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 // Copyright Christopher Kormanyos 2002 - 2011.
 // Copyright 2011 John Maddock.
 // Distributed under 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)
 
 // This work is based on an earlier work:
 // "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
 // in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
 //
 // This file has no include guards or namespaces - it's expanded inline inside default_ops.hpp
 //
 
 #include <boost/multiprecision/detail/standalone_config.hpp>
 #include <boost/multiprecision/detail/no_exceptions_support.hpp>
 #include <boost/multiprecision/detail/assert.hpp>
 
 #ifdef BOOST_MSVC
 #pragma warning(push)
 #pragma warning(disable : 6326) // comparison of two constants
 #pragma warning(disable : 4127) // conditional expression is constant
 #endif
 
 template <class T>
 void hyp0F1(T& result, const T& b, const T& x)
 {
    using si_type = typename boost::multiprecision::detail::canonical<std::int32_t, T>::type ;
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
 
    // Compute the series representation of Hypergeometric0F1 taken from
    // http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/06/01/01/
    // There are no checks on input range or parameter boundaries.
 
    T x_pow_n_div_n_fact(x);
    T pochham_b(b);
    T bp(b);
 
    eval_divide(result, x_pow_n_div_n_fact, pochham_b);
    eval_add(result, ui_type(1));
 
    si_type n;
 
    T tol;
    tol = ui_type(1);
    eval_ldexp(tol, tol, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
    eval_multiply(tol, result);
    if (eval_get_sign(tol) < 0)
       tol.negate();
    T term;
 
    const int series_limit =
        boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
            ? 100
            : boost::multiprecision::detail::digits2<number<T, et_on> >::value();
    // Series expansion of hyperg_0f1(; b; x).
    for (n = 2; n < series_limit; ++n)
    {
       eval_multiply(x_pow_n_div_n_fact, x);
       eval_divide(x_pow_n_div_n_fact, n);
       eval_increment(bp);
       eval_multiply(pochham_b, bp);
 
       eval_divide(term, x_pow_n_div_n_fact, pochham_b);
       eval_add(result, term);
 
       bool neg_term = eval_get_sign(term) < 0;
       if (neg_term)
          term.negate();
       if (term.compare(tol) <= 0)
          break;
    }
 
    if (n >= series_limit)
       BOOST_MP_THROW_EXCEPTION(std::runtime_error("H0F1 Failed to Converge"));
 }
 
 template <class T, unsigned N, bool b = boost::multiprecision::detail::is_variable_precision<boost::multiprecision::number<T> >::value>
 struct scoped_N_precision
 {
    template <class U>
    scoped_N_precision(U const&) {}
    template <class U>
    void reduce(U&) {}
 };
 
 template <class T, unsigned N>
 struct scoped_N_precision<T, N, true>
 {
    unsigned old_precision, old_arg_precision;
    scoped_N_precision(T& arg)
    {
       old_precision     = T::thread_default_precision();
       old_arg_precision = arg.precision();
       T::thread_default_precision(old_arg_precision * N);
       arg.precision(old_arg_precision * N);
    }
    ~scoped_N_precision()
    {
       T::thread_default_precision(old_precision);
    }
    void reduce(T& arg) 
    {
       arg.precision(old_arg_precision);
    }
 };
 
 template <class T>
 void reduce_n_half_pi(T& arg, const T& n, bool go_down)
 {
    //
    // We need to perform argument reduction at 3 times the precision of arg
    // in order to ensure a correct result up to arg = 1/epsilon.  Beyond that
    // the value of n will have been incorrectly calculated anyway since it will
    // have a value greater than 1/epsilon and no longer be an exact integer value.
    //
    // More information in ARGUMENT REDUCTION FOR HUGE ARGUMENTS. K C Ng.
    //
    // There are two mutually exclusive ways to achieve this, both of which are 
    // supported here:
    // 1) To define a fixed precision type with 3 times the precision for the calculation.
    // 2) To dynamically increase the precision of the variables.
    //
    using reduction_type = typename boost::multiprecision::detail::transcendental_reduction_type<T>::type;
    //
    // Make a copy of the arg at higher precision:
    //
    reduction_type big_arg(arg);
    //
    // Dynamically increase precision when supported, this increases the default
    // and ups the precision of big_arg to match:
    //
    scoped_N_precision<T, 3> scoped_precision(big_arg);
    //
    // High precision PI:
    //
    reduction_type reduction = get_constant_pi<reduction_type>();
    eval_ldexp(reduction, reduction, -1); // divide by 2
    eval_multiply(reduction, n);
 
    BOOST_MATH_INSTRUMENT_CODE(big_arg.str(10, std::ios_base::scientific));
    BOOST_MATH_INSTRUMENT_CODE(reduction.str(10, std::ios_base::scientific));
 
    if (go_down)
       eval_subtract(big_arg, reduction, big_arg);
    else
       eval_subtract(big_arg, reduction);
    arg = T(big_arg);
    //
    // If arg is a variable precision type, then we have just copied the
    // precision of big_arg s well it's value.  Reduce the precision now:
    //
    scoped_precision.reduce(arg);
    BOOST_MATH_INSTRUMENT_CODE(big_arg.str(10, std::ios_base::scientific));
    BOOST_MATH_INSTRUMENT_CODE(arg.str(10, std::ios_base::scientific));
 }
 
 template <class T>
 void eval_sin(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The sin function is only valid for floating point types.");
    BOOST_MATH_INSTRUMENT_CODE(x.str(0, std::ios_base::scientific));
    if (&result == &x)
    {
       T temp;
       eval_sin(temp, x);
       result = temp;
       return;
    }
 
    using si_type = typename boost::multiprecision::detail::canonical<std::int32_t, T>::type ;
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
    using fp_type = typename std::tuple_element<0, typename T::float_types>::type                         ;
 
    switch (eval_fpclassify(x))
    {
    case FP_INFINITE:
    case FP_NAN:
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    case FP_ZERO:
       result = x;
       return;
    default:;
    }
 
    // Local copy of the argument
    T xx = x;
 
    // Analyze and prepare the phase of the argument.
    // Make a local, positive copy of the argument, xx.
    // The argument xx will be reduced to 0 <= xx <= pi/2.
    bool b_negate_sin = false;
 
    if (eval_get_sign(x) < 0)
    {
       xx.negate();
       b_negate_sin = !b_negate_sin;
    }
 
    T n_pi, t;
    T half_pi = get_constant_pi<T>();
    eval_ldexp(half_pi, half_pi, -1); // divide by 2
    // Remove multiples of pi/2.
    if (xx.compare(half_pi) > 0)
    {
       eval_divide(n_pi, xx, half_pi);
       eval_trunc(n_pi, n_pi);
       t = ui_type(4);
       eval_fmod(t, n_pi, t);
       bool b_go_down = false;
       if (t.compare(ui_type(1)) == 0)
       {
          b_go_down = true;
       }
       else if (t.compare(ui_type(2)) == 0)
       {
          b_negate_sin = !b_negate_sin;
       }
       else if (t.compare(ui_type(3)) == 0)
       {
          b_negate_sin = !b_negate_sin;
          b_go_down    = true;
       }
 
       if (b_go_down)
          eval_increment(n_pi);
       //
       // If n_pi is > 1/epsilon, then it is no longer an exact integer value
       // but an approximation.  As a result we can no longer reliably reduce
       // xx to 0 <= xx < pi/2, nor can we tell the sign of the result as we need
       // n_pi % 4 for that, but that will always be zero in this situation.
       // We could use a higher precision type for n_pi, along with division at
       // higher precision, but that's rather expensive.  So for now we do not support
       // this, and will see if anyone complains and has a legitimate use case.
       //
       if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
       {
          result = ui_type(0);
          return;
       }
 
       reduce_n_half_pi(xx, n_pi, b_go_down);
       //
       // Post reduction we may be a few ulp below zero or above pi/2
       // given that n_pi was calculated at working precision and not
       // at the higher precision used for reduction.  Correct that now:
       //
       if (eval_get_sign(xx) < 0)
       {
          xx.negate();
          b_negate_sin = !b_negate_sin;
       }
       if (xx.compare(half_pi) > 0)
       {
          eval_ldexp(half_pi, half_pi, 1);
          eval_subtract(xx, half_pi, xx);
          eval_ldexp(half_pi, half_pi, -1);
          b_go_down = !b_go_down;
       }
 
       BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
       BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
       BOOST_MP_ASSERT(xx.compare(half_pi) <= 0);
       BOOST_MP_ASSERT(xx.compare(ui_type(0)) >= 0);
    }
 
    t = half_pi;
    eval_subtract(t, xx);
 
    const bool b_zero    = eval_get_sign(xx) == 0;
    const bool b_pi_half = eval_get_sign(t) == 0;
 
    BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
    BOOST_MATH_INSTRUMENT_CODE(t.str(0, std::ios_base::scientific));
 
    // Check if the reduced argument is very close to 0 or pi/2.
    const bool b_near_zero    = xx.compare(fp_type(1e-1)) < 0;
    const bool b_near_pi_half = t.compare(fp_type(1e-1)) < 0;
 
    if (b_zero)
    {
       result = ui_type(0);
    }
    else if (b_pi_half)
    {
       result = ui_type(1);
    }
    else if (b_near_zero)
    {
       eval_multiply(t, xx, xx);
       eval_divide(t, si_type(-4));
       T t2;
       t2 = fp_type(1.5);
       hyp0F1(result, t2, t);
       BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
       eval_multiply(result, xx);
    }
    else if (b_near_pi_half)
    {
       eval_multiply(t, t);
       eval_divide(t, si_type(-4));
       T t2;
       t2 = fp_type(0.5);
       hyp0F1(result, t2, t);
       BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
    }
    else
    {
       // Scale to a small argument for an efficient Taylor series,
       // implemented as a hypergeometric function. Use a standard
       // divide by three identity a certain number of times.
       // Here we use division by 3^9 --> (19683 = 3^9).
 
       constexpr si_type n_scale           = 9;
       constexpr si_type n_three_pow_scale = static_cast<si_type>(19683L);
 
       eval_divide(xx, n_three_pow_scale);
 
       // Now with small arguments, we are ready for a series expansion.
       eval_multiply(t, xx, xx);
       eval_divide(t, si_type(-4));
       T t2;
       t2 = fp_type(1.5);
       hyp0F1(result, t2, t);
       BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
       eval_multiply(result, xx);
 
       // Convert back using multiple angle identity.
       for (std::int32_t k = static_cast<std::int32_t>(0); k < n_scale; k++)
       {
          // Rescale the cosine value using the multiple angle identity.
          eval_multiply(t2, result, ui_type(3));
          eval_multiply(t, result, result);
          eval_multiply(t, result);
          eval_multiply(t, ui_type(4));
          eval_subtract(result, t2, t);
       }
    }
 
    if (b_negate_sin)
       result.negate();
    BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
 }
 
 template <class T>
 void eval_cos(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The cos function is only valid for floating point types.");
    if (&result == &x)
    {
       T temp;
       eval_cos(temp, x);
       result = temp;
       return;
    }
 
    using si_type = typename boost::multiprecision::detail::canonical<std::int32_t, T>::type ;
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
 
    switch (eval_fpclassify(x))
    {
    case FP_INFINITE:
    case FP_NAN:
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    case FP_ZERO:
       result = ui_type(1);
       return;
    default:;
    }
 
    // Local copy of the argument
    T xx = x;
 
    // Analyze and prepare the phase of the argument.
    // Make a local, positive copy of the argument, xx.
    // The argument xx will be reduced to 0 <= xx <= pi/2.
    bool b_negate_cos = false;
 
    if (eval_get_sign(x) < 0)
    {
       xx.negate();
    }
    BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
 
    T n_pi, t;
    T half_pi = get_constant_pi<T>();
    eval_ldexp(half_pi, half_pi, -1); // divide by 2
    // Remove even multiples of pi.
    if (xx.compare(half_pi) > 0)
    {
       eval_divide(t, xx, half_pi);
       eval_trunc(n_pi, t);
       //
       // If n_pi is > 1/epsilon, then it is no longer an exact integer value
       // but an approximation.  As a result we can no longer reliably reduce
       // xx to 0 <= xx < pi/2, nor can we tell the sign of the result as we need
       // n_pi % 4 for that, but that will always be zero in this situation.
       // We could use a higher precision type for n_pi, along with division at
       // higher precision, but that's rather expensive.  So for now we do not support
       // this, and will see if anyone complains and has a legitimate use case.
       //
       if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
       {
          result = ui_type(1);
          return;
       }
       BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
       t = ui_type(4);
       eval_fmod(t, n_pi, t);
 
       bool b_go_down = false;
       if (t.compare(ui_type(0)) == 0)
       {
          b_go_down = true;
       }
       else if (t.compare(ui_type(1)) == 0)
       {
          b_negate_cos = true;
       }
       else if (t.compare(ui_type(2)) == 0)
       {
          b_go_down    = true;
          b_negate_cos = true;
       }
       else
       {
          BOOST_MP_ASSERT(t.compare(ui_type(3)) == 0);
       }
 
       if (b_go_down)
          eval_increment(n_pi);
 
       reduce_n_half_pi(xx, n_pi, b_go_down);
       //
       // Post reduction we may be a few ulp below zero or above pi/2
       // given that n_pi was calculated at working precision and not
       // at the higher precision used for reduction.  Correct that now:
       //
       if (eval_get_sign(xx) < 0)
       {
          xx.negate();
          b_negate_cos = !b_negate_cos;
       }
       if (xx.compare(half_pi) > 0)
       {
          eval_ldexp(half_pi, half_pi, 1);
          eval_subtract(xx, half_pi, xx);
          eval_ldexp(half_pi, half_pi, -1);
       }
       BOOST_MP_ASSERT(xx.compare(half_pi) <= 0);
       BOOST_MP_ASSERT(xx.compare(ui_type(0)) >= 0);
    }
    else
    {
       n_pi = ui_type(1);
       reduce_n_half_pi(xx, n_pi, true);
    }
 
    const bool b_zero = eval_get_sign(xx) == 0;
 
    if (b_zero)
    {
       result = si_type(0);
    }
    else
    {
       eval_sin(result, xx);
    }
    if (b_negate_cos)
       result.negate();
    BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
 }
 
 template <class T>
 void eval_tan(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The tan function is only valid for floating point types.");
    if (&result == &x)
    {
       T temp;
       eval_tan(temp, x);
       result = temp;
       return;
    }
    T t;
    eval_sin(result, x);
    eval_cos(t, x);
    eval_divide(result, t);
 }
 
 template <class T>
 void hyp2F1(T& result, const T& a, const T& b, const T& c, const T& x)
 {
    // Compute the series representation of hyperg_2f1 taken from
    // Abramowitz and Stegun 15.1.1.
    // There are no checks on input range or parameter boundaries.
 
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
 
    T x_pow_n_div_n_fact(x);
    T pochham_a(a);
    T pochham_b(b);
    T pochham_c(c);
    T ap(a);
    T bp(b);
    T cp(c);
 
    eval_multiply(result, pochham_a, pochham_b);
    eval_divide(result, pochham_c);
    eval_multiply(result, x_pow_n_div_n_fact);
    eval_add(result, ui_type(1));
 
    T lim;
    eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
 
    if (eval_get_sign(lim) < 0)
       lim.negate();
 
    ui_type n;
    T       term;
 
    const unsigned series_limit =
        boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
            ? 100
            : boost::multiprecision::detail::digits2<number<T, et_on> >::value();
    // Series expansion of hyperg_2f1(a, b; c; x).
    for (n = 2; n < series_limit; ++n)
    {
       eval_multiply(x_pow_n_div_n_fact, x);
       eval_divide(x_pow_n_div_n_fact, n);
 
       eval_increment(ap);
       eval_multiply(pochham_a, ap);
       eval_increment(bp);
       eval_multiply(pochham_b, bp);
       eval_increment(cp);
       eval_multiply(pochham_c, cp);
 
       eval_multiply(term, pochham_a, pochham_b);
       eval_divide(term, pochham_c);
       eval_multiply(term, x_pow_n_div_n_fact);
       eval_add(result, term);
 
       if (eval_get_sign(term) < 0)
          term.negate();
       if (lim.compare(term) >= 0)
          break;
    }
    if (n > series_limit)
       BOOST_MP_THROW_EXCEPTION(std::runtime_error("H2F1 failed to converge."));
 }
 
 template <class T>
 void eval_asin(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The asin function is only valid for floating point types.");
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
    using fp_type = typename std::tuple_element<0, typename T::float_types>::type                         ;
 
    if (&result == &x)
    {
       T t(x);
       eval_asin(result, t);
       return;
    }
 
    switch (eval_fpclassify(x))
    {
    case FP_NAN:
    case FP_INFINITE:
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    case FP_ZERO:
       result = x;
       return;
    default:;
    }
 
    const bool b_neg = eval_get_sign(x) < 0;
 
    T xx(x);
    if (b_neg)
       xx.negate();
 
    int c = xx.compare(ui_type(1));
    if (c > 0)
    {
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    }
    else if (c == 0)
    {
       result = get_constant_pi<T>();
       eval_ldexp(result, result, -1);
       if (b_neg)
          result.negate();
       return;
    }
 
    if (xx.compare(fp_type(1e-3)) < 0)
    {
       // http://functions.wolfram.com/ElementaryFunctions/ArcSin/26/01/01/
       eval_multiply(xx, xx);
       T t1, t2;
       t1 = fp_type(0.5f);
       t2 = fp_type(1.5f);
       hyp2F1(result, t1, t1, t2, xx);
       eval_multiply(result, x);
       return;
    }
    else if (xx.compare(fp_type(1 - 5e-2f)) > 0)
    {
       // http://functions.wolfram.com/ElementaryFunctions/ArcSin/26/01/01/
       // This branch is simlilar in complexity to Newton iterations down to
       // the above limit.  It is *much* more accurate.
       T dx1;
       T t1, t2;
       eval_subtract(dx1, ui_type(1), xx);
       t1 = fp_type(0.5f);
       t2 = fp_type(1.5f);
       eval_ldexp(dx1, dx1, -1);
       hyp2F1(result, t1, t1, t2, dx1);
       eval_ldexp(dx1, dx1, 2);
       eval_sqrt(t1, dx1);
       eval_multiply(result, t1);
       eval_ldexp(t1, get_constant_pi<T>(), -1);
       result.negate();
       eval_add(result, t1);
       if (b_neg)
          result.negate();
       return;
    }
 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
    using guess_type = typename boost::multiprecision::detail::canonical<long double, T>::type;
 #else
    using guess_type = fp_type;
 #endif
    // Get initial estimate using standard math function asin.
    guess_type dd;
    eval_convert_to(&dd, xx);
 
    result = (guess_type)(std::asin(dd));
 
    // Newton-Raphson iteration, we should double our precision with each iteration,
    // in practice this seems to not quite work in all cases... so terminate when we
    // have at least 2/3 of the digits correct on the assumption that the correction
    // we've just added will finish the job...
 
    std::intmax_t current_precision = eval_ilogb(result);
    std::intmax_t target_precision  = std::numeric_limits<number<T> >::is_specialized ? 
       current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
       : current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
 
    // Newton-Raphson iteration
    while (current_precision > target_precision)
    {
       T sine, cosine;
       eval_sin(sine, result);
       eval_cos(cosine, result);
       eval_subtract(sine, xx);
       eval_divide(sine, cosine);
       eval_subtract(result, sine);
       current_precision = eval_ilogb(sine);
       if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
          break;
    }
    if (b_neg)
       result.negate();
 }
 
 template <class T>
 inline void eval_acos(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The acos function is only valid for floating point types.");
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
 
    switch (eval_fpclassify(x))
    {
    case FP_NAN:
    case FP_INFINITE:
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    case FP_ZERO:
       result = get_constant_pi<T>();
       eval_ldexp(result, result, -1); // divide by two.
       return;
    }
 
    T xx;
    eval_abs(xx, x);
    int c = xx.compare(ui_type(1));
 
    if (c > 0)
    {
       BOOST_IF_CONSTEXPR(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
       {
          result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
          errno  = EDOM;
       }
       else
          BOOST_MP_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
       return;
    }
    else if (c == 0)
    {
       if (eval_get_sign(x) < 0)
          result = get_constant_pi<T>();
       else
          result = ui_type(0);
       return;
    }
 
    using fp_type = typename std::tuple_element<0, typename T::float_types>::type;
 
    if (xx.compare(fp_type(1e-3)) < 0)
    {
       // https://functions.wolfram.com/ElementaryFunctions/ArcCos/26/01/01/
       eval_multiply(xx, xx);
       T t1, t2;
       t1 = fp_type(0.5f);
       t2 = fp_type(1.5f);
       hyp2F1(result, t1, t1, t2, xx);
       eval_multiply(result, x);
       eval_ldexp(t1, get_constant_pi<T>(), -1);
       result.negate();
       eval_add(result, t1);
       return;
    }
    if (eval_get_sign(x) < 0)
    {
       eval_acos(result, xx);
       result.negate();
       eval_add(result, get_constant_pi<T>());
       return;
    }
    else if (xx.compare(fp_type(0.85)) > 0)
    {
       // https://functions.wolfram.com/ElementaryFunctions/ArcCos/26/01/01/
       // This branch is simlilar in complexity to Newton iterations down to
       // the above limit.  It is *much* more accurate.
       T dx1;
       T t1, t2;
       eval_subtract(dx1, ui_type(1), xx);
       t1 = fp_type(0.5f);
       t2 = fp_type(1.5f);
       eval_ldexp(dx1, dx1, -1);
       hyp2F1(result, t1, t1, t2, dx1);
       eval_ldexp(dx1, dx1, 2);
       eval_sqrt(t1, dx1);
       eval_multiply(result, t1);
       return;
    }
 
 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
    using guess_type = typename boost::multiprecision::detail::canonical<long double, T>::type;
 #else
    using guess_type = fp_type;
 #endif
    // Get initial estimate using standard math function asin.
    guess_type dd;
    eval_convert_to(&dd, xx);
 
    result = (guess_type)(std::acos(dd));
 
    // Newton-Raphson iteration, we should double our precision with each iteration,
    // in practice this seems to not quite work in all cases... so terminate when we
    // have at least 2/3 of the digits correct on the assumption that the correction
    // we've just added will finish the job...
 
    std::intmax_t current_precision = eval_ilogb(result);
    std::intmax_t target_precision = std::numeric_limits<number<T> >::is_specialized ?
       current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
       : current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
 
    // Newton-Raphson iteration
    while (current_precision > target_precision)
    {
       T sine, cosine;
       eval_sin(sine, result);
       eval_cos(cosine, result);
       eval_subtract(cosine, xx);
       cosine.negate();
       eval_divide(cosine, sine);
       eval_subtract(result, cosine);
       current_precision = eval_ilogb(cosine);
       if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
          break;
    }
 }
 
 template <class T>
 void eval_atan(T& result, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The atan function is only valid for floating point types.");
    using si_type = typename boost::multiprecision::detail::canonical<std::int32_t, T>::type ;
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
    using fp_type = typename std::tuple_element<0, typename T::float_types>::type                         ;
 
    switch (eval_fpclassify(x))
    {
    case FP_NAN:
       result = x;
       errno  = EDOM;
       return;
    case FP_ZERO:
       result = x;
       return;
    case FP_INFINITE:
       if (eval_get_sign(x) < 0)
       {
          eval_ldexp(result, get_constant_pi<T>(), -1);
          result.negate();
       }
       else
          eval_ldexp(result, get_constant_pi<T>(), -1);
       return;
    default:;
    }
 
    const bool b_neg = eval_get_sign(x) < 0;
 
    T xx(x);
    if (b_neg)
       xx.negate();
 
    if (xx.compare(fp_type(0.1)) < 0)
    {
       T t1, t2, t3;
       t1 = ui_type(1);
       t2 = fp_type(0.5f);
       t3 = fp_type(1.5f);
       eval_multiply(xx, xx);
       xx.negate();
       hyp2F1(result, t1, t2, t3, xx);
       eval_multiply(result, x);
       return;
    }
 
    if (xx.compare(fp_type(10)) > 0)
    {
       T t1, t2, t3;
       t1 = fp_type(0.5f);
       t2 = ui_type(1u);
       t3 = fp_type(1.5f);
       eval_multiply(xx, xx);
       eval_divide(xx, si_type(-1), xx);
       hyp2F1(result, t1, t2, t3, xx);
       eval_divide(result, x);
       if (!b_neg)
          result.negate();
       eval_ldexp(t1, get_constant_pi<T>(), -1);
       eval_add(result, t1);
       if (b_neg)
          result.negate();
       return;
    }
 
    // Get initial estimate using standard math function atan.
    fp_type d;
    eval_convert_to(&d, xx);
    result = fp_type(std::atan(d));
 
    // Newton-Raphson iteration, we should double our precision with each iteration,
    // in practice this seems to not quite work in all cases... so terminate when we
    // have at least 2/3 of the digits correct on the assumption that the correction
    // we've just added will finish the job...
 
    std::intmax_t current_precision = eval_ilogb(result);
    std::intmax_t target_precision  = std::numeric_limits<number<T> >::is_specialized ?
       current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
       : current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
 
    T s, c, t;
    while (current_precision > target_precision)
    {
       eval_sin(s, result);
       eval_cos(c, result);
       eval_multiply(t, xx, c);
       eval_subtract(t, s);
       eval_multiply(s, t, c);
       eval_add(result, s);
       current_precision = eval_ilogb(s);
       if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
          break;
    }
    if (b_neg)
       result.negate();
 }
 
 template <class T>
 void eval_atan2(T& result, const T& y, const T& x)
 {
    static_assert(number_category<T>::value == number_kind_floating_point, "The atan2 function is only valid for floating point types.");
    if (&result == &y)
    {
       T temp(y);
       eval_atan2(result, temp, x);
       return;
    }
    else if (&result == &x)
    {
       T temp(x);
       eval_atan2(result, y, temp);
       return;
    }
 
    using ui_type = typename boost::multiprecision::detail::canonical<std::uint32_t, T>::type;
 
    switch (eval_fpclassify(y))
    {
    case FP_NAN:
       result = y;
       errno  = EDOM;
       return;
    case FP_ZERO:
    {
       if (eval_signbit(x))
       {
          result = get_constant_pi<T>();
          if (eval_signbit(y))
             result.negate();
       }
       else
       {
          result = y; // Note we allow atan2(0,0) to be +-zero, even though it's mathematically undefined
       }
       return;
    }
    case FP_INFINITE:
    {
       if (eval_fpclassify(x) == FP_INFINITE)
       {
          if (eval_signbit(x))
          {
             // 3Pi/4
             eval_ldexp(result, get_constant_pi<T>(), -2);
             eval_subtract(result, get_constant_pi<T>());
             if (eval_get_sign(y) >= 0)
                result.negate();
          }
          else
          {
             // Pi/4
             eval_ldexp(result, get_constant_pi<T>(), -2);
             if (eval_get_sign(y) < 0)
                result.negate();
          }
       }
       else
       {
          eval_ldexp(result, get_constant_pi<T>(), -1);
          if (eval_get_sign(y) < 0)
             result.negate();
       }
       return;
    }
    }
 
    switch (eval_fpclassify(x))
    {
    case FP_NAN:
       result = x;
       errno  = EDOM;
       return;
    case FP_ZERO:
    {
       eval_ldexp(result, get_constant_pi<T>(), -1);
       if (eval_get_sign(y) < 0)
          result.negate();
       return;
    }
    case FP_INFINITE:
       if (eval_get_sign(x) > 0)
          result = ui_type(0);
       else
          result = get_constant_pi<T>();
       if (eval_get_sign(y) < 0)
          result.negate();
       return;
    }
 
    T xx;
    eval_divide(xx, y, x);
    if (eval_get_sign(xx) < 0)
       xx.negate();
 
    eval_atan(result, xx);
 
    // Determine quadrant (sign) based on signs of x, y
    const bool y_neg = eval_get_sign(y) < 0;
    const bool x_neg = eval_get_sign(x) < 0;
 
    if (y_neg != x_neg)
       result.negate();
 
    if (x_neg)
    {
       if (y_neg)
          eval_subtract(result, get_constant_pi<T>());
       else
          eval_add(result, get_constant_pi<T>());
    }
 }
 template <class T, class A>
 inline typename std::enable_if<boost::multiprecision::detail::is_arithmetic<A>::value, void>::type eval_atan2(T& result, const T& x, const A& a)
 {
    using canonical_type = typename boost::multiprecision::detail::canonical<A, T>::type         ;
    using cast_type = typename std::conditional<std::is_same<A, canonical_type>::value, T, canonical_type>::type;
    cast_type                                                                      c;
    c = a;
    eval_atan2(result, x, c);
 }
 
 template <class T, class A>
 inline typename std::enable_if<boost::multiprecision::detail::is_arithmetic<A>::value, void>::type eval_atan2(T& result, const A& x, const T& a)
 {
    using canonical_type = typename boost::multiprecision::detail::canonical<A, T>::type         ;
    using cast_type = typename std::conditional<std::is_same<A, canonical_type>::value, T, canonical_type>::type;
    cast_type                                                                      c;
    c = x;
    eval_atan2(result, c, a);
 }
 
 #ifdef BOOST_MSVC
 #pragma warning(pop)
 #endif

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