/* 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"
/* 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 );
}