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/* ndtrf.c
*
* Normal distribution function
*
*
*
* SYNOPSIS:
*
* float x, y, ndtrf();
*
* y = ndtrf( 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 50000 1.5e-5 2.6e-6
*
*
* ERROR MESSAGES:
*
* See erfcf().
*
*/
/* erff.c
*
* Error function
*
*
*
* SYNOPSIS:
*
* float x, y, erff();
*
* y = erff( x );
*
*
*
* DESCRIPTION:
*
* The integral is
*
* x
* -
* 2 | | 2
* erf(x) = -------- | exp( - t ) dt.
* sqrt(pi) | |
* -
* 0
*
* The magnitude of x is limited to 9.231948545 for DEC
* arithmetic; 1 or -1 is returned outside this range.
*
* For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise
* erf(x) = 1 - erfc(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -9.3,9.3 50000 1.7e-7 2.8e-8
*
*/
/* erfcf.c
*
* Complementary error function
*
*
*
* SYNOPSIS:
*
* float x, y, erfcf();
*
* y = erfcf( 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 polynomial
* approximations 1/x P(1/x**2) are computed.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -9.3,9.3 50000 3.9e-6 7.2e-7
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfcf underflow x**2 > MAXLOGF 0.0
*
*
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1984, 1987, 1988 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
extern float MAXLOGF, SQRTHF;
/* erfc(x) = exp(-x^2) P(1/x), 1 < x < 2 */
static float P[] = {
2.326819970068386E-002,
-1.387039388740657E-001,
3.687424674597105E-001,
-5.824733027278666E-001,
6.210004621745983E-001,
-4.944515323274145E-001,
3.404879937665872E-001,
-2.741127028184656E-001,
5.638259427386472E-001
};
/* erfc(x) = exp(-x^2) 1/x P(1/x^2), 2 < x < 14 */
static float R[] = {
-1.047766399936249E+001,
1.297719955372516E+001,
-7.495518717768503E+000,
2.921019019210786E+000,
-1.015265279202700E+000,
4.218463358204948E-001,
-2.820767439740514E-001,
5.641895067754075E-001
};
/* erf(x) = x P(x^2), 0 < x < 1 */
static float T[] = {
7.853861353153693E-005,
-8.010193625184903E-004,
5.188327685732524E-003,
-2.685381193529856E-002,
1.128358514861418E-001,
-3.761262582423300E-001,
1.128379165726710E+000
};
/*#define UTHRESH 37.519379347*/
#define UTHRESH 14.0
#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
#ifdef ANSIC
float polevlf(float, float *, int);
float expf(float), logf(float), erff(float), erfcf(float);
#else
float polevlf(), expf(), logf(), erff(), erfcf();
#endif
float ndtrf(float aa)
{
float x, y, z;
x = aa;
x *= SQRTHF;
z = fabsf(x);
if( z < SQRTHF )
y = 0.5 + 0.5 * erff(x);
else
{
y = 0.5 * erfcf(z);
if( x > 0 )
y = 1.0 - y;
}
return(y);
}
float erfcf(float aa)
{
float a, p,q,x,y,z;
a = aa;
x = fabsf(a);
if( x < 1.0 )
return( 1.0 - erff(a) );
z = -a * a;
if( z < -MAXLOGF )
{
under:
mtherr( "erfcf", UNDERFLOW );
if( a < 0 )
return( 2.0 );
else
return( 0.0 );
}
z = expf(z);
q = 1.0/x;
y = q * q;
if( x < 2.0 )
{
p = polevlf( y, P, 8 );
}
else
{
p = polevlf( y, R, 7 );
}
y = z * q * p;
if( a < 0 )
y = 2.0 - y;
if( y == 0.0 )
goto under;
return(y);
}
float erff(float xx)
{
float x, y, z;
x = xx;
if( fabsf(x) > 1.0 )
return( 1.0 - erfcf(x) );
z = x * x;
y = x * polevlf( z, T, 6 );
return( y );
}
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