3
* Sine and cosine integrals
9
* double x, Ci, Si, sici();
11
* sici( x, &Si, &Ci );
16
* Evaluates the integrals
21
* Ci(x) = eul + ln x + | --------- dt,
33
* where eul = 0.57721566490153286061 is Euler's constant.
34
* The integrals are approximated by rational functions.
35
* For x > 8 auxiliary functions f(x) and g(x) are employed
38
* Ci(x) = f(x) sin(x) - g(x) cos(x)
39
* Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
43
* Test interval = [0,50].
44
* Absolute error, except relative when > 1:
45
* arithmetic function # trials peak rms
46
* IEEE Si 30000 4.4e-16 7.3e-17
47
* IEEE Ci 30000 6.9e-16 5.1e-17
48
* DEC Si 5000 4.4e-17 9.0e-18
49
* DEC Ci 5300 7.9e-17 5.2e-18
53
Cephes Math Library Release 2.1: January, 1989
54
Copyright 1984, 1987, 1989 by Stephen L. Moshier
55
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
62
static double SN[] = {
63
-8.39167827910303881427E-11,
64
4.62591714427012837309E-8,
65
-9.75759303843632795789E-6,
66
9.76945438170435310816E-4,
67
-4.13470316229406538752E-2,
68
1.00000000000000000302E0,
70
static double SD[] = {
71
2.03269266195951942049E-12,
72
1.27997891179943299903E-9,
73
4.41827842801218905784E-7,
74
9.96412122043875552487E-5,
75
1.42085239326149893930E-2,
76
9.99999999999999996984E-1,
80
static unsigned short SN[] = {
81
0127670,0104362,0167505,0035161,
82
0032106,0127177,0032131,0056461,
83
0134043,0132213,0000476,0172351,
84
0035600,0006331,0064761,0032665,
85
0137051,0055601,0044667,0017645,
86
0040200,0000000,0000000,0000000,
88
static unsigned short SD[] = {
89
0026417,0004674,0052064,0001573,
90
0030657,0165501,0014666,0131526,
91
0032755,0032133,0034147,0024124,
92
0034720,0173167,0166624,0154477,
93
0036550,0145336,0063534,0063220,
94
0040200,0000000,0000000,0000000,
98
static unsigned short SN[] = {
99
0xa74e,0x5de8,0x111e,0xbdd7,
100
0x2ba6,0xe68b,0xd5cf,0x3e68,
101
0xde9d,0x6027,0x7691,0xbee4,
102
0x26b7,0x2d3e,0x019b,0x3f50,
103
0xe3f5,0x2936,0x2b70,0xbfa5,
104
0x0000,0x0000,0x0000,0x3ff0,
106
static unsigned short SD[] = {
107
0x806f,0x8a86,0xe137,0x3d81,
108
0xd66b,0x2336,0xfd68,0x3e15,
109
0xe50a,0x670c,0xa68b,0x3e9d,
110
0x9b28,0xfdb2,0x1ece,0x3f1a,
111
0x8cd2,0xcceb,0x195b,0x3f8d,
112
0x0000,0x0000,0x0000,0x3ff0,
116
static unsigned short SN[] = {
117
0xbdd7,0x111e,0x5de8,0xa74e,
118
0x3e68,0xd5cf,0xe68b,0x2ba6,
119
0xbee4,0x7691,0x6027,0xde9d,
120
0x3f50,0x019b,0x2d3e,0x26b7,
121
0xbfa5,0x2b70,0x2936,0xe3f5,
122
0x3ff0,0x0000,0x0000,0x0000,
124
static unsigned short SD[] = {
125
0x3d81,0xe137,0x8a86,0x806f,
126
0x3e15,0xfd68,0x2336,0xd66b,
127
0x3e9d,0xa68b,0x670c,0xe50a,
128
0x3f1a,0x1ece,0xfdb2,0x9b28,
129
0x3f8d,0x195b,0xcceb,0x8cd2,
130
0x3ff0,0x0000,0x0000,0x0000,
134
static double CN[] = {
135
2.02524002389102268789E-11,
136
-1.35249504915790756375E-8,
137
3.59325051419993077021E-6,
138
-4.74007206873407909465E-4,
139
2.89159652607555242092E-2,
140
-1.00000000000000000080E0,
142
static double CD[] = {
143
4.07746040061880559506E-12,
144
3.06780997581887812692E-9,
145
1.23210355685883423679E-6,
146
3.17442024775032769882E-4,
147
5.10028056236446052392E-2,
148
4.00000000000000000080E0,
152
static unsigned short CN[] = {
153
0027262,0022131,0160257,0020166,
154
0131550,0055534,0077637,0000557,
155
0033561,0021622,0161463,0026575,
156
0135370,0102053,0116333,0000466,
157
0036754,0160454,0122022,0024622,
158
0140200,0000000,0000000,0000000,
160
static unsigned short CD[] = {
161
0026617,0073177,0107543,0104425,
162
0031122,0150573,0156453,0041517,
163
0033245,0057301,0077706,0110510,
164
0035246,0067130,0165424,0044543,
165
0037120,0164121,0061206,0053657,
166
0040600,0000000,0000000,0000000,
170
static unsigned short CN[] = {
171
0xe40f,0x3c15,0x448b,0x3db6,
172
0xe02e,0x8ff3,0x0b6b,0xbe4d,
173
0x65b0,0x5c66,0x2472,0x3ece,
174
0x6027,0x739b,0x1085,0xbf3f,
175
0x4532,0x9482,0x9c25,0x3f9d,
176
0x0000,0x0000,0x0000,0xbff0,
178
static unsigned short CD[] = {
179
0x7123,0xf1ec,0xeecf,0x3d91,
180
0x686a,0x7ba5,0x5a2f,0x3e2a,
181
0xd229,0x2ff8,0xabd8,0x3eb4,
182
0x892c,0x1d62,0xcdcb,0x3f34,
183
0xcaf6,0x2c50,0x1d0a,0x3faa,
184
0x0000,0x0000,0x0000,0x4010,
188
static unsigned short CN[] = {
189
0x3db6,0x448b,0x3c15,0xe40f,
190
0xbe4d,0x0b6b,0x8ff3,0xe02e,
191
0x3ece,0x2472,0x5c66,0x65b0,
192
0xbf3f,0x1085,0x739b,0x6027,
193
0x3f9d,0x9c25,0x9482,0x4532,
194
0xbff0,0x0000,0x0000,0x0000,
196
static unsigned short CD[] = {
197
0x3d91,0xeecf,0xf1ec,0x7123,
198
0x3e2a,0x5a2f,0x7ba5,0x686a,
199
0x3eb4,0xabd8,0x2ff8,0xd229,
200
0x3f34,0xcdcb,0x1d62,0x892c,
201
0x3faa,0x1d0a,0x2c50,0xcaf6,
202
0x4010,0x0000,0x0000,0x0000,
208
static double FN4[] = {
209
4.23612862892216586994E0,
210
5.45937717161812843388E0,
211
1.62083287701538329132E0,
212
1.67006611831323023771E-1,
213
6.81020132472518137426E-3,
214
1.08936580650328664411E-4,
215
5.48900223421373614008E-7,
217
static double FD4[] = {
218
/* 1.00000000000000000000E0,*/
219
8.16496634205391016773E0,
220
7.30828822505564552187E0,
221
1.86792257950184183883E0,
222
1.78792052963149907262E-1,
223
7.01710668322789753610E-3,
224
1.10034357153915731354E-4,
225
5.48900252756255700982E-7,
229
static unsigned short FN4[] = {
230
0040607,0107135,0120133,0153471,
231
0040656,0131467,0140424,0017567,
232
0040317,0073563,0121610,0002511,
233
0037453,0001710,0000040,0006334,
234
0036337,0024033,0176003,0171425,
235
0034744,0072341,0121657,0126035,
236
0033023,0054042,0154652,0000451,
238
static unsigned short FD4[] = {
239
/*0040200,0000000,0000000,0000000,*/
240
0041002,0121663,0137500,0177450,
241
0040751,0156577,0042213,0061552,
242
0040357,0014026,0045465,0147265,
243
0037467,0012503,0110413,0131772,
244
0036345,0167701,0155706,0160551,
245
0034746,0141076,0162250,0123547,
246
0033023,0054043,0056706,0151050,
250
static unsigned short FN4[] = {
251
0x7ae7,0xb40b,0xf1cb,0x4010,
252
0x83ef,0xf822,0xd666,0x4015,
253
0x00a9,0x7471,0xeeee,0x3ff9,
254
0x019c,0x0004,0x6079,0x3fc5,
255
0x7e63,0x7f80,0xe503,0x3f7b,
256
0xf584,0x3475,0x8e9c,0x3f1c,
257
0x4025,0x5b35,0x6b04,0x3ea2,
259
static unsigned short FD4[] = {
260
/*0x0000,0x0000,0x0000,0x3ff0,*/
261
0x1fe5,0x77e8,0x5476,0x4020,
262
0x6c6d,0xe891,0x3baf,0x401d,
263
0xb9d7,0xc966,0xe302,0x3ffd,
264
0x767f,0x7221,0xe2a8,0x3fc6,
265
0xdc2d,0x3b78,0xbdf8,0x3f7c,
266
0x14ed,0xdc95,0xd847,0x3f1c,
267
0xda45,0x6bb8,0x6b04,0x3ea2,
271
static unsigned short FN4[] = {
272
0x4010,0xf1cb,0xb40b,0x7ae7,
273
0x4015,0xd666,0xf822,0x83ef,
274
0x3ff9,0xeeee,0x7471,0x00a9,
275
0x3fc5,0x6079,0x0004,0x019c,
276
0x3f7b,0xe503,0x7f80,0x7e63,
277
0x3f1c,0x8e9c,0x3475,0xf584,
278
0x3ea2,0x6b04,0x5b35,0x4025,
280
static unsigned short FD4[] = {
281
/* 0x3ff0,0x0000,0x0000,0x0000,*/
282
0x4020,0x5476,0x77e8,0x1fe5,
283
0x401d,0x3baf,0xe891,0x6c6d,
284
0x3ffd,0xe302,0xc966,0xb9d7,
285
0x3fc6,0xe2a8,0x7221,0x767f,
286
0x3f7c,0xbdf8,0x3b78,0xdc2d,
287
0x3f1c,0xd847,0xdc95,0x14ed,
288
0x3ea2,0x6b04,0x6bb8,0xda45,
293
static double FN8[] = {
294
4.55880873470465315206E-1,
295
7.13715274100146711374E-1,
296
1.60300158222319456320E-1,
297
1.16064229408124407915E-2,
298
3.49556442447859055605E-4,
299
4.86215430826454749482E-6,
300
3.20092790091004902806E-8,
301
9.41779576128512936592E-11,
302
9.70507110881952024631E-14,
304
static double FD8[] = {
305
/* 1.00000000000000000000E0,*/
306
9.17463611873684053703E-1,
307
1.78685545332074536321E-1,
308
1.22253594771971293032E-2,
309
3.58696481881851580297E-4,
310
4.92435064317881464393E-6,
311
3.21956939101046018377E-8,
312
9.43720590350276732376E-11,
313
9.70507110881952025725E-14,
317
static unsigned short FN8[] = {
318
0037751,0064467,0142332,0164573,
319
0040066,0133013,0050352,0071102,
320
0037444,0022671,0102157,0013535,
321
0036476,0024335,0136423,0146444,
322
0035267,0042253,0164110,0110460,
323
0033643,0022626,0062535,0060320,
324
0032011,0075223,0010110,0153413,
325
0027717,0014572,0011360,0014034,
326
0025332,0104755,0004563,0152354,
328
static unsigned short FD8[] = {
329
/*0040200,0000000,0000000,0000000,*/
330
0040152,0157345,0030104,0075616,
331
0037466,0174527,0172740,0071060,
332
0036510,0046337,0144272,0156552,
333
0035274,0007555,0042537,0015572,
334
0033645,0035731,0112465,0026474,
335
0032012,0043612,0030613,0030123,
336
0027717,0103277,0004564,0151000,
337
0025332,0104755,0004563,0152354,
341
static unsigned short FN8[] = {
342
0x5d2f,0xf89b,0x2d26,0x3fdd,
343
0x4e48,0x6a1d,0xd6c1,0x3fe6,
344
0xe2ec,0x308d,0x84b7,0x3fc4,
345
0x79a4,0xb7a2,0xc51b,0x3f87,
346
0x1226,0x7d09,0xe895,0x3f36,
347
0xac1a,0xccab,0x64b2,0x3ed4,
348
0x1ae1,0x6209,0x2f52,0x3e61,
349
0x0304,0x425e,0xe32f,0x3dd9,
350
0x7a9d,0xa12e,0x513d,0x3d3b,
352
static unsigned short FD8[] = {
353
/*0x0000,0x0000,0x0000,0x3ff0,*/
354
0x8f72,0xa608,0x5bdc,0x3fed,
355
0x0e46,0xfebc,0xdf2a,0x3fc6,
356
0x5bad,0xf917,0x099b,0x3f89,
357
0xe36f,0xa8ab,0x81ed,0x3f37,
358
0xa5a8,0x32a6,0xa77b,0x3ed4,
359
0x660a,0x4631,0x48f1,0x3e61,
360
0x9a40,0xe12e,0xf0d7,0x3dd9,
361
0x7a9d,0xa12e,0x513d,0x3d3b,
365
static unsigned short FN8[] = {
366
0x3fdd,0x2d26,0xf89b,0x5d2f,
367
0x3fe6,0xd6c1,0x6a1d,0x4e48,
368
0x3fc4,0x84b7,0x308d,0xe2ec,
369
0x3f87,0xc51b,0xb7a2,0x79a4,
370
0x3f36,0xe895,0x7d09,0x1226,
371
0x3ed4,0x64b2,0xccab,0xac1a,
372
0x3e61,0x2f52,0x6209,0x1ae1,
373
0x3dd9,0xe32f,0x425e,0x0304,
374
0x3d3b,0x513d,0xa12e,0x7a9d,
376
static unsigned short FD8[] = {
377
/*0x3ff0,0x0000,0x0000,0x0000,*/
378
0x3fed,0x5bdc,0xa608,0x8f72,
379
0x3fc6,0xdf2a,0xfebc,0x0e46,
380
0x3f89,0x099b,0xf917,0x5bad,
381
0x3f37,0x81ed,0xa8ab,0xe36f,
382
0x3ed4,0xa77b,0x32a6,0xa5a8,
383
0x3e61,0x48f1,0x4631,0x660a,
384
0x3dd9,0xf0d7,0xe12e,0x9a40,
385
0x3d3b,0x513d,0xa12e,0x7a9d,
390
static double GN4[] = {
391
8.71001698973114191777E-2,
392
6.11379109952219284151E-1,
393
3.97180296392337498885E-1,
394
7.48527737628469092119E-2,
395
5.38868681462177273157E-3,
396
1.61999794598934024525E-4,
397
1.97963874140963632189E-6,
398
7.82579040744090311069E-9,
400
static double GD4[] = {
401
/* 1.00000000000000000000E0,*/
402
1.64402202413355338886E0,
403
6.66296701268987968381E-1,
404
9.88771761277688796203E-2,
405
6.22396345441768420760E-3,
406
1.73221081474177119497E-4,
407
2.02659182086343991969E-6,
408
7.82579218933534490868E-9,
412
static unsigned short GN4[] = {
413
0037262,0060622,0164572,0157515,
414
0040034,0101527,0061263,0147204,
415
0037713,0055467,0037475,0144512,
416
0037231,0046151,0035234,0045261,
417
0036260,0111624,0150617,0053536,
418
0035051,0157175,0016675,0155456,
419
0033404,0154757,0041211,0000055,
420
0031406,0071060,0130322,0033322,
422
static unsigned short GD4[] = {
423
/* 0040200,0000000,0000000,0000000,*/
424
0040322,0067520,0046707,0053275,
425
0040052,0111153,0126542,0005516,
426
0037312,0100035,0167121,0014552,
427
0036313,0171143,0137176,0014213,
428
0035065,0121256,0012033,0150603,
429
0033410,0000225,0013121,0071643,
430
0031406,0071062,0131152,0150454,
434
static unsigned short GN4[] = {
435
0x5bea,0x5d2f,0x4c32,0x3fb6,
436
0x79d1,0xec56,0x906a,0x3fe3,
437
0xb929,0xe7e7,0x6b66,0x3fd9,
438
0x8956,0x2753,0x298d,0x3fb3,
439
0xeaec,0x9a31,0x1272,0x3f76,
440
0xbb66,0xa3b7,0x3bcf,0x3f25,
441
0x2006,0xe851,0x9b3d,0x3ec0,
442
0x46da,0x161a,0xce46,0x3e40,
444
static unsigned short GD4[] = {
445
/* 0x0000,0x0000,0x0000,0x3ff0,*/
446
0xead8,0x09b8,0x4dea,0x3ffa,
447
0x416a,0x75ac,0x524d,0x3fe5,
448
0x232d,0xbdca,0x5003,0x3fb9,
449
0xc311,0x77cf,0x7e4c,0x3f79,
450
0x7a30,0xc283,0xb455,0x3f26,
451
0x2e74,0xa2ca,0x0012,0x3ec1,
452
0x5a26,0x564d,0xce46,0x3e40,
456
static unsigned short GN4[] = {
457
0x3fb6,0x4c32,0x5d2f,0x5bea,
458
0x3fe3,0x906a,0xec56,0x79d1,
459
0x3fd9,0x6b66,0xe7e7,0xb929,
460
0x3fb3,0x298d,0x2753,0x8956,
461
0x3f76,0x1272,0x9a31,0xeaec,
462
0x3f25,0x3bcf,0xa3b7,0xbb66,
463
0x3ec0,0x9b3d,0xe851,0x2006,
464
0x3e40,0xce46,0x161a,0x46da,
466
static unsigned short GD4[] = {
467
/*0x3ff0,0x0000,0x0000,0x0000,*/
468
0x3ffa,0x4dea,0x09b8,0xead8,
469
0x3fe5,0x524d,0x75ac,0x416a,
470
0x3fb9,0x5003,0xbdca,0x232d,
471
0x3f79,0x7e4c,0x77cf,0xc311,
472
0x3f26,0xb455,0xc283,0x7a30,
473
0x3ec1,0x0012,0xa2ca,0x2e74,
474
0x3e40,0xce46,0x564d,0x5a26,
479
static double GN8[] = {
480
6.97359953443276214934E-1,
481
3.30410979305632063225E-1,
482
3.84878767649974295920E-2,
483
1.71718239052347903558E-3,
484
3.48941165502279436777E-5,
485
3.47131167084116673800E-7,
486
1.70404452782044526189E-9,
487
3.85945925430276600453E-12,
488
3.14040098946363334640E-15,
490
static double GD8[] = {
491
/* 1.00000000000000000000E0,*/
492
1.68548898811011640017E0,
493
4.87852258695304967486E-1,
494
4.67913194259625806320E-2,
495
1.90284426674399523638E-3,
496
3.68475504442561108162E-5,
497
3.57043223443740838771E-7,
498
1.72693748966316146736E-9,
499
3.87830166023954706752E-12,
500
3.14040098946363335242E-15,
504
static unsigned short GN8[] = {
505
0040062,0103056,0110624,0033123,
506
0037651,0025640,0136266,0145647,
507
0037035,0122566,0137770,0061777,
508
0035741,0011424,0065311,0013370,
509
0034422,0055505,0134324,0016755,
510
0032672,0056530,0022565,0014747,
511
0030752,0031674,0114735,0013162,
512
0026607,0145353,0022020,0123625,
513
0024142,0045054,0060033,0016505,
515
static unsigned short GD8[] = {
516
/*0040200,0000000,0000000,0000000,*/
517
0040327,0137032,0064331,0136425,
518
0037771,0143705,0070300,0105711,
519
0037077,0124101,0025275,0035356,
520
0035771,0064333,0145103,0105357,
521
0034432,0106301,0105311,0010713,
522
0032677,0127645,0120034,0157551,
523
0030755,0054466,0010743,0105566,
524
0026610,0072242,0142530,0135744,
525
0024142,0045054,0060033,0016505,
529
static unsigned short GN8[] = {
530
0x86ca,0xd232,0x50c5,0x3fe6,
531
0xd975,0x1796,0x2574,0x3fd5,
532
0x0c80,0xd7ff,0xb4ae,0x3fa3,
533
0x22df,0x8d59,0x2262,0x3f5c,
534
0x83be,0xb71a,0x4b68,0x3f02,
535
0xa33d,0x04ae,0x4bab,0x3e97,
536
0xa2ce,0x933b,0x4677,0x3e1d,
537
0x14f3,0x6482,0xf95d,0x3d90,
538
0x63a9,0x8c03,0x4945,0x3cec,
540
static unsigned short GD8[] = {
541
/*0x0000,0x0000,0x0000,0x3ff0,*/
542
0x37a3,0x4d1b,0xf7c3,0x3ffa,
543
0x1179,0xae18,0x38f8,0x3fdf,
544
0xa75e,0x2557,0xf508,0x3fa7,
545
0x715e,0x7948,0x2d1b,0x3f5f,
546
0x2239,0x3159,0x5198,0x3f03,
547
0x9bed,0xb403,0xf5f4,0x3e97,
548
0x716f,0xc23c,0xab26,0x3e1d,
549
0x177c,0x58ab,0x0e94,0x3d91,
550
0x63a9,0x8c03,0x4945,0x3cec,
554
static unsigned short GN8[] = {
555
0x3fe6,0x50c5,0xd232,0x86ca,
556
0x3fd5,0x2574,0x1796,0xd975,
557
0x3fa3,0xb4ae,0xd7ff,0x0c80,
558
0x3f5c,0x2262,0x8d59,0x22df,
559
0x3f02,0x4b68,0xb71a,0x83be,
560
0x3e97,0x4bab,0x04ae,0xa33d,
561
0x3e1d,0x4677,0x933b,0xa2ce,
562
0x3d90,0xf95d,0x6482,0x14f3,
563
0x3cec,0x4945,0x8c03,0x63a9,
565
static unsigned short GD8[] = {
566
/*0x3ff0,0x0000,0x0000,0x0000,*/
567
0x3ffa,0xf7c3,0x4d1b,0x37a3,
568
0x3fdf,0x38f8,0xae18,0x1179,
569
0x3fa7,0xf508,0x2557,0xa75e,
570
0x3f5f,0x2d1b,0x7948,0x715e,
571
0x3f03,0x5198,0x3159,0x2239,
572
0x3e97,0xf5f4,0xb403,0x9bed,
573
0x3e1d,0xab26,0xc23c,0x716f,
574
0x3d91,0x0e94,0x58ab,0x177c,
575
0x3cec,0x4945,0x8c03,0x63a9,
579
#define EUL 0.57721566490153286061
580
extern double MAXNUM, PIO2, MACHEP;
583
int sici( x, si, ci )
587
double z, c, s, f, g;
609
*si = PIO2 - cos(x)/x;
619
s = x * polevl( z, SN, 5 ) / polevl( z, SD, 5 );
620
c = z * polevl( z, CN, 5 ) / polevl( z, CD, 5 );
625
*ci = EUL + log(x) + c; /* real part if x < 0 */
630
/* The auxiliary functions are:
637
* t = *ci * s - *si * c;
638
* a = *ci * c + *si * s;
652
f = polevl( z, FN4, 6 ) / (x * p1evl( z, FD4, 7 ));
653
g = z * polevl( z, GN4, 7 ) / p1evl( z, GD4, 7 );
657
f = polevl( z, FN8, 8 ) / (x * p1evl( z, FD8, 8 ));
658
g = z * polevl( z, GN8, 8 ) / p1evl( z, GD8, 9 );
660
*si = PIO2 - f * c - g * s;
added
removed
cephes/sici.c
