/* 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. * * A special function expx2l.c is used to suppress error amplification * in computing exp(-x^2). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,13 50000 8.4e-19 9.7e-20 * IEEE 6,106.56 20000 2.9e-19 7.1e-20 * * * ERROR MESSAGES: * * message condition value returned * erfcl underflow x^2 > MAXLOGL 0.0 * * */ /* Modified from file ndtrl.c Cephes Math Library Release 2.3: January, 1995 Copyright 1984, 1995 by Stephen L. Moshier */ #include #include "cephes_mconf.h" /* erfc(x) = exp(-x^2) P(1/x)/Q(1/x) 1/8 <= 1/x <= 1 Peak relative error 5.8e-21 */ static const unsigned 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 const unsigned 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 }; /* 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 */ static const unsigned 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 const unsigned 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 }; /* erf(x) = x T(x^2)/U(x^2) 0 <= x <= 1 Peak relative error 7.6e-23 */ static const unsigned 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 const unsigned 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 }; /* expx2l.c * * Exponential of squared argument * * * * SYNOPSIS: * * long double x, y, expmx2l(); * int sign; * * y = expx2l( x ); * * * * DESCRIPTION: * * Computes y = exp(x*x) while suppressing error amplification * that would ordinarily arise from the inexactness of the * exponential argument x*x. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -106.566, 106.566 10^5 1.6e-19 4.4e-20 * */ #define M 32768.0L #define MINV 3.0517578125e-5L static long double expx2l (long double x) { long double u, u1, m, f; x = fabsl (x); /* Represent x as an exact multiple of M plus a residual. M is a power of 2 chosen so that exp(m * m) does not overflow or underflow and so that |x - m| is small. */ m = MINV * floorl(M * x + 0.5L); f = x - m; /* x^2 = m^2 + 2mf + f^2 */ u = m * m; u1 = 2 * m * f + f * f; if ((u+u1) > MAXLOGL) return (INFINITYL); /* u is exact, u1 is small. */ u = expl(u) * expl(u1); return(u); } long double erfcl(long double a) { long double p,q,x,y,z; if (isinf (a)) return (signbit (a) ? 2.0 : 0.0); x = fabsl (a); if (x < 1.0L) return (1.0L - erfl(a)); z = a * a; if( z > MAXLOGL ) { under: mtherr( "erfcl", UNDERFLOW ); errno = ERANGE; return (signbit (a) ? 2.0 : 0.0); } /* Compute z = expl(a * a). */ z = expx2l (a); 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 = p/(q * z); if (a < 0.0L) y = 2.0L - y; if (y == 0.0L) goto under; return (y); } long double erfl(long double x) { long double y, z; if( x == 0.0L ) return (x); if (isinf (x)) return (signbit (x) ? -1.0L : 1.0L); 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 ); }