscanf(buffer, "%lf", &res)) {
free(buffer);
return res;
} else {
free(buffer);
//mp->arith_error = 1;
return 0.0; // whatever
}
}
@ Borrowed code from libdfp:
x^3 x^5 x^7
arctan(x) = x - --- + --- - --- + ...
3 5 7
This power series works well, if x is close to zero (|x|<0.5).
If x is larger, the series converges too slowly,
so in order to get a smaller x, we apply the identity
sqrt(1+x^2) - 1
arctan(x) = 2*arctan ---------------
x
twice. The first application gives us a new x with x < 1.
The second application gives us a new x with x < 0.4142136.
For that x, we use the power series and multiply the result by four.
@c
static void decNumberAtan (decNumber *result, decNumber *x_orig, decContext *set)
{
decNumber x, f, g, mx2, term;
int i;
decNumberCopy(&x, x_orig);
if (decNumberIsZero (&x)) {
decNumberCopy (result, &x);
return;
}
for (i=0; i<2; i++) {
decNumber y;
decNumberMultiply (&y, &x, &x, set); // y = x^2
decNumberAdd (&y, &y, &one, set); // y = y+1
decNumberSquareRoot (&y, &y, set); // y = sqrt(y)
decNumberSubtract (&y, &y, &one, set); // y = y-1
decNumberDivide (&x, &y, &x, set); // x = y/x
if (decNumberIsZero (&x)) {
decNumberCopy (result, &x);
return;
}
}
decNumberCopy (&f, &x); // f(0) = x
decNumberCopy (&g, &one); // g(0) = 1
decNumberCopy (&term, &x); // term = x
decNumberCopy (result, &x); // sum = x
decNumberMultiply (&mx2, &x, &x, set); // mx2 = x^2
decNumberMinus (&mx2, &mx2, set); // mx2 = -x^2
for (i=0; i<2*set->digits; i++) {
decNumberMultiply (&f, &f, &mx2, set);
decNumberAdd (&g, &g, &two_decNumber, set);
decNumberDivide (&term, &f, &g, set);
decNumberAdd (result, result, &term, set);
}
decNumberAdd (result, result, result, set);
decNumberAdd (result, result, result, set);
return;
}
static void decNumberAtan2 (decNumber *result, decNumber *y, decNumber *x, decContext *set)
{
decNumber temp;
if (!decNumberIsInfinite (x) && !decNumberIsZero (y)
&& !decNumberIsInfinite (y) && !decNumberIsZero (x)) {
decNumberDivide (&temp, y, x, set);
decNumberAtan (result, &temp, set);
/* decNumberAtan doesn't quite return the values in the ranges we
* want for x < 0. So we need to do some correction */
if (decNumberIsNegative (x)) {
if (decNumberIsNegative (y)) {
decNumberSubtract(result, result, &PI_decNumber, set);
} else {
decNumberAdd(result, result, &PI_decNumber, set);
}
}
return;
}
if (decNumberIsInfinite (y) && decNumberIsInfinite (x)) {
/* If x and y are both inf, the result depends on the sign of x */
decNumberDivide(result, &PI_decNumber, &four_decNumber, set);
if (decNumberIsNegative (x) ) {
decNumber a;
decNumberFromDouble(&a, 3.0);
decNumberMultiply(result, result, &a, set);
}
} else if (!decNumberIsZero (y) && !decNumberIsInfinite (x) ) {
/* If y is non-zero and x is non-inf, the result is +-pi/2 */
decNumberDivide(result, &PI_decNumber, &two_decNumber, set);
} else { /* Otherwise it is +0 if x is positive, +pi if x is neg */
if (decNumberIsNegative (x)) {
decNumberCopy(result, &PI_decNumber);
} else {
decNumberZero(result);
}
}
/* Atan2 will be negative if y<0 */
if (decNumberIsNegative (y)) {
decNumberMinus(result, result, set);
}
}
@ And these are the ones that {\it are} used elsewhere
@<Internal library declarations@>=
void * mp_initialize_decimal_math (MP mp);
@
@d unity 1
@d two 2
@d three 3
@d four 4
@d half_unit 0.5
@d three_quarter_unit 0.75
@d coef_bound ((7.0/3.0)*fraction_multiplier) /* |fraction| approximation to 7/3 */
@d fraction_threshold 0.04096 /* a |fraction| coefficient less than this is zeroed */
@d half_fraction_threshold (fraction_threshold/2) /* half of |fraction_threshold| */
@d scaled_threshold 0.000122 /* a |scaled| coefficient less than this is zeroed */
@d half_scaled_threshold (scaled_threshold/2) /* half of |scaled_threshold| */
@d near_zero_angle (0.0256*angle_multiplier) /* an angle of about 0.0256 */
@d p_over_v_threshold 0x80000 /* TODO */
@d equation_threshold 0.001
@d tfm_warn_threshold 0.0625
@d epsilon "1E-52"
@d epsilonf pow(2.0,-52.0)
@d EL_GORDO "1E1000000" /* the largest value that \MP\ likes. */
@d warning_limit "1E1000000" /* this is a large value that can just be expressed without loss of precision */
@d DECPRECISION_DEFAULT 34
@<Declarations@>=
static decNumber zero;
static decNumber one;
static decNumber minusone;
static decNumber two_decNumber;
static decNumber three_decNumber;
static decNumber four_decNumber;
static decNumber fraction_multiplier_decNumber;
static decNumber angle_multiplier_decNumber;
static decNumber fraction_one_decNumber;
static decNumber fraction_one_plus_decNumber;
static decNumber PI_decNumber;
static decNumber epsilon_decNumber;
static decNumber EL_GORDO_decNumber;
static decNumber **factorials = NULL;
static int last_cached_factorial = 0;
@ @c
void * mp_initialize_decimal_math (MP mp) {
math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
// various decNumber initializations
decContextDefault(&set, DEC_INIT_BASE); // initialize
set.traps=0; // no traps, thank you
decContextDefault(&limitedset, DEC_INIT_BASE); // initialize
limitedset.traps=0; // no traps, thank you
limitedset.emax = 999999;
limitedset.emin = -999999;
set.digits = DECPRECISION_DEFAULT;
limitedset.digits = DECPRECISION_DEFAULT;
decNumberFromInt32(&one, 1);
decNumberFromInt32(&minusone, -1);
decNumberFromInt32(&zero, 0);
decNumberFromInt32(&two_decNumber, two);
decNumberFromInt32(&three_decNumber, three);
decNumberFromInt32(&four_decNumber, four);
decNumberFromInt32(&fraction_multiplier_decNumber, fraction_multiplier);
decNumberFromInt32(&fraction_one_decNumber, fraction_one);
decNumberFromInt32(&fraction_one_plus_decNumber, (fraction_one+1));
decNumberFromInt32(&angle_multiplier_decNumber, angle_multiplier);
decNumberFromString(&PI_decNumber, PI_STRING, &set);
decNumberFromString(&epsilon_decNumber, epsilon, &set);
decNumberFromString(&EL_GORDO_decNumber, EL_GORDO, &set);
factorials = (decNumber **)mp_xmalloc(mp,PRECALC_FACTORIALS_CACHESIZE,sizeof(decNumber *));
factorials[0] = (decNumber *)mp_xmalloc(mp,1,sizeof(decNumber));
decNumberCopy(factorials[0], &one);
/* alloc */
math->allocate = mp_new_number;
math->free = mp_free_number;
mp_new_number (mp, &math->precision_default, mp_scaled_type);
decNumberFromInt32(math->precision_default.data.num, DECPRECISION_DEFAULT);
mp_new_number (mp, &math->precision_max, mp_scaled_type);
decNumberFromInt32(math->precision_max.data.num, DECNUMDIGITS);
mp_new_number (mp, &math->precision_min, mp_scaled_type);
decNumberFromInt32(math->precision_min.data.num, 1);
/* here are the constants for |scaled| objects */
mp_new_number (mp, &math->epsilon_t, mp_scaled_type);
decNumberCopy(math->epsilon_t.data.num, &epsilon_decNumber);
mp_new_number (mp, &math->inf_t, mp_scaled_type);
decNumberCopy(math->inf_t.data.num, &EL_GORDO_decNumber);
mp_new_number (mp, &math->warning_limit_t, mp_scaled_type);
decNumberFromString(math->warning_limit_t.data.num, warning_limit, &set);
mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type);
decNumberDivide(math->one_third_inf_t.data.num, math->inf_t.data.num, &three_decNumber, &set);
mp_new_number (mp, &math->unity_t, mp_scaled_type);
decNumberCopy(math->unity_t.data.num, &one);
mp_new_number (mp, &math->two_t, mp_scaled_type);
decNumberFromInt32(math->two_t.data.num, two);
mp_new_number (mp, &math->three_t, mp_scaled_type);
decNumberFromInt32(math->three_t.data.num, three);
mp_new_number (mp, &math->half_unit_t, mp_scaled_type);
decNumberFromString(math->half_unit_t.data.num, "0.5", &set);
mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type);
decNumberFromString(math->three_quarter_unit_t.data.num, "0.75", &set);
mp_new_number (mp, &math->zero_t, mp_scaled_type);
decNumberZero(math->zero_t.data.num);
/* |fractions| */
mp_new_number (mp, &math->arc_tol_k, mp_fraction_type);
{
decNumber fourzeroninesix;
decNumberFromInt32(&fourzeroninesix, 4096);
decNumberDivide(math->arc_tol_k.data.num, &one, &fourzeroninesix, &set);
/* quit when change in arc length estimate reaches this */
}
mp_new_number (mp, &math->fraction_one_t, mp_fraction_type);
decNumberFromInt32(math->fraction_one_t.data.num, fraction_one);
mp_new_number (mp, &math->fraction_half_t, mp_fraction_type);
decNumberFromInt32(math->fraction_half_t.data.num, fraction_half);
mp_new_number (mp, &math->fraction_three_t, mp_fraction_type);
decNumberFromInt32(math->fraction_three_t.data.num, fraction_three);
mp_new_number (mp, &math->fraction_four_t, mp_fraction_type);
decNumberFromInt32(math->fraction_four_t.data.num, fraction_four);
/* |angles| */
mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type);
decNumberFromInt32(math->three_sixty_deg_t.data.num, 360 * angle_multiplier);
mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type);
decNumberFromInt32(math->one_eighty_deg_t.data.num, 180 * angle_multiplier);
/* various approximations */
mp_new_number (mp, &math->one_k, mp_scaled_type);
decNumberFromInt32(math->one_k.data.num, 1024);
mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type);
{
decNumberFromDouble(math->sqrt_8_e_k.data.num, 112428.82793 / 65536.0);
/* $2^{16}\sqrt{8/e}\approx 112428.82793$ */
}
mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type);
{
decNumberFromDouble(math->twelve_ln_2_k.data.num, 139548959.6165 / 65536.0);
/* $2^{24}\cdot12\ln2\approx139548959.6165$ */
}
mp_new_number (mp, &math->coef_bound_k, mp_fraction_type);
decNumberFromDouble(math->coef_bound_k.data.num,coef_bound);
mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type);
decNumberFromDouble(math->coef_bound_minus_1.data.num,coef_bound - 1 / 65536.0);
mp_new_number (mp, &math->twelvebits_3, mp_scaled_type);
{
decNumberFromDouble(math->twelvebits_3.data.num, 1365 / 65536.0);
/* $1365\approx 2^{12}/3$ */
}
mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type);
{
decNumberFromDouble(math->twentysixbits_sqrt2_t.data.num, 94906265.62 / 65536.0);
/* $2^{26}\sqrt2\approx94906265.62$ */
}
mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type);
{
decNumberFromDouble(math->twentyeightbits_d_t.data.num, 35596754.69 / 65536.0);
/* $2^{28}d\approx35596754.69$ */
}
mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type);
{
decNumberFromDouble(math->twentysevenbits_sqrt2_d_t.data.num, 25170706.63 / 65536.0);
/* $2^{27}\sqrt2\,d\approx25170706.63$ */
}
/* thresholds */
mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type);
decNumberFromDouble(math->fraction_threshold_t.data.num, fraction_threshold);
mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type);
decNumberFromDouble(math->half_fraction_threshold_t.data.num, half_fraction_threshold);
mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type);
decNumberFromDouble(math->scaled_threshold_t.data.num, scaled_threshold);
mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type);
decNumberFromDouble(math->half_scaled_threshold_t.data.num, half_scaled_threshold);
mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type);
decNumberFromDouble(math->near_zero_angle_t.data.num, near_zero_angle);
mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type);
decNumberFromDouble(math->p_over_v_threshold_t.data.num, p_over_v_threshold);
mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type);
decNumberFromDouble(math->equation_threshold_t.data.num, equation_threshold);
mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type);
decNumberFromDouble(math->tfm_warn_threshold_t.data.num, tfm_warn_threshold);
/* functions */
math->from_int = mp_set_decimal_from_int;
math->from_boolean = mp_set_decimal_from_boolean;
math->from_scaled = mp_set_decimal_from_scaled;
math->from_double = mp_set_decimal_from_double;
math->from_addition = mp_set_decimal_from_addition;
math->from_substraction = mp_set_decimal_from_substraction;
math->from_oftheway = mp_set_decimal_from_of_the_way;
math->from_div = mp_set_decimal_from_div;
math->from_mul = mp_set_decimal_from_mul;
math->from_int_div = mp_set_decimal_from_int_div;
math->from_int_mul = mp_set_decimal_from_int_mul;
math->negate = mp_number_negate;
math->add = mp_number_add;
math->substract = mp_number_substract;
math->half = mp_number_half;
math->halfp = mp_number_halfp;
math->do_double = mp_number_double;
math->abs = mp_decimal_abs;
math->clone = mp_number_clone;
math->swap = mp_number_swap;
math->add_scaled = mp_number_add_scaled;
math->multiply_int = mp_number_multiply_int;
math->divide_int = mp_number_divide_int;
math->to_boolean = mp_number_to_boolean;
math->to_scaled = mp_number_to_scaled;
math->to_double = mp_number_to_double;
math->to_int = mp_number_to_int;
math->odd = mp_number_odd;
math->equal = mp_number_equal;
math->less = mp_number_less;
math->greater = mp_number_greater;
math->nonequalabs = mp_number_nonequalabs;
math->round_unscaled = mp_round_unscaled;
math->floor_scaled = mp_number_floor;
math->fraction_to_round_scaled = mp_decimal_fraction_to_round_scaled;
math->make_scaled = mp_decimal_number_make_scaled;
math->make_fraction = mp_decimal_number_make_fraction;
math->take_fraction = mp_decimal_number_take_fraction;
math->take_scaled = mp_decimal_number_take_scaled;
math->velocity = mp_decimal_velocity;
math->n_arg = mp_decimal_n_arg;
math->m_log = mp_decimal_m_log;
math->m_exp = mp_decimal_m_exp;
math->pyth_add = mp_decimal_pyth_add;
math->pyth_sub = mp_decimal_pyth_sub;
math->fraction_to_scaled = mp_number_fraction_to_scaled;
math->scaled_to_fraction = mp_number_scaled_to_fraction;
math->scaled_to_angle = mp_number_scaled_to_angle;
math->angle_to_scaled = mp_number_angle_to_scaled;
math->init_randoms = mp_init_randoms;
math->sin_cos = mp_decimal_sin_cos;
math->slow_add = mp_decimal_slow_add;
math->sqrt = mp_decimal_square_rt;
math->print = mp_decimal_print_number;
math->tostring = mp_decimal_number_tostring;
math->modulo = mp_decimal_number_modulo;
math->ab_vs_cd = mp_ab_vs_cd;
math->crossing_point = mp_decimal_crossing_point;
math->scan_numeric = mp_decimal_scan_numeric_token;
math->scan_fractional = mp_decimal_scan_fractional_token;
math->free_math = mp_free_decimal_math;
math->set_precision = mp_decimal_set_precision;
return (void *)math;
}
void mp_decimal_set_precision (MP mp) {
int i;
i = decNumberToInt32((decNumber *)internal_value (mp_number_precision).data.num, &set);
set.digits = i;
limitedset.digits = i;
}
void mp_free_decimal_math (MP mp) {
int i;
free_number (((math_data *)mp->math)->three_sixty_deg_t);
free_number (((math_data *)mp->math)->one_eighty_deg_t);
free_number (((math_data *)mp->math)->fraction_one_t);
free_number (((math_data *)mp->math)->zero_t);
free_number (((math_data *)mp->math)->half_unit_t);
free_number (((math_data *)mp->math)->three_quarter_unit_t);
free_number (((math_data *)mp->math)->unity_t);
free_number (((math_data *)mp->math)->two_t);
free_number (((math_data *)mp->math)->three_t);
free_number (((math_data *)mp->math)->one_third_inf_t);
free_number (((math_data *)mp->math)->inf_t);
free_number (((math_data *)mp->math)->warning_limit_t);
free_number (((math_data *)mp->math)->one_k);
free_number (((math_data *)mp->math)->sqrt_8_e_k);
free_number (((math_data *)mp->math)->twelve_ln_2_k);
free_number (((math_data *)mp->math)->coef_bound_k);
free_number (((math_data *)mp->math)->coef_bound_minus_1);
free_number (((math_data *)mp->math)->fraction_threshold_t);
free_number (((math_data *)mp->math)->half_fraction_threshold_t);
free_number (((math_data *)mp->math)->scaled_threshold_t);
free_number (((math_data *)mp->math)->half_scaled_threshold_t);
free_number (((math_data *)mp->math)->near_zero_angle_t);
free_number (((math_data *)mp->math)->p_over_v_threshold_t);
free_number (((math_data *)mp->math)->equation_threshold_t);
free_number (((math_data *)mp->math)->tfm_warn_threshold_t);
for (i = 0; i <= last_cached_factorial; i++) {
free(factorials[i]);
}
free(factorials);
free(mp->math);
}
@ Creating an destroying |mp_number| objects
@ @c
void mp_new_number (MP mp, mp_number *n, mp_number_type t) {
(void)mp;
n->data.num = mp_xmalloc(mp,1,sizeof(decNumber));
decNumberZero(n->data.num);
n->type = t;
}
@
@c
void mp_free_number (MP mp, mp_number *n) {
(void)mp;
free(n->data.num);
n->data.num = NULL;
n->type = mp_nan_type;
}
@ Here are the low-level functions on |mp_number| items, setters first.
@c
void mp_set_decimal_from_int(mp_number *A, int B) {
decNumberFromInt32(A->data.num,B);
}
void mp_set_decimal_from_boolean(mp_number *A, int B) {
decNumberFromInt32(A->data.num,B);
}
void mp_set_decimal_from_scaled(mp_number *A, int B) {
decNumber c;
decNumberFromInt32(&c, 65536);
decNumberFromInt32(A->data.num,B);
decNumberDivide(A->data.num,A->data.num,&c, &set);
}
void mp_set_decimal_from_double(mp_number *A, double B) {
decNumberFromDouble(A->data.num, B);
}
void mp_set_decimal_from_addition(mp_number *A, mp_number B, mp_number C) {
decNumberAdd(A->data.num,B.data.num,C.data.num, &set);
}
void mp_set_decimal_from_substraction (mp_number *A, mp_number B, mp_number C) {
decNumberSubtract(A->data.num,B.data.num,C.data.num, &set);
}
void mp_set_decimal_from_div(mp_number *A, mp_number B, mp_number C) {
decNumberDivide(A->data.num,B.data.num,C.data.num, &set);
}
void mp_set_decimal_from_mul(mp_number *A, mp_number B, mp_number C) {
decNumberMultiply(A->data.num,B.data.num,C.data.num, &set);
}
void mp_set_decimal_from_int_div(mp_number *A, mp_number B, int C) {
decNumber c;
decNumberFromInt32(&c, C);
decNumberDivide(A->data.num,B.data.num,&c, &set);
}
void mp_set_decimal_from_int_mul(mp_number *A, mp_number B, int C) {
decNumber c;
decNumberFromInt32(&c, C);
decNumberMultiply(A->data.num,B.data.num,&c, &set);
}
void mp_set_decimal_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) {
decNumber c;
decNumber r1;
decNumberSubtract(&c,B.data.num, C.data.num, &set);
mp_decimal_take_fraction(mp, &r1, &c, t.data.num);
decNumberSubtract(A->data.num, B.data.num, &r1, &set);
mp_check_decNumber(mp, A->data.num, &set);
}
void mp_number_negate(mp_number *A) {
decNumberCopyNegate(A->data.num, A->data.num);
checkZero(A->data.num);
}
void mp_number_add(mp_number *A, mp_number B) {
decNumberAdd(A->data.num,A->data.num,B.data.num, &set);
}
void mp_number_substract(mp_number *A, mp_number B) {
decNumberSubtract(A->data.num,A->data.num,B.data.num, &set);
}
void mp_number_half(mp_number *A) {
decNumber c;
decNumberFromInt32(&c, 2);
decNumberDivide(A->data.num,A->data.num, &c, &set);
}
void mp_number_halfp(mp_number *A) {
decNumber c;
decNumberFromInt32(&c, 2);
decNumberDivide(A->data.num,A->data.num, &c, &set);
}
void mp_number_double(mp_number *A) {
decNumber c;
decNumberFromInt32(&c, 2);
decNumberMultiply(A->data.num,A->data.num, &c, &set);
}
void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */
decNumber b,c;
decNumberFromInt32(&c, 65536);
decNumberFromInt32(&b, B);
decNumberDivide(&b,&b, &c, &set);
decNumberAdd(A->data.num,A->data.num, &b, &set);
}
void mp_number_multiply_int(mp_number *A, int B) {
decNumber b;
decNumberFromInt32(&b, B);
decNumberMultiply(A->data.num,A->data.num, &b, &set);
}
void mp_number_divide_int(mp_number *A, int B) {
decNumber b;
decNumberFromInt32(&b, B);
decNumberDivide(A->data.num,A->data.num,&b, &set);
}
void mp_decimal_abs(mp_number *A) {
decNumberAbs(A->data.num, A->data.num, &set);
}
void mp_number_clone(mp_number *A, mp_number B) {
decNumberCopy(A->data.num, B.data.num);
}
void mp_number_swap(mp_number *A, mp_number *B) {
decNumber swap_tmp;
decNumberCopy(&swap_tmp, A->data.num);
decNumberCopy(A->data.num, B->data.num);
decNumberCopy(B->data.num, &swap_tmp);
}
void mp_number_fraction_to_scaled (mp_number *A) {
A->type = mp_scaled_type;
decNumberDivide(A->data.num, A->data.num, &fraction_multiplier_decNumber, &set);
}
void mp_number_angle_to_scaled (mp_number *A) {
A->type = mp_scaled_type;
decNumberDivide(A->data.num, A->data.num, &angle_multiplier_decNumber, &set);
}
void mp_number_scaled_to_fraction (mp_number *A) {
A->type = mp_fraction_type;
decNumberMultiply(A->data.num, A->data.num, &fraction_multiplier_decNumber, &set);
}
void mp_number_scaled_to_angle (mp_number *A) {
A->type = mp_angle_type;
decNumberMultiply(A->data.num, A->data.num, &angle_multiplier_decNumber, &set);
}
@* Query functions
@ Convert a number to a scaled value. |decNumberToInt32| is not
able to make this conversion properly, so instead we are using
|decNumberToDouble| and a typecast. Bad!
@c
int mp_number_to_scaled(mp_number A) {
int32_t result;
decNumber corrected;
decNumberFromInt32(&corrected, 65536);
decNumberMultiply(&corrected,&corrected,A.data.num, &set);
decNumberReduce(&corrected, &corrected, &set);
result = (int)floor(decNumberToDouble(&corrected)+0.5);
return result;
}
@
@d odd(A) ((A)%2==1)
@c
int mp_number_to_int(mp_number A) {
int32_t result;
set.status = 0;
result = decNumberToInt32(A.data.num, &set);
if (set.status == DEC_Invalid_operation) {
set.status = 0;
// mp->arith_error = 1;
return 0; // whatever
} else {
return result;
}
}
int mp_number_to_boolean(mp_number A) {
uint32_t result;
set.status = 0;
result = decNumberToUInt32(A.data.num, &set);
if (set.status == DEC_Invalid_operation) {
set.status = 0;
// mp->arith_error = 1;
return 0; // whatever
} else {
return (result ? 1 : 0);
}
}
double mp_number_to_double(mp_number A) {
char *buffer = malloc(((decNumber *)A.data.num)->digits + 14);
double res = 0.0;
assert (buffer);
decNumberToString(A.data.num, buffer);
if (sscanf(buffer, "%lf", &res)) {
free(buffer);
return res;
} else {
free(buffer);
//mp->arith_error = 1;
return 0.0; // whatever
}
}
int mp_number_odd(mp_number A) {
return odd(mp_number_to_int(A));
}
int mp_number_equal(mp_number A, mp_number B) {
decNumber res;
decNumberCompare(&res,A.data.num,B.data.num, &set);
return decNumberIsZero(&res);
}
int mp_number_greater(mp_number A, mp_number B) {
decNumber res;
decNumberCompare(&res,A.data.num,B.data.num, &set);
return decNumberIsPositive(&res);
}
int mp_number_less(mp_number A, mp_number B) {
decNumber res;
decNumberCompare(&res,A.data.num,B.data.num, &set);
return decNumberIsNegative(&res);
}
int mp_number_nonequalabs(mp_number A, mp_number B) {
decNumber res, a, b;
decNumberCopyAbs(&a, A.data.num);
decNumberCopyAbs(&b, B.data.num);
decNumberCompare(&res, &a, &b, &set);
return !decNumberIsZero(&res);
}
@ Fixed-point arithmetic is done on {\sl scaled integers\/} that are multiples
of $2^{-16}$. In other words, a binary point is assumed to be sixteen bit
positions from the right end of a binary computer word.
@ One of \MP's most common operations is the calculation of
$\lfloor{a+b\over2}\rfloor$,
the midpoint of two given integers |a| and~|b|. The most decent way to do
this is to write `|(a+b)/2|'; but on many machines it is more efficient
to calculate `|(a+b)>>1|'.
Therefore the midpoint operation will always be denoted by `|half(a+b)|'
in this program. If \MP\ is being implemented with languages that permit
binary shifting, the |half| macro should be changed to make this operation
as efficient as possible. Since some systems have shift operators that can
only be trusted to work on positive numbers, there is also a macro |halfp|
that is used only when the quantity being halved is known to be positive
or zero.
@ Here is a procedure analogous to |print_int|. The current version
is fairly stupid, and it is not round-trip safe, but this is good
enough for a beta test.
@c
char * mp_decnumber_tostring (decNumber *n) {
decNumber corrected;
char *buffer = malloc(((decNumber *)n)->digits + 14);
assert (buffer);
decNumberCopy(&corrected,n);
decNumberTrim(&corrected);
decNumberToString(&corrected, buffer);
return buffer;
}
char * mp_decimal_number_tostring (MP mp, mp_number n) {
return mp_decnumber_tostring(n.data.num);
}
@ @c
void mp_decimal_print_number (MP mp, mp_number n) {
char *str = mp_decimal_number_tostring(mp, n);
mp_print (mp, str);
free (str);
}
@ Addition is not always checked to make sure that it doesn't overflow,
but in places where overflow isn't too unlikely the |slow_add| routine
is used.
@c
void mp_decimal_slow_add (MP mp, mp_number *ret, mp_number A, mp_number B) {
decNumberAdd(ret->data.num,A.data.num,B.data.num, &set);
}
@ The |make_fraction| routine produces the |fraction| equivalent of
|p/q|, given integers |p| and~|q|; it computes the integer
$f=\lfloor2^{28}p/q+{1\over2}\rfloor$, when $p$ and $q$ are
positive. If |p| and |q| are both of the same scaled type |t|,
the ``type relation'' |make_fraction(t,t)=fraction| is valid;
and it's also possible to use the subroutine ``backwards,'' using
the relation |make_fraction(t,fraction)=t| between scaled types.
If the result would have magnitude $2^{31}$ or more, |make_fraction|
sets |arith_error:=true|. Most of \MP's internal computations have
been designed to avoid this sort of error.
If this subroutine were programmed in assembly language on a typical
machine, we could simply compute |(@t$2^{28}$@>*p)div q|, since a
double-precision product can often be input to a fixed-point division
instruction. But when we are restricted to int-eger arithmetic it
is necessary either to resort to multiple-precision maneuvering
or to use a simple but slow iteration. The multiple-precision technique
would be about three times faster than the code adopted here, but it
would be comparatively long and tricky, involving about sixteen
additional multiplications and divisions.
This operation is part of \MP's ``inner loop''; indeed, it will
consume nearly 10\pct! of the running time (exclusive of input and output)
if the code below is left unchanged. A machine-dependent recoding
will therefore make \MP\ run faster. The present implementation
is highly portable, but slow; it avoids multiplication and division
except in the initial stage. System wizards should be careful to
replace it with a routine that is guaranteed to produce identical
results in all cases.
@^system dependencies@>
As noted below, a few more routines should also be replaced by machine-dependent
code, for efficiency. But when a procedure is not part of the ``inner loop,''
such changes aren't advisable; simplicity and robustness are
preferable to trickery, unless the cost is too high.
@^inner loop@>
@c
void mp_decimal_make_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q) {
decNumberDivide(ret, p, q, &set);
mp_check_decNumber(mp, ret, &set);
decNumberMultiply(ret, ret, &fraction_multiplier_decNumber, &set);
}
void mp_decimal_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
mp_decimal_make_fraction (mp, ret->data.num, p.data.num, q.data.num);
}
@ @<Declarations@>=
void mp_decimal_make_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q);
@ The dual of |make_fraction| is |take_fraction|, which multiplies a
given integer~|q| by a fraction~|f|. When the operands are positive, it
computes $p=\lfloor qf/2^{28}+{1\over2}\rfloor$, a symmetric function
of |q| and~|f|.
This routine is even more ``inner loopy'' than |make_fraction|;
the present implementation consumes almost 20\pct! of \MP's computation
time during typical jobs, so a machine-language substitute is advisable.
@^inner loop@> @^system dependencies@>
@c
void mp_decimal_take_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q) {
decNumberMultiply(ret, p, q, &set);
decNumberDivide(ret, ret, &fraction_multiplier_decNumber, &set);
}
void mp_decimal_number_take_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
mp_decimal_take_fraction (mp, ret->data.num, p.data.num, q.data.num);
}
@ @<Declarations@>=
void mp_decimal_take_fraction (MP mp, decNumber *ret, decNumber *p, decNumber *q);
@ When we want to multiply something by a |scaled| quantity, we use a scheme
analogous to |take_fraction| but with a different scaling.
Given positive operands, |take_scaled|
computes the quantity $p=\lfloor qf/2^{16}+{1\over2}\rfloor$.
Once again it is a good idea to use a machine-language replacement if
possible; otherwise |take_scaled| will use more than 2\pct! of the running time
when the Computer Modern fonts are being generated.
@^inner loop@>
@c
void mp_decimal_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
decNumberMultiply(ret->data.num, p_orig.data.num, q_orig.data.num, &set);
}
@ For completeness, there's also |make_scaled|, which computes a
quotient as a |scaled| number instead of as a |fraction|.
In other words, the result is $\lfloor2^{16}p/q+{1\over2}\rfloor$, if the
operands are positive. \ (This procedure is not used especially often,
so it is not part of \MP's inner loop.)
@c
void mp_decimal_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
decNumberDivide(ret->data.num, p_orig.data.num, q_orig.data.num, &set);
mp_check_decNumber(mp, ret->data.num, &set);
}
@
@d halfp(A) (integer)((unsigned)(A) >> 1)
@* Scanning numbers in the input
The definitions below are temporarily here
@d set_cur_cmd(A) mp->cur_mod_->type=(A)
@d set_cur_mod(A) decNumberCopy((decNumber *)(mp->cur_mod_->data.n.data.num),&A)
@<Declarations...@>=
static void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop);
@
@d too_precise(a) (a == (DEC_Inexact+DEC_Rounded))
@d too_large(a) (a & DEC_Overflow)
@c
void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop) {
decNumber result;
size_t l = stop-start+1;
char *buf = mp_xmalloc(mp, l+1, 1);
buf[l] = '\0';
(void)strncpy(buf,(const char *)start, l);
set.status = 0;
decNumberFromString(&result,buf, &set);
free(buf);
if (set.status == 0) {
set_cur_mod(result);
} else if (mp->scanner_status != tex_flushing) {
if (too_large(set.status)) {
const char *hlp[] = {"I could not handle this number specification",
"because it is out of range.",
NULL };
decNumber_check (&result, &set);
set_cur_mod(result);
mp_error (mp, "Enormous number has been reduced", hlp, false);
} else if (too_precise(set.status)) {
set_cur_mod(result);
if (decNumberIsPositive((decNumber *)internal_value (mp_warning_check).data.num) &&
(mp->scanner_status != tex_flushing)) {
char msg[256];
const char *hlp[] = {"Continue and I'll round the value until it fits the current numberprecision",
"(Set warningcheck:=0 to suppress this message.)",
NULL };
mp_snprintf (msg, 256, "Number is too precise (numberprecision = %d)", set.digits);
mp_error (mp, msg, hlp, true);
}
} else { // this also captures underflow
const char *hlp[] = {"I could not handle this number specification",
"Error:",
"",
NULL };
hlp[2] = decContextStatusToString(&set);
mp_error (mp, "Erroneous number specification changed to zero", hlp, false);
decNumberZero(&result);
set_cur_mod(result);
}
}
set_cur_cmd((mp_variable_type)mp_numeric_token);
}
@ @c
static void find_exponent (MP mp) {
if (mp->buffer[mp->cur_input.loc_field] == 'e' ||
mp->buffer[mp->cur_input.loc_field] == 'E') {
mp->cur_input.loc_field++;
if (!(mp->buffer[mp->cur_input.loc_field] == '+' ||
mp->buffer[mp->cur_input.loc_field] == '-' ||
mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class)) {
mp->cur_input.loc_field--;
return;
}
if (mp->buffer[mp->cur_input.loc_field] == '+' ||
mp->buffer[mp->cur_input.loc_field] == '-') {
mp->cur_input.loc_field++;
}
while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
mp->cur_input.loc_field++;
}
}
}
void mp_decimal_scan_fractional_token (MP mp, int n) { /* n: scaled */
unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
unsigned char *stop;
while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
mp->cur_input.loc_field++;
}
find_exponent(mp);
stop = &mp->buffer[mp->cur_input.loc_field-1];
mp_wrapup_numeric_token (mp, start, stop);
}
@ We just have to collect bytes.
@c
void mp_decimal_scan_numeric_token (MP mp, int n) { /* n: scaled */
unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
unsigned char *stop;
while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
mp->cur_input.loc_field++;
}
if (mp->buffer[mp->cur_input.loc_field] == '.' &&
mp->buffer[mp->cur_input.loc_field+1] != '.') {
mp->cur_input.loc_field++;
while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
mp->cur_input.loc_field++;
}
}
find_exponent(mp);
stop = &mp->buffer[mp->cur_input.loc_field-1];
mp_wrapup_numeric_token (mp, start, stop);
}
@ The |scaled| quantities in \MP\ programs are generally supposed to be
less than $2^{12}$ in absolute value, so \MP\ does much of its internal
arithmetic with 28~significant bits of precision. A |fraction| denotes
a scaled integer whose binary point is assumed to be 28 bit positions
from the right.
@d fraction_half (fraction_multiplier/2)
@d fraction_one (1*fraction_multiplier)
@d fraction_two (2*fraction_multiplier)
@d fraction_three (3*fraction_multiplier)
@d fraction_four (4*fraction_multiplier)
@ Here is a typical example of how the routines above can be used.
It computes the function
$${1\over3\tau}f(\theta,\phi)=
{\tau^{-1}\bigl(2+\sqrt2\,(\sin\theta-{1\over16}\sin\phi)
(\sin\phi-{1\over16}\sin\theta)(\cos\theta-\cos\phi)\bigr)\over
3\,\bigl(1+{1\over2}(\sqrt5-1)\cos\theta+{1\over2}(3-\sqrt5\,)\cos\phi\bigr)},$$
where $\tau$ is a |scaled| ``tension'' parameter. This is \MP's magic
fudge factor for placing the first control point of a curve that starts
at an angle $\theta$ and ends at an angle $\phi$ from the straight path.
(Actually, if the stated quantity exceeds 4, \MP\ reduces it to~4.)
The trigonometric quantity to be multiplied by $\sqrt2$ is less than $\sqrt2$.
(It's a sum of eight terms whose absolute values can be bounded using
relations such as $\sin\theta\cos\theta\L{1\over2}$.) Thus the numerator
is positive; and since the tension $\tau$ is constrained to be at least
$3\over4$, the numerator is less than $16\over3$. The denominator is
nonnegative and at most~6.
The angles $\theta$ and $\phi$ are given implicitly in terms of |fraction|
arguments |st|, |ct|, |sf|, and |cf|, representing $\sin\theta$, $\cos\theta$,
$\sin\phi$, and $\cos\phi$, respectively.
@c
void mp_decimal_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,
mp_number cf, mp_number t) {
decNumber acc, num, denom; /* registers for intermediate calculations */
decNumber r1, r2;
decNumber arg1, arg2;
decNumber i16, fone, fhalf, ftwo, sqrtfive;
decNumberFromInt32(&i16, 16);
decNumberFromInt32(&fone, fraction_one);
decNumberFromInt32(&fhalf, fraction_half);
decNumberFromInt32(&ftwo, fraction_two);
decNumberFromInt32(&sqrtfive, 5); // sqrt(5)
decNumberSquareRoot(&sqrtfive, &sqrtfive, &set);
decNumberDivide(&arg1,sf.data.num, &i16, &set); // arg1 = sf / 16
decNumberSubtract(&arg1,st.data.num,&arg1, &set); // arg1 = st - arg1
decNumberDivide(&arg2,st.data.num, &i16, &set); // arg2 = st / 16
decNumberSubtract(&arg2,sf.data.num,&arg2, &set); // arg2 = sf - arg2
mp_decimal_take_fraction (mp, &acc, &arg1, &arg2); // acc = (arg1 * arg2) / fmul
decNumberCopy(&arg1, &acc);
decNumberSubtract(&arg2, ct.data.num, cf.data.num, &set); // arg2 = ct - cf
mp_decimal_take_fraction (mp, &acc, &arg1, &arg2); // acc = (arg1 * arg2 ) / fmul
decNumberSquareRoot(&arg1, &two_decNumber, &set); // arg1 = sqrt(2)
decNumberMultiply(&arg1, &arg1, &fone, &set); // arg1 = arg1 * fmul
mp_decimal_take_fraction (mp, &r1, &acc, &arg1); // r1 = (acc * arg1) / fmul
decNumberAdd(&num, &ftwo, &r1, &set); // num = ftwo + r1
decNumberSubtract(&arg1,&sqrtfive, &one, &set); // arg1 = sqrt(5) - 1
decNumberMultiply(&arg1,&arg1,&fhalf, &set); // arg1 = arg1 * fmul/2
decNumberMultiply(&arg1,&arg1,&three_decNumber, &set); // arg1 = arg1 * 3
decNumberSubtract(&arg2,&three_decNumber, &sqrtfive, &set); // arg2 = 3 - sqrt(5)
decNumberMultiply(&arg2,&arg2,&fhalf, &set); // arg2 = arg2 * fmul/2
decNumberMultiply(&arg2,&arg2,&three_decNumber, &set); // arg2 = arg2 * 3
mp_decimal_take_fraction (mp, &r1, ct.data.num, &arg1) ; // r1 = (ct * arg1) / fmul
mp_decimal_take_fraction (mp, &r2, cf.data.num, &arg2); // r2 = (cf * arg2) / fmul
decNumberFromInt32(&denom, fraction_three); // denom = 3fmul
decNumberAdd(&denom, &denom, &r1, &set); // denom = denom + r1
decNumberAdd(&denom, &denom, &r2, &set); // denom = denom + r1
decNumberCompare(&arg1, t.data.num, &one, &set);
if (!decNumberIsZero(&arg1)) { // t != r1
decNumberDivide(&num, &num, t.data.num, &set); // num = num / t
}
decNumberCopy(&r2, &num); // r2 = num / 4
decNumberDivide(&r2, &r2, &four_decNumber, &set);
if (decNumberLess(&denom,&r2)) { // num/4 >= denom => denom < num/4
decNumberFromInt32(ret->data.num,fraction_four);
} else {
mp_decimal_make_fraction (mp, ret->data.num, &num, &denom);
}
#if DEBUG
fprintf(stdout, "\n%f = velocity(%f,%f,%f,%f,%f)", mp_number_to_double(*ret),
mp_number_to_double(st),mp_number_to_double(ct),
mp_number_to_double(sf),mp_number_to_double(cf),
mp_number_to_double(t));
#endif
mp_check_decNumber(mp, ret->data.num, &set);
}
@ The following somewhat different subroutine tests rigorously if $ab$ is
greater than, equal to, or less than~$cd$,
given integers $(a,b,c,d)$. In most cases a quick decision is reached.
The result is $+1$, 0, or~$-1$ in the three respective cases.
@c
void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {
decNumber q, r, test; /* temporary registers */
decNumber a, b, c, d;
(void)mp;
decNumberCopy(&a, (decNumber *)a_orig.data.num);
decNumberCopy(&b, (decNumber *)b_orig.data.num);
decNumberCopy(&c, (decNumber *)c_orig.data.num);
decNumberCopy(&d, (decNumber *)d_orig.data.num);
@<Reduce to the case that |a,c>=0|, |b,d>0|@>;
while (1) {
decNumberDivide(&q,&a,&d, &set);
decNumberDivide(&r,&c,&b, &set);
decNumberCompare(&test,&q,&r, &set);
if (!decNumberIsZero(&test)) {
if (decNumberIsPositive(&test)) {
decNumberCopy(ret->data.num, &one);
} else {
decNumberCopy(ret->data.num, &minusone);
}
goto RETURN;
}
decNumberRemainder(&q,&a,&d, &set);
decNumberRemainder(&r,&c,&b, &set);
if (decNumberIsZero(&r)) {
if (decNumberIsZero(&q)) {
decNumberCopy(ret->data.num, &zero);
} else {
decNumberCopy(ret->data.num, &one);
}
goto RETURN;
}
if (decNumberIsZero(&q)) {
decNumberCopy(ret->data.num, &minusone);
goto RETURN;
}
decNumberCopy(&a,&b);
decNumberCopy(&b,&q);
decNumberCopy(&c,&d);
decNumberCopy(&d,&r);
} /* now |a>d>0| and |c>b>0| */
RETURN:
#if DEBUG
fprintf(stdout, "\n%f = ab_vs_cd(%f,%f,%f,%f)", mp_number_to_double(*ret),
mp_number_to_double(a_orig),mp_number_to_double(b_orig),
mp_number_to_double(c_orig),mp_number_to_double(d_orig));
#endif
mp_check_decNumber(mp, ret->data.num, &set);
return;
}
@ @<Reduce to the case that |a...@>=
if (decNumberIsNegative(&a)) {
decNumberCopyNegate(&a, &a);
decNumberCopyNegate(&b, &b);
}
if (decNumberIsNegative(&c)) {
decNumberCopyNegate(&c, &c);
decNumberCopyNegate(&d, &d);
}
if (!decNumberIsPositive(&d)) {
if (!decNumberIsNegative(&b)) {
if ((decNumberIsZero(&a) || decNumberIsZero(&b)) && (decNumberIsZero(&c) || decNumberIsZero(&d)))
decNumberCopy(ret->data.num, &zero);
else
decNumberCopy(ret->data.num, &one);
goto RETURN;
}
if (decNumberIsZero(&d)) {
if (decNumberIsZero(&a))
decNumberCopy(ret->data.num, &zero);
else
decNumberCopy(ret->data.num, &minusone);
goto RETURN;
}
decNumberCopy(&q, &a);
decNumberCopy(&a, &c);
decNumberCopy(&c, &q);
decNumberCopyNegate(&q, &b);
decNumberCopyNegate(&b, &d);
decNumberCopy(&d, &q);
} else if (!decNumberIsPositive(&b)) {
if (decNumberIsNegative(&b) && decNumberIsPositive(&a)) {
decNumberCopy(ret->data.num, &minusone);
goto RETURN;
}
if (decNumberIsZero(&c))
decNumberCopy(ret->data.num, &zero);
else
decNumberCopy(ret->data.num, &minusone);
goto RETURN;
}
@ Now here's a subroutine that's handy for all sorts of path computations:
Given a quadratic polynomial $B(a,b,c;t)$, the |crossing_point| function
returns the unique |fraction| value |t| between 0 and~1 at which
$B(a,b,c;t)$ changes from positive to negative, or returns
|t=fraction_one+1| if no such value exists. If |a<0| (so that $B(a,b,c;t)$
is already negative at |t=0|), |crossing_point| returns the value zero.
The general bisection method is quite simple when $n=2$, hence
|crossing_point| does not take much time. At each stage in the
recursion we have a subinterval defined by |l| and~|j| such that
$B(a,b,c;2^{-l}(j+t))=B(x_0,x_1,x_2;t)$, and we want to ``zero in'' on
the subinterval where $x_0\G0$ and $\min(x_1,x_2)<0$.
It is convenient for purposes of calculation to combine the values
of |l| and~|j| in a single variable $d=2^l+j$, because the operation
of bisection then corresponds simply to doubling $d$ and possibly
adding~1. Furthermore it proves to be convenient to modify
our previous conventions for bisection slightly, maintaining the
variables $X_0=2^lx_0$, $X_1=2^l(x_0-x_1)$, and $X_2=2^l(x_1-x_2)$.
With these variables the conditions $x_0\ge0$ and $\min(x_1,x_2)<0$ are
equivalent to $\max(X_1,X_1+X_2)>X_0\ge0$.
The following code maintains the invariant relations
$0\L|x0|<\max(|x1|,|x1|+|x2|)$,
$\vert|x1|\vert<2^{30}$, $\vert|x2|\vert<2^{30}$;
it has been constructed in such a way that no arithmetic overflow
will occur if the inputs satisfy
$a<2^{30}$, $\vert a-b\vert<2^{30}$, and $\vert b-c\vert<2^{30}$.
@d no_crossing { decNumberCopy(ret->data.num, &fraction_one_plus_decNumber); goto RETURN; }
@d one_crossing { decNumberCopy(ret->data.num, &fraction_one_decNumber); goto RETURN; }
@d zero_crossing { decNumberCopy(ret->data.num, &zero); goto RETURN; }
@c
static void mp_decimal_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) {
decNumber a,b,c;
double d; /* recursive counter */
decNumber x, xx, x0, x1, x2; /* temporary registers for bisection */
decNumber scratch, scratch2;
decNumberCopy(&a, (decNumber *)aa.data.num);
decNumberCopy(&b, (decNumber *)bb.data.num);
decNumberCopy(&c, (decNumber *)cc.data.num);
if (decNumberIsNegative(&a))
zero_crossing;
if (!decNumberIsNegative(&c)) {
if (!decNumberIsNegative(&b)) {
if (decNumberIsPositive(&c)) {
no_crossing;
} else if (decNumberIsZero(&a) && decNumberIsZero(&b)) {
no_crossing;
} else {
one_crossing;
}
}
if (decNumberIsZero(&a))
zero_crossing;
} else if (decNumberIsZero(&a)) {
if (!decNumberIsPositive(&b))
zero_crossing;
}
/* Use bisection to find the crossing point... */
d = epsilonf;
decNumberCopy(&x0, &a);
decNumberSubtract(&x1,&a, &b, &set);
decNumberSubtract(&x2,&b, &c, &set);
/* not sure why the error correction has to be >= 1E-12 */
decNumberFromDouble(&scratch2, 1E-12);
do {
decNumberAdd(&x, &x1, &x2, &set);
decNumberDivide(&x, &x, &two_decNumber, &set);
decNumberAdd(&x, &x, &scratch2, &set);
decNumberSubtract(&scratch, &x1, &x0, &set);
if (decNumberGreater(&scratch, &x0)) {
decNumberCopy(&x2, &x);
decNumberAdd(&x0, &x0, &x0, &set);
d += d;
} else {
decNumberAdd(&xx, &scratch, &x, &set);
if (decNumberGreater(&xx,&x0)) {
decNumberCopy(&x2,&x);
decNumberAdd(&x0, &x0, &x0, &set);
d += d;
} else {
decNumberSubtract(&x0, &x0, &xx, &set);
if (!decNumberGreater(&x,&x0)) {
decNumberAdd(&scratch, &x, &x2, &set);
if (!decNumberGreater(&scratch, &x0))
no_crossing;
}
decNumberCopy(&x1,&x);
d = d + d + epsilonf;
}
}
} while (d < fraction_one);
decNumberFromDouble(&scratch, d);
decNumberSubtract(ret->data.num,&scratch, &fraction_one_decNumber, &set);
RETURN:
#if DEBUG
fprintf(stdout, "\n%f = crossing_point(%f,%f,%f)", mp_number_to_double(*ret),
mp_number_to_double(aa),mp_number_to_double(bb),mp_number_to_double(cc));
#endif
mp_check_decNumber(mp, ret->data.num, &set);
return;
}
@ We conclude this set of elementary routines with some simple rounding
and truncation operations.
@ |round_unscaled| rounds a |scaled| and converts it to |int|
@c
int mp_round_unscaled(mp_number x_orig) {
double xx = mp_number_to_double(x_orig);
int x = (int)ROUND(xx);
return x;
}
@ |number_floor| floors a number
@c
void mp_number_floor (mp_number *i) {
int round = set.round;
set.round = DEC_ROUND_DOWN;
decNumberToIntegralValue(i->data.num, i->data.num, &set);
set.round = round;
}
@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled|
@c
void mp_decimal_fraction_to_round_scaled (mp_number *x_orig) {
x_orig->type = mp_scaled_type;
decNumberDivide(x_orig->data.num, x_orig->data.num, &fraction_multiplier_decNumber, &set);
}
@* Algebraic and transcendental functions.
\MP\ computes all of the necessary special functions from scratch, without
relying on |real| arithmetic or system subroutines for sines, cosines, etc.
@
@c
void mp_decimal_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
decNumber x;
decNumberCopy(&x, x_orig.data.num);
if (!decNumberIsPositive(&x)) {
@<Handle square root of zero or negative argument@>;
} else {
decNumberSquareRoot(ret->data.num, &x, &set);
}
mp_check_decNumber(mp, ret->data.num, &set);
}
@ @<Handle square root of zero...@>=
{
if (decNumberIsNegative(&x)) {
char msg[256];
const char *hlp[] = {
"Since I don't take square roots of negative numbers,",
"I'm zeroing this one. Proceed, with fingers crossed.",
NULL };
char *xstr = mp_decimal_number_tostring (mp, x_orig);
mp_snprintf(msg, 256, "Square root of %s has been replaced by 0", xstr);
free(xstr);
@.Square root...replaced by 0@>;
mp_error (mp, msg, hlp, true);
}
decNumberZero(ret->data.num);
return;
}
@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by a quick hack
@c
void mp_decimal_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
decNumber a, b;
decNumber asq, bsq;
decNumberCopyAbs(&a, a_orig.data.num);
decNumberCopyAbs(&b, b_orig.data.num);
decNumberMultiply(&asq, &a, &a, &set);
decNumberMultiply(&bsq, &b, &b, &set);
decNumberAdd(&a, &asq, &bsq, &set);
decNumberSquareRoot(ret->data.num, &a, &set);
//if (set.status != 0) {
// mp->arith_error = true;
// decNumberCopy(ret->data.num, &EL_GORDO_decNumber);
//}
mp_check_decNumber(mp, ret->data.num, &set);
}
@ Here is a similar algorithm for $\psqrt{a^2-b^2}$. Same quick hack, also.
@c
void mp_decimal_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
decNumber a, b;
decNumberCopyAbs(&a, a_orig.data.num);
decNumberCopyAbs(&b, b_orig.data.num);
if (!decNumberGreater(&a,&b)) {
@<Handle erroneous |pyth_sub| and set |a:=0|@>;
} else {
decNumber asq, bsq;
decNumberMultiply(&asq, &a, &a, &set);
decNumberMultiply(&bsq, &b, &b, &set);
decNumberSubtract(&a, &asq, &bsq, &set);
decNumberSquareRoot(&a, &a, &set);
}
decNumberCopy(ret->data.num, &a);
mp_check_decNumber(mp, ret->data.num, &set);
}
@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
{
if (decNumberLess(&a, &b)) {
char msg[256];
const char *hlp[] = {
"Since I don't take square roots of negative numbers,",
"I'm zeroing this one. Proceed, with fingers crossed.",
NULL };
char *astr = mp_decimal_number_tostring (mp, a_orig);
char *bstr = mp_decimal_number_tostring (mp, b_orig);
mp_snprintf (msg, 256, "Pythagorean subtraction %s+-+%s has been replaced by 0", astr, bstr);
free(astr);
free(bstr);
@.Pythagorean...@>;
mp_error (mp, msg, hlp, true);
}
decNumberZero(&a);
}
@ Here is the routine that calculates $2^8$ times the natural logarithm
of a |scaled| quantity;
@c
void mp_decimal_m_log (MP mp, mp_number *ret, mp_number x_orig) {
if (!decNumberIsPositive((decNumber *)x_orig.data.num)) {
@<Handle non-positive logarithm@>;
} else {
decNumber twofivesix;
decNumberFromInt32(&twofivesix, 256);
decNumberLn(ret->data.num, x_orig.data.num, &limitedset);
mp_check_decNumber(mp, ret->data.num, &limitedset);
decNumberMultiply(ret->data.num, ret->data.num, &twofivesix, &set);
}
mp_check_decNumber(mp, ret->data.num, &set);
}
@ @<Handle non-positive logarithm@>=
{
char msg[256];
const char *hlp[] = {
"Since I don't take logs of non-positive numbers,",
"I'm zeroing this one. Proceed, with fingers crossed.",
NULL };
char *xstr = mp_decimal_number_tostring (mp, x_orig);
mp_snprintf (msg, 256, "Logarithm of %s has been replaced by 0", xstr);
free (xstr);
@.Logarithm...replaced by 0@>;
mp_error (mp, msg, hlp, true);
decNumberZero(ret->data.num);
}
@ Conversely, the exponential routine calculates $\exp(x/2^8)$,
when |x| is |scaled|.
@c
void mp_decimal_m_exp (MP mp, mp_number *ret, mp_number x_orig) {
decNumber temp, twofivesix;
decNumberFromInt32(&twofivesix, 256);
decNumberDivide(&temp, x_orig.data.num, &twofivesix, &set);
limitedset.status = 0;
decNumberExp(ret->data.num, &temp, &limitedset);
if (limitedset.status & DEC_Clamped) {
if (decNumberIsPositive((decNumber *)x_orig.data.num)) {
mp->arith_error = true;
decNumberCopy(ret->data.num, &EL_GORDO_decNumber);
} else {
decNumberZero(ret->data.num);
}
}
mp_check_decNumber(mp, ret->data.num, &limitedset);
limitedset.status = 0;
}
@ Given integers |x| and |y|, not both zero, the |n_arg| function
returns the |angle| whose tangent points in the direction $(x,y)$.
@c
void mp_decimal_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
if (decNumberIsZero((decNumber *)x_orig.data.num) && decNumberIsZero((decNumber *)y_orig.data.num)) {
@<Handle undefined arg@>;
} else {
decNumber atan2val, oneeighty_angle;
ret->type = mp_angle_type;
decNumberFromInt32(&oneeighty_angle, 180 * angle_multiplier);
decNumberDivide(&oneeighty_angle, &oneeighty_angle, &PI_decNumber, &set);
checkZero(y_orig.data.num);
checkZero(x_orig.data.num);
decNumberAtan2(&atan2val, y_orig.data.num, x_orig.data.num, &set);
#if DEBUG
fprintf(stdout, "\n%g = atan2(%g,%g)", decNumberToDouble(&atan2val),mp_number_to_double(x_orig),mp_number_to_double(y_orig));
#endif
decNumberMultiply(ret->data.num,&atan2val, &oneeighty_angle, &set);
checkZero(ret->data.num);
#if DEBUG
fprintf(stdout, "\nn_arg(%g,%g,%g)", mp_number_to_double(*ret),
mp_number_to_double(x_orig),mp_number_to_double(y_orig));
#endif
}
mp_check_decNumber(mp, ret->data.num, &set);
}
@ @<Handle undefined arg@>=
{
const char *hlp[] = {
"The `angle' between two identical points is undefined.",
"I'm zeroing this one. Proceed, with fingers crossed.",
NULL };
mp_error (mp, "angle(0,0) is taken as zero", hlp, true);
@.angle(0,0)...zero@>;
decNumberZero(ret->data.num);
}
@ Conversely, the |n_sin_cos| routine takes an |angle| and produces the sine
and cosine of that angle. The results of this routine are
stored in global integer variables |n_sin| and |n_cos|.
First, we need a decNumber function that calculates sines and cosines
using the Taylor series. This function is fairly optimized.
@d PRECALC_FACTORIALS_CACHESIZE 50
@c
static void sinecosine(decNumber *theangle, decNumber *c, decNumber *s)
{
int n, i, prec;
decNumber p, pxa, fac, cc;
decNumber n1, n2, p1;
decNumberZero(c);
decNumberZero(s);
prec = (set.digits/2);
if (prec < DECPRECISION_DEFAULT) prec = DECPRECISION_DEFAULT;
for (n=0;n<prec;n++)
{
decNumberFromInt32(&p1, n);
decNumberFromInt32(&n1, 2*n);
decNumberPower(&p, &minusone, &p1, &limitedset);
if (n==0) {
decNumberCopy(&pxa, &one);
} else {
decNumberPower(&pxa, theangle, &n1, &limitedset);
}
if (2*n<last_cached_factorial) {
decNumberCopy(&fac,factorials[2*n]);
} else {
decNumberCopy(&fac,factorials[last_cached_factorial]);
for (i = last_cached_factorial+1; i <= 2*n; i++) {
decNumberFromInt32(&cc, i);
decNumberMultiply (&fac, &fac, &cc, &set);
if (i<PRECALC_FACTORIALS_CACHESIZE) {
factorials[i] = malloc(sizeof(decNumber));
decNumberCopy(factorials[i],&fac);
last_cached_factorial = i;
}
}
}
decNumberDivide (&pxa, &pxa, &fac, &set);
decNumberMultiply (&pxa, &pxa, &p, &set);
decNumberAdd (s, s, &pxa, &set);
decNumberFromInt32(&n2, 2*n+1);
decNumberMultiply (&fac, &fac, &n2, &set); // fac = fac * (2*n+1)
decNumberPower(&pxa, theangle, &n2, &limitedset);
decNumberDivide (&pxa, &pxa, &fac, &set);
decNumberMultiply (&pxa, &pxa, &p, &set);
decNumberAdd (c, c, &pxa, &set);
// printf("\niteration %2d: %-42s %-42s",n,tostring(c), tostring(s));
}
}
@ Calculate sines and cosines.
@c
void mp_decimal_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) {
decNumber rad;
decNumber one_eighty;
decNumberFromInt32(&one_eighty, 180 * 16);
#if DEBUG
fprintf(stdout, "\nsin_cos(%f)", mp_number_to_double(z_orig));
#endif
decNumberMultiply(&rad, z_orig.data.num, &PI_decNumber, &set);
decNumberDivide(&rad, &rad, &one_eighty, &set);
#if 0
if (decNumberIsNegative(&rad)) {
while (decNumberLess(&rad,&PI_decNumber))
decNumberAdd(&rad, &rad, &PI_decNumber, &set);
} else {
while (decNumberGreater(&rad,&PI_decNumber))
decNumberSubtract(&rad, &rad, &PI_decNumber, &set);
}
#endif
sinecosine(&rad, n_sin->data.num, n_cos->data.num);
decNumberMultiply(n_cos->data.num,n_cos->data.num,&fraction_multiplier_decNumber, &set);
decNumberMultiply(n_sin->data.num,n_sin->data.num,&fraction_multiplier_decNumber, &set);
#if DEBUG
fprintf(stdout, "\nsin_cos(%f,%f,%f)", decNumberToDouble(&rad),
mp_number_to_double(*n_cos), mp_number_to_double(*n_sin));
#endif
mp_check_decNumber(mp, n_cos->data.num, &set);
mp_check_decNumber(mp, n_sin->data.num, &set);
}
@ To initialize the |randoms| table, we call the following routine.
@c
void mp_init_randoms (MP mp, int seed) {
int j, jj, k; /* more or less random integers */
int i; /* index into |randoms| */
j = abs (seed);
while (j >= fraction_one) {
j = j/2;
}
k = 1;
for (i = 0; i <= 54; i++) {
jj = k;
k = j - k;
j = jj;
if (k<0)
k += fraction_one;
decNumberFromInt32(mp->randoms[(i * 21) % 55].data.num, j);
}
mp_new_randoms (mp);
mp_new_randoms (mp);
mp_new_randoms (mp); /* ``warm up'' the array */
}
@ @c
void mp_decimal_number_modulo (mp_number *a, mp_number b) {
decNumberRemainder(a->data.num, a->data.num, b.data.num, &set);
}
|