From 0e48617447ee5c03023b3351f1dd2bf9d3973f61 Mon Sep 17 00:00:00 2001
From: Jacob Hass <jacobhass8@gmail.com>
Date: Tue, 10 Feb 2026 11:02:47 -0800
Subject: [PATCH] Reverted to checking the relative error for small nc

---
 .../math/distributions/non_central_f.hpp      | 21 ++++++++-----------
 1 file changed, 9 insertions(+), 12 deletions(-)

diff --git a/include/boost/math/distributions/non_central_f.hpp b/include/boost/math/distributions/non_central_f.hpp
index a6df68b8f..60d5926a9 100644
--- a/include/boost/math/distributions/non_central_f.hpp
+++ b/include/boost/math/distributions/non_central_f.hpp
@@ -46,7 +46,8 @@ namespace boost
          };
          
          template <class RealType, class Policy>
-         BOOST_MATH_GPU_ENABLED RealType find_non_centrality_f(const RealType x, const RealType v1, const RealType v2, const RealType p, const RealType q, const RealType p_q_precision, const Policy& pol)         {
+         BOOST_MATH_GPU_ENABLED RealType find_non_centrality_f(const RealType x, const RealType v1, const RealType v2, const RealType p, const RealType q, const Policy& pol)         
+         {
             constexpr auto function = "non_central_f<%1%>::find_non_centrality";
 
             if ( p == 0 || q == 0) {
@@ -54,16 +55,14 @@ namespace boost
                   RealType(boost::math::numeric_limits<RealType>::quiet_NaN()), Policy()); // LCOV_EXCL_LINE
             }
 
-            // Check if nc = 0 (which is just the F-distribution)
             non_centrality_finder_f<RealType, Policy> f(x, v1, v2, p < q ? p : q, p < q ? false : true);
-            // This occurs when the root finder would need to find a result smaller than
-            // tools::min_value (which it cannot do).  Note that we have to add in a small
-            // amount of "tolerance" since the subtraction in our termination condition
-            // implies a small amount of wobble in the result which should be of the
-            // order p * eps (note not q * eps, since q is calculated as 1-p).
-            // Also note that p_q_precision is passed down from our caller as the
-            // epsilon of the original called values, and not after possible promotion.
-            if (f(tools::min_value<RealType>()) <= 3 * p_q_precision * p){
+
+            // Check if nc = 0 (which is just the F-distribution)
+            // When the relative difference between the cdf with nc=0
+            // and p drops below 1e-7, the function f becomes 
+            // numerically unstable. Thus, we can't find nc below this point. 
+            // See PR 1345 for more details. 
+            if (abs(f(tools::min_value<RealType>()) / p) <= 1e-7){
                return 0;
             }
 
@@ -136,7 +135,6 @@ namespace boost
                static_cast<eval_type>(v2),
                static_cast<eval_type>(p),
                static_cast<eval_type>(1-p),
-               static_cast<eval_type>(tools::epsilon<RealType>()),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
@@ -159,7 +157,6 @@ namespace boost
                static_cast<eval_type>(c.param2),
                static_cast<eval_type>(1-c.param3),
                static_cast<eval_type>(c.param3),
-               static_cast<eval_type>(tools::epsilon<RealType>()),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,