/* ndtr.c * * Normal distribution function * * * * SYNOPSIS: * * double x, y, ndtr(); * * y = ndtr( x ); * * * * DESCRIPTION: * * Returns the area under the Gaussian probability density * function, integrated from minus infinity to x: * * x * - * 1 | | 2 * ndtr(x) = --------- | exp( - t /2 ) dt * sqrt(2pi) | | * - * -inf. * * = ( 1 + our_erf(z) ) / 2 * = our_our_erfc(z) / 2 * * where z = x/sqrt(2). Computation is via the functions * our_erf and our_our_erfc. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -13,0 8000 2.1e-15 4.8e-16 * IEEE -13,0 30000 3.4e-14 6.7e-15 * * * ERROR MESSAGES: * * message condition value returned * our_our_erfc undour_erflow x > 37.519379347 0.0 * */ /* our_erf.c * * Error function * * * * SYNOPSIS: * * double x, y, our_erf(); * * y = our_erf( x ); * * * * DESCRIPTION: * * The integral is * * x * - * 2 | | 2 * our_erf(x) = -------- | exp( - t ) dt. * sqrt(pi) | | * - * 0 * * The magnitude of x is limited to 9.231948545 for DEC * arithmetic; 1 or -1 is returned outside this range. * * For 0 <= |x| < 1, our_erf(x) = x * P4(x**2)/Q5(x**2); otherwise * our_erf(x) = 1 - our_our_erfc(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0,1 14000 4.7e-17 1.5e-17 * IEEE 0,1 30000 3.7e-16 1.0e-16 * */ /* our_our_erfc.c * * Complementary error function * * * * SYNOPSIS: * * double x, y, our_our_erfc(); * * y = our_our_erfc( x ); * * * * DESCRIPTION: * * * 1 - our_erf(x) = * * inf. * - * 2 | | 2 * our_our_erfc(x) = -------- | exp( - t ) dt * sqrt(pi) | | * - * x * * * For small x, our_our_erfc(x) = 1 - our_erf(x); otherwise rational * approximations are computed. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0, 9.2319 12000 5.1e-16 1.2e-16 * IEEE 0,26.6417 30000 5.7e-14 1.5e-14 * * * ERROR MESSAGES: * * message condition value returned * our_our_erfc undour_erflow x > 9.231948545 (DEC) 0.0 * * */ /* Cephes Math Library Release 2.2: June, 1992 Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ #include #include "mconf.h" extern double SQRTH; extern double MAXLOG; #ifdef UNK static double P[] = { 2.46196981473530512524E-10, 5.64189564831068821977E-1, 7.46321056442269912687E0, 4.86371970985681366614E1, 1.96520832956077098242E2, 5.26445194995477358631E2, 9.34528527171957607540E2, 1.02755188689515710272E3, 5.57535335369399327526E2 }; static double Q[] = { /* 1.00000000000000000000E0,*/ 1.32281951154744992508E1, 8.67072140885989742329E1, 3.54937778887819891062E2, 9.75708501743205489753E2, 1.82390916687909736289E3, 2.24633760818710981792E3, 1.65666309194161350182E3, 5.57535340817727675546E2 }; static double R[] = { 5.64189583547755073984E-1, 1.27536670759978104416E0, 5.01905042251180477414E0, 6.16021097993053585195E0, 7.40974269950448939160E0, 2.97886665372100240670E0 }; static double S[] = { /* 1.00000000000000000000E0,*/ 2.26052863220117276590E0, 9.39603524938001434673E0, 1.20489539808096656605E1, 1.70814450747565897222E1, 9.60896809063285878198E0, 3.36907645100081516050E0 }; static double T[] = { 9.60497373987051638749E0, 9.00260197203842689217E1, 2.23200534594684319226E3, 7.00332514112805075473E3, 5.55923013010394962768E4 }; static double U[] = { /* 1.00000000000000000000E0,*/ 3.35617141647503099647E1, 5.21357949780152679795E2, 4.59432382970980127987E3, 2.26290000613890934246E4, 4.92673942608635921086E4 }; #define UTHRESH 37.519379347 #endif #ifdef DEC static unsigned short P[] = { 0030207, 0054445, 0011173, 0021706, 0040020, 0067272, 0030661, 0122075, 0040756, 0151236, 0173053, 0067042, 0041502, 0106175, 0062555, 0151457, 0042104, 0102525, 0047401, 0003667, 0042403, 0116176, 0011446, 0075303, 0042551, 0120723, 0061641, 0123275, 0042600, 0070651, 0007264, 0134516, 0042413, 0061102, 0167507, 0176625 }; static unsigned short Q[] = { /*0040200,0000000,0000000,0000000,*/ 0041123, 0123257, 0165741, 0017142, 0041655, 0065027, 0173413, 0115450, 0042261, 0074011, 0021573, 0004150, 0042563, 0166530, 0013662, 0007200, 0042743, 0176427, 0162443, 0105214, 0043014, 0062546, 0153727, 0123772, 0042717, 0012470, 0006227, 0067424, 0042413, 0061103, 0003042, 0013254 }; static unsigned short R[] = { 0040020, 0067272, 0101024, 0155421, 0040243, 0037467, 0056706, 0026462, 0040640, 0116017, 0120665, 0034315, 0040705, 0020162, 0143350, 0060137, 0040755, 0016234, 0134304, 0130157, 0040476, 0122700, 0051070, 0015473 }; static unsigned short S[] = { /*0040200,0000000,0000000,0000000,*/ 0040420, 0126200, 0044276, 0070413, 0041026, 0053051, 0007302, 0063746, 0041100, 0144203, 0174051, 0061151, 0041210, 0123314, 0126343, 0177646, 0041031, 0137125, 0051431, 0033011, 0040527, 0117362, 0152661, 0066201 }; static unsigned short T[] = { 0041031, 0126770, 0170672, 0166101, 0041664, 0006522, 0072360, 0031770, 0043013, 0100025, 0162641, 0126671, 0043332, 0155231, 0161627, 0076200, 0044131, 0024115, 0021020, 0117343 }; static unsigned short U[] = { /*0040200,0000000,0000000,0000000,*/ 0041406, 0037461, 0177575, 0032714, 0042402, 0053350, 0123061, 0153557, 0043217, 0111227, 0032007, 0164217, 0043660, 0145000, 0004013, 0160114, 0044100, 0071544, 0167107, 0125471 }; #define UTHRESH 14.0 #endif #ifdef IBMPC static unsigned short P[] = { 0x6479, 0xa24f, 0xeb24, 0x3df0, 0x3488, 0x4636, 0x0dd7, 0x3fe2, 0x6dc4, 0xdec5, 0xda53, 0x401d, 0xba66, 0xacad, 0x518f, 0x4048, 0x20f7, 0xa9e0, 0x90aa, 0x4068, 0xcf58, 0xc264, 0x738f, 0x4080, 0x34d8, 0x6c74, 0x343a, 0x408d, 0x972a, 0x21d6, 0x0e35, 0x4090, 0xffb3, 0x5de8, 0x6c48, 0x4081 }; static unsigned short Q[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0x23cc, 0xfd7c, 0x74d5, 0x402a, 0x7365, 0xfee1, 0xad42, 0x4055, 0x610d, 0x246f, 0x2f01, 0x4076, 0x41d0, 0x02f6, 0x7dab, 0x408e, 0x7151, 0xfca4, 0x7fa2, 0x409c, 0xf4ff, 0xdafa, 0x8cac, 0x40a1, 0xede2, 0x0192, 0xe2a7, 0x4099, 0x42d6, 0x60c4, 0x6c48, 0x4081 }; static unsigned short R[] = { 0x9b62, 0x5042, 0x0dd7, 0x3fe2, 0xc5a6, 0xebb8, 0x67e6, 0x3ff4, 0xa71a, 0xf436, 0x1381, 0x4014, 0x0c0c, 0x58dd, 0xa40e, 0x4018, 0x960e, 0x9718, 0xa393, 0x401d, 0x0367, 0x0a47, 0xd4b8, 0x4007 }; static unsigned short S[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0xce21, 0x0917, 0x1590, 0x4002, 0x4cfd, 0x21d8, 0xcac5, 0x4022, 0x2c4d, 0x7f05, 0x1910, 0x4028, 0x7ff5, 0x959c, 0x14d9, 0x4031, 0x26c1, 0xaa63, 0x37ca, 0x4023, 0x2d90, 0x5ab6, 0xf3de, 0x400a }; static unsigned short T[] = { 0x5d88, 0x1e37, 0x35bf, 0x4023, 0x067f, 0x4e9e, 0x81aa, 0x4056, 0x35b7, 0xbcb4, 0x7002, 0x40a1, 0xef90, 0x3c72, 0x5b53, 0x40bb, 0x13dc, 0xa442, 0x2509, 0x40eb }; static unsigned short U[] = { /*0x0000,0x0000,0x0000,0x3ff0,*/ 0xa6ba, 0x3fef, 0xc7e6, 0x4040, 0x3aee, 0x14c6, 0x4add, 0x4080, 0xfd12, 0xe680, 0xf252, 0x40b1, 0x7c0a, 0x0101, 0x1940, 0x40d6, 0xf567, 0x9dc8, 0x0e6c, 0x40e8 }; #define UTHRESH 37.519379347 #endif #ifdef MIEEE static unsigned short P[] = { 0x3df0, 0xeb24, 0xa24f, 0x6479, 0x3fe2, 0x0dd7, 0x4636, 0x3488, 0x401d, 0xda53, 0xdec5, 0x6dc4, 0x4048, 0x518f, 0xacad, 0xba66, 0x4068, 0x90aa, 0xa9e0, 0x20f7, 0x4080, 0x738f, 0xc264, 0xcf58, 0x408d, 0x343a, 0x6c74, 0x34d8, 0x4090, 0x0e35, 0x21d6, 0x972a, 0x4081, 0x6c48, 0x5de8, 0xffb3 }; static unsigned short Q[] = { 0x402a, 0x74d5, 0xfd7c, 0x23cc, 0x4055, 0xad42, 0xfee1, 0x7365, 0x4076, 0x2f01, 0x246f, 0x610d, 0x408e, 0x7dab, 0x02f6, 0x41d0, 0x409c, 0x7fa2, 0xfca4, 0x7151, 0x40a1, 0x8cac, 0xdafa, 0xf4ff, 0x4099, 0xe2a7, 0x0192, 0xede2, 0x4081, 0x6c48, 0x60c4, 0x42d6 }; static unsigned short R[] = { 0x3fe2, 0x0dd7, 0x5042, 0x9b62, 0x3ff4, 0x67e6, 0xebb8, 0xc5a6, 0x4014, 0x1381, 0xf436, 0xa71a, 0x4018, 0xa40e, 0x58dd, 0x0c0c, 0x401d, 0xa393, 0x9718, 0x960e, 0x4007, 0xd4b8, 0x0a47, 0x0367 }; static unsigned short S[] = { 0x4002, 0x1590, 0x0917, 0xce21, 0x4022, 0xcac5, 0x21d8, 0x4cfd, 0x4028, 0x1910, 0x7f05, 0x2c4d, 0x4031, 0x14d9, 0x959c, 0x7ff5, 0x4023, 0x37ca, 0xaa63, 0x26c1, 0x400a, 0xf3de, 0x5ab6, 0x2d90 }; static unsigned short T[] = { 0x4023, 0x35bf, 0x1e37, 0x5d88, 0x4056, 0x81aa, 0x4e9e, 0x067f, 0x40a1, 0x7002, 0xbcb4, 0x35b7, 0x40bb, 0x5b53, 0x3c72, 0xef90, 0x40eb, 0x2509, 0xa442, 0x13dc }; static unsigned short U[] = { 0x4040, 0xc7e6, 0x3fef, 0xa6ba, 0x4080, 0x4add, 0x14c6, 0x3aee, 0x40b1, 0xf252, 0xe680, 0xfd12, 0x40d6, 0x1940, 0x0101, 0x7c0a, 0x40e8, 0x0e6c, 0x9dc8, 0xf567 }; #define UTHRESH 37.519379347 #endif double our_erf (double); double our_erfc (double); double ndtr (double a) { double x, y, z; x = a * SQRTH; z = fabs (x); if (z < SQRTH) y = 0.5 + 0.5 * our_erf (x); else { y = 0.5 * our_erfc (z); if (x > 0) y = 1.0 - y; } return (y); } double our_erfc (double a) { double p, q, x, y, z; if (a < 0.0) x = -a; else x = a; if (x < 1.0) return (1.0 - our_erf (a)); z = -a * a; if (z < -MAXLOG) { under: mtherr ("our_erfc", MATHERR_UNDERFLOW); if (a < 0) return (2.0); else return (0.0); } z = exp (z); if (x < 8.0) { p = polevl (x, P, 8); q = p1evl (x, Q, 8); } else { p = polevl (x, R, 5); q = p1evl (x, S, 6); } y = (z * p) / q; if (a < 0) y = 2.0 - y; if (y == 0.0) goto under; return (y); } double our_erf (double x) { double y, z; if (fabs (x) > 1.0) return (1.0 - our_erfc (x)); z = x * x; y = x * polevl (z, T, 4) / p1evl (z, U, 5); return (y); }