From 6f59f3852acd161d56eaa04540bcc0b59b5bdd4f Mon Sep 17 00:00:00 2001
From: jzmaddock <john@johnmaddock.co.uk>
Date: Wed, 11 Mar 2026 16:39:39 +0000
Subject: [PATCH 1/2] Correct ellint_2 for N * PI / 2 phi argument. Fixes
 https://github.com/boostorg/math/issues/1377.

---
 include/boost/math/special_functions/ellint_2.hpp | 12 +++++++++---
 test/test_ellint_2.hpp                            |  6 +++++-
 2 files changed, 14 insertions(+), 4 deletions(-)

diff --git a/include/boost/math/special_functions/ellint_2.hpp b/include/boost/math/special_functions/ellint_2.hpp
index 0cc1fa0944..76aff168de 100644
--- a/include/boost/math/special_functions/ellint_2.hpp
+++ b/include/boost/math/special_functions/ellint_2.hpp
@@ -136,9 +136,15 @@ BOOST_MATH_GPU_ENABLED T ellint_e_imp(T phi, T k, const Policy& pol)
              return policies::raise_domain_error<T>("boost::math::ellint_2<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol);
           }
           T cosp = cos(rphi);
-          T c = 1 / (sinp * sinp);
-          T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
-          result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2))));
+          T c = sinp * sinp;
+          if (c > tools::min_value<T>())
+          {
+             c = 1 / c;
+             T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
+             result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2))));
+          }
+          else
+             result = 0;
        }
        if (m != 0)
        {
diff --git a/test/test_ellint_2.hpp b/test/test_ellint_2.hpp
index 29a73c9961..3b565a493d 100644
--- a/test/test_ellint_2.hpp
+++ b/test/test_ellint_2.hpp
@@ -97,7 +97,7 @@ void test_spots(T, const char* type_name)
     // called a second time.
     //
 #ifndef TEST_UDT
-    static const std::array<std::array<T, 3>, 28> data1 = {{
+    static const std::array<std::array<T, 3>, 31> data1 = {{
         {{ SC_(0.0), SC_(0.0), SC_(0.0) }},
         {{ SC_(-10.0), SC_(0.0), SC_(-10.0) }},
         {{ SC_(-1.0), SC_(-1.0), SC_(-0.84147098480789650665250232163029899962256306079837) }},
@@ -127,6 +127,10 @@ void test_spots(T, const char* type_name)
         {{ -2 * boost::math::constants::pi<T>(), SC_(1.0), SC_(-4.0) }},
         {{ -(20 * boost::math::constants::pi<T>()) / 21, SC_(1.0), SC_(-1.85095773382382555307064528472278244309033056100177750424044959888147151556780639619867349409562910808166870200808) }},
         {{ -(20 * boost::math::constants::pi<T>()) / 19, SC_(1.0), SC_(-2.16459459028073389414365205908793841951217248335965412335147127404907690285247170629778731438009399303864259295133) }},
+        // Test at n * pi / 2, see https://github.com/boostorg/math/issues/1377
+        {{ boost::math::constants::half_pi<T>(), SC_(0.25), boost::math::ellint_2(static_cast<T>(0.25)) }},
+        {{ 3 * boost::math::constants::half_pi<T>(), SC_(0.25), 3 * boost::math::ellint_2(static_cast<T>(0.25)) }},
+        {{ 6 * boost::math::constants::half_pi<T>(), SC_(0.25), 6 * boost::math::ellint_2(static_cast<T>(0.25)) }},
     }};
 
     do_test_ellint_e2<T>(data1, type_name, "Elliptic Integral E: Mathworld Data");

From c793e1cd1b4ad84dfdb5bcd551857a951ca4556d Mon Sep 17 00:00:00 2001
From: jzmaddock <john@johnmaddock.co.uk>
Date: Thu, 12 Mar 2026 10:22:45 +0000
Subject: [PATCH 2/2] Ellint2: Simplify fix.

---
 include/boost/math/special_functions/ellint_2.hpp | 14 ++++----------
 1 file changed, 4 insertions(+), 10 deletions(-)

diff --git a/include/boost/math/special_functions/ellint_2.hpp b/include/boost/math/special_functions/ellint_2.hpp
index 76aff168de..501d36b66b 100644
--- a/include/boost/math/special_functions/ellint_2.hpp
+++ b/include/boost/math/special_functions/ellint_2.hpp
@@ -122,7 +122,7 @@ BOOST_MATH_GPU_ENABLED T ellint_e_imp(T phi, T k, const Policy& pol)
           rphi = constants::half_pi<T>() - rphi;
        }
        T k2 = k * k;
-       if(boost::math::pow<3>(rphi) * k2 / 6 < tools::epsilon<T>() * fabs(rphi))
+       if(boost::math::pow<3>(rphi) * k2 / 6 <= tools::epsilon<T>() * fabs(rphi))
        {
           // See http://functions.wolfram.com/EllipticIntegrals/EllipticE2/06/01/03/0001/
           result = s * rphi;
@@ -136,15 +136,9 @@ BOOST_MATH_GPU_ENABLED T ellint_e_imp(T phi, T k, const Policy& pol)
              return policies::raise_domain_error<T>("boost::math::ellint_2<%1%>(%1%, %1%)", "The parameter k is out of range, got k = %1%", k, pol);
           }
           T cosp = cos(rphi);
-          T c = sinp * sinp;
-          if (c > tools::min_value<T>())
-          {
-             c = 1 / c;
-             T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
-             result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2))));
-          }
-          else
-             result = 0;
+          T c = 1 / (sinp * sinp);
+          T cm1 = cosp * cosp / (sinp * sinp);  // c - 1
+          result = s * ((1 - k2) * ellint_rf_imp(cm1, T(c - k2), c, pol) + k2 * (1 - k2) * ellint_rd(cm1, c, T(c - k2), pol) / 3 + k2 * sqrt(cm1 / (c * (c - k2))));
        }
        if (m != 0)
        {