/* ndtrl.c * * Normal distribution function * * * * SYNOPSIS: * * long double x, y, ndtrl(); * * y = ndtrl( 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. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -13,0 30000 1.6e-17 2.9e-18 * IEEE -150.7,0 2000 1.6e-15 3.8e-16 * Accuracy is limited by error amplification in computing exp(-x^2). * * * ERROR MESSAGES: * * message condition value returned * erfcl underflow x^2 / 2 > MAXLOGL 0.0 * */ /* erfl.c * * Error function * * * * SYNOPSIS: * * long double x, y, erfl(); * * y = erfl( x ); * * * * DESCRIPTION: * * The integral is * * x * - * 2 | | 2 * erf(x) = -------- | exp( - t ) dt. * sqrt(pi) | | * - * 0 * * The magnitude of x is limited to about 106.56 for IEEE * arithmetic; 1 or -1 is returned outside this range. * * For 0 <= |x| < 1, erf(x) = x * P6(x^2)/Q6(x^2); otherwise * erf(x) = 1 - erfc(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,1 50000 2.0e-19 5.7e-20 * */ /* erfcl.c * * Complementary error function * * * * SYNOPSIS: * * long double x, y, erfcl(); * * y = erfcl( 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. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,13 20000 7.0e-18 1.8e-18 * IEEE 0,106.56 10000 4.4e-16 1.2e-16 * Accuracy is limited by error amplification in computing exp(-x^2). * * * ERROR MESSAGES: * * message condition value returned * erfcl underflow x^2 > MAXLOGL 0.0 * * */ /* Cephes Math Library Release 2.3: January, 1995 Copyright 1984, 1995 by Stephen L. Moshier */ #include extern long double MAXLOGL; static long double SQRTHL = 7.071067811865475244008e-1L; /* erfc(x) = exp(-x^2) P(1/x)/Q(1/x) 1/8 <= 1/x <= 1 Peak relative error 5.8e-21 */ #if UNK static long double P[10] = { 1.130609921802431462353E9L, 2.290171954844785638925E9L, 2.295563412811856278515E9L, 1.448651275892911637208E9L, 6.234814405521647580919E8L, 1.870095071120436715930E8L, 3.833161455208142870198E7L, 4.964439504376477951135E6L, 3.198859502299390825278E5L, -9.085943037416544232472E-6L, }; static long double Q[10] = { /* 1.000000000000000000000E0L, */ 1.130609910594093747762E9L, 3.565928696567031388910E9L, 5.188672873106859049556E9L, 4.588018188918609726890E9L, 2.729005809811924550999E9L, 1.138778654945478547049E9L, 3.358653716579278063988E8L, 6.822450775590265689648E7L, 8.799239977351261077610E6L, 5.669830829076399819566E5L, }; #endif #if IBMPC static short P[] = { 0x4bf0,0x9ad8,0x7a03,0x86c7,0x401d, XPD 0xdf23,0xd843,0x4032,0x8881,0x401e, XPD 0xd025,0xcfd5,0x8494,0x88d3,0x401e, XPD 0xb6d0,0xc92b,0x5417,0xacb1,0x401d, XPD 0xada8,0x356a,0x4982,0x94a6,0x401c, XPD 0x4e13,0xcaee,0x9e31,0xb258,0x401a, XPD 0x5840,0x554d,0x37a3,0x9239,0x4018, XPD 0x3b58,0x3da2,0xaf02,0x9780,0x4015, XPD 0x0144,0x489e,0xbe68,0x9c31,0x4011, XPD 0x333b,0xd9e6,0xd404,0x986f,0xbfee, XPD }; static short Q[] = { /* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ 0x0e43,0x302d,0x79ed,0x86c7,0x401d, XPD 0xf817,0x9128,0xc0f8,0xd48b,0x401e, XPD 0x8eae,0x8dad,0x6eb4,0x9aa2,0x401f, XPD 0x00e7,0x7595,0xcd06,0x88bb,0x401f, XPD 0x4991,0xcfda,0x52f1,0xa2a9,0x401e, XPD 0xc39d,0xe415,0xc43d,0x87c0,0x401d, XPD 0xa75d,0x436f,0x30dd,0xa027,0x401b, XPD 0xc4cb,0x305a,0xbf78,0x8220,0x4019, XPD 0x3708,0x33b1,0x07fa,0x8644,0x4016, XPD 0x24fa,0x96f6,0x7153,0x8a6c,0x4012, XPD }; #endif #if MIEEE static long P[30] = { 0x401d0000,0x86c77a03,0x9ad84bf0, 0x401e0000,0x88814032,0xd843df23, 0x401e0000,0x88d38494,0xcfd5d025, 0x401d0000,0xacb15417,0xc92bb6d0, 0x401c0000,0x94a64982,0x356aada8, 0x401a0000,0xb2589e31,0xcaee4e13, 0x40180000,0x923937a3,0x554d5840, 0x40150000,0x9780af02,0x3da23b58, 0x40110000,0x9c31be68,0x489e0144, 0xbfee0000,0x986fd404,0xd9e6333b, }; static long Q[30] = { /* 0x3fff0000,0x80000000,0x00000000, */ 0x401d0000,0x86c779ed,0x302d0e43, 0x401e0000,0xd48bc0f8,0x9128f817, 0x401f0000,0x9aa26eb4,0x8dad8eae, 0x401f0000,0x88bbcd06,0x759500e7, 0x401e0000,0xa2a952f1,0xcfda4991, 0x401d0000,0x87c0c43d,0xe415c39d, 0x401b0000,0xa02730dd,0x436fa75d, 0x40190000,0x8220bf78,0x305ac4cb, 0x40160000,0x864407fa,0x33b13708, 0x40120000,0x8a6c7153,0x96f624fa, }; #endif /* erfc(x) = exp(-x^2) 1/x R(1/x^2) / S(1/x^2) 1/128 <= 1/x < 1/8 Peak relative error 1.9e-21 */ #if UNK static long double R[5] = { 3.621349282255624026891E0L, 7.173690522797138522298E0L, 3.445028155383625172464E0L, 5.537445669807799246891E-1L, 2.697535671015506686136E-2L, }; static long double S[5] = { /* 1.000000000000000000000E0L, */ 1.072884067182663823072E1L, 1.533713447609627196926E1L, 6.572990478128949439509E0L, 1.005392977603322982436E0L, 4.781257488046430019872E-2L, }; #endif #if IBMPC static short R[] = { 0x260a,0xab95,0x2fc7,0xe7c4,0x4000, XPD 0x4761,0x613e,0xdf6d,0xe58e,0x4001, XPD 0x0615,0x4b00,0x575f,0xdc7b,0x4000, XPD 0x521d,0x8527,0x3435,0x8dc2,0x3ffe, XPD 0x22cf,0xc711,0x6c5b,0xdcfb,0x3ff9, XPD }; static short S[] = { /* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ 0x5de6,0x17d7,0x54d6,0xaba9,0x4002, XPD 0x55d5,0xd300,0xe71e,0xf564,0x4002, XPD 0xb611,0x8f76,0xf020,0xd255,0x4001, XPD 0x3684,0x3798,0xb793,0x80b0,0x3fff, XPD 0xf5af,0x2fb2,0x1e57,0xc3d7,0x3ffa, XPD }; #endif #if MIEEE static long R[15] = { 0x40000000,0xe7c42fc7,0xab95260a, 0x40010000,0xe58edf6d,0x613e4761, 0x40000000,0xdc7b575f,0x4b000615, 0x3ffe0000,0x8dc23435,0x8527521d, 0x3ff90000,0xdcfb6c5b,0xc71122cf, }; static long S[15] = { /* 0x3fff0000,0x80000000,0x00000000, */ 0x40020000,0xaba954d6,0x17d75de6, 0x40020000,0xf564e71e,0xd30055d5, 0x40010000,0xd255f020,0x8f76b611, 0x3fff0000,0x80b0b793,0x37983684, 0x3ffa0000,0xc3d71e57,0x2fb2f5af, }; #endif /* erf(x) = x P(x^2)/Q(x^2) 0 <= x <= 1 Peak relative error 7.6e-23 */ #if UNK static long double T[7] = { 1.097496774521124996496E-1L, 5.402980370004774841217E0L, 2.871822526820825849235E2L, 2.677472796799053019985E3L, 4.825977363071025440855E4L, 1.549905740900882313773E5L, 1.104385395713178565288E6L, }; static long double U[6] = { /* 1.000000000000000000000E0L, */ 4.525777638142203713736E1L, 9.715333124857259246107E2L, 1.245905812306219011252E4L, 9.942956272177178491525E4L, 4.636021778692893773576E5L, 9.787360737578177599571E5L, }; #endif #if IBMPC static short T[] = { 0xfd7a,0x3a1a,0x705b,0xe0c4,0x3ffb, XPD 0x3128,0xc337,0x3716,0xace5,0x4001, XPD 0x9517,0x4e93,0x540e,0x8f97,0x4007, XPD 0x6118,0x6059,0x9093,0xa757,0x400a, XPD 0xb954,0xa987,0xc60c,0xbc83,0x400e, XPD 0x7a56,0xe45a,0xa4bd,0x975b,0x4010, XPD 0xc446,0x6bab,0x0b2a,0x86d0,0x4013, XPD }; static short U[] = { /* 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD */ 0x3453,0x1f8e,0xf688,0xb507,0x4004, XPD 0x71ac,0xb12f,0x21ca,0xf2e2,0x4008, XPD 0xffe8,0x9cac,0x3b84,0xc2ac,0x400c, XPD 0x481d,0x445b,0xc807,0xc232,0x400f, XPD 0x9ad5,0x1aef,0x45b1,0xe25e,0x4011, XPD 0x71a7,0x1cad,0x012e,0xeef3,0x4012, XPD }; #endif #if MIEEE static long T[21] = { 0x3ffb0000,0xe0c4705b,0x3a1afd7a, 0x40010000,0xace53716,0xc3373128, 0x40070000,0x8f97540e,0x4e939517, 0x400a0000,0xa7579093,0x60596118, 0x400e0000,0xbc83c60c,0xa987b954, 0x40100000,0x975ba4bd,0xe45a7a56, 0x40130000,0x86d00b2a,0x6babc446, }; static long U[18] = { /* 0x3fff0000,0x80000000,0x00000000, */ 0x40040000,0xb507f688,0x1f8e3453, 0x40080000,0xf2e221ca,0xb12f71ac, 0x400c0000,0xc2ac3b84,0x9cacffe8, 0x400f0000,0xc232c807,0x445b481d, 0x40110000,0xe25e45b1,0x1aef9ad5, 0x40120000,0xeef3012e,0x1cad71a7, }; #endif #ifdef ANSIPROT extern long double polevll ( long double, void *, int ); extern long double p1evll ( long double, void *, int ); extern long double expl ( long double ); extern long double logl ( long double ); extern long double erfl ( long double ); extern long double erfcl ( long double ); extern long double fabsl ( long double ); #else long double polevll(), p1evll(), expl(), logl(), erfl(), erfcl(), fabsl(); #endif #ifdef INFINITIES extern long double INFINITYL; #endif long double ndtrl(a) long double a; { long double x, y, z; x = a * SQRTHL; z = fabsl(x); if( z < SQRTHL ) y = 0.5L + 0.5L * erfl(x); else { y = 0.5L * erfcl(z); if( x > 0.0L ) y = 1.0L - y; } return(y); } long double erfcl(a) long double a; { long double p,q,x,y,z; #ifdef INFINITIES if( a == INFINITYL ) return(0.0L); if( a == -INFINITYL ) return(2.0L); #endif if( a < 0.0L ) x = -a; else x = a; if( x < 1.0L ) return( 1.0L - erfl(a) ); z = -a * a; if( z < -MAXLOGL ) { under: mtherr( "erfcl", UNDERFLOW ); if( a < 0 ) return( 2.0L ); else return( 0.0L ); } z = expl(z); y = 1.0L/x; if( x < 8.0L ) { p = polevll( y, P, 9 ); q = p1evll( y, Q, 10 ); } else { q = y * y; p = y * polevll( q, R, 4 ); q = p1evll( q, S, 5 ); } y = (z * p)/q; if( a < 0.0L ) y = 2.0L - y; if( y == 0.0L ) goto under; return(y); } long double erfl(x) long double x; { long double y, z; #if MINUSZERO if( x == 0.0L ) return(x); #endif #ifdef INFINITIES if( x == -INFINITYL ) return(-1.0L); if( x == INFINITYL ) return(1.0L); #endif if( fabsl(x) > 1.0L ) return( 1.0L - erfcl(x) ); z = x * x; y = x * polevll( z, T, 6 ) / p1evll( z, U, 6 ); return( y ); }