src/nmath/qbeta.c

/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 1998--2018  The R Core Team
 *  Copyright (C) 1995, 1996  Robert Gentleman and Ross Ihaka
 *  based on code (C) 1979 and later Royal Statistical Society
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, a copy is available at
 *  https://www.R-project.org/Licenses/
 *

 * Reference:
 * Cran, G. W., K. J. Martin and G. E. Thomas (1977).
 *	Remark AS R19 and Algorithm AS 109,
 *	Applied Statistics, 26(1), 111-114.
 * Remark AS R83 (v.39, 309-310) and the correction (v.40(1) p.236)
 *	have been incorporated in this version.
 */

#include "nmath.h"
#include "dpq.h"

#ifdef DEBUG_qbeta
# define R_ifDEBUG_printf(...) REprintf(__VA_ARGS__)
#else
# define R_ifDEBUG_printf(...)
#endif

#define USE_LOG_X_CUTOFF -5.
//                       --- based on some testing; had = -10

#define n_NEWTON_FREE 4
//                   --- based on some testing; had = 10

#define MLOGICAL_NA -1
// an "NA_LOGICAL" substitute for Mathlib {only used here, for now}

//attribute_hidden
static void
qbeta_raw(double alpha, double p, double q, int lower_tail, int log_p,
	  int swap_01, double log_q_cut, int n_N, double* qb);

double qbeta(double alpha, double p, double q, int lower_tail, int log_p)
{

    /* test for admissibility of parameters */
#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(q) || ISNAN(alpha))
	return p + q + alpha;
#endif
    if(p < 0. || q < 0.) ML_ERR_return_NAN;
    // allowing p==0 and q==0  <==> treat as one- or two-point mass

    double qbet[2];// = { qbeta(), 1 - qbeta() }
    qbeta_raw(alpha, p, q, lower_tail, log_p,
	      MLOGICAL_NA, USE_LOG_X_CUTOFF, n_NEWTON_FREE, qbet);
    return qbet[0];
}

static const double
#ifdef IEEE_754
// CARE: assumes subnormal numbers, i.e., no underflow at DBL_MIN:
    DBL_very_MIN  = DBL_MIN / 4.,
    DBL_log_v_MIN = M_LN2*(DBL_MIN_EXP - 2),
// Too extreme: inaccuracy in pbeta(); e.g for  qbeta(0.95, 1e-9, 20):
// -> in pbeta() --> bgrat(..... b*z == 0 underflow, hence inaccurate pbeta()
    /* DBL_very_MIN  = 0x0.0000001p-1022, // = 2^-1050 = 2^(-1022 - 28) */
    /* DBL_log_v_MIN = -1050. * M_LN2, // = log(DBL_very_MIN) */
// the most extreme -- not ok, as pbeta() then behaves strangely,
// e.g., for  qbeta(0.95, 1e-8, 20):
    /* DBL_very_MIN  = 0x0.0000000000001p-1022, // = 2^-1074 = 2^(-1022 -52) */
    /* DBL_log_v_MIN = -1074. * M_LN2, // = log(DBL_very_MIN) */

    DBL_1__eps    = 0x1.fffffffffffffp-1; // = 1 - 2^-53
#else // untested :
    DBL_1__eps    = 1 - DBL_EPSILON;     // or rather (1 - DBL_EPSILON/2) (??)
#endif

/* set the exponent of acu to -2r-2 for r digits of accuracy */
/*---- NEW ---- -- still fails for p = 1e11, q=.5*/

#define fpu 3e-308
/* acu_min:  Minimal value for accuracy 'acu' which will depend on (a,p);
	     acu_min >= fpu ! */
#define acu_min 1e-300
#define p_lo fpu
#define p_hi 1-2.22e-16

#define const1 2.30753
#define const2 0.27061
#define const3 0.99229
#define const4 0.04481

// Returns both qbeta() and its "mirror" 1-qbeta(). Useful notably when qbeta() ~= 1
attribute_hidden void
qbeta_raw(double alpha, double p, double q, int lower_tail, int log_p,
	  int swap_01, // {TRUE, NA, FALSE}: if NA, algorithm decides swap_tail
	  double log_q_cut, /* if == Inf: return log(qbeta(..));
			       otherwise, if finite: the bound for
			       switching to log(x)-scale; see use_log_x */
	  int n_N,  // number of "unconstrained" Newton steps before switching to constrained
	  double *qb) // = qb[0:1] = { qbeta(), 1 - qbeta() }
{
    Rboolean
	swap_choose = (swap_01 == MLOGICAL_NA),
	swap_tail,
	log_, give_log_q = (log_q_cut == ML_POSINF),
	use_log_x = give_log_q, // or u < log_q_cut  below
	warned = FALSE, add_N_step = TRUE;
    int i_pb, i_inn;
    double a, la, logbeta, g, h, pp, p_, qq, r, s, t, w, y = -1.;
    volatile double u, xinbta;

    // Assuming p >= 0, q >= 0  here ...

    // Deal with boundary cases here:
    if(alpha == R_DT_0) {
#define return_q_0						\
	if(give_log_q) { qb[0] = ML_NEGINF; qb[1] = 0; }	\
	else {           qb[0] = 0;         qb[1] = 1; }	\
	return

	return_q_0;
    }
    if(alpha == R_DT_1) {
#define return_q_1						\
	if(give_log_q) { qb[0] = 0; qb[1] = ML_NEGINF; }	\
	else {           qb[0] = 1; qb[1] = 0;         }	\
	return

	return_q_1;
    }

    // check alpha {*before* transformation which may all accuracy}:
    if((log_p && alpha > 0) ||
       (!log_p && (alpha < 0 || alpha > 1))) { // alpha is outside
	R_ifDEBUG_printf("qbeta(alpha=%g, %g, %g, .., log_p=%d): %s%s\n",
			 alpha, p,q, log_p, "alpha not in ",
			 log_p ? "[-Inf, 0]" : "[0,1]");
	// ML_ERR_return_NAN :
	ML_ERROR(ME_DOMAIN, "");
	qb[0] = qb[1] = ML_NAN; return;
    }

    //  p==0, q==0, p = Inf, q = Inf  <==> treat as one- or two-point mass
    if(p == 0 || q == 0 || !R_FINITE(p) || !R_FINITE(q)) {
	// We know 0 < T(alpha) < 1 : pbeta() is constant and trivial in {0, 1/2, 1}
	R_ifDEBUG_printf(
	    "qbeta(%g, %g, %g, lower_t=%d, log_p=%d): (p,q)-boundary: trivial\n",
	    alpha, p,q, lower_tail, log_p);
	if(p == 0 && q == 0) { // point mass 1/2 at each of {0,1} :
	    if(alpha < R_D_half) { return_q_0; }
	    if(alpha > R_D_half) { return_q_1; }
	    // else:  alpha == "1/2"
#define return_q_half					\
	    if(give_log_q) qb[0] = qb[1] = -M_LN2;	\
	    else	   qb[0] = qb[1] = 0.5;		\
	    return

	    return_q_half;
	} else if (p == 0 || p/q == 0) { // point mass 1 at 0 - "flipped around"
	    return_q_0;
	} else if (q == 0 || q/p == 0) { // point mass 1 at 0 - "flipped around"
	    return_q_1;
	}
	// else:  p = q = Inf : point mass 1 at 1/2
	return_q_half;
    }

    /* initialize */
    p_ = R_DT_qIv(alpha);/* lower_tail prob (in any case) */
    // Conceptually,  0 < p_ < 1  (but can be 0 or 1 because of cancellation!)
    logbeta = lbeta(p, q);

    swap_tail = (swap_choose) ? (p_ > 0.5) : swap_01;
    // change tail; default (swap_01 = NA): afterwards 0 < a <= 1/2
    if(swap_tail) { /* change tail, swap  p <-> q :*/
	a = R_DT_CIv(alpha); // = 1 - p_ < 1/2
	/* la := log(a), but without numerical cancellation: */
	la = R_DT_Clog(alpha);
	pp = q; qq = p;
    }
    else {
	a = p_;
	la = R_DT_log(alpha);
	pp = p; qq = q;
    }

    /* calculate the initial approximation */

    /* Desired accuracy for Newton iterations (below) should depend on  (a,p)
     * This is from Remark .. on AS 109, adapted.
     * However, it's not clear if this is "optimal" for IEEE double prec.

     * acu = fmax2(acu_min, pow(10., -25. - 5./(pp * pp) - 1./(a * a)));

     * NEW: 'acu' accuracy NOT for squared adjustment, but simple;
     * ---- i.e.,  "new acu" = sqrt(old acu)
     */
    double acu = fmax2(acu_min, pow(10., -13. - 2.5/(pp * pp) - 0.5/(a * a)));
    // try to catch  "extreme left tail" early
    double tx, u0 = (la + log(pp) + logbeta) / pp; // = log(x_0)
    static const double
	log_eps_c = M_LN2 * (1. - DBL_MANT_DIG);// = log(DBL_EPSILON) = -36.04..
    r = pp*(1.-qq)/(pp+1.);

    t = 0.2;
    // FIXME: Factor 0.2 is a bit arbitrary;  '1' is clearly much too much.

    R_ifDEBUG_printf(
	"qbeta(%g, %g, %g, lower_t=%d, log_p=%d):%s\n"
	"  swap_tail=%d, la=%#8g, u0=%#8g (bnd: %g (%g)) ",
	alpha, p,q, lower_tail, log_p,
	(log_p && (p_ == 0. || p_ == 1.)) ? (p_==0.?" p_=0":" p_=1") : "",
	swap_tail, la, u0,
	(t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.,
	 t*log_eps_c - log(fabs(r))
	);

    if(M_LN2 * DBL_MIN_EXP < u0 && // cannot allow exp(u0) = 0 ==> exp(u1) = exp(u0) = 0
       u0 < -0.01 && // (must: u0 < 0, but too close to 0 <==> x = exp(u0) = 0.99..)
       // qq <= 2 && // <--- "arbitrary"
       // u0 <  t*log_eps_c - log(fabs(r)) &&
       u0 < (t*log_eps_c - log(fabs(pp*(1.-qq)*(2.-qq)/(2.*(pp+2.)))))/2.)
    {
// TODO: maybe jump here from below, when initial u "fails" ?
// L_tail_u:
	// MM's one-step correction (cheaper than 1 Newton!)
	r = r*exp(u0);// = r*x0
	if(r > -1.) {
	    u = u0 - log1p(r)/pp;
	    R_ifDEBUG_printf("u1-u0=%9.3g --> choosing u = u1\n", u-u0);
	} else {
	    u = u0;
	    R_ifDEBUG_printf("cannot cheaply improve u0\n");
	}
	tx = xinbta = exp(u);
	use_log_x = TRUE; // or (u < log_q_cut)  ??
	goto L_Newton;
    }

    // y := y_\alpha in AS 64 := Hastings(1955) approximation of qnorm(1 - a) :
    r = sqrt(-2 * la);
    y = r - (const1 + const2 * r) / (1. + (const3 + const4 * r) * r);

    if (pp > 1 && qq > 1) { // use  Carter(1947), see AS 109, remark '5.'
	r = (y * y - 3.) / 6.;
	s = 1. / (pp + pp - 1.);
	t = 1. / (qq + qq - 1.);
	h = 2. / (s + t);
	w = y * sqrt(h + r) / h - (t - s) * (r + 5. / 6. - 2. / (3. * h));
	R_ifDEBUG_printf("p,q > 1 => w=%g", w);
	if(w > 300) { // exp(w+w) is huge or overflows
	    t = w+w + log(qq) - log(pp); // = argument of log1pexp(.)
	    u = // log(xinbta) = - log1p(qq/pp * exp(w+w)) = -log(1 + exp(t))
		(t <= 18) ? -log1p(exp(t)) : -t - exp(-t);
	    xinbta = exp(u);
	} else {
	    xinbta = pp / (pp + qq * exp(w + w));
	    u = // log(xinbta)
		- log1p(qq/pp * exp(w+w));
	}
    } else { // use the original AS 64 proposal, Scheffé-Tukey (1944) and Wilson-Hilferty
	r = qq + qq;
	/* A slightly more stable version of  t := \chi^2_{alpha} of AS 64
	 * t = 1. / (9. * qq); t = r * R_pow_di(1. - t + y * sqrt(t), 3);  */
	t = 1. / (3. * sqrt(qq));
	t = r * R_pow_di(1. + t*(-t + y), 3);// = \chi^2_{alpha} of AS 64
	s = 4. * pp + r - 2.;// 4p + 2q - 2 = numerator of new t = (...) / chi^2
	R_ifDEBUG_printf("min(p,q) <= 1: t=%g", t);
	if (t == 0 || (t < 0. && s >= t)) { // cannot use chisq approx
	    // x0 = 1 - { (1-a)*q*B(p,q) } ^{1/q}    {AS 65}
	    // xinbta = 1. - exp((log(1-a)+ log(qq) + logbeta) / qq);
	    double l1ma;/* := log(1-a), directly from alpha (as 'la' above):
			 * FIXME: not worth it? log1p(-a) always the same ?? */
	    if(swap_tail)
		l1ma = R_DT_log(alpha);
	    else
		l1ma = R_DT_Clog(alpha);
	    R_ifDEBUG_printf(" t <= 0 : log1p(-a)=%.15g, better l1ma=%.15g\n", log1p(-a), l1ma);
	    double xx = (l1ma + log(qq) + logbeta) / qq;
	    if(xx <= 0.) {
		xinbta = -expm1(xx);
		u = R_Log1_Exp (xx);// =  log(xinbta) = log(1 - exp(...A...))
	    } else { // xx > 0 ==> 1 - e^xx < 0 .. is nonsense
		R_ifDEBUG_printf(" xx=%g > 0: xinbta:= 1-e^xx < 0\n", xx);
		xinbta = 0; u = ML_NEGINF; /// FIXME can do better?
	    }
	} else {
	    t = s / t;
	    R_ifDEBUG_printf(" t > 0 or s < t < 0:  new t = %g ( > 1 ?)\n", t);
	    if (t <= 1.) { // cannot use chisq, either
		u = (la + log(pp) + logbeta) / pp;
		xinbta = exp(u);
	    } else { // (1+x0)/(1-x0) = t,  solved for x0 :
		xinbta = 1. - 2. / (t + 1.);
		u = log1p(-2. / (t + 1.));
	    }
	}
    }

    // Problem: If initial u is completely wrong, we make a wrong decision here
    if(swap_choose &&
       (( swap_tail && u >= -exp(  log_q_cut)) || // ==> "swap back"
	(!swap_tail && u >= -exp(4*log_q_cut) && pp / qq < 1000.) // ==> "swap now"
	   )) {
	// "revert swap" -- and use_log_x
	swap_tail = !swap_tail;
	R_ifDEBUG_printf(" u = %g (e^u = xinbta = %.16g) ==> ", u, xinbta);
	if(swap_tail) { // "swap now" (much less easily)
	    a = R_DT_CIv(alpha); // needed ?
	    la = R_DT_Clog(alpha);
	    pp = q; qq = p;
	}
	else { // swap back :
	    a = p_;
	    la = R_DT_log(alpha);
	    pp = p; qq = q;
	}
	R_ifDEBUG_printf("\"%s\"; la = %g\n",
			 (swap_tail ? "swap now" : "swap back"), la);
	// we could redo computations above, but this should be stable
	u = R_Log1_Exp(u);
	xinbta = exp(u);

/* Careful: "swap now"  should not fail if
   1) the above initial xinbta is "completely wrong"
   2) The correction step can go outside (u_n > 0 ==>  e^u > 1 is illegal)
   e.g., for  qbeta(0.2066, 0.143891, 0.05)
*/
    } else R_ifDEBUG_printf("\n");

    if(!use_log_x)
	use_log_x = (u < log_q_cut);// <==> xinbta = e^u < exp(log_q_cut)
    Rboolean
	bad_u = !R_FINITE(u),
	bad_init = bad_u || xinbta > p_hi;

    R_ifDEBUG_printf(" -> u = %g, e^u = xinbta = %.16g, (Newton acu=%g%s%s%s)\n",
		     u, xinbta, acu, (bad_u ? ", ** bad u **" : ""),
		     ((bad_init && !bad_u) ? ", ** bad_init **" : ""),
		     (use_log_x ? ", on u = LOG(x) SCALE" : ""));

    double u_n = 1.; // -Wall
    tx = xinbta; // keeping "original initial x" (for now)

    if(bad_u || u < log_q_cut) {
	/* e.g.
	   qbeta(0.21, .001, 0.05)
	   try "left border" quickly, i.e.,
	   try at smallest positive number: */
	w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_p);
	if(w > (log_p ? la : a)) {
	    R_ifDEBUG_printf(
		" quantile is left of %g; \"convergence\"\n", DBL_very_MIN);
	    if(log_p || fabs(w - a) < fabs(0 - a)) { // DBL_very_MIN is better than 0
		tx   = DBL_very_MIN;
		u_n  = DBL_log_v_MIN;// = log(DBL_very_MIN)
	    } else {
		tx   = 0.;
		u_n  = ML_NEGINF;
	    }
	    use_log_x = log_p; add_N_step = FALSE; goto L_return;
	}
	else {
	    R_ifDEBUG_printf(" pbeta(%g, *) = %g <= %g (= %s) --> continuing\n",
			     DBL_log_v_MIN, w, (log_p ? la : a), (log_p ? "la" : "a"));
	    if(u  < DBL_log_v_MIN) {
		u = DBL_log_v_MIN;// = log(DBL_very_MIN)
		xinbta = DBL_very_MIN;
	    }
	}
    }


    /* Sometimes the approximation is negative (and == 0 is also not "ok") */
    if (bad_init && !(use_log_x && tx > 0)) {
	if(u == ML_NEGINF) {
	    R_ifDEBUG_printf("  u = -Inf;");
	    u = M_LN2 * DBL_MIN_EXP;
	    xinbta = DBL_MIN;
	} else {
	    R_ifDEBUG_printf(" bad_init: u=%g, xinbta=%g;", u,xinbta);
	    xinbta = (xinbta > 1.1) // i.e. "way off"
		? 0.5 // otherwise, keep the respective boundary:
		: ((xinbta < p_lo) ? exp(u) : p_hi);
	    if(bad_u)
		u = log(xinbta);
	    // otherwise: not changing "potentially better" u than the above
	}
	R_ifDEBUG_printf(" -> (partly)new u=%g, xinbta=%g\n", u,xinbta);
    }

L_Newton:
    /* --------------------------------------------------------------------

     * Solve for x by a modified Newton-Raphson method, using pbeta_raw()
     */
    r = 1 - pp;
    t = 1 - qq;
    double wprev = 0., prev = 1., adj = 1.; // -Wall

    if(use_log_x) { // find  log(xinbta) -- work in  u := log(x) scale
	// if(bad_init && tx > 0) xinbta = tx;// may have been better

	for (i_pb=0; i_pb < 1000; i_pb++) {
	    // using log_p == TRUE  unconditionally here
	    /* FIXME: if exp(u) = xinbta underflows to 0,
	     *  want different formula pbeta_log(u, ..) */
	    y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, TRUE);

	    /* w := Newton step size for   L(u) = log F(e^u)  =!= 0;   u := log(x)
	     *   =  (L(.) - la) / L'(.);  L'(u)= (F'(e^u) * e^u ) / F(e^u)
	     *   =  (L(.) - la)*F(.) / {F'(e^u) * e^u } =
	     *   =  (L(.) - la) * e^L(.) * e^{-log F'(e^u) - u}
	     *   =  ( y   - la) * e^{ y - u -log F'(e^u)}
	     and  -log F'(x)= -log f(x) = - -logbeta + (1-p) log(x) + (1-q) log(1-x)
	                                = logbeta + (1-p) u + (1-q) log(1-e^u)
	    */
	    w = (y == ML_NEGINF) // y = -Inf  well possible: we are on log scale!
		? 0. : (y - la) * exp(y - u + logbeta + r * u + t * R_Log1_Exp(u));
	    if(!R_FINITE(w))
		break;
	    if (i_pb >= n_N && w * wprev <= 0.)
		prev = fmax2(fabs(adj),fpu);
	    R_ifDEBUG_printf(
		"N(i=%2d): u=%#20.16g, pb(e^u)=%#15.9g, w=%#15.9g, %s prev=%g,",
		i_pb, u, y, w,
		(i_pb >= n_N && w * wprev <= 0.) ? "new" : "old", prev);
	    g = 1;
	    for (i_inn=0; i_inn < 1000; i_inn++) {
		adj = g * w;
		// safe guard (here, from the very beginning)
		if (fabs(adj) < prev) {
		    u_n = u - adj; // u_{n+1} = u_n - g*w
		    if (u_n <= 0.) { // <==> 0 <  xinbta := e^u  <= 1
			if (prev <= acu || fabs(w) <= acu) {
		 	    R_ifDEBUG_printf(
				" it{in}=%d, -adj=%g, %s <= acu  ==> convergence\n",
				i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
			    goto L_converged;
			}
			// if (u_n != ML_NEGINF && u_n != 1)
			break;
		    }
		}
		g /= 3;
	    }
	    // (cancellation in (u_n -u) => may differ from adj:
	    double D = fmin2(fabs(adj), fabs(u_n - u));
	    /* R_ifDEBUG_printf(" delta(u)=%g\n", u_n - u); */
	    R_ifDEBUG_printf(" it{in}=%d, delta(u)=%9.3g, D/|.|=%.3g\n",
			     i_inn, u_n - u, D/fabs(u_n + u));
	    if (D <= 4e-16 * fabs(u_n + u))
		goto L_converged;
	    u = u_n;
	    xinbta = exp(u);
	    wprev = w;
	} // for(i )

    } else { // "normal scale" Newton

	for (i_pb=0; i_pb < 1000; i_pb++) {
	    y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
	    // delta{y} :   d_y = y - (log_p ? la : a);
#ifdef IEEE_754
	    if(!R_FINITE(y) && !(log_p && y == ML_NEGINF))// y = -Inf  is ok if(log_p)
#else
	    if (errno)
#endif
		{ // ML_ERR_return_NAN :
		    ML_ERROR(ME_DOMAIN, "");
		    qb[0] = qb[1] = ML_NAN; return;
		}


	    /* w := Newton step size  (F(.) - a) / F'(.)  or,
	     * --   log: (lF - la) / (F' / F) = exp(lF) * (lF - la) / F'
	     */
	    w = log_p
		? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
		: (y - a)  * exp(    logbeta + r * log(xinbta) + t * log1p(-xinbta));
	    if (i_pb >= n_N && w * wprev <= 0.)
		prev = fmax2(fabs(adj),fpu);
	    R_ifDEBUG_printf(
		"N(i=%2d): x0=%#17.15g, pb(x0)=%#15.9g, w=%#15.9g, %s prev=%g,",
		i_pb, xinbta, y, w,
		(i_pb >= n_N && w * wprev <= 0.) ? "new" : "old", prev);
	    g = 1;
	    for (i_inn=0; i_inn < 1000;i_inn++) {
		adj = g * w;
		// take full Newton steps at the beginning; only then safe guard:
		if (i_pb < n_N || fabs(adj) < prev) {
		    tx = xinbta - adj; // x_{n+1} = x_n - g*w
		    if (0. <= tx && tx <= 1.) {
			if (prev <= acu || fabs(w) <= acu) {
			    R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g, %s <= acu  ==> convergence\n",
					     i_inn, -adj, (prev <= acu) ? "prev" : "|w|");
			    goto L_converged;
			}
			if (tx != 0. && tx != 1)
			    break;
		    }
		}
		g /= 3;
	    }
	    R_ifDEBUG_printf(" it{in}=%d, delta(x)=%g\n", i_inn, tx - xinbta);
	    if (fabs(tx - xinbta) <= 4e-16 * (tx + xinbta)) // "<=" : (.) == 0
		goto L_converged;
	    xinbta = tx;
	    if(tx == 0) // "we have lost"
		break;
	    wprev = w;
	} // for( i_pb ..)

    } // end{else : normal scale Newton}

    /*-- NOT converged: Iteration count --*/
    warned = TRUE;
    ML_ERROR(ME_PRECISION, "qbeta");

L_converged:
    log_ = log_p || use_log_x; // only for printing
    R_ifDEBUG_printf(" %s: Final delta(y) = %g%s\n",
		     warned ? "_NO_ convergence" : "converged",
		     y - (log_ ? la : a), (log_ ? " (log_)" : ""));
    if((log_ && y == ML_NEGINF) || (!log_ && y == 0)) {
	// stuck at left, try if smallest positive number is "better"
	w = pbeta_raw(DBL_very_MIN, pp, qq, TRUE, log_);
	if(log_ || fabs(w - a) <= fabs(y - a)) {
	    tx  = DBL_very_MIN;
	    u_n = DBL_log_v_MIN;// = log(DBL_very_MIN)
	}
	add_N_step = FALSE; // not trying to do better anymore
    }
    else if(!warned && (log_ ? fabs(y - la) > 3 : fabs(y - a) > 1e-4)) {
	if(!(log_ && y == ML_NEGINF &&
	     // e.g. qbeta(-1e-10, .2, .03, log=TRUE) cannot get accurate ==> do NOT warn
	     pbeta_raw(DBL_1__eps, // = 1 - eps
		       pp, qq, TRUE, TRUE) > la + 2))
	    MATHLIB_WARNING2( // low accuracy for more platform independent output:
		"qbeta(a, *) =: x0 with |pbeta(x0,*%s) - alpha| = %.5g is not accurate",
		(log_ ? ", log_" : ""), fabs(y - (log_ ? la : a)));
    }
L_return:
    if(give_log_q) { // ==> use_log_x , too
	if(!use_log_x) // (see if claim above is true)
	    MATHLIB_WARNING(
		"qbeta() L_return, u_n=%g;  give_log_q=TRUE but use_log_x=FALSE -- please report!",
		u_n);
	double r = R_Log1_Exp(u_n);
	if(swap_tail) {
	    qb[0] = r;	 qb[1] = u_n;
	} else {
	    qb[0] = u_n; qb[1] = r;
	}
    } else {
	if(use_log_x) {
	    if(add_N_step) {
		/* add one last Newton step on original x scale, e.g., for
		   qbeta(2^-98, 0.125, 2^-96) */
		xinbta = exp(u_n);
		y = pbeta_raw(xinbta, pp, qq, /*lower_tail = */ TRUE, log_p);
		w = log_p
		    ? (y - la) * exp(y + logbeta + r * log(xinbta) + t * log1p(-xinbta))
		    : (y - a)  * exp(    logbeta + r * log(xinbta) + t * log1p(-xinbta));
		tx = xinbta - w;
		R_ifDEBUG_printf(" Final Newton correction(non-log scale):\n"
								   //   \n  xinbta=%.16g
				 "  xinbta=%.16g, y=%g, w=-Delta(x)=%g. \n=> new x=%.16g\n",
		    xinbta, y, w, tx);
	    } else {
		if(swap_tail) {
		    qb[0] = -expm1(u_n); qb[1] =  exp  (u_n);
		} else {
		    qb[0] =  exp  (u_n); qb[1] = -expm1(u_n);
		}
		return;
	    }
	}
	if(swap_tail) {
	    qb[0] = 1 - tx; qb[1] = tx;
	} else {
	    qb[0] = tx;	qb[1] = 1 - tx;
	}
    }
    return;
}