From c3656f941c8e962d364be082f77798e68d3e9cff Mon Sep 17 00:00:00 2001
From: jzmaddock <john@johnmaddock.co.uk>
Date: Mon, 10 Aug 2026 13:14:52 +0100
Subject: [PATCH] Correct ibeta when it's an arcsine distribution. So we don't
 take the asin of values very close to 1. Fixes:
 https://github.com/boostorg/math/issues/1429.

---
 include/boost/math/special_functions/beta.hpp | 10 +++++++++-
 test/test_students_t.cpp                      |  1 +
 2 files changed, 10 insertions(+), 1 deletion(-)

diff --git a/include/boost/math/special_functions/beta.hpp b/include/boost/math/special_functions/beta.hpp
index dc1519c76d..d136fd8ad3 100644
--- a/include/boost/math/special_functions/beta.hpp
+++ b/include/boost/math/special_functions/beta.hpp
@@ -1285,7 +1285,15 @@ BOOST_MATH_GPU_ENABLED T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, b
       {
          *p_derivative = 1 / (constants::pi<T>() * sqrt(y * x));
       }
-      T p = invert ? asin(sqrt(y)) / constants::half_pi<T>() : asin(sqrt(x)) / constants::half_pi<T>();
+      T p;
+      if (invert)
+      {
+          p = (y < 0.75) ? asin(sqrt(y)) / constants::half_pi<T>() : 1 - asin(sqrt(x)) / constants::half_pi<T>();
+      }
+      else
+      {
+          p = (x > 0.75) ? 1 - asin(sqrt(y)) / constants::half_pi<T>() : asin(sqrt(x)) / constants::half_pi<T>();
+      }
       if(!normalised)
          p *= constants::pi<T>();
       return p;
diff --git a/test/test_students_t.cpp b/test/test_students_t.cpp
index 3c63d3d639..dfc0921bb2 100644
--- a/test/test_students_t.cpp
+++ b/test/test_students_t.cpp
@@ -421,6 +421,7 @@ void test_spots(RealType)
          {
             BOOST_CHECK_THROW(boost::math::quantile(students_t_distribution<RealType>((std::numeric_limits<RealType>::min)() / 2), static_cast<RealType>(0.0025f)), std::overflow_error);
          }
+         BOOST_CHECK_CLOSE(boost::math::cdf(students_t_distribution<RealType>(static_cast<RealType>(1)), ldexp(RealType(1), -30)), static_cast<RealType>(0.5000000002964491827262478614651948102240210317156775562512071908L), tolerance);
       }
 
   // Student's t pdf tests.