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/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998 Ross Ihaka
 *  Copyright (C) 2000-2007 The R Development Core Team
 *  Copyright (C) 2003      The R Foundation
 *
 *  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
 *  http://www.r-project.org/Licenses/
 *
 *  DESCRIPTION
 *
 *  The "Student" t distribution quantile function.
 *
 *  NOTES
 *
 *  This is a C translation of the Fortran routine given in:
 *  Hill, G.W (1970) "Algorithm 396: Student's t-quantiles"
 *  CACM 13(10), 619-620.
 *
 *  Supplemented by inversion for 0 < ndf < 1.
 *
 *  ADDITIONS:
 *  - lower_tail, log_p
 *  - using  expm1() : takes care of  Lozy (1979) "Remark on Algo.", TOMS
 *  - Apply 2-term Taylor expansion as in
 *    Hill, G.W (1981) "Remark on Algo.396", ACM TOMS 7, 250-1
 *  - Improve the formula decision for 1 < df < 2
 */

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

double qt(double p, double ndf, int lower_tail, int log_p)
{
    const static double eps = 1.e-12;

    double P, q;
    Rboolean neg;

#ifdef IEEE_754
    if (ISNAN(p) || ISNAN(ndf))
    return p + ndf;
#endif

    R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);

    if (ndf <= 0) ML_ERR_return_NAN;

    if (ndf < 1) { /* based on qnt */
    const static double accu = 1e-13;
    const static double Eps = 1e-11; /* must be > accu */

    double ux, lx, nx, pp;
    
    int iter = 0;

    p = R_DT_qIv(p);

    /* Invert pt(.) :
     * 1. finding an upper and lower bound */
    if(p > 1 - DBL_EPSILON) return ML_POSINF;
    pp = fmin2(1 - DBL_EPSILON, p * (1 + Eps));
    for(ux = 1.; ux < DBL_MAX && pt(ux, ndf, TRUE, FALSE) < pp; ux *= 2);
    pp = p * (1 - Eps);
    for(lx =-1.; lx > -DBL_MAX && pt(lx, ndf, TRUE, FALSE) > pp; lx *= 2);

    /* 2. interval (lx,ux)  halving
       regula falsi failed on qt(0.1, 0.1)
     */
    do {
        nx = 0.5 * (lx + ux);
        if (pt(nx, ndf, TRUE, FALSE) > p) ux = nx; else lx = nx;
    } while ((ux - lx) / fabs(nx) > accu && ++iter < 1000);

    if(iter >= 1000) ML_ERROR(ME_PRECISION, "qt");

    return 0.5 * (lx + ux);
    }

    /* Old comment:
     * FIXME: "This test should depend on  ndf  AND p  !!
     * -----  and in fact should be replaced by
     * something like Abramowitz & Stegun 26.7.5 (p.949)"
     *
     * That would say that if the qnorm value is x then
     * the result is about x + (x^3+x)/4df + (5x^5+16x^3+3x)/96df^2
     * The differences are tiny even if x ~ 1e5, and qnorm is not
     * that accurate in the extreme tails.
     */
    if (ndf > 1e20) return qnorm(p, 0., 1., lower_tail, log_p);

    P = R_D_qIv(p); /* if exp(p) underflows, we fix below */

    neg = (!lower_tail || P < 0.5) && (lower_tail || P > 0.5);
    if(neg)
    P = 2 * (log_p ? (lower_tail ? P : -expm1(p)) : R_D_Lval(p));
    else
    P = 2 * (log_p ? (lower_tail ? -expm1(p) : P) : R_D_Cval(p));
    /* 0 <= P <= 1 ; P = 2*min(P', 1 - P')  in all cases */

/* Use this if(log_p) only : */
#define P_is_exp_2p (lower_tail == neg) /* both TRUE or FALSE == !xor */

     if (fabs(ndf - 2) < eps) { /* df ~= 2 */
    if(P > DBL_MIN) {
        if(3* P < DBL_EPSILON) /* P ~= 0 */
        q = 1 / sqrt(P);
        else if (P > 0.9)      /* P ~= 1 */
        q = (1 - P) * sqrt(2 /(P * (2 - P)));
        else /* eps/3 <= P <= 0.9 */
        q = sqrt(2 / (P * (2 - P)) - 2);
    }
    else { /* P << 1, q = 1/sqrt(P) = ... */
        if(log_p)
        q = P_is_exp_2p ? exp(- p/2) / M_SQRT2 : 1/sqrt(-expm1(p));
        else
        q = ML_POSINF;
    }
    }
    else if (ndf < 1 + eps) { /* df ~= 1  (df < 1 excluded above): Cauchy */
    if(P > 0)
        q = 1/tan(P * M_PI_2);/* == - tan((P+1) * M_PI_2) -- suffers for P ~= 0 */

    else { /* P = 0, but maybe = 2*exp(p) ! */
        if(log_p) /* 1/tan(e) ~ 1/e */
        q = P_is_exp_2p ? M_1_PI * exp(-p) : -1./(M_PI * expm1(p));
        else
        q = ML_POSINF;
    }
    }
    else {      /*-- usual case;  including, e.g.,  df = 1.1 */
    double x = 0., y, log_P2 = 0./* -Wall */,
        a = 1 / (ndf - 0.5),
        b = 48 / (a * a),
        c = ((20700 * a / b - 98) * a - 16) * a + 96.36,
        d = ((94.5 / (b + c) - 3) / b + 1) * sqrt(a * M_PI_2) * ndf;

    Rboolean P_ok1 = P > DBL_MIN || !log_p,  P_ok = P_ok1;
    if(P_ok1) {
        y = pow(d * P, 2 / ndf);
        P_ok = (y >= DBL_EPSILON);
    }
    if(!P_ok) { /* log_p && P very small */
        log_P2 = P_is_exp_2p ? p : R_Log1_Exp(p); /* == log(P / 2) */
        x = (log(d) + M_LN2 + log_P2) / ndf;
        y = exp(2 * x);
    }

    if ((ndf < 2.1 && P > 0.5) || y > 0.05 + a) { /* P > P0(df) */
        /* Asymptotic inverse expansion about normal */
        if(P_ok)
        x = qnorm(0.5 * P, 0., 1., /*lower_tail*/TRUE,  /*log_p*/FALSE);
        else /* log_p && P underflowed */
        x = qnorm(log_P2,  0., 1., lower_tail,          /*log_p*/ TRUE);

        y = x * x;
        if (ndf < 5)
        c += 0.3 * (ndf - 4.5) * (x + 0.6);
        c = (((0.05 * d * x - 5) * x - 7) * x - 2) * x + b + c;
        y = (((((0.4 * y + 6.3) * y + 36) * y + 94.5) / c
          - y - 3) / b + 1) * x;
        y = expm1(a * y * y);
        q = sqrt(ndf * y);
    } else { /* re-use 'y' from above */

        if(!P_ok && x < - M_LN2 * DBL_MANT_DIG) {/* 0.5* log(DBL_EPSILON) */
        /* y above might have underflown */
        q = sqrt(ndf) * exp(-x);
        }
        else {
        y = ((1 / (((ndf + 6) / (ndf * y) - 0.089 * d - 0.822)
               * (ndf + 2) * 3) + 0.5 / (ndf + 4))
             * y - 1) * (ndf + 1) / (ndf + 2) + 1 / y;
        q = sqrt(ndf * y);
        }
    }


    /* Now apply 2-term Taylor expansion improvement (1-term = Newton):
     * as by Hill (1981) [ref.above] */

    /* FIXME: This can be far from optimal when log_p = TRUE
     *      but is still needed, e.g. for qt(-2, df=1.01, log=TRUE).
     *  Probably also improvable when  lower_tail = FALSE */

    if(P_ok1) {
        int it=0;
        while(it++ < 10 && (y = dt(q, ndf, FALSE)) > 0 &&
          R_FINITE(x = (pt(q, ndf, FALSE, FALSE) - P/2) / y) &&
          fabs(x) > 1e-14*fabs(q))
        /* Newton (=Taylor 1 term):
         *  q += x;
         * Taylor 2-term : */
        q += x * (1. + x * q * (ndf + 1) / (2 * (q * q + ndf)));
    }
    }
    if(neg) q = -q;
    return q;
}