// Copyright (c) 2006 Xiaogang Zhang, 2015 John Maddock.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// History:
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// XZ wrote the original of this file as part of the Google
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// Summer of Code 2006. JM modified it slightly to fit into the
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// Boost.Math conceptual framework better.
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// Updated 2015 to use Carlson's latest methods.
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#ifndef BOOST_MATH_ELLINT_RD_HPP
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#define BOOST_MATH_ELLINT_RD_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/special_functions/ellint_rc.hpp>
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#include <boost/math/special_functions/pow.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/policies/error_handling.hpp>
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// Carlson's elliptic integral of the second kind
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// R_D(x, y, z) = R_J(x, y, z, z) = 1.5 * \int_{0}^{\infty} [(t+x)(t+y)]^{-1/2} (t+z)^{-3/2} dt
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// Carlson, Numerische Mathematik, vol 33, 1 (1979)
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namespace boost { namespace math { namespace detail{
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template <typename T, typename Policy>
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T ellint_rd_imp(T x, T y, T z, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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using std::swap;
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static const char* function = "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)";
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if(x < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument x must be >= 0, but got %1%", x, pol);
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}
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if(y < 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument y must be >= 0, but got %1%", y, pol);
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}
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if(z <= 0)
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{
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return policies::raise_domain_error<T>(function,
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"Argument z must be > 0, but got %1%", z, pol);
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}
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if(x + y == 0)
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{
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return policies::raise_domain_error<T>(function,
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"At most one argument can be zero, but got, x + y = %1%", x + y, pol);
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}
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//
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// Special cases from http://dlmf.nist.gov/19.20#iv
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//
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using std::swap;
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if(x == z)
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swap(x, y);
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if(y == z)
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{
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if(x == y)
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{
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return 1 / (x * sqrt(x));
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}
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else if(x == 0)
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{
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return 3 * constants::pi<T>() / (4 * y * sqrt(y));
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}
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else
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{
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if((std::min)(x, y) / (std::max)(x, y) > 1.3)
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return 3 * (ellint_rc_imp(x, y, pol) - sqrt(x) / y) / (2 * (y - x));
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// Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
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}
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}
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if(x == y)
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{
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if((std::min)(x, z) / (std::max)(x, z) > 1.3)
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return 3 * (ellint_rc_imp(z, x, pol) - 1 / sqrt(z)) / (z - x);
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// Otherwise fall through to avoid cancellation in the above (RC(x,y) -> 1/x^0.5 as x -> y)
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}
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if(y == 0)
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swap(x, y);
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if(x == 0)
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{
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//
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// Special handling for common case, from
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// Numerical Computation of Real or Complex Elliptic Integrals, eq.47
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//
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T xn = sqrt(y);
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T yn = sqrt(z);
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T x0 = xn;
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T y0 = yn;
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T sum = 0;
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T sum_pow = 0.25f;
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while(fabs(xn - yn) >= 2.7 * tools::root_epsilon<T>() * fabs(xn))
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{
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T t = sqrt(xn * yn);
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xn = (xn + yn) / 2;
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yn = t;
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sum_pow *= 2;
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sum += sum_pow * boost::math::pow<2>(xn - yn);
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}
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T RF = constants::pi<T>() / (xn + yn);
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//
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// This following calculation suffers from serious cancellation when y ~ z
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// unless we combine terms. We have:
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//
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// ( ((x0 + y0)/2)^2 - z ) / (z(y-z))
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//
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// Substituting y = x0^2 and z = y0^2 and simplifying we get the following:
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//
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T pt = (x0 + 3 * y0) / (4 * z * (x0 + y0));
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//
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// Since we've moved the denominator from eq.47 inside the expression, we
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// need to also scale "sum" by the same value:
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//
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pt -= sum / (z * (y - z));
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return pt * RF * 3;
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}
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T xn = x;
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T yn = y;
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T zn = z;
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T An = (x + y + 3 * z) / 5;
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T A0 = An;
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// This has an extra 1.2 fudge factor which is really only needed when x, y and z are close in magnitude:
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T Q = pow(tools::epsilon<T>() / 4, -T(1) / 8) * (std::max)((std::max)(An - x, An - y), An - z) * 1.2f;
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BOOST_MATH_INSTRUMENT_VARIABLE(Q);
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T lambda, rx, ry, rz;
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unsigned k = 0;
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T fn = 1;
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T RD_sum = 0;
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for(; k < policies::get_max_series_iterations<Policy>(); ++k)
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{
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rx = sqrt(xn);
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ry = sqrt(yn);
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rz = sqrt(zn);
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lambda = rx * ry + rx * rz + ry * rz;
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RD_sum += fn / (rz * (zn + lambda));
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An = (An + lambda) / 4;
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xn = (xn + lambda) / 4;
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yn = (yn + lambda) / 4;
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zn = (zn + lambda) / 4;
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fn /= 4;
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Q /= 4;
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BOOST_MATH_INSTRUMENT_VARIABLE(k);
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BOOST_MATH_INSTRUMENT_VARIABLE(RD_sum);
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BOOST_MATH_INSTRUMENT_VARIABLE(Q);
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if(Q < An)
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break;
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}
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policies::check_series_iterations<T, Policy>(function, k, pol);
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T X = fn * (A0 - x) / An;
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T Y = fn * (A0 - y) / An;
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T Z = -(X + Y) / 3;
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T E2 = X * Y - 6 * Z * Z;
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T E3 = (3 * X * Y - 8 * Z * Z) * Z;
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T E4 = 3 * (X * Y - Z * Z) * Z * Z;
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T E5 = X * Y * Z * Z * Z;
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T result = fn * pow(An, T(-3) / 2) *
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(1 - 3 * E2 / 14 + E3 / 6 + 9 * E2 * E2 / 88 - 3 * E4 / 22 - 9 * E2 * E3 / 52 + 3 * E5 / 26 - E2 * E2 * E2 / 16
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+ 3 * E3 * E3 / 40 + 3 * E2 * E4 / 20 + 45 * E2 * E2 * E3 / 272 - 9 * (E3 * E4 + E2 * E5) / 68);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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result += 3 * RD_sum;
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return result;
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}
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} // namespace detail
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template <class T1, class T2, class T3, class Policy>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rd(T1 x, T2 y, T3 z, const Policy& pol)
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{
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typedef typename tools::promote_args<T1, T2, T3>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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return policies::checked_narrowing_cast<result_type, Policy>(
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detail::ellint_rd_imp(
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static_cast<value_type>(x),
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static_cast<value_type>(y),
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static_cast<value_type>(z), pol), "boost::math::ellint_rd<%1%>(%1%,%1%,%1%)");
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}
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template <class T1, class T2, class T3>
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inline typename tools::promote_args<T1, T2, T3>::type
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ellint_rd(T1 x, T2 y, T3 z)
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{
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return ellint_rd(x, y, z, policies::policy<>());
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}
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}} // namespaces
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#endif // BOOST_MATH_ELLINT_RD_HPP
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