/* 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 + erf(z) ) / 2 * = erfc(z) / 2 * * where z = x/sqrt(2). Computation is via the functions * erf and erfc with care to avoid error amplification in computing exp(-x^2). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -13,0 30000 1.3e-15 2.2e-16 * * * ERROR MESSAGES: * * message condition value returned * erfc underflow x > 37.519379347 0.0 * */ /* erf.c * * Error function * * * * SYNOPSIS: * * double x, y, erf(); * * y = erf( x ); * * * * DESCRIPTION: * * The integral is * * x * - * 2 | | 2 * 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, erf(x) = x * P4(x**2)/Q5(x**2); otherwise * erf(x) = 1 - 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 * */ /* erfc.c * * Complementary error function * * * * SYNOPSIS: * * double x, y, erfc(); * * y = erfc( x ); * * * * DESCRIPTION: * * * 1 - erf(x) = * * inf. * - * 2 | | 2 * erfc(x) = -------- | exp( - t ) dt * sqrt(pi) | | * - * x * * * For small x, erfc(x) = 1 - erf(x); otherwise rational * approximations are computed. * * A special function expx2.c is used to suppress error amplification * in computing exp(-x^2). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,26.6417 30000 1.3e-15 2.2e-16 * * * ERROR MESSAGES: * * message condition value returned * erfc underflow x > 9.231948545 (DEC) 0.0 * * */ /* Cephes Math Library Release 2.9: November, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier */ #include "mconf.h" extern double SQRTH; extern double MAXLOG; /* Define this macro to suppress error propagation in exp(x^2) by using the expx2 function. The tradeoff is that doing so generates two calls to the exponential function instead of one. */ #define USE_EXPXSQ 1 #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 #ifdef ANSIPROT extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); extern double exp ( double ); extern double log ( double ); extern double fabs ( double ); extern double sqrt ( double ); extern double expx2 ( double, int ); double erf ( double ); double erfc ( double ); static double erfce ( double ); #else double polevl(), p1evl(), exp(), log(), fabs(); double erf(), erfc(), expx2(), sqrt(); static double erfce(); #endif double ndtr(a) double a; { double x, y, z; x = a * SQRTH; z = fabs(x); /* if( z < SQRTH ) */ if( z < 1.0 ) y = 0.5 + 0.5 * erf(x); else { #ifdef USE_EXPXSQ /* See below for erfce. */ y = 0.5 * erfce(z); /* Multiply by exp(-x^2 / 2) */ z = expx2(a, -1); y = y * sqrt(z); #else y = 0.5 * erfc(z); #endif if( x > 0 ) y = 1.0 - y; } return(y); } double erfc(a) double a; { double p,q,x,y,z; if( a < 0.0 ) x = -a; else x = a; if( x < 1.0 ) return( 1.0 - erf(a) ); z = -a * a; if( z < -MAXLOG ) { under: mtherr( "erfc", UNDERFLOW ); if( a < 0 ) return( 2.0 ); else return( 0.0 ); } #ifdef USE_EXPXSQ /* Compute z = exp(z). */ z = expx2(a, -1); #else z = exp(z); #endif 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); } /* Exponentially scaled erfc function exp(x^2) erfc(x) valid for x > 1. Use with ndtr and expx2. */ static double erfce(x) double x; { double p,q; 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 ); } return (p/q); } double erf(x) double x; { double y, z; if( fabs(x) > 1.0 ) return( 1.0 - erfc(x) ); z = x * x; y = x * polevl( z, T, 4 ) / p1evl( z, U, 5 ); return( y ); }