Fix lower incomplete gamma functions with x = 0

In this case, the errno error handling did not work correctly,
as internal functions where accidently setting it, although no
overflow happens.

Fixes boostorg#1249.

Author: cohomology

include/boost/math/special_functions/gamma.hpp

@@ -908,7 +908,7 @@ BOOST_MATH_GPU_ENABLED T full_igamma_prefix(T a, T z, const Policy& pol)
 {
    BOOST_MATH_STD_USING
 
-   if (z > tools::max_value<T>())
+   if (z > tools::max_value<T>() || (a > 0 && z == 0))
       return 0;
 
    T alz = a * log(z);
@@ -962,7 +962,7 @@ template <class T, class Policy, class Lanczos>
 BOOST_MATH_GPU_ENABLED T regularised_gamma_prefix(T a, T z, const Policy& pol, const Lanczos& l)
 {
    BOOST_MATH_STD_USING
-   if (z >= tools::max_value<T>())
+   if (z >= tools::max_value<T>() || (a > 0 && z == 0))
       return 0;
    T agh = a + static_cast<T>(Lanczos::g()) - T(0.5);
    T prefix;
@@ -1297,7 +1297,11 @@ BOOST_MATH_GPU_ENABLED T gamma_incomplete_imp_final(T a, T x, bool normalised, b
 
    int eval_method;
 
-   if(is_int && (x > 0.6))
+   if (x == 0)
+   {
+      eval_method = 8; // do nothing
+   }
+   else if(is_int && (x > 0.6))
    {
       // calculate Q via finite sum:
       invert = !invert;
@@ -1606,6 +1610,11 @@ BOOST_MATH_GPU_ENABLED T gamma_incomplete_imp_final(T a, T x, bool normalised, b
          result *= incomplete_tgamma_large_x(a, x, pol);
       break;
    }
+   case 8:
+      // x is zero, result is zero.
+      if (p_derivative)
+         *p_derivative = 0;
+      break;
    }
 
    if(normalised && (result > 1))
@@ -1627,7 +1636,7 @@ BOOST_MATH_GPU_ENABLED T gamma_incomplete_imp_final(T a, T x, bool normalised, b
       #endif
       result = gam - result;
    }
-   if(p_derivative)
+   if(p_derivative && x > 0)
    {
       //
       // Need to convert prefix term to derivative:
@@ -1660,7 +1669,7 @@ BOOST_MATH_GPU_ENABLED T gamma_incomplete_imp(T a, T x, bool normalised, bool in
 
    T result = 0; // Just to avoid warning C4701: potentially uninitialized local variable 'result' used
 
-   if(a >= max_factorial<T>::value && !normalised)
+   if(x > 0 && a >= max_factorial<T>::value && !normalised)
    {
       //
       // When we're computing the non-normalized incomplete gamma

test/Jamfile.v2

@@ -194,6 +194,7 @@ test-suite special_fun :
    [ run git_issue_1139.cpp ]
    [ run git_issue_1175.cpp ]
    [ run git_issue_1194.cpp ]
+   [ run git_issue_1249.cpp /boost/test//boost_unit_test_framework  ]
    [ run special_functions_test.cpp /boost/test//boost_unit_test_framework  ]
    [ run test_airy.cpp test_instances//test_instances pch_light /boost/test//boost_unit_test_framework  ]
    [ run test_bessel_j.cpp test_instances//test_instances pch_light /boost/test//boost_unit_test_framework  ]

test/git_issue_1249.cpp

@@ -0,0 +1,123 @@
+//  (C) Copyright Kilian Kilger 2025.
+//  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)
+
+#define BOOST_TEST_MAIN
+
+#include <boost/test/unit_test.hpp>
+#include <boost/test/tools/floating_point_comparison.hpp>
+#include <boost/test/unit_test.hpp>
+#include <boost/test/results_collector.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+
+using namespace std;
+using namespace boost::math;
+using namespace boost::math::policies;
+
+typedef policy<
+   policies::domain_error<errno_on_error>,
+   policies::pole_error<errno_on_error>,
+   policies::overflow_error<errno_on_error>,
+   policies::evaluation_error<errno_on_error>
+> c_policy;
+
+template<typename T>
+struct test_lower
+{
+   T operator()(T a, T x) const
+   {
+      return tgamma_lower(a, x, c_policy());
+   }
+
+   T expected(T a) const
+   {
+      return T(0.0);
+   }
+};
+
+template<typename T>
+struct test_upper
+{
+   T operator()(T a, T x) const
+   {
+      return tgamma(a, x, c_policy());
+   }
+   T expected(T a) const
+   {
+      return tgamma(a, c_policy());
+   }
+};
+
+template<typename T>
+struct test_gamma_p
+{
+   T operator()(T a, T x) const
+   {
+      return gamma_p(a, x, c_policy());
+   }
+   T expected(T) const
+   {
+      return T(0.0);
+   }
+};
+
+template<typename T>
+struct test_gamma_q
+{
+   T operator()(T a, T x) const
+   {
+      return gamma_q(a, x, c_policy());
+   }
+   T expected(T) const
+   {
+      return T(1.0);
+   }
+};
+
+template<typename T, template<typename> class Fun>
+void test_impl(T a)
+{
+   Fun<T> fn;
+   errno = 0;
+   T x = T(0.0);
+   T result = fn(a, x);
+   int saveErrno = errno;
+
+   errno = 0;
+
+   T expected = fn.expected(a);
+
+   BOOST_CHECK(errno == saveErrno);
+   BOOST_CHECK_EQUAL(result, expected);
+}
+
+template<template<typename> class Fun>
+void test_type_dispatch(float a)
+{
+   test_impl<float, Fun>(a);
+   test_impl<double, Fun>(double(a));
+   test_impl<long double, Fun>(static_cast<long double>(a));
+}
+
+template<template<typename> class Fun>
+void test_impl()
+{
+   test_type_dispatch<Fun>(1.0);
+   test_type_dispatch<Fun>(0.1);
+   test_type_dispatch<Fun>(0.5);
+   test_type_dispatch<Fun>(0.6);
+   test_type_dispatch<Fun>(1.3);
+   test_type_dispatch<Fun>(1.5);
+   test_type_dispatch<Fun>(2);
+   test_type_dispatch<Fun>(100);
+   test_type_dispatch<Fun>(std::numeric_limits<float>::max());
+}
+
+BOOST_AUTO_TEST_CASE( test_main )
+{
+   test_impl<test_lower>();
+   test_impl<test_upper>();
+   test_impl<test_gamma_p>();
+   test_impl<test_gamma_q>();
+}