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 //  Copyright John Maddock 2006.
 //  Copyright Paul A. Bristow 2007
 //  Use, modification and distribution are subject to the
 //  Boost Software License, Version 1.0. (See accompanying file
 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 
 #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP
 #define BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP
 
 #ifdef _MSC_VER
 #pragma once
 #endif
 
 #include <boost/math/tools/config.hpp>
 #include <boost/math/tools/precision.hpp>
 #include <boost/math/tools/roots.hpp>
 #include <boost/math/tools/tuple.hpp>
 #include <boost/math/special_functions/beta.hpp>
 #include <boost/math/special_functions/erf.hpp>
 #include <boost/math/special_functions/detail/t_distribution_inv.hpp>
 #include <boost/math/special_functions/fpclassify.hpp>
 
 namespace boost{ namespace math{ namespace detail{
 
 //
 // Helper object used by root finding
 // code to convert eta to x.
 //
 template <class T>
 struct temme_root_finder
 {
    BOOST_MATH_GPU_ENABLED temme_root_finder(const T t_, const T a_) : t(t_), a(a_) {
       BOOST_MATH_ASSERT(
          math::tools::epsilon<T>() <= a && !(boost::math::isinf)(a));
    }
 
    BOOST_MATH_GPU_ENABLED boost::math::tuple<T, T> operator()(T x)
    {
       BOOST_MATH_STD_USING // ADL of std names
 
       T y = 1 - x;
       T f = log(x) + a * log(y) + t;
       T f1 = (1 / x) - (a / (y));
       return boost::math::make_tuple(f, f1);
    }
 private:
    T t, a;
 };
 //
 // See:
 // "Asymptotic Inversion of the Incomplete Beta Function"
 // N.M. Temme
 // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
 // Section 2.
 //
 template <class T, class Policy>
 BOOST_MATH_GPU_ENABLED T temme_method_1_ibeta_inverse(T a, T b, T z, const Policy& pol)
 {
    BOOST_MATH_STD_USING // ADL of std names
 
    const T r2 = sqrt(T(2));
    //
    // get the first approximation for eta from the inverse
    // error function (Eq: 2.9 and 2.10).
    //
    T eta0 = boost::math::erfc_inv(2 * z, pol);
    eta0 /= -sqrt(a / 2);
 
    T terms[4] = { eta0 };
    T workspace[7];
    //
    // calculate powers:
    //
    T B = b - a;
    T B_2 = B * B;
    T B_3 = B_2 * B;
    //
    // Calculate correction terms:
    //
 
    // See eq following 2.15:
    workspace[0] = -B * r2 / 2;
    workspace[1] = (1 - 2 * B) / 8;
    workspace[2] = -(B * r2 / 48);
    workspace[3] = T(-1) / 192;
    workspace[4] = -B * r2 / 3840;
    terms[1] = tools::evaluate_polynomial(workspace, eta0, 5);
    // Eq Following 2.17:
    workspace[0] = B * r2 * (3 * B - 2) / 12;
    workspace[1] = (20 * B_2 - 12 * B + 1) / 128;
    workspace[2] = B * r2 * (20 * B - 1) / 960;
    workspace[3] = (16 * B_2 + 30 * B - 15) / 4608;
    workspace[4] = B * r2 * (21 * B + 32) / 53760;
    workspace[5] = (-32 * B_2 + 63) / 368640;
    workspace[6] = -B * r2 * (120 * B + 17) / 25804480;
    terms[2] = tools::evaluate_polynomial(workspace, eta0, 7);
    // Eq Following 2.17:
    workspace[0] = B * r2 * (-75 * B_2 + 80 * B - 16) / 480;
    workspace[1] = (-1080 * B_3 + 868 * B_2 - 90 * B - 45) / 9216;
    workspace[2] = B * r2 * (-1190 * B_2 + 84 * B + 373) / 53760;
    workspace[3] = (-2240 * B_3 - 2508 * B_2 + 2100 * B - 165) / 368640;
    terms[3] = tools::evaluate_polynomial(workspace, eta0, 4);
    //
    // Bring them together to get a final estimate for eta:
    //
    T eta = tools::evaluate_polynomial(terms, T(1/a), 4);
    //
    // now we need to convert eta to x, by solving the appropriate
    // quadratic equation:
    //
    T eta_2 = eta * eta;
    T c = -exp(-eta_2 / 2);
    T x;
    if(eta_2 == 0)
       x = static_cast<T>(0.5f);
    else
       x = (1 + eta * sqrt((1 + c) / eta_2)) / 2;
    //
    // These are post-conditions of the method, but the addition above
    // may result in us being out by 1ulp either side of the boundary,
    // so just check that we're in bounds and adjust as needed.
    // See https://github.com/boostorg/math/issues/961
    //
    if (x < 0)
       x = 0;
    else if (x > 1)
       x = 1;
    
    BOOST_MATH_ASSERT(eta * (x - 0.5) >= 0);
 #ifdef BOOST_INSTRUMENT
    std::cout << "Estimating x with Temme method 1: " << x << std::endl;
 #endif
    return x;
 }
 //
 // See:
 // "Asymptotic Inversion of the Incomplete Beta Function"
 // N.M. Temme
 // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
 // Section 3.
 //
 template <class T, class Policy>
 BOOST_MATH_GPU_ENABLED T temme_method_2_ibeta_inverse(T /*a*/, T /*b*/, T z, T r, T theta, const Policy& pol)
 {
    BOOST_MATH_STD_USING // ADL of std names
 
    //
    // Get first estimate for eta, see Eq 3.9 and 3.10,
    // but note there is a typo in Eq 3.10:
    //
    T eta0 = boost::math::erfc_inv(2 * z, pol);
    eta0 /= -sqrt(r / 2);
 
    T s = sin(theta);
    T c = cos(theta);
    //
    // Now we need to perturb eta0 to get eta, which we do by
    // evaluating the polynomial in 1/r at the bottom of page 151,
    // to do this we first need the error terms e1, e2 e3
    // which we'll fill into the array "terms".  Since these
    // terms are themselves polynomials, we'll need another
    // array "workspace" to calculate those...
    //
    T terms[4] = { eta0 };
    T workspace[6];
    //
    // some powers of sin(theta)cos(theta) that we'll need later:
    //
    T sc = s * c;
    T sc_2 = sc * sc;
    T sc_3 = sc_2 * sc;
    T sc_4 = sc_2 * sc_2;
    T sc_5 = sc_2 * sc_3;
    T sc_6 = sc_3 * sc_3;
    T sc_7 = sc_4 * sc_3;
    //
    // Calculate e1 and put it in terms[1], see the middle of page 151:
    //
    workspace[0] = (2 * s * s - 1) / (3 * s * c);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co1[] = { -1, -5, 5 };
    workspace[1] = -tools::evaluate_even_polynomial(co1, s, 3) / (36 * sc_2);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co2[] = { 1, 21, -69, 46 };
    workspace[2] = tools::evaluate_even_polynomial(co2, s, 4) / (1620 * sc_3);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co3[] = { 7, -2, 33, -62, 31 };
    workspace[3] = -tools::evaluate_even_polynomial(co3, s, 5) / (6480 * sc_4);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co4[] = { 25, -52, -17, 88, -115, 46 };
    workspace[4] = tools::evaluate_even_polynomial(co4, s, 6) / (90720 * sc_5);
    terms[1] = tools::evaluate_polynomial(workspace, eta0, 5);
    //
    // Now evaluate e2 and put it in terms[2]:
    //
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co5[] = { 7, 12, -78, 52 };
    workspace[0] = -tools::evaluate_even_polynomial(co5, s, 4) / (405 * sc_3);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co6[] = { -7, 2, 183, -370, 185 };
    workspace[1] = tools::evaluate_even_polynomial(co6, s, 5) / (2592 * sc_4);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co7[] = { -533, 776, -1835, 10240, -13525, 5410 };
    workspace[2] = -tools::evaluate_even_polynomial(co7, s, 6) / (204120 * sc_5);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co8[] = { -1579, 3747, -3372, -15821, 45588, -45213, 15071 };
    workspace[3] = -tools::evaluate_even_polynomial(co8, s, 7) / (2099520 * sc_6);
    terms[2] = tools::evaluate_polynomial(workspace, eta0, 4);
    //
    // And e3, and put it in terms[3]:
    //
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co9[] = {449, -1259, -769, 6686, -9260, 3704 };
    workspace[0] = tools::evaluate_even_polynomial(co9, s, 6) / (102060 * sc_5);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co10[] = { 63149, -151557, 140052, -727469, 2239932, -2251437, 750479 };
    workspace[1] = -tools::evaluate_even_polynomial(co10, s, 7) / (20995200 * sc_6);
    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co11[] = { 29233, -78755, 105222, 146879, -1602610, 3195183, -2554139, 729754 };
    workspace[2] = tools::evaluate_even_polynomial(co11, s, 8) / (36741600 * sc_7);
    terms[3] = tools::evaluate_polynomial(workspace, eta0, 3);
    //
    // Bring the correction terms together to evaluate eta,
    // this is the last equation on page 151:
    //
    T eta = tools::evaluate_polynomial(terms, T(1/r), 4);
    //
    // Now that we have eta we need to back solve for x,
    // we seek the value of x that gives eta in Eq 3.2.
    // The two methods used are described in section 5.
    //
    // Begin by defining a few variables we'll need later:
    //
    T x;
    T s_2 = s * s;
    T c_2 = c * c;
    T alpha = c / s;
    alpha *= alpha;
    T lu = (-(eta * eta) / (2 * s_2) + log(s_2) + c_2 * log(c_2) / s_2);
    //
    // Temme doesn't specify what value to switch on here,
    // but this seems to work pretty well:
    //
    if(fabs(eta) < 0.7)
    {
       //
       // Small eta use the expansion Temme gives in the second equation
       // of section 5, it's a polynomial in eta:
       //
       workspace[0] = s * s;
       workspace[1] = s * c;
       workspace[2] = (1 - 2 * workspace[0]) / 3;
       static const BOOST_MATH_INT_TABLE_TYPE(T, int) co12[] = { 1, -13, 13 };
       workspace[3] = tools::evaluate_polynomial(co12, workspace[0], 3) / (36 * s * c);
       static const BOOST_MATH_INT_TABLE_TYPE(T, int) co13[] = { 1, 21, -69, 46 };
       workspace[4] = tools::evaluate_polynomial(co13, workspace[0], 4) / (270 * workspace[0] * c * c);
       x = tools::evaluate_polynomial(workspace, eta, 5);
 #ifdef BOOST_INSTRUMENT
       std::cout << "Estimating x with Temme method 2 (small eta): " << x << std::endl;
 #endif
    }
    else
    {
       //
       // If eta is large we need to solve Eq 3.2 more directly,
       // begin by getting an initial approximation for x from
       // the last equation on page 155, this is a polynomial in u:
       //
       T u = exp(lu);
       workspace[0] = u;
       workspace[1] = alpha;
       workspace[2] = 0;
       workspace[3] = 3 * alpha * (3 * alpha + 1) / 6;
       workspace[4] = 4 * alpha * (4 * alpha + 1) * (4 * alpha + 2) / 24;
       workspace[5] = 5 * alpha * (5 * alpha + 1) * (5 * alpha + 2) * (5 * alpha + 3) / 120;
       x = tools::evaluate_polynomial(workspace, u, 6);
       //
       // At this point we may or may not have the right answer, Eq-3.2 has
       // two solutions for x for any given eta, however the mapping in 3.2
       // is 1:1 with the sign of eta and x-sin^2(theta) being the same.
       // So we can check if we have the right root of 3.2, and if not
       // switch x for 1-x.  This transformation is motivated by the fact
       // that the distribution is *almost* symmetric so 1-x will be in the right
       // ball park for the solution:
       //
       if((x - s_2) * eta < 0)
          x = 1 - x;
 #ifdef BOOST_INSTRUMENT
       std::cout << "Estimating x with Temme method 2 (large eta): " << x << std::endl;
 #endif
    }
    //
    // The final step is a few Newton-Raphson iterations to
    // clean up our approximation for x, this is pretty cheap
    // in general, and very cheap compared to an incomplete beta
    // evaluation.  The limits set on x come from the observation
    // that the sign of eta and x-sin^2(theta) are the same.
    //
    T lower, upper;
    if(eta < 0)
    {
       lower = 0;
       upper = s_2;
    }
    else
    {
       lower = s_2;
       upper = 1;
    }
    //
    // If our initial approximation is out of bounds then bisect:
    //
    if((x < lower) || (x > upper))
       x = (lower+upper) / 2;
    //
    // And iterate:
    //
 #ifndef BOOST_MATH_NO_EXCEPTIONS
    try {
 #endif
       x = tools::newton_raphson_iterate(
          temme_root_finder<T>(-lu, alpha), x, lower, upper, policies::digits<T, Policy>() / 2);
 #ifndef BOOST_MATH_NO_EXCEPTIONS
    }
    catch (const boost::math::evaluation_error&)
    {
       // Due to numerical instability we may have cases where no root is found when
       // in fact we should just touch the origin.  We simply ignore the error here
       // and return our best guess for x so far...
       // Maybe we should special case the symmetrical parameter case, but it's not clear 
       // whether that is the only situation when problems can occur.
       // See https://github.com/boostorg/math/issues/1169
    }
 #endif
    return x;
 }
 //
 // See:
 // "Asymptotic Inversion of the Incomplete Beta Function"
 // N.M. Temme
 // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
 // Section 4.
 //
 template <class T, class Policy>
 BOOST_MATH_GPU_ENABLED T temme_method_3_ibeta_inverse(T a, T b, T p, T q, const Policy& pol)
 {
    BOOST_MATH_STD_USING // ADL of std names
 
 
    //
    // Begin by getting an initial approximation for the quantity
    // eta from the dominant part of the incomplete beta:
    //
    T eta0;
    if(p < q)
       eta0 = boost::math::gamma_q_inv(b, p, pol);
    else
       eta0 = boost::math::gamma_p_inv(b, q, pol);
    eta0 /= a;
    //
    // Define the variables and powers we'll need later on:
    //
    T mu = b / a;
    T w = sqrt(1 + mu);
    T w_2 = w * w;
    T w_3 = w_2 * w;
    T w_4 = w_2 * w_2;
    T w_5 = w_3 * w_2;
    T w_6 = w_3 * w_3;
    T w_7 = w_4 * w_3;
    T w_8 = w_4 * w_4;
    T w_9 = w_5 * w_4;
    T w_10 = w_5 * w_5;
    T d = eta0 - mu;
    T d_2 = d * d;
    T d_3 = d_2 * d;
    T d_4 = d_2 * d_2;
    T w1 = w + 1;
    T w1_2 = w1 * w1;
    T w1_3 = w1 * w1_2;
    T w1_4 = w1_2 * w1_2;
    //
    // Now we need to compute the perturbation error terms that
    // convert eta0 to eta, these are all polynomials of polynomials.
    // Probably these should be re-written to use tabulated data
    // (see examples above), but it's less of a win in this case as we
    // need to calculate the individual powers for the denominator terms
    // anyway, so we might as well use them for the numerator-polynomials
    // as well....
    //
    // Refer to p154-p155 for the details of these expansions:
    //
    T e1 = (w + 2) * (w - 1) / (3 * w);
    e1 += (w_3 + 9 * w_2 + 21 * w + 5) * d / (36 * w_2 * w1);
    e1 -= (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2 / (1620 * w1_2 * w_3);
    e1 -= (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3 / (6480 * w1_3 * w_4);
    e1 -= (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4 / (272160 * w1_4 * w_5);
 
    T e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1) / (1620 * w1 * w_3);
    e2 -= (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d / (12960 * w1_2 * w_4);
    e2 -= (2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3 + 141183 * w_2 + 95993 * w + 21640) * d_2  / (816480 * w_5 * w1_3);
    e2 -= (11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4 - 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497) * d_3 / (T(14696640) * w1_4 * w_6);
 
    T e3 = -((3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3 - 154413 * w_2 - 116063 * w - 29632) * (w - 1)) / (816480 * w_5 * w1_2);
    e3 -= (442043 * w_9 + T(2054169) * w_8 + T(3803094) * w_7 + T(3470754) * w_6 + T(2141568) * w_5 - T(2393568) * w_4 - T(19904934) * w_3 - T(34714674) * w_2 - T(23128299) * w - T(5253353)) * d / (T(146966400) * w_6 * w1_3);
    e3 -= (116932 * w_10 + 819281 * w_9 + T(2378172) * w_8 + T(4341330) * w_7 + T(6806004) * w_6 + T(10622748) * w_5 + T(18739500) * w_4 + T(30651894) * w_3 + T(30869976) * w_2 + T(15431867) * w + T(2919016)) * d_2 / (T(146966400) * w1_4 * w_7);
    //
    // Combine eta0 and the error terms to compute eta (Second equation p155):
    //
    T eta = eta0 + e1 / a + e2 / (a * a) + e3 / (a * a * a);
    //
    // Now we need to solve Eq 4.2 to obtain x.  For any given value of
    // eta there are two solutions to this equation, and since the distribution
    // may be very skewed, these are not related by x ~ 1-x we used when
    // implementing section 3 above.  However we know that:
    //
    //  cross < x <= 1       ; iff eta < mu
    //          x == cross   ; iff eta == mu
    //     0 <= x < cross    ; iff eta > mu
    //
    // Where cross == 1 / (1 + mu)
    // Many thanks to Prof Temme for clarifying this point.
    //
    // Therefore we'll just jump straight into Newton iterations
    // to solve Eq 4.2 using these bounds, and simple bisection
    // as the first guess, in practice this converges pretty quickly
    // and we only need a few digits correct anyway:
    //
    if(eta <= 0)
       eta = tools::min_value<T>();
    T u = eta - mu * log(eta) + (1 + mu) * log(1 + mu) - mu;
    T cross = 1 / (1 + mu);
    T lower = eta < mu ? cross : 0;
    T upper = eta < mu ? 1 : cross;
    T x = (lower + upper) / 2;
 
    // Early exit for cases with numerical precision issues.
    if (cross == 0 || cross == 1) { return cross; }
    
    x = tools::newton_raphson_iterate(
       temme_root_finder<T>(u, mu), x, lower, upper, policies::digits<T, Policy>() / 2);
 #ifdef BOOST_INSTRUMENT
    std::cout << "Estimating x with Temme method 3: " << x << std::endl;
 #endif
    return x;
 }
 
 template <class T, class Policy>
 struct ibeta_roots
 {
    BOOST_MATH_GPU_ENABLED ibeta_roots(T _a, T _b, T t, bool inv = false)
       : a(_a), b(_b), target(t), invert(inv) {}
 
    BOOST_MATH_GPU_ENABLED boost::math::tuple<T, T, T> operator()(T x)
    {
       BOOST_MATH_STD_USING // ADL of std names
 
       BOOST_FPU_EXCEPTION_GUARD
 
       T f1;
       T y = 1 - x;
       T f = ibeta_imp(a, b, x, Policy(), invert, true, &f1) - target;
       if(invert)
          f1 = -f1;
       if(y == 0)
          y = tools::min_value<T>() * 64;
       if(x == 0)
          x = tools::min_value<T>() * 64;
 
       T f2 = f1 * (-y * a + (b - 2) * x + 1);
       if(fabs(f2) < y * x * tools::max_value<T>())
          f2 /= (y * x);
       if(invert)
          f2 = -f2;
 
       // make sure we don't have a zero derivative:
       if(f1 == 0)
          f1 = (invert ? -1 : 1) * tools::min_value<T>() * 64;
 
       return boost::math::make_tuple(f, f1, f2);
    }
 private:
    T a, b, target;
    bool invert;
 };
 
 template <class T, class Policy>
 BOOST_MATH_GPU_ENABLED T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py)
 {
    BOOST_MATH_STD_USING  // For ADL of math functions.
 
    //
    // The flag invert is set to true if we swap a for b and p for q,
    // in which case the result has to be subtracted from 1:
    //
    bool invert = false;
    //
    // Handle trivial cases first:
    //
    if(q == 0)
    {
       if(py) *py = 0;
       return 1;
    }
    else if(p == 0)
    {
       if(py) *py = 1;
       return 0;
    }
    else if(a == 1)
    {
       if(b == 1)
       {
          if(py) *py = 1 - p;
          return p;
       }
       // Change things around so we can handle as b == 1 special case below:
       BOOST_MATH_GPU_SAFE_SWAP(a, b);
       BOOST_MATH_GPU_SAFE_SWAP(p, q);
       invert = true;
    }
    //
    // Depending upon which approximation method we use, we may end up
    // calculating either x or y initially (where y = 1-x):
    //
    T x = 0; // Set to a safe zero to avoid a
    // MSVC 2005 warning C4701: potentially uninitialized local variable 'x' used
    // But code inspection appears to ensure that x IS assigned whatever the code path.
    T y;
 
    // For some of the methods we can put tighter bounds
    // on the result than simply [0,1]:
    //
    T lower = 0;
    T upper = 1;
    //
    // Student's T with b = 0.5 gets handled as a special case, swap
    // around if the arguments are in the "wrong" order:
    //
    if(a == 0.5f)
    {
       if(b == 0.5f)
       {
          x = sin(p * constants::half_pi<T>());
          x *= x;
          if(py)
          {
             *py = sin(q * constants::half_pi<T>());
             *py *= *py;
          }
          return x;
       }
       else if(b > 0.5f)
       {
          BOOST_MATH_GPU_SAFE_SWAP(a, b);
          BOOST_MATH_GPU_SAFE_SWAP(p, q);
          invert = !invert;
       }
    }
    //
    // Select calculation method for the initial estimate:
    //
    if((b == 0.5f) && (a >= 0.5f) && (p != 1))
    {
       //
       // We have a Student's T distribution:
       x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol);
    }
    else if(b == 1)
    {
       if(p < q)
       {
          if(a > 1)
          {
             x = pow(p, 1 / a);
             y = -boost::math::expm1(log(p) / a, pol);
          }
          else
          {
             x = pow(p, 1 / a);
             y = 1 - x;
          }
       }
       else
       {
          x = exp(boost::math::log1p(-q, pol) / a);
          y = -boost::math::expm1(boost::math::log1p(-q, pol) / a, pol);
       }
       if(invert)
          BOOST_MATH_GPU_SAFE_SWAP(x, y);
       if(py)
          *py = y;
       return x;
    }
    else if(a + b > 5)
    {
       //
       // When a+b is large then we can use one of Prof Temme's
       // asymptotic expansions, begin by swapping things around
       // so that p < 0.5, we do this to avoid cancellations errors
       // when p is large.
       //
       if(p > 0.5)
       {
          BOOST_MATH_GPU_SAFE_SWAP(a, b);
          BOOST_MATH_GPU_SAFE_SWAP(p, q);
          invert = !invert;
       }
       T minv = BOOST_MATH_GPU_SAFE_MIN(a, b);
       T maxv = BOOST_MATH_GPU_SAFE_MAX(a, b);
       if((sqrt(minv) > (maxv - minv)) && (minv > 5))
       {
          //
          // When a and b differ by a small amount
          // the curve is quite symmetrical and we can use an error
          // function to approximate the inverse. This is the cheapest
          // of the three Temme expansions, and the calculated value
          // for x will never be much larger than p, so we don't have
          // to worry about cancellation as long as p is small.
          //
          x = temme_method_1_ibeta_inverse(a, b, p, pol);
          y = 1 - x;
       }
       else
       {
          T r = a + b;
          T theta = asin(sqrt(a / r));
          T lambda = minv / r;
          if((lambda >= 0.2) && (lambda <= 0.8) && (r >= 10))
          {
             //
             // The second error function case is the next cheapest
             // to use, it brakes down when the result is likely to be
             // very small, if a+b is also small, but we can use a
             // cheaper expansion there in any case.  As before x won't
             // be much larger than p, so as long as p is small we should
             // be free of cancellation error.
             //
             T ppa = pow(p, 1/a);
             if((ppa < 0.0025) && (a + b < 200))
             {
                x = ppa * pow(a * boost::math::beta(a, b, pol), 1/a);
             }
             else
                x = temme_method_2_ibeta_inverse(a, b, p, r, theta, pol);
             y = 1 - x;
          }
          else
          {
             //
             // If we get here then a and b are very different in magnitude
             // and we need to use the third of Temme's methods which
             // involves inverting the incomplete gamma.  This is much more
             // expensive than the other methods.  We also can only use this
             // method when a > b, which can lead to cancellation errors
             // if we really want y (as we will when x is close to 1), so
             // a different expansion is used in that case.
             //
             if(a < b)
             {
                BOOST_MATH_GPU_SAFE_SWAP(a, b);
                BOOST_MATH_GPU_SAFE_SWAP(p, q);
                invert = !invert;
             }
             //
             // Try and compute the easy way first:
             //
             T bet = 0;
             if (b < 2)
             {
 #ifndef BOOST_MATH_NO_EXCEPTIONS
                try
 #endif
                {
                   bet = boost::math::beta(a, b, pol);
 
                   typedef typename Policy::overflow_error_type overflow_type;
 
                   BOOST_MATH_IF_CONSTEXPR(overflow_type::value != boost::math::policies::throw_on_error)
                      if(bet > tools::max_value<T>())
                         bet = tools::max_value<T>();
                }
 #ifndef BOOST_MATH_NO_EXCEPTIONS
                catch (const std::overflow_error&)
                {
                   bet = tools::max_value<T>();
                }
 #endif
             }
             if(bet != 0)
             {
                y = pow(b * q * bet, 1/b);
                x = 1 - y;
             }
             else
                y = 1;
             if(y > 1e-5)
             {
                x = temme_method_3_ibeta_inverse(a, b, p, q, pol);
                y = 1 - x;
             }
          }
       }
    }
    else if((a < 1) && (b < 1))
    {
       //
       // Both a and b less than 1,
       // there is a point of inflection at xs:
       //
       T xs = (1 - a) / (2 - a - b);
       //
       // Now we need to ensure that we start our iteration from the
       // right side of the inflection point:
       //
       T fs = boost::math::ibeta(a, b, xs, pol) - p;
       if(fabs(fs) / p < tools::epsilon<T>() * 3)
       {
          // The result is at the point of inflection, best just return it:
          *py = invert ? xs : 1 - xs;
          return invert ? 1-xs : xs;
       }
       if(fs < 0)
       {
          BOOST_MATH_GPU_SAFE_SWAP(a, b);
          BOOST_MATH_GPU_SAFE_SWAP(p, q);
          invert = !invert;
          xs = 1 - xs;
       }
       if ((a < tools::min_value<T>()) && (b > tools::min_value<T>()))
       {
          if (py)
          {
             *py = invert ? 0 : 1;
          }
          return invert ? 1 : 0; // nothing interesting going on here.
       }
       //
       // The call to beta may overflow, plus the alternative using lgamma may do the same
       // if T is a type where 1/T is infinite for small values (denorms for example).
       //
       T bet = 0;
       T xg;
       bool overflow = false;
 #ifndef BOOST_MATH_NO_EXCEPTIONS
       try {
 #endif
          bet = boost::math::beta(a, b, pol);
 #ifndef BOOST_MATH_NO_EXCEPTIONS
       }
       catch (const std::runtime_error&)
       {
          overflow = true;
       }
 #endif
       if (overflow || !(boost::math::isfinite)(bet))
       {
          xg = exp((boost::math::lgamma(a + 1, pol) + boost::math::lgamma(b, pol) - boost::math::lgamma(a + b, pol) + log(p)) / a);
          if (xg > 2 / tools::epsilon<T>())
             xg = 2 / tools::epsilon<T>();
       }
       else
          xg = pow(a * p * bet, 1/a);
       x = xg / (1 + xg);
       y = 1 / (1 + xg);
       //
       // And finally we know that our result is below the inflection
       // point, so set an upper limit on our search:
       //
       if(x > xs)
          x = xs;
       upper = xs;
    }
    else if((a > 1) && (b > 1))
    {
       //
       // Small a and b, both greater than 1,
       // there is a point of inflection at xs,
       // and it's complement is xs2, we must always
       // start our iteration from the right side of the
       // point of inflection.
       //
       T xs = (a - 1) / (a + b - 2);
       T xs2 = (b - 1) / (a + b - 2);
       T ps = boost::math::ibeta(a, b, xs, pol) - p;
 
       if(ps < 0)
       {
          BOOST_MATH_GPU_SAFE_SWAP(a, b);
          BOOST_MATH_GPU_SAFE_SWAP(p, q);
          BOOST_MATH_GPU_SAFE_SWAP(xs, xs2);
          invert = !invert;
       }
       //
       // Estimate x and y, using expm1 to get a good estimate
       // for y when it's very small:
       //
       T lx = log(p * a * boost::math::beta(a, b, pol)) / a;
       x = exp(lx);
       y = x < 0.9 ? T(1 - x) : (T)(-boost::math::expm1(lx, pol));
 
       if((b < a) && (x < 0.2))
       {
          //
          // Under a limited range of circumstances we can improve
          // our estimate for x, frankly it's clear if this has much effect!
          //
          T ap1 = a - 1;
          T bm1 = b - 1;
          T a_2 = a * a;
          T a_3 = a * a_2;
          T b_2 = b * b;
          T terms[5] = { 0, 1 };
          terms[2] = bm1 / ap1;
          ap1 *= ap1;
          terms[3] = bm1 * (3 * a * b + 5 * b + a_2 - a - 4) / (2 * (a + 2) * ap1);
          ap1 *= (a + 1);
          terms[4] = bm1 * (33 * a * b_2 + 31 * b_2 + 8 * a_2 * b_2 - 30 * a * b - 47 * b + 11 * a_2 * b + 6 * a_3 * b + 18 + 4 * a - a_3 + a_2 * a_2 - 10 * a_2)
                     / (3 * (a + 3) * (a + 2) * ap1);
          x = tools::evaluate_polynomial(terms, x, 5);
       }
       //
       // And finally we know that our result is below the inflection
       // point, so set an upper limit on our search:
       //
       if(x > xs)
          x = xs;
       upper = xs;
    }
    else /*if((a <= 1) != (b <= 1))*/
    {
       //
       // If all else fails we get here, only one of a and b
       // is above 1, and a+b is small.  Start by swapping
       // things around so that we have a concave curve with b > a
       // and no points of inflection in [0,1].  As long as we expect
       // x to be small then we can use the simple (and cheap) power
       // term to estimate x, but when we expect x to be large then
       // this greatly underestimates x and leaves us trying to
       // iterate "round the corner" which may take almost forever...
       //
       // We could use Temme's inverse gamma function case in that case,
       // this works really rather well (albeit expensively) even though
       // strictly speaking we're outside it's defined range.
       //
       // However it's expensive to compute, and an alternative approach
       // which models the curve as a distorted quarter circle is much
       // cheaper to compute, and still keeps the number of iterations
       // required down to a reasonable level.  With thanks to Prof Temme
       // for this suggestion.
       //
       if(b < a)
       {
          BOOST_MATH_GPU_SAFE_SWAP(a, b);
          BOOST_MATH_GPU_SAFE_SWAP(p, q);
          invert = !invert;
       }
       if (a < tools::min_value<T>())
       {
          // Avoid spurious overflows for denorms:
          if (p < 1)
          {
             x = 1;
             y = 0;
          }
          else
          {
             x = 0;
             y = 1;
          }
       }
       else if(pow(p, 1/a) < 0.5)
       {
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          try 
          {
 #endif
             x = pow(p * a * boost::math::beta(a, b, pol), 1 / a);
             if ((x > 1) || !(boost::math::isfinite)(x))
                x = 1;
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          }
          catch (const std::overflow_error&)
          {
             x = 1;
          }
 #endif
          if(x == 0)
             x = boost::math::tools::min_value<T>();
          y = 1 - x;
       }
       else /*if(pow(q, 1/b) < 0.1)*/
       {
          // model a distorted quarter circle:
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          try 
          {
 #endif
             y = pow(1 - pow(p, b * boost::math::beta(a, b, pol)), 1/b);
             if ((y > 1) || !(boost::math::isfinite)(y))
                y = 1;
 #ifndef BOOST_MATH_NO_EXCEPTIONS
          }
          catch (const std::overflow_error&)
          {
             y = 1;
          }
 #endif
          if(y == 0)
             y = boost::math::tools::min_value<T>();
          x = 1 - y;
       }
    }
 
    //
    // Now we have a guess for x (and for y) we can set things up for
    // iteration.  If x > 0.5 it pays to swap things round:
    //
    if(x > 0.5)
    {
       BOOST_MATH_GPU_SAFE_SWAP(a, b);
       BOOST_MATH_GPU_SAFE_SWAP(p, q);
       BOOST_MATH_GPU_SAFE_SWAP(x, y);
       invert = !invert;
       T l = 1 - upper;
       T u = 1 - lower;
       lower = l;
       upper = u;
    }
    //
    // lower bound for our search:
    //
    // We're not interested in denormalised answers as these tend to
    // these tend to take up lots of iterations, given that we can't get
    // accurate derivatives in this area (they tend to be infinite).
    //
    if(lower == 0)
    {
       if(invert && (py == 0))
       {
          //
          // We're not interested in answers smaller than machine epsilon:
          //
          lower = boost::math::tools::epsilon<T>();
          if(x < lower)
             x = lower;
       }
       else
          lower = boost::math::tools::min_value<T>();
       if(x < lower)
          x = lower;
    }
    boost::math::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
    boost::math::uintmax_t max_iter_used = 0;
    //
    // Figure out how many digits to iterate towards:
    //
    int digits = boost::math::policies::digits<T, Policy>() / 2;
    if((x < 1e-50) && ((a < 1) || (b < 1)))
    {
       //
       // If we're in a region where the first derivative is very
       // large, then we have to take care that the root-finder
       // doesn't terminate prematurely.  We'll bump the precision
       // up to avoid this, but we have to take care not to set the
       // precision too high or the last few iterations will just
       // thrash around and convergence may be slow in this case.
       // Try 3/4 of machine epsilon:
       //
       digits *= 3;
       digits /= 2;
    }
    //
    // Now iterate, we can use either p or q as the target here
    // depending on which is smaller:
    //
    // Since we can't use halley_iterate on device we use newton raphson
    //
    #ifndef BOOST_MATH_HAS_GPU_SUPPORT
    x = boost::math::tools::halley_iterate(
    #else
    x = boost::math::tools::newton_raphson_iterate(
    #endif
       boost::math::detail::ibeta_roots<T, Policy>(a, b, (p < q ? p : q), (p < q ? false : true)), x, lower, upper, digits, max_iter);
    policies::check_root_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%)", max_iter + max_iter_used, pol);
    //
    // We don't really want these asserts here, but they are useful for sanity
    // checking that we have the limits right, uncomment if you suspect bugs *only*.
    //
    //BOOST_MATH_ASSERT(x != upper);
    //BOOST_MATH_ASSERT((x != lower) || (x == boost::math::tools::min_value<T>()) || (x == boost::math::tools::epsilon<T>()));
    //
    // Tidy up, if we "lower" was too high then zero is the best answer we have:
    //
    if(x == lower)
       x = 0;
    if(py)
       *py = invert ? x : 1 - x;
    return invert ? 1-x : x;
 }
 
 } // namespace detail
 
 template <class T1, class T2, class T3, class T4, class Policy>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
    ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy& pol)
 {
    constexpr auto function = "boost::math::ibeta_inv<%1%>(%1%,%1%,%1%)";
    BOOST_FPU_EXCEPTION_GUARD
    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
    typedef typename policies::evaluation<result_type, Policy>::type value_type;
    typedef typename policies::normalise<
       Policy,
       policies::promote_float<false>,
       policies::promote_double<false>,
       policies::discrete_quantile<>,
       policies::assert_undefined<> >::type forwarding_policy;
 
    if(a <= 0)
       return policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol);
    if(b <= 0)
       return policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol);
    if((p < 0) || (p > 1))
       return policies::raise_domain_error<result_type>(function, "Argument p outside the range [0,1] in the incomplete beta function inverse (got p=%1%).", p, pol);
 
    value_type rx, ry;
 
    rx = detail::ibeta_inv_imp(
          static_cast<value_type>(a),
          static_cast<value_type>(b),
          static_cast<value_type>(p),
          static_cast<value_type>(1 - p),
          forwarding_policy(), &ry);
 
    if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function);
    return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function);
 }
 
 template <class T1, class T2, class T3, class T4>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
    ibeta_inv(T1 a, T2 b, T3 p, T4* py)
 {
    return ibeta_inv(a, b, p, py, policies::policy<>());
 }
 
 template <class T1, class T2, class T3>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3>::type
    ibeta_inv(T1 a, T2 b, T3 p)
 {
    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
    return ibeta_inv(a, b, p, static_cast<result_type*>(nullptr), policies::policy<>());
 }
 
 template <class T1, class T2, class T3, class Policy>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3>::type
    ibeta_inv(T1 a, T2 b, T3 p, const Policy& pol)
 {
    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
    return ibeta_inv(a, b, p, static_cast<result_type*>(nullptr), pol);
 }
 
 template <class T1, class T2, class T3, class T4, class Policy>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
    ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy& pol)
 {
    constexpr auto function = "boost::math::ibetac_inv<%1%>(%1%,%1%,%1%)";
    BOOST_FPU_EXCEPTION_GUARD
    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
    typedef typename policies::evaluation<result_type, Policy>::type value_type;
    typedef typename policies::normalise<
       Policy,
       policies::promote_float<false>,
       policies::promote_double<false>,
       policies::discrete_quantile<>,
       policies::assert_undefined<> >::type forwarding_policy;
 
    if(a <= 0)
       return policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol);
    if(b <= 0)
       return policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol);
    if((q < 0) || (q > 1))
       return policies::raise_domain_error<result_type>(function, "Argument q outside the range [0,1] in the incomplete beta function inverse (got q=%1%).", q, pol);
 
    value_type rx, ry;
 
    rx = detail::ibeta_inv_imp(
          static_cast<value_type>(a),
          static_cast<value_type>(b),
          static_cast<value_type>(1 - q),
          static_cast<value_type>(q),
          forwarding_policy(), &ry);
 
    if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function);
    return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function);
 }
 
 template <class T1, class T2, class T3, class T4>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<T1, T2, T3, T4>::type
    ibetac_inv(T1 a, T2 b, T3 q, T4* py)
 {
    return ibetac_inv(a, b, q, py, policies::policy<>());
 }
 
 template <class RT1, class RT2, class RT3>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<RT1, RT2, RT3>::type
    ibetac_inv(RT1 a, RT2 b, RT3 q)
 {
    typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
    return ibetac_inv(a, b, q, static_cast<result_type*>(nullptr), policies::policy<>());
 }
 
 template <class RT1, class RT2, class RT3, class Policy>
 BOOST_MATH_GPU_ENABLED inline typename tools::promote_args<RT1, RT2, RT3>::type
    ibetac_inv(RT1 a, RT2 b, RT3 q, const Policy& pol)
 {
    typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
    return ibetac_inv(a, b, q, static_cast<result_type*>(nullptr), pol);
 }
 
 } // namespace math
 } // namespace boost
 
 #endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP
 
 
 
 

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