/* fresnl.c * * Fresnel integral * * * * SYNOPSIS: * * double x, S, C; * void fresnl(); * * fresnl( x, _&S, _&C ); * * * DESCRIPTION: * * Evaluates the Fresnel integrals * * x * - * | | * C(x) = | cos(pi/2 t**2) dt, * | | * - * 0 * * x * - * | | * S(x) = | sin(pi/2 t**2) dt. * | | * - * 0 * * * The integrals are evaluated by a power series for x < 1. * For x >= 1 auxiliary functions f(x) and g(x) are employed * such that * * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) * * * * ACCURACY: * * Relative error. * * Arithmetic function domain # trials peak rms * IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16 * IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 * DEC S(x) 0, 10 6000 2.2e-16 3.9e-17 * DEC C(x) 0, 10 5000 2.3e-16 3.9e-17 */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier */ #include /* S(x) for small x */ #ifdef UNK static double sn[6] = { -2.99181919401019853726E3, 7.08840045257738576863E5, -6.29741486205862506537E7, 2.54890880573376359104E9, -4.42979518059697779103E10, 3.18016297876567817986E11, }; static double sd[6] = { /* 1.00000000000000000000E0,*/ 2.81376268889994315696E2, 4.55847810806532581675E4, 5.17343888770096400730E6, 4.19320245898111231129E8, 2.24411795645340920940E10, 6.07366389490084639049E11, }; #endif #ifdef DEC static unsigned short sn[24] = { 0143072,0176433,0065455,0127034, 0045055,0007200,0134540,0026661, 0146560,0035061,0023667,0127545, 0050027,0166503,0002673,0153756, 0151045,0002721,0121737,0102066, 0051624,0013177,0033451,0021271, }; static unsigned short sd[24] = { /*0040200,0000000,0000000,0000000,*/ 0042214,0130051,0112070,0101617, 0044062,0010307,0172346,0152510, 0045635,0160575,0143200,0136642, 0047307,0171215,0127457,0052361, 0050647,0031447,0032621,0013510, 0052015,0064733,0117362,0012653, }; #endif #ifdef IBMPC static unsigned short sn[24] = { 0xb5c3,0x6d65,0x5fa3,0xc0a7, 0x05b6,0x172c,0xa1d0,0x4125, 0xf5ed,0x24f6,0x0746,0xc18e, 0x7afe,0x60b7,0xfda8,0x41e2, 0xf087,0x347b,0xa0ba,0xc224, 0x2457,0xe6e5,0x82cf,0x4252, }; static unsigned short sd[24] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x1072,0x3287,0x9605,0x4071, 0xdaa9,0xfe9c,0x4218,0x40e6, 0x17b4,0xb8d0,0xbc2f,0x4153, 0xea9e,0xb5e5,0xfe51,0x41b8, 0x22e9,0xe6b2,0xe664,0x4214, 0x42b5,0x73de,0xad3b,0x4261, }; #endif #ifdef MIEEE static unsigned short sn[24] = { 0xc0a7,0x5fa3,0x6d65,0xb5c3, 0x4125,0xa1d0,0x172c,0x05b6, 0xc18e,0x0746,0x24f6,0xf5ed, 0x41e2,0xfda8,0x60b7,0x7afe, 0xc224,0xa0ba,0x347b,0xf087, 0x4252,0x82cf,0xe6e5,0x2457, }; static unsigned short sd[24] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x4071,0x9605,0x3287,0x1072, 0x40e6,0x4218,0xfe9c,0xdaa9, 0x4153,0xbc2f,0xb8d0,0x17b4, 0x41b8,0xfe51,0xb5e5,0xea9e, 0x4214,0xe664,0xe6b2,0x22e9, 0x4261,0xad3b,0x73de,0x42b5, }; #endif /* C(x) for small x */ #ifdef UNK static double cn[6] = { -4.98843114573573548651E-8, 9.50428062829859605134E-6, -6.45191435683965050962E-4, 1.88843319396703850064E-2, -2.05525900955013891793E-1, 9.99999999999999998822E-1, }; static double cd[7] = { 3.99982968972495980367E-12, 9.15439215774657478799E-10, 1.25001862479598821474E-7, 1.22262789024179030997E-5, 8.68029542941784300606E-4, 4.12142090722199792936E-2, 1.00000000000000000118E0, }; #endif #ifdef DEC static unsigned short cn[24] = { 0132126,0040141,0063733,0013231, 0034037,0072223,0010200,0075637, 0135451,0021020,0073264,0036057, 0036632,0131520,0101316,0060233, 0137522,0072541,0136124,0132202, 0040200,0000000,0000000,0000000, }; static unsigned short cd[28] = { 0026614,0135503,0051776,0032631, 0030573,0121116,0154033,0126712, 0032406,0034100,0012442,0106212, 0034115,0017567,0150520,0164623, 0035543,0106171,0177336,0146351, 0037050,0150073,0000607,0171635, 0040200,0000000,0000000,0000000, }; #endif #ifdef IBMPC static unsigned short cn[24] = { 0x62d3,0x2cfb,0xc80c,0xbe6a, 0x0f74,0x6210,0xee92,0x3ee3, 0x8786,0x0ed6,0x2442,0xbf45, 0xcc13,0x1059,0x566a,0x3f93, 0x9690,0x378a,0x4eac,0xbfca, 0x0000,0x0000,0x0000,0x3ff0, }; static unsigned short cd[28] = { 0xc6b3,0x6a7f,0x9768,0x3d91, 0x75b9,0xdb03,0x7449,0x3e0f, 0x5191,0x02a4,0xc708,0x3e80, 0x1d32,0xfa2a,0xa3ee,0x3ee9, 0xd99d,0x3fdb,0x718f,0x3f4c, 0xfe74,0x6030,0x1a07,0x3fa5, 0x0000,0x0000,0x0000,0x3ff0, }; #endif #ifdef MIEEE static unsigned short cn[24] = { 0xbe6a,0xc80c,0x2cfb,0x62d3, 0x3ee3,0xee92,0x6210,0x0f74, 0xbf45,0x2442,0x0ed6,0x8786, 0x3f93,0x566a,0x1059,0xcc13, 0xbfca,0x4eac,0x378a,0x9690, 0x3ff0,0x0000,0x0000,0x0000, }; static unsigned short cd[28] = { 0x3d91,0x9768,0x6a7f,0xc6b3, 0x3e0f,0x7449,0xdb03,0x75b9, 0x3e80,0xc708,0x02a4,0x5191, 0x3ee9,0xa3ee,0xfa2a,0x1d32, 0x3f4c,0x718f,0x3fdb,0xd99d, 0x3fa5,0x1a07,0x6030,0xfe74, 0x3ff0,0x0000,0x0000,0x0000, }; #endif /* Auxiliary function f(x) */ #ifdef UNK static double fn[10] = { 4.21543555043677546506E-1, 1.43407919780758885261E-1, 1.15220955073585758835E-2, 3.45017939782574027900E-4, 4.63613749287867322088E-6, 3.05568983790257605827E-8, 1.02304514164907233465E-10, 1.72010743268161828879E-13, 1.34283276233062758925E-16, 3.76329711269987889006E-20, }; static double fd[10] = { /* 1.00000000000000000000E0,*/ 7.51586398353378947175E-1, 1.16888925859191382142E-1, 6.44051526508858611005E-3, 1.55934409164153020873E-4, 1.84627567348930545870E-6, 1.12699224763999035261E-8, 3.60140029589371370404E-11, 5.88754533621578410010E-14, 4.52001434074129701496E-17, 1.25443237090011264384E-20, }; #endif #ifdef DEC static unsigned short fn[40] = { 0037727,0152216,0106601,0016214, 0037422,0154606,0112710,0071355, 0036474,0143453,0154253,0166545, 0035264,0161606,0022250,0073743, 0033633,0110036,0024653,0136246, 0032003,0036652,0041164,0036413, 0027740,0174122,0046305,0036726, 0025501,0125270,0121317,0167667, 0023032,0150555,0076175,0047443, 0020061,0133570,0070130,0027657, }; static unsigned short fd[40] = { /*0040200,0000000,0000000,0000000,*/ 0040100,0063767,0054413,0151452, 0037357,0061566,0007243,0065754, 0036323,0005365,0033552,0133625, 0035043,0101123,0000275,0165402, 0033367,0146614,0110623,0023647, 0031501,0116644,0125222,0144263, 0027436,0062051,0117235,0001411, 0025204,0111543,0056370,0036201, 0022520,0071351,0015227,0122144, 0017554,0172240,0112713,0005006, }; #endif #ifdef IBMPC static unsigned short fn[40] = { 0x2391,0xd1b0,0xfa91,0x3fda, 0x0e5e,0xd2b9,0x5b30,0x3fc2, 0x7dad,0x7b15,0x98e5,0x3f87, 0x0efc,0xc495,0x9c70,0x3f36, 0x7795,0xc535,0x7203,0x3ed3, 0x87a1,0x484e,0x67b5,0x3e60, 0xa7bb,0x4998,0x1f0a,0x3ddc, 0xfdf7,0x1459,0x3557,0x3d48, 0xa9e4,0xaf8f,0x5a2d,0x3ca3, 0x05f6,0x0e0b,0x36ef,0x3be6, }; static unsigned short fd[40] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x7a65,0xeb21,0x0cfe,0x3fe8, 0x6d7d,0xc1d4,0xec6e,0x3fbd, 0x56f3,0xa6ed,0x615e,0x3f7a, 0xbd60,0x6017,0x704a,0x3f24, 0x64f5,0x9232,0xf9b1,0x3ebe, 0x5916,0x9552,0x33b4,0x3e48, 0xa061,0x33d3,0xcc85,0x3dc3, 0x0790,0x6b9f,0x926c,0x3d30, 0xf48d,0x2352,0x0e5d,0x3c8a, 0x6141,0x12b9,0x9e94,0x3bcd, }; #endif #ifdef MIEEE static unsigned short fn[40] = { 0x3fda,0xfa91,0xd1b0,0x2391, 0x3fc2,0x5b30,0xd2b9,0x0e5e, 0x3f87,0x98e5,0x7b15,0x7dad, 0x3f36,0x9c70,0xc495,0x0efc, 0x3ed3,0x7203,0xc535,0x7795, 0x3e60,0x67b5,0x484e,0x87a1, 0x3ddc,0x1f0a,0x4998,0xa7bb, 0x3d48,0x3557,0x1459,0xfdf7, 0x3ca3,0x5a2d,0xaf8f,0xa9e4, 0x3be6,0x36ef,0x0e0b,0x05f6, }; static unsigned short fd[40] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x3fe8,0x0cfe,0xeb21,0x7a65, 0x3fbd,0xec6e,0xc1d4,0x6d7d, 0x3f7a,0x615e,0xa6ed,0x56f3, 0x3f24,0x704a,0x6017,0xbd60, 0x3ebe,0xf9b1,0x9232,0x64f5, 0x3e48,0x33b4,0x9552,0x5916, 0x3dc3,0xcc85,0x33d3,0xa061, 0x3d30,0x926c,0x6b9f,0x0790, 0x3c8a,0x0e5d,0x2352,0xf48d, 0x3bcd,0x9e94,0x12b9,0x6141, }; #endif /* Auxiliary function g(x) */ #ifdef UNK static double gn[11] = { 5.04442073643383265887E-1, 1.97102833525523411709E-1, 1.87648584092575249293E-2, 6.84079380915393090172E-4, 1.15138826111884280931E-5, 9.82852443688422223854E-8, 4.45344415861750144738E-10, 1.08268041139020870318E-12, 1.37555460633261799868E-15, 8.36354435630677421531E-19, 1.86958710162783235106E-22, }; static double gd[11] = { /* 1.00000000000000000000E0,*/ 1.47495759925128324529E0, 3.37748989120019970451E-1, 2.53603741420338795122E-2, 8.14679107184306179049E-4, 1.27545075667729118702E-5, 1.04314589657571990585E-7, 4.60680728146520428211E-10, 1.10273215066240270757E-12, 1.38796531259578871258E-15, 8.39158816283118707363E-19, 1.86958710162783236342E-22, }; #endif #ifdef DEC static unsigned short gn[44] = { 0040001,0021435,0120406,0053123, 0037511,0152523,0037703,0122011, 0036631,0134302,0122721,0110235, 0035463,0051712,0043215,0114732, 0034101,0025677,0147725,0057630, 0032323,0010342,0067523,0002206, 0030364,0152247,0110007,0054107, 0026230,0057654,0035464,0047124, 0023706,0036401,0167705,0045440, 0021166,0154447,0105632,0142461, 0016142,0002353,0011175,0170530, }; static unsigned short gd[44] = { /*0040200,0000000,0000000,0000000,*/ 0040274,0145551,0016742,0127005, 0037654,0166557,0076416,0015165, 0036717,0140217,0030675,0050111, 0035525,0110060,0076405,0070502, 0034125,0176061,0060120,0031730, 0032340,0001615,0054343,0120501, 0030375,0041414,0070747,0107060, 0026233,0031034,0160757,0074526, 0023710,0003341,0137100,0144664, 0021167,0126414,0023774,0015435, 0016142,0002353,0011175,0170530, }; #endif #ifdef IBMPC static unsigned short gn[44] = { 0xcaca,0xb420,0x2463,0x3fe0, 0x7481,0x67f8,0x3aaa,0x3fc9, 0x3214,0x54ba,0x3718,0x3f93, 0xb33b,0x48d1,0x6a79,0x3f46, 0xabf3,0xf9fa,0x2577,0x3ee8, 0x6091,0x4dea,0x621c,0x3e7a, 0xeb09,0xf200,0x9a94,0x3dfe, 0x89cb,0x8766,0x0bf5,0x3d73, 0xa964,0x3df8,0xc7a0,0x3cd8, 0x58a6,0xf173,0xdb24,0x3c2e, 0xbe2b,0x624f,0x409d,0x3b6c, }; static unsigned short gd[44] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x55c1,0x23bc,0x996d,0x3ff7, 0xc34f,0xefa1,0x9dad,0x3fd5, 0xaa09,0xe637,0xf811,0x3f99, 0xae28,0x0fa0,0xb206,0x3f4a, 0x067b,0x2c0a,0xbf86,0x3eea, 0x7428,0xab1c,0x0071,0x3e7c, 0xf1c6,0x8e3c,0xa861,0x3dff, 0xef2b,0x9c3d,0x6643,0x3d73, 0x1936,0x37c8,0x00dc,0x3cd9, 0x8364,0x84ff,0xf5a1,0x3c2e, 0xbe2b,0x624f,0x409d,0x3b6c, }; #endif #ifdef MIEEE static unsigned short gn[44] = { 0x3fe0,0x2463,0xb420,0xcaca, 0x3fc9,0x3aaa,0x67f8,0x7481, 0x3f93,0x3718,0x54ba,0x3214, 0x3f46,0x6a79,0x48d1,0xb33b, 0x3ee8,0x2577,0xf9fa,0xabf3, 0x3e7a,0x621c,0x4dea,0x6091, 0x3dfe,0x9a94,0xf200,0xeb09, 0x3d73,0x0bf5,0x8766,0x89cb, 0x3cd8,0xc7a0,0x3df8,0xa964, 0x3c2e,0xdb24,0xf173,0x58a6, 0x3b6c,0x409d,0x624f,0xbe2b, }; static unsigned short gd[44] = { /*0x3ff0,0x0000,0x0000,0x0000,*/ 0x3ff7,0x996d,0x23bc,0x55c1, 0x3fd5,0x9dad,0xefa1,0xc34f, 0x3f99,0xf811,0xe637,0xaa09, 0x3f4a,0xb206,0x0fa0,0xae28, 0x3eea,0xbf86,0x2c0a,0x067b, 0x3e7c,0x0071,0xab1c,0x7428, 0x3dff,0xa861,0x8e3c,0xf1c6, 0x3d73,0x6643,0x9c3d,0xef2b, 0x3cd9,0x00dc,0x37c8,0x1936, 0x3c2e,0xf5a1,0x84ff,0x8364, 0x3b6c,0x409d,0x624f,0xbe2b, }; #endif #ifdef ANSIPROT extern double fabs ( double ); extern double cos ( double ); extern double sin ( double ); extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); #else double fabs(), cos(), sin(), polevl(), p1evl(); #endif extern double PI, PIO2, MACHEP; int fresnl( xxa, ssa, cca ) double xxa, *ssa, *cca; { double f, g, cc, ss, c, s, t, u; double x, x2; x = fabs(xxa); x2 = x * x; if( x2 < 2.5625 ) { t = x2 * x2; ss = x * x2 * polevl( t, sn, 5)/p1evl( t, sd, 6 ); cc = x * polevl( t, cn, 5)/polevl(t, cd, 6 ); goto done; } if( x > 36974.0 ) { cc = 0.5; ss = 0.5; goto done; } /* Asymptotic power series auxiliary functions * for large argument */ x2 = x * x; t = PI * x2; u = 1.0/(t * t); t = 1.0/t; f = 1.0 - u * polevl( u, fn, 9)/p1evl(u, fd, 10); g = t * polevl( u, gn, 10)/p1evl(u, gd, 11); t = PIO2 * x2; c = cos(t); s = sin(t); t = PI * x; cc = 0.5 + (f * s - g * c)/t; ss = 0.5 - (f * c + g * s)/t; done: if( xxa < 0.0 ) { cc = -cc; ss = -ss; } *cca = cc; *ssa = ss; return(0); }