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ndtr.c

/*                                        ndtr.c
 *
 *    Normal distribution function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, ndtr();
 *
 * y = ndtr( 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 with care to avoid error amplification in computing exp(-x^2).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE     -13,0        30000       1.3e-15     2.2e-16
 *
 *
 * ERROR MESSAGES:
 *
 *   message         condition         value returned
 * erfc underflow    x > 37.519379347       0.0
 *
 */
/*                                       erf.c
 *
 *    Error function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, erf();
 *
 * y = erf( 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 * P4(x**2)/Q5(x**2); otherwise
 * erf(x) = 1 - erfc(x).
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    DEC       0,1         14000       4.7e-17     1.5e-17
 *    IEEE      0,1         30000       3.7e-16     1.0e-16
 *
 */
/*                                       erfc.c
 *
 *    Complementary error function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, erfc();
 *
 * y = erfc( 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 expx2.c is used to suppress error amplification
 * in computing exp(-x^2).
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,26.6417   30000       1.3e-15     2.2e-16
 *
 *
 * ERROR MESSAGES:
 *
 *   message         condition              value returned
 * erfc underflow    x > 9.231948545 (DEC)       0.0
 *
 *
 */


/*
Cephes Math Library Release 2.9:  November, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

/* Define this macro to suppress error propagation in exp(x^2)
   by using the expx2 function.  The tradeoff is that doing so
   generates two calls to the exponential function instead of one.  */
#define USE_EXPXSQ 0

static double P[] = {
    2.46196981473530512524E-10,
    5.64189564831068821977E-1,
    7.46321056442269912687E0,
    4.86371970985681366614E1,
    1.96520832956077098242E2,
    5.26445194995477358631E2,
    9.34528527171957607540E2,
    1.02755188689515710272E3,
    5.57535335369399327526E2
};
static double Q[] = {
    /* 1.00000000000000000000E0,*/
    1.32281951154744992508E1,
    8.67072140885989742329E1,
    3.54937778887819891062E2,
    9.75708501743205489753E2,
    1.82390916687909736289E3,
    2.24633760818710981792E3,
    1.65666309194161350182E3,
    5.57535340817727675546E2
};
static double R[] = {
    5.64189583547755073984E-1,
    1.27536670759978104416E0,
    5.01905042251180477414E0,
    6.16021097993053585195E0,
    7.40974269950448939160E0,
    2.97886665372100240670E0
};
static double S[] = {
    /* 1.00000000000000000000E0,*/
    2.26052863220117276590E0,
    9.39603524938001434673E0,
    1.20489539808096656605E1,
    1.70814450747565897222E1,
    9.60896809063285878198E0,
    3.36907645100081516050E0
};
static double T[] = {
    9.60497373987051638749E0,
    9.00260197203842689217E1,
    2.23200534594684319226E3,
    7.00332514112805075473E3,
    5.55923013010394962768E4
};
static double U[] = {
    /* 1.00000000000000000000E0,*/
    3.35617141647503099647E1,
    5.21357949780152679795E2,
    4.59432382970980127987E3,
    2.26290000613890934246E4,
    4.92673942608635921086E4
};

#define UTHRESH 37.519379347

static double cephes_erfc (double a);
static double cephes_erf (double x);
#if USE_EXPXSQ
static double erfce (double a);
#endif

double ndtr (double a)
{
    double x, y, z;

    x = a * SQRTH;
    z = fabs(x);

    /* if (z < SQRTH) */
    if (z < 1.0) {
      y = 0.5 + 0.5 * cephes_erf(x);
    } else {
#if USE_EXPXSQ
      /* See below for erfce */
      y = 0.5 * erfce(z);
      /* Multiply by exp(-x^2 / 2) */
      z = expx2(a, -1);
      y = y * sqrt(z);
#else
      y = 0.5 * cephes_erfc(z);
#endif
      if (x > 0) {
          y = 1.0 - y;
      }
    }

    return y;
}

static double cephes_erfc (double a)
{
    double p, q, x, y, z;

    if (a < 0.0) {
      x = -a;
    } else {
      x = a;
    }

    if (x < 1.0) {
      return 1.0 - cephes_erf(a);
    }

    z = -a * a;

    if (z < -MAXLOG) {
    under:
#if 0
      mtherr("erfc", CEPHES_UNDERFLOW);
#endif
      if (a < 0) {
          return 2.0;
      } else {
          return 0.0;
      }
    }

#ifdef USE_EXPXSQ
    /* Compute z = exp(z).  */
    z = expx2(a, -1);
#else
    z = exp(z);
#endif

    if (x < 8.0) {
      p = polevl(x, P, 8);
      q = p1evl(x, Q, 8);
    } else {
      p = polevl(x, R, 5);
      q = p1evl(x, S, 6);
    }

    y = (z * p)/q;

    if (a < 0) {
      y = 2.0 - y;
    }

    if (y == 0.0) {
      goto under;
    }

    return y;
}

#if USE_EXPXSQ
/* Exponentially scaled erfc function
   exp(x^2) erfc(x)
   valid for x > 1.
   Use with ndtr and expx2.  */

static double erfce (double x)
{
    double p,q;

    if (x < 8.0) {
      p = polevl(x, P, 8);
      q = p1evl(x, Q, 8);
    } else {
      p = polevl(x, R, 5);
      q = p1evl(x, S, 6);
    }

    return p/q;
}
#endif

static double cephes_erf (double x)
{
    double y, z;

    if (fabs(x) > 1.0) {
      return 1.0 - cephes_erfc(x);
    }

    z = x * x;
    y = x * polevl(z, T, 4) / p1evl(z, U, 5);

    return y;
}

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