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diff --git a/boost/math/special_functions/ellint_rf.hpp b/boost/math/special_functions/ellint_rf.hpp
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+//  Copyright (c) 2006 Xiaogang Zhang
+//  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)
+//
+//  History:
+//  XZ wrote the original of this file as part of the Google
+//  Summer of Code 2006.  JM modified it to fit into the
+//  Boost.Math conceptual framework better, and to handle
+//  types longer than 80-bit reals.
+//
+#ifndef BOOST_MATH_ELLINT_RF_HPP
+#define BOOST_MATH_ELLINT_RF_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/tools/config.hpp>
+
+#include <boost/math/policies/error_handling.hpp>
+
+// Carlson's elliptic integral of the first kind
+// R_F(x, y, z) = 0.5 * \int_{0}^{\infty} [(t+x)(t+y)(t+z)]^{-1/2} dt
+// Carlson, Numerische Mathematik, vol 33, 1 (1979)
+
+namespace boost { namespace math { namespace detail{
+
+template <typename T, typename Policy>
+T ellint_rf_imp(T x, T y, T z, const Policy& pol)
+{
+    T value, X, Y, Z, E2, E3, u, lambda, tolerance;
+    unsigned long k;
+
+    BOOST_MATH_STD_USING
+    using namespace boost::math::tools;
+
+    static const char* function = "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)";
+
+    if (x < 0 || y < 0 || z < 0)
+    {
+       return policies::raise_domain_error<T>(function,
+            "domain error, all arguments must be non-negative, "
+            "only sensible result is %1%.",
+            std::numeric_limits<T>::quiet_NaN(), pol);
+    }
+    if (x + y == 0 || y + z == 0 || z + x == 0)
+    {
+       return policies::raise_domain_error<T>(function,
+            "domain error, at most one argument can be zero, "
+            "only sensible result is %1%.",
+            std::numeric_limits<T>::quiet_NaN(), pol);
+    }
+
+    // Carlson scales error as the 6th power of tolerance,
+    // but this seems not to work for types larger than
+    // 80-bit reals, this heuristic seems to work OK:
+    if(policies::digits<T, Policy>() > 64)
+    {
+      tolerance = pow(tools::epsilon<T>(), T(1)/4.25f);
+      BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
+    }
+    else
+    {
+      tolerance = pow(4*tools::epsilon<T>(), T(1)/6);
+      BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
+    }
+
+    // duplication
+    k = 1;
+    do
+    {
+        u = (x + y + z) / 3;
+        X = (u - x) / u;
+        Y = (u - y) / u;
+        Z = (u - z) / u;
+
+        // Termination condition: 
+        if ((tools::max)(abs(X), abs(Y), abs(Z)) < tolerance) 
+           break; 
+
+        T sx = sqrt(x);
+        T sy = sqrt(y);
+        T sz = sqrt(z);
+        lambda = sy * (sx + sz) + sz * sx;
+        x = (x + lambda) / 4;
+        y = (y + lambda) / 4;
+        z = (z + lambda) / 4;
+        ++k;
+    }
+    while(k < policies::get_max_series_iterations<Policy>());
+
+    // Check to see if we gave up too soon:
+    policies::check_series_iterations<T>(function, k, pol);
+    BOOST_MATH_INSTRUMENT_VARIABLE(k);
+
+    // Taylor series expansion to the 5th order
+    E2 = X * Y - Z * Z;
+    E3 = X * Y * Z;
+    value = (1 + E2*(E2/24 - E3*T(3)/44 - T(0.1)) + E3/14) / sqrt(u);
+    BOOST_MATH_INSTRUMENT_VARIABLE(value);
+
+    return value;
+}
+
+} // namespace detail
+
+template <class T1, class T2, class T3, class Policy>
+inline typename tools::promote_args<T1, T2, T3>::type 
+   ellint_rf(T1 x, T2 y, T3 z, const Policy& pol)
+{
+   typedef typename tools::promote_args<T1, T2, T3>::type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   return policies::checked_narrowing_cast<result_type, Policy>(
+      detail::ellint_rf_imp(
+         static_cast<value_type>(x),
+         static_cast<value_type>(y),
+         static_cast<value_type>(z), pol), "boost::math::ellint_rf<%1%>(%1%,%1%,%1%)");
+}
+
+template <class T1, class T2, class T3>
+inline typename tools::promote_args<T1, T2, T3>::type 
+   ellint_rf(T1 x, T2 y, T3 z)
+{
+   return ellint_rf(x, y, z, policies::policy<>());
+}
+
+}} // namespaces
+
+#endif // BOOST_MATH_ELLINT_RF_HPP
+