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Diffstat (limited to 'winsup/cygwin/math/erfl.c')
| -rw-r--r-- | winsup/cygwin/math/erfl.c | 303 |
1 files changed, 303 insertions, 0 deletions
diff --git a/winsup/cygwin/math/erfl.c b/winsup/cygwin/math/erfl.c new file mode 100644 index 0000000..3832fe9 --- /dev/null +++ b/winsup/cygwin/math/erfl.c @@ -0,0 +1,303 @@ +/** + * 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); +} |