The R Project SVN
R
(root)/branches/ALTREP/src/nmath/qbeta.c – Rev 74267
Rev 72049 | Blame | Compare with Previous | Last modification | View Log | Download | RSS feed
/** 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_hiddenstatic voidqbeta_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_754if (ISNAN(p) || ISNAN(q) || ISNAN(alpha))return p + q + alpha;#endifif(p < 0. || q < 0.) ML_ERR_return_NAN;// allowing p==0 and q==0 <==> treat as one- or two-point massdouble 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() ~= 1attribute_hidden voidqbeta_raw(double alpha, double p, double q, int lower_tail, int log_p,int swap_01, // {TRUE, NA, FALSE}: if NA, algorithm decides swap_taildouble log_q_cut, /* if == Inf: return log(qbeta(..));otherwise, if finite: the bound forswitching to log(x)-scale; see use_log_x */int n_N, // number of "unconstrained" Newton steps before switching to constraineddouble *qb) // = qb[0:1] = { qbeta(), 1 - qbeta() }{Rbooleanswap_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 belowwarned = 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; } \returnreturn_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; } \returnreturn_q_1;}// check alpha {*before* transformation which may all accuracy}:if((log_p && alpha > 0) ||(!log_p && (alpha < 0 || alpha > 1))) { // alpha is outsideR_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 massif(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; \returnreturn_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 1/2return_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 <= 1/2if(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" earlydouble tx, u0 = (la + log(pp) + logbeta) / pp; // = log(x_0)static const doublelog_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) = 0u0 < -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*x0if(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 overflowst = 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-Hilfertyr = 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 64s = 4. * pp + r - 2.;// 4p + 2q - 2 = numerator of new t = (...) / chi^2R_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);elsel1ma = 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 nonsenseR_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, eitheru = (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 hereif(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_xswap_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 stableu = R_Log1_Exp(u);xinbta = exp(u);/* Careful: "swap now" should not fail if1) 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)Rbooleanbad_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.; // -Walltx = 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 0tx = 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.; // -Wallif(use_log_x) { // find log(xinbta) -- work in u := log(x) scale// if(bad_init && tx > 0) xinbta = tx;// may have been betterfor (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*wif (u_n <= 0.) { // <==> 0 < xinbta := e^u <= 1if (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" Newtonfor (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_754if(!R_FINITE(y) && !(log_p && y == ML_NEGINF))// y = -Inf is ok if(log_p)#elseif (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*wif (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)) // "<=" : (.) == 0goto 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 printingR_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 warnpbeta_raw(DBL_1__eps, // = 1 - epspp, 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 , tooif(!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., forqbeta(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;}