| /** |
| * This file has no copyright assigned and is placed in the Public Domain. |
| * This file is part of the mingw-w64 runtime package. |
| * No warranty is given; refer to the file DISCLAIMER.PD within this package. |
| */ |
| /* 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 <math.h> |
| #include "cephes_mconf.h" |
| |
| long double erfl(long double x); |
| |
| /* erfc(x) = exp(-x^2) P(1/x)/Q(1/x) |
| 1/8 <= 1/x <= 1 |
| Peak relative error 5.8e-21 */ |
| |
| static const uLD P[10] = { |
| { { 0x4bf0,0x9ad8,0x7a03,0x86c7,0x401d, 0, 0, 0 } }, |
| { { 0xdf23,0xd843,0x4032,0x8881,0x401e, 0, 0, 0 } }, |
| { { 0xd025,0xcfd5,0x8494,0x88d3,0x401e, 0, 0, 0 } }, |
| { { 0xb6d0,0xc92b,0x5417,0xacb1,0x401d, 0, 0, 0 } }, |
| { { 0xada8,0x356a,0x4982,0x94a6,0x401c, 0, 0, 0 } }, |
| { { 0x4e13,0xcaee,0x9e31,0xb258,0x401a, 0, 0, 0 } }, |
| { { 0x5840,0x554d,0x37a3,0x9239,0x4018, 0, 0, 0 } }, |
| { { 0x3b58,0x3da2,0xaf02,0x9780,0x4015, 0, 0, 0 } }, |
| { { 0x0144,0x489e,0xbe68,0x9c31,0x4011, 0, 0, 0 } }, |
| { { 0x333b,0xd9e6,0xd404,0x986f,0xbfee, 0, 0, 0 } } |
| }; |
| static const uLD Q[] = { |
| { { 0x0e43,0x302d,0x79ed,0x86c7,0x401d, 0, 0, 0 } }, |
| { { 0xf817,0x9128,0xc0f8,0xd48b,0x401e, 0, 0, 0 } }, |
| { { 0x8eae,0x8dad,0x6eb4,0x9aa2,0x401f, 0, 0, 0 } }, |
| { { 0x00e7,0x7595,0xcd06,0x88bb,0x401f, 0, 0, 0 } }, |
| { { 0x4991,0xcfda,0x52f1,0xa2a9,0x401e, 0, 0, 0 } }, |
| { { 0xc39d,0xe415,0xc43d,0x87c0,0x401d, 0, 0, 0 } }, |
| { { 0xa75d,0x436f,0x30dd,0xa027,0x401b, 0, 0, 0 } }, |
| { { 0xc4cb,0x305a,0xbf78,0x8220,0x4019, 0, 0, 0 } }, |
| { { 0x3708,0x33b1,0x07fa,0x8644,0x4016, 0, 0, 0 } }, |
| { { 0x24fa,0x96f6,0x7153,0x8a6c,0x4012, 0, 0, 0 } } |
| }; |
| |
| /* 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 uLD R[] = { |
| { { 0x260a,0xab95,0x2fc7,0xe7c4,0x4000, 0, 0, 0 } }, |
| { { 0x4761,0x613e,0xdf6d,0xe58e,0x4001, 0, 0, 0 } }, |
| { { 0x0615,0x4b00,0x575f,0xdc7b,0x4000, 0, 0, 0 } }, |
| { { 0x521d,0x8527,0x3435,0x8dc2,0x3ffe, 0, 0, 0 } }, |
| { { 0x22cf,0xc711,0x6c5b,0xdcfb,0x3ff9, 0, 0, 0 } } |
| }; |
| static const uLD S[] = { |
| { { 0x5de6,0x17d7,0x54d6,0xaba9,0x4002, 0, 0, 0 } }, |
| { { 0x55d5,0xd300,0xe71e,0xf564,0x4002, 0, 0, 0 } }, |
| { { 0xb611,0x8f76,0xf020,0xd255,0x4001, 0, 0, 0 } }, |
| { { 0x3684,0x3798,0xb793,0x80b0,0x3fff, 0, 0, 0 } }, |
| { { 0xf5af,0x2fb2,0x1e57,0xc3d7,0x3ffa, 0, 0, 0 } } |
| }; |
| |
| /* erf(x) = x T(x^2)/U(x^2) |
| 0 <= x <= 1 |
| Peak relative error 7.6e-23 */ |
| |
| static const uLD T[] = { |
| { { 0xfd7a,0x3a1a,0x705b,0xe0c4,0x3ffb, 0, 0, 0 } }, |
| { { 0x3128,0xc337,0x3716,0xace5,0x4001, 0, 0, 0 } }, |
| { { 0x9517,0x4e93,0x540e,0x8f97,0x4007, 0, 0, 0 } }, |
| { { 0x6118,0x6059,0x9093,0xa757,0x400a, 0, 0, 0 } }, |
| { { 0xb954,0xa987,0xc60c,0xbc83,0x400e, 0, 0, 0 } }, |
| { { 0x7a56,0xe45a,0xa4bd,0x975b,0x4010, 0, 0, 0 } }, |
| { { 0xc446,0x6bab,0x0b2a,0x86d0,0x4013, 0, 0, 0 } } |
| }; |
| |
| static const uLD U[] = { |
| { { 0x3453,0x1f8e,0xf688,0xb507,0x4004, 0, 0, 0 } }, |
| { { 0x71ac,0xb12f,0x21ca,0xf2e2,0x4008, 0, 0, 0 } }, |
| { { 0xffe8,0x9cac,0x3b84,0xc2ac,0x400c, 0, 0, 0 } }, |
| { { 0x481d,0x445b,0xc807,0xc232,0x400f, 0, 0, 0 } }, |
| { { 0x9ad5,0x1aef,0x45b1,0xe25e,0x4011, 0, 0, 0 } }, |
| { { 0x71a7,0x1cad,0x012e,0xeef3,0x4012, 0, 0, 0 } } |
| }; |
| |
| /* 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); |
| } |