CocoRoboDesktop/pdf.js/node_modules/cephes/cephes/incbet.c

/*							incbet.c
 *
 *	Incomplete beta integral
 *
 *
 * SYNOPSIS:
 *
 * double a, b, x, y, incbet();
 *
 * y = incbet( a, b, x );
 *
 *
 * DESCRIPTION:
 *
 * Returns incomplete beta integral of the arguments, evaluated
 * from zero to x.  The function is defined as
 *
 *                  x
 *     -            -
 *    | (a+b)      | |  a-1     b-1
 *  -----------    |   t   (1-t)   dt.
 *   -     -     | |
 *  | (a) | (b)   -
 *                 0
 *
 * The domain of definition is 0 <= x <= 1.  In this
 * implementation a and b are restricted to positive values.
 * The integral from x to 1 may be obtained by the symmetry
 * relation
 *
 *    1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).
 *
 * The integral is evaluated by a continued fraction expansion
 * or, when b*x is small, by a power series.
 *
 * ACCURACY:
 *
 * Tested at uniformly distributed random points (a,b,x) with a and b
 * in "domain" and x between 0 and 1.
 *                                        Relative error
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,5         10000       6.9e-15     4.5e-16
 *    IEEE      0,85       250000       2.2e-13     1.7e-14
 *    IEEE      0,1000      30000       5.3e-12     6.3e-13
 *    IEEE      0,10000    250000       9.3e-11     7.1e-12
 *    IEEE      0,100000    10000       8.7e-10     4.8e-11
 * Outputs smaller than the IEEE gradual underflow threshold
 * were excluded from these statistics.
 *
 * ERROR MESSAGES:
 *   message         condition      value returned
 * incbet domain      x<0, x>1          0.0
 * incbet underflow                     0.0
 */

/*
Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

#ifdef DEC
#define MAXGAM 34.84425627277176174
#else
#define MAXGAM 171.624376956302725
#endif

extern double MACHEP, MINLOG, MAXLOG;
#ifdef ANSIPROT
extern double gamma(double);
extern double lgam(double);
extern double exp(double);
extern double log(double);
extern double pow(double, double);
extern double fabs(double);
static double incbcf(double, double, double);
static double incbd(double, double, double);
static double pseries(double, double, double);
#else
double gamma(), lgam(), exp(), log(), pow(), fabs();
static double incbcf(), incbd(), pseries();
#endif

static double big = 4.503599627370496e15;
static double biginv = 2.22044604925031308085e-16;

double incbet(aa, bb, xx) double aa, bb, xx;
{
  double a, b, t, x, xc, w, y;
  int flag;

  if (aa <= 0.0 || bb <= 0.0)
    goto domerr;

  if ((xx <= 0.0) || (xx >= 1.0)) {
    if (xx == 0.0)
      return (0.0);
    if (xx == 1.0)
      return (1.0);
  domerr:
    mtherr("incbet", DOMAIN);
    return (0.0);
  }

  flag = 0;
  if ((bb * xx) <= 1.0 && xx <= 0.95) {
    t = pseries(aa, bb, xx);
    goto done;
  }

  w = 1.0 - xx;

  /* Reverse a and b if x is greater than the mean. */
  if (xx > (aa / (aa + bb))) {
    flag = 1;
    a = bb;
    b = aa;
    xc = xx;
    x = w;
  } else {
    a = aa;
    b = bb;
    xc = w;
    x = xx;
  }

  if (flag == 1 && (b * x) <= 1.0 && x <= 0.95) {
    t = pseries(a, b, x);
    goto done;
  }

  /* Choose expansion for better convergence. */
  y = x * (a + b - 2.0) - (a - 1.0);
  if (y < 0.0)
    w = incbcf(a, b, x);
  else
    w = incbd(a, b, x) / xc;

  /* Multiply w by the factor
       a      b   _             _     _
      x  (1-x)   | (a+b) / ( a | (a) | (b) ) .   */

  y = a * log(x);
  t = b * log(xc);
  if ((a + b) < MAXGAM && fabs(y) < MAXLOG && fabs(t) < MAXLOG) {
    t = pow(xc, b);
    t *= pow(x, a);
    t /= a;
    t *= w;
    t *= gamma(a + b) / (gamma(a) * gamma(b));
    goto done;
  }
  /* Resort to logarithms.  */
  y += t + lgam(a + b) - lgam(a) - lgam(b);
  y += log(w / a);
  if (y < MINLOG)
    t = 0.0;
  else
    t = exp(y);

done:

  if (flag == 1) {
    if (t <= MACHEP)
      t = 1.0 - MACHEP;
    else
      t = 1.0 - t;
  }
  return (t);
}

/* Continued fraction expansion #1
 * for incomplete beta integral
 */

static double incbcf(a, b, x) double a, b, x;
{
  double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
  double k1, k2, k3, k4, k5, k6, k7, k8;
  double r, t, ans, thresh;
  int n;

  k1 = a;
  k2 = a + b;
  k3 = a;
  k4 = a + 1.0;
  k5 = 1.0;
  k6 = b - 1.0;
  k7 = k4;
  k8 = a + 2.0;

  pkm2 = 0.0;
  qkm2 = 1.0;
  pkm1 = 1.0;
  qkm1 = 1.0;
  ans = 1.0;
  r = 1.0;
  n = 0;
  thresh = 3.0 * MACHEP;
  do {

    xk = -(x * k1 * k2) / (k3 * k4);
    pk = pkm1 + pkm2 * xk;
    qk = qkm1 + qkm2 * xk;
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;

    xk = (x * k5 * k6) / (k7 * k8);
    pk = pkm1 + pkm2 * xk;
    qk = qkm1 + qkm2 * xk;
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;

    if (qk != 0)
      r = pk / qk;
    if (r != 0) {
      t = fabs((ans - r) / r);
      ans = r;
    } else
      t = 1.0;

    if (t < thresh)
      goto cdone;

    k1 += 1.0;
    k2 += 1.0;
    k3 += 2.0;
    k4 += 2.0;
    k5 += 1.0;
    k6 -= 1.0;
    k7 += 2.0;
    k8 += 2.0;

    if ((fabs(qk) + fabs(pk)) > big) {
      pkm2 *= biginv;
      pkm1 *= biginv;
      qkm2 *= biginv;
      qkm1 *= biginv;
    }
    if ((fabs(qk) < biginv) || (fabs(pk) < biginv)) {
      pkm2 *= big;
      pkm1 *= big;
      qkm2 *= big;
      qkm1 *= big;
    }
  } while (++n < 300);

cdone:
  return (ans);
}

/* Continued fraction expansion #2
 * for incomplete beta integral
 */

static double incbd(a, b, x) double a, b, x;
{
  double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
  double k1, k2, k3, k4, k5, k6, k7, k8;
  double r, t, ans, z, thresh;
  int n;

  k1 = a;
  k2 = b - 1.0;
  k3 = a;
  k4 = a + 1.0;
  k5 = 1.0;
  k6 = a + b;
  k7 = a + 1.0;
  ;
  k8 = a + 2.0;

  pkm2 = 0.0;
  qkm2 = 1.0;
  pkm1 = 1.0;
  qkm1 = 1.0;
  z = x / (1.0 - x);
  ans = 1.0;
  r = 1.0;
  n = 0;
  thresh = 3.0 * MACHEP;
  do {

    xk = -(z * k1 * k2) / (k3 * k4);
    pk = pkm1 + pkm2 * xk;
    qk = qkm1 + qkm2 * xk;
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;

    xk = (z * k5 * k6) / (k7 * k8);
    pk = pkm1 + pkm2 * xk;
    qk = qkm1 + qkm2 * xk;
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;

    if (qk != 0)
      r = pk / qk;
    if (r != 0) {
      t = fabs((ans - r) / r);
      ans = r;
    } else
      t = 1.0;

    if (t < thresh)
      goto cdone;

    k1 += 1.0;
    k2 -= 1.0;
    k3 += 2.0;
    k4 += 2.0;
    k5 += 1.0;
    k6 += 1.0;
    k7 += 2.0;
    k8 += 2.0;

    if ((fabs(qk) + fabs(pk)) > big) {
      pkm2 *= biginv;
      pkm1 *= biginv;
      qkm2 *= biginv;
      qkm1 *= biginv;
    }
    if ((fabs(qk) < biginv) || (fabs(pk) < biginv)) {
      pkm2 *= big;
      pkm1 *= big;
      qkm2 *= big;
      qkm1 *= big;
    }
  } while (++n < 300);
cdone:
  return (ans);
}

/* Power series for incomplete beta integral.
   Use when b*x is small and x not too close to 1.  */

static double pseries(a, b, x) double a, b, x;
{
  double s, t, u, v, n, t1, z, ai;

  ai = 1.0 / a;
  u = (1.0 - b) * x;
  v = u / (a + 1.0);
  t1 = v;
  t = u;
  n = 2.0;
  s = 0.0;
  z = MACHEP * ai;
  while (fabs(v) > z) {
    u = (n - b) * x / n;
    t *= u;
    v = t / (a + n);
    s += v;
    n += 1.0;
  }
  s += t1;
  s += ai;

  u = a * log(x);
  if ((a + b) < MAXGAM && fabs(u) < MAXLOG) {
    t = gamma(a + b) / (gamma(a) * gamma(b));
    s = s * t * pow(x, a);
  } else {
    t = lgam(a + b) - lgam(a) - lgam(b) + u + log(s);
    if (t < MINLOG)
      s = 0.0;
    else
      s = exp(t);
  }
  return (s);
}