/* sici.c * * Sine and cosine integrals * * * * SYNOPSIS: * * double x, Ci, Si, sici(); * * sici( x, &Si, &Ci ); * * * DESCRIPTION: * * Evaluates the integrals * * x * - * | cos t - 1 * Ci(x) = eul + ln x + | --------- dt, * | t * - * 0 * x * - * | sin t * Si(x) = | ----- dt * | t * - * 0 * * where eul = 0.57721566490153286061 is Euler's constant. * The integrals are approximated by rational functions. * For x > 8 auxiliary functions f(x) and g(x) are employed * such that * * Ci(x) = f(x) sin(x) - g(x) cos(x) * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) * * * ACCURACY: * Test interval = [0,50]. * Absolute error, except relative when > 1: * arithmetic function # trials peak rms * IEEE Si 30000 4.4e-16 7.3e-17 * IEEE Ci 30000 6.9e-16 5.1e-17 * DEC Si 5000 4.4e-17 9.0e-18 * DEC Ci 5300 7.9e-17 5.2e-18 */ /* Cephes Math Library Release 2.1: January, 1989 Copyright 1984, 1987, 1989 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ #include #ifdef UNK static double SN[] = { -8.39167827910303881427E-11, 4.62591714427012837309E-8, -9.75759303843632795789E-6, 9.76945438170435310816E-4, -4.13470316229406538752E-2, 1.00000000000000000302E0, }; static double SD[] = { 2.03269266195951942049E-12, 1.27997891179943299903E-9, 4.41827842801218905784E-7, 9.96412122043875552487E-5, 1.42085239326149893930E-2, 9.99999999999999996984E-1, }; #endif #ifdef DEC static unsigned short SN[] = { 0127670,0104362,0167505,0035161, 0032106,0127177,0032131,0056461, 0134043,0132213,0000476,0172351, 0035600,0006331,0064761,0032665, 0137051,0055601,0044667,0017645, 0040200,0000000,0000000,0000000, }; static unsigned short SD[] = { 0026417,0004674,0052064,0001573, 0030657,0165501,0014666,0131526, 0032755,0032133,0034147,0024124, 0034720,0173167,0166624,0154477, 0036550,0145336,0063534,0063220, 0040200,0000000,0000000,0000000, }; #endif #ifdef IBMPC static unsigned short SN[] = { 0xa74e,0x5de8,0x111e,0xbdd7, 0x2ba6,0xe68b,0xd5cf,0x3e68, 0xde9d,0x6027,0x7691,0xbee4, 0x26b7,0x2d3e,0x019b,0x3f50, 0xe3f5,0x2936,0x2b70,0xbfa5, 0x0000,0x0000,0x0000,0x3ff0, }; static unsigned short SD[] = { 0x806f,0x8a86,0xe137,0x3d81, 0xd66b,0x2336,0xfd68,0x3e15, 0xe50a,0x670c,0xa68b,0x3e9d, 0x9b28,0xfdb2,0x1ece,0x3f1a, 0x8cd2,0xcceb,0x195b,0x3f8d, 0x0000,0x0000,0x0000,0x3ff0, }; #endif #ifdef MIEEE static unsigned short SN[] = { 0xbdd7,0x111e,0x5de8,0xa74e, 0x3e68,0xd5cf,0xe68b,0x2ba6, 0xbee4,0x7691,0x6027,0xde9d, 0x3f50,0x019b,0x2d3e,0x26b7, 0xbfa5,0x2b70,0x2936,0xe3f5, 0x3ff0,0x0000,0x0000,0x0000, }; static unsigned short SD[] = { 0x3d81,0xe137,0x8a86,0x806f, 0x3e15,0xfd68,0x2336,0xd66b, 0x3e9d,0xa68b,0x670c,0xe50a, 0x3f1a,0x1ece,0xfdb2,0x9b28, 0x3f8d,0x195b,0xcceb,0x8cd2, 0x3ff0,0x0000,0x0000,0x0000, }; #endif #ifdef UNK static double CN[] = { 2.02524002389102268789E-11, -1.35249504915790756375E-8, 3.59325051419993077021E-6, -4.74007206873407909465E-4, 2.89159652607555242092E-2, -1.00000000000000000080E0, }; static double CD[] = { 4.07746040061880559506E-12, 3.06780997581887812692E-9, 1.23210355685883423679E-6, 3.17442024775032769882E-4, 5.10028056236446052392E-2, 4.00000000000000000080E0, }; #endif #ifdef DEC static unsigned short CN[] = { 0027262,0022131,0160257,0020166, 0131550,0055534,0077637,0000557, 0033561,0021622,0161463,0026575, 0135370,0102053,0116333,0000466, 0036754,0160454,0122022,0024622, 0140200,0000000,0000000,0000000, }; static unsigned short CD[] = { 0026617,0073177,0107543,0104425, 0031122,0150573,0156453,0041517, 0033245,0057301,0077706,0110510, 0035246,0067130,0165424,0044543, 0037120,0164121,0061206,0053657, 0040600,0000000,0000000,0000000, }; #endif #ifdef IBMPC static unsigned short CN[] = { 0xe40f,0x3c15,0x448b,0x3db6, 0xe02e,0x8ff3,0x0b6b,0xbe4d, 0x65b0,0x5c66,0x2472,0x3ece, 0x6027,0x739b,0x1085,0xbf3f, 0x4532,0x9482,0x9c25,0x3f9d, 0x0000,0x0000,0x0000,0xbff0, }; static unsigned short CD[] = { 0x7123,0xf1ec,0xeecf,0x3d91, 0x686a,0x7ba5,0x5a2f,0x3e2a, 0xd229,0x2ff8,0xabd8,0x3eb4, 0x892c,0x1d62,0xcdcb,0x3f34, 0xcaf6,0x2c50,0x1d0a,0x3faa, 0x0000,0x0000,0x0000,0x4010, }; #endif #ifdef MIEEE static unsigned short CN[] = { 0x3db6,0x448b,0x3c15,0xe40f, 0xbe4d,0x0b6b,0x8ff3,0xe02e, 0x3ece,0x2472,0x5c66,0x65b0, 0xbf3f,0x1085,0x739b,0x6027, 0x3f9d,0x9c25,0x9482,0x4532, 0xbff0,0x0000,0x0000,0x0000, }; static unsigned short CD[] = { 0x3d91,0xeecf,0xf1ec,0x7123, 0x3e2a,0x5a2f,0x7ba5,0x686a, 0x3eb4,0xabd8,0x2ff8,0xd229, 0x3f34,0xcdcb,0x1d62,0x892c, 0x3faa,0x1d0a,0x2c50,0xcaf6, 0x4010,0x0000,0x0000,0x0000, }; #endif #ifdef UNK static double FN4[] = { 4.23612862892216586994E0, 5.45937717161812843388E0, 1.62083287701538329132E0, 1.67006611831323023771E-1, 6.81020132472518137426E-3, 1.08936580650328664411E-4, 5.48900223421373614008E-7, }; static double FD4[] = { /* 1.00000000000000000000E0,*/ 8.16496634205391016773E0, 7.30828822505564552187E0, 1.86792257950184183883E0, 1.78792052963149907262E-1, 7.01710668322789753610E-3, 1.10034357153915731354E-4, 5.48900252756255700982E-7, }; #endif #ifdef DEC static unsigned short FN4[] = { 0040607,0107135,0120133,0153471, 0040656,0131467,0140424,0017567, 0040317,0073563,0121610,0002511, 0037453,0001710,0000040,0006334, 0036337,0024033,0176003,0171425, 0034744,0072341,0121657,0126035, 0033023,0054042,0154652,0000451, }; static unsigned short FD4[] = { /*0040200,0000000,0000000,0000000,*/ 0041002,0121663,0137500,0177450, 0040751,0156577,0042213,0061552, 0040357,0014026,0045465,0147265, 0037467,0012503,0110413,0131772, 0036345,0167701,0155706,0160551, 0034746,0141076,0162250,0123547, 0033023,0054043,0056706,0151050, }; #endif #ifdef IBMPC static unsigned short FN4[] = { 0x7ae7,0xb40b,0xf1cb,0x4010, 0x83ef,0xf822,0xd666,0x4015, 0x00a9,0x7471,0xeeee,0x3ff9, 0x019c,0x0004,0x6079,0x3fc5, 0x7e63,0x7f80,0xe503,0x3f7b, 0xf584,0x3475,0x8e9c,0x3f1c, 0x4025,0x5b35,0x6b04,0x3ea2, }; static unsigned short FD4[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x1fe5,0x77e8,0x5476,0x4020, 0x6c6d,0xe891,0x3baf,0x401d, 0xb9d7,0xc966,0xe302,0x3ffd, 0x767f,0x7221,0xe2a8,0x3fc6, 0xdc2d,0x3b78,0xbdf8,0x3f7c, 0x14ed,0xdc95,0xd847,0x3f1c, 0xda45,0x6bb8,0x6b04,0x3ea2, }; #endif #ifdef MIEEE static unsigned short FN4[] = { 0x4010,0xf1cb,0xb40b,0x7ae7, 0x4015,0xd666,0xf822,0x83ef, 0x3ff9,0xeeee,0x7471,0x00a9, 0x3fc5,0x6079,0x0004,0x019c, 0x3f7b,0xe503,0x7f80,0x7e63, 0x3f1c,0x8e9c,0x3475,0xf584, 0x3ea2,0x6b04,0x5b35,0x4025, }; static unsigned short FD4[] = { /* 0x3ff0,0x0000,0x0000,0x0000,*/ 0x4020,0x5476,0x77e8,0x1fe5, 0x401d,0x3baf,0xe891,0x6c6d, 0x3ffd,0xe302,0xc966,0xb9d7, 0x3fc6,0xe2a8,0x7221,0x767f, 0x3f7c,0xbdf8,0x3b78,0xdc2d, 0x3f1c,0xd847,0xdc95,0x14ed, 0x3ea2,0x6b04,0x6bb8,0xda45, }; #endif #ifdef UNK static double FN8[] = { 4.55880873470465315206E-1, 7.13715274100146711374E-1, 1.60300158222319456320E-1, 1.16064229408124407915E-2, 3.49556442447859055605E-4, 4.86215430826454749482E-6, 3.20092790091004902806E-8, 9.41779576128512936592E-11, 9.70507110881952024631E-14, }; static double FD8[] = { /* 1.00000000000000000000E0,*/ 9.17463611873684053703E-1, 1.78685545332074536321E-1, 1.22253594771971293032E-2, 3.58696481881851580297E-4, 4.92435064317881464393E-6, 3.21956939101046018377E-8, 9.43720590350276732376E-11, 9.70507110881952025725E-14, }; #endif #ifdef DEC static unsigned short FN8[] = { 0037751,0064467,0142332,0164573, 0040066,0133013,0050352,0071102, 0037444,0022671,0102157,0013535, 0036476,0024335,0136423,0146444, 0035267,0042253,0164110,0110460, 0033643,0022626,0062535,0060320, 0032011,0075223,0010110,0153413, 0027717,0014572,0011360,0014034, 0025332,0104755,0004563,0152354, }; static unsigned short FD8[] = { /*0040200,0000000,0000000,0000000,*/ 0040152,0157345,0030104,0075616, 0037466,0174527,0172740,0071060, 0036510,0046337,0144272,0156552, 0035274,0007555,0042537,0015572, 0033645,0035731,0112465,0026474, 0032012,0043612,0030613,0030123, 0027717,0103277,0004564,0151000, 0025332,0104755,0004563,0152354, }; #endif #ifdef IBMPC static unsigned short FN8[] = { 0x5d2f,0xf89b,0x2d26,0x3fdd, 0x4e48,0x6a1d,0xd6c1,0x3fe6, 0xe2ec,0x308d,0x84b7,0x3fc4, 0x79a4,0xb7a2,0xc51b,0x3f87, 0x1226,0x7d09,0xe895,0x3f36, 0xac1a,0xccab,0x64b2,0x3ed4, 0x1ae1,0x6209,0x2f52,0x3e61, 0x0304,0x425e,0xe32f,0x3dd9, 0x7a9d,0xa12e,0x513d,0x3d3b, }; static unsigned short FD8[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x8f72,0xa608,0x5bdc,0x3fed, 0x0e46,0xfebc,0xdf2a,0x3fc6, 0x5bad,0xf917,0x099b,0x3f89, 0xe36f,0xa8ab,0x81ed,0x3f37, 0xa5a8,0x32a6,0xa77b,0x3ed4, 0x660a,0x4631,0x48f1,0x3e61, 0x9a40,0xe12e,0xf0d7,0x3dd9, 0x7a9d,0xa12e,0x513d,0x3d3b, }; #endif #ifdef MIEEE static unsigned short FN8[] = { 0x3fdd,0x2d26,0xf89b,0x5d2f, 0x3fe6,0xd6c1,0x6a1d,0x4e48, 0x3fc4,0x84b7,0x308d,0xe2ec, 0x3f87,0xc51b,0xb7a2,0x79a4, 0x3f36,0xe895,0x7d09,0x1226, 0x3ed4,0x64b2,0xccab,0xac1a, 0x3e61,0x2f52,0x6209,0x1ae1, 0x3dd9,0xe32f,0x425e,0x0304, 0x3d3b,0x513d,0xa12e,0x7a9d, }; static unsigned short FD8[] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x3fed,0x5bdc,0xa608,0x8f72, 0x3fc6,0xdf2a,0xfebc,0x0e46, 0x3f89,0x099b,0xf917,0x5bad, 0x3f37,0x81ed,0xa8ab,0xe36f, 0x3ed4,0xa77b,0x32a6,0xa5a8, 0x3e61,0x48f1,0x4631,0x660a, 0x3dd9,0xf0d7,0xe12e,0x9a40, 0x3d3b,0x513d,0xa12e,0x7a9d, }; #endif #ifdef UNK static double GN4[] = { 8.71001698973114191777E-2, 6.11379109952219284151E-1, 3.97180296392337498885E-1, 7.48527737628469092119E-2, 5.38868681462177273157E-3, 1.61999794598934024525E-4, 1.97963874140963632189E-6, 7.82579040744090311069E-9, }; static double GD4[] = { /* 1.00000000000000000000E0,*/ 1.64402202413355338886E0, 6.66296701268987968381E-1, 9.88771761277688796203E-2, 6.22396345441768420760E-3, 1.73221081474177119497E-4, 2.02659182086343991969E-6, 7.82579218933534490868E-9, }; #endif #ifdef DEC static unsigned short GN4[] = { 0037262,0060622,0164572,0157515, 0040034,0101527,0061263,0147204, 0037713,0055467,0037475,0144512, 0037231,0046151,0035234,0045261, 0036260,0111624,0150617,0053536, 0035051,0157175,0016675,0155456, 0033404,0154757,0041211,0000055, 0031406,0071060,0130322,0033322, }; static unsigned short GD4[] = { /* 0040200,0000000,0000000,0000000,*/ 0040322,0067520,0046707,0053275, 0040052,0111153,0126542,0005516, 0037312,0100035,0167121,0014552, 0036313,0171143,0137176,0014213, 0035065,0121256,0012033,0150603, 0033410,0000225,0013121,0071643, 0031406,0071062,0131152,0150454, }; #endif #ifdef IBMPC static unsigned short GN4[] = { 0x5bea,0x5d2f,0x4c32,0x3fb6, 0x79d1,0xec56,0x906a,0x3fe3, 0xb929,0xe7e7,0x6b66,0x3fd9, 0x8956,0x2753,0x298d,0x3fb3, 0xeaec,0x9a31,0x1272,0x3f76, 0xbb66,0xa3b7,0x3bcf,0x3f25, 0x2006,0xe851,0x9b3d,0x3ec0, 0x46da,0x161a,0xce46,0x3e40, }; static unsigned short GD4[] = { /* 0x0000,0x0000,0x0000,0x3ff0,*/ 0xead8,0x09b8,0x4dea,0x3ffa, 0x416a,0x75ac,0x524d,0x3fe5, 0x232d,0xbdca,0x5003,0x3fb9, 0xc311,0x77cf,0x7e4c,0x3f79, 0x7a30,0xc283,0xb455,0x3f26, 0x2e74,0xa2ca,0x0012,0x3ec1, 0x5a26,0x564d,0xce46,0x3e40, }; #endif #ifdef MIEEE static unsigned short GN4[] = { 0x3fb6,0x4c32,0x5d2f,0x5bea, 0x3fe3,0x906a,0xec56,0x79d1, 0x3fd9,0x6b66,0xe7e7,0xb929, 0x3fb3,0x298d,0x2753,0x8956, 0x3f76,0x1272,0x9a31,0xeaec, 0x3f25,0x3bcf,0xa3b7,0xbb66, 0x3ec0,0x9b3d,0xe851,0x2006, 0x3e40,0xce46,0x161a,0x46da, }; static unsigned short GD4[] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x3ffa,0x4dea,0x09b8,0xead8, 0x3fe5,0x524d,0x75ac,0x416a, 0x3fb9,0x5003,0xbdca,0x232d, 0x3f79,0x7e4c,0x77cf,0xc311, 0x3f26,0xb455,0xc283,0x7a30, 0x3ec1,0x0012,0xa2ca,0x2e74, 0x3e40,0xce46,0x564d,0x5a26, }; #endif #ifdef UNK static double GN8[] = { 6.97359953443276214934E-1, 3.30410979305632063225E-1, 3.84878767649974295920E-2, 1.71718239052347903558E-3, 3.48941165502279436777E-5, 3.47131167084116673800E-7, 1.70404452782044526189E-9, 3.85945925430276600453E-12, 3.14040098946363334640E-15, }; static double GD8[] = { /* 1.00000000000000000000E0,*/ 1.68548898811011640017E0, 4.87852258695304967486E-1, 4.67913194259625806320E-2, 1.90284426674399523638E-3, 3.68475504442561108162E-5, 3.57043223443740838771E-7, 1.72693748966316146736E-9, 3.87830166023954706752E-12, 3.14040098946363335242E-15, }; #endif #ifdef DEC static unsigned short GN8[] = { 0040062,0103056,0110624,0033123, 0037651,0025640,0136266,0145647, 0037035,0122566,0137770,0061777, 0035741,0011424,0065311,0013370, 0034422,0055505,0134324,0016755, 0032672,0056530,0022565,0014747, 0030752,0031674,0114735,0013162, 0026607,0145353,0022020,0123625, 0024142,0045054,0060033,0016505, }; static unsigned short GD8[] = { /*0040200,0000000,0000000,0000000,*/ 0040327,0137032,0064331,0136425, 0037771,0143705,0070300,0105711, 0037077,0124101,0025275,0035356, 0035771,0064333,0145103,0105357, 0034432,0106301,0105311,0010713, 0032677,0127645,0120034,0157551, 0030755,0054466,0010743,0105566, 0026610,0072242,0142530,0135744, 0024142,0045054,0060033,0016505, }; #endif #ifdef IBMPC static unsigned short GN8[] = { 0x86ca,0xd232,0x50c5,0x3fe6, 0xd975,0x1796,0x2574,0x3fd5, 0x0c80,0xd7ff,0xb4ae,0x3fa3, 0x22df,0x8d59,0x2262,0x3f5c, 0x83be,0xb71a,0x4b68,0x3f02, 0xa33d,0x04ae,0x4bab,0x3e97, 0xa2ce,0x933b,0x4677,0x3e1d, 0x14f3,0x6482,0xf95d,0x3d90, 0x63a9,0x8c03,0x4945,0x3cec, }; static unsigned short GD8[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x37a3,0x4d1b,0xf7c3,0x3ffa, 0x1179,0xae18,0x38f8,0x3fdf, 0xa75e,0x2557,0xf508,0x3fa7, 0x715e,0x7948,0x2d1b,0x3f5f, 0x2239,0x3159,0x5198,0x3f03, 0x9bed,0xb403,0xf5f4,0x3e97, 0x716f,0xc23c,0xab26,0x3e1d, 0x177c,0x58ab,0x0e94,0x3d91, 0x63a9,0x8c03,0x4945,0x3cec, }; #endif #ifdef MIEEE static unsigned short GN8[] = { 0x3fe6,0x50c5,0xd232,0x86ca, 0x3fd5,0x2574,0x1796,0xd975, 0x3fa3,0xb4ae,0xd7ff,0x0c80, 0x3f5c,0x2262,0x8d59,0x22df, 0x3f02,0x4b68,0xb71a,0x83be, 0x3e97,0x4bab,0x04ae,0xa33d, 0x3e1d,0x4677,0x933b,0xa2ce, 0x3d90,0xf95d,0x6482,0x14f3, 0x3cec,0x4945,0x8c03,0x63a9, }; static unsigned short GD8[] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x3ffa,0xf7c3,0x4d1b,0x37a3, 0x3fdf,0x38f8,0xae18,0x1179, 0x3fa7,0xf508,0x2557,0xa75e, 0x3f5f,0x2d1b,0x7948,0x715e, 0x3f03,0x5198,0x3159,0x2239, 0x3e97,0xf5f4,0xb403,0x9bed, 0x3e1d,0xab26,0xc23c,0x716f, 0x3d91,0x0e94,0x58ab,0x177c, 0x3cec,0x4945,0x8c03,0x63a9, }; #endif #ifdef ANSIPROT extern double log ( double ); extern double sin ( double ); extern double cos ( double ); extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); #else double log(), sin(), cos(), polevl(), p1evl(); #endif #define EUL 0.57721566490153286061 extern double MAXNUM, PIO2, MACHEP; int sici( x, si, ci ) double x; double *si, *ci; { double z, c, s, f, g; short sign; if( x < 0.0 ) { sign = -1; x = -x; } else sign = 0; if( x == 0.0 ) { *si = 0.0; *ci = -MAXNUM; return( 0 ); } if( x > 1.0e9 ) { *si = PIO2 - cos(x)/x; *ci = sin(x)/x; return( 0 ); } if( x > 4.0 ) goto asympt; z = x * x; s = x * polevl( z, SN, 5 ) / polevl( z, SD, 5 ); c = z * polevl( z, CN, 5 ) / polevl( z, CD, 5 ); if( sign ) s = -s; *si = s; *ci = EUL + log(x) + c; /* real part if x < 0 */ return(0); /* The auxiliary functions are: * * * *si = *si - PIO2; * c = cos(x); * s = sin(x); * * t = *ci * s - *si * c; * a = *ci * c + *si * s; * * *si = t; * *ci = -a; */ asympt: s = sin(x); c = cos(x); z = 1.0/(x*x); if( x < 8.0 ) { f = polevl( z, FN4, 6 ) / (x * p1evl( z, FD4, 7 )); g = z * polevl( z, GN4, 7 ) / p1evl( z, GD4, 7 ); } else { f = polevl( z, FN8, 8 ) / (x * p1evl( z, FD8, 8 )); g = z * polevl( z, GN8, 8 ) / p1evl( z, GD8, 9 ); } *si = PIO2 - f * c - g * s; if( sign ) *si = -( *si ); *ci = f * s - g * c; return(0); }