Fix noncentral t degrees-of-freedom domain checking

include/boost/math/distributions/non_central_t.hpp

@@ -772,6 +772,26 @@ namespace boost
                return policies::raise_evaluation_error<RealType>(function, "Can't find degrees of freedom when the abscissa value is very close to zero as all degrees of freedom generate the same CDF at x=0: try again further out in the tails!!",
                   RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy()); // LCOV_EXCL_LINE
             }
+
+            // The limiting CDF values are
+            //
+            //   v -> 0:   Phi(-delta)
+            //   v -> inf: Phi(x - delta)
+            //
+            // Outside the interval between these limits, a monotone CDF(v)
+            // has no inverse.  In a non-monotone case there may instead be
+            // multiple inverses, so we cannot select a unique degree of freedom.
+            normal_distribution<RealType, Policy> normal;
+            RealType p0 = cdf(normal, -delta);
+            RealType pinf = cdf(normal, x - delta);
+            RealType lower = p0 < pinf ? p0 : pinf;
+            RealType upper = p0 < pinf ? pinf : p0;
+            if((p < lower) || (p > upper))
+            {
+               return policies::raise_evaluation_error<RealType>(function,
+                  "Probability %1% lies outside the interval between the limiting CDF values; the inverse is absent or non-unique", p, Policy());
+            }
+
             t_degrees_of_freedom_finder<RealType, Policy> f(delta, x, p < q ? p : q, p < q ? false : true);
             tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
             std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();

test/test_nc_t.hpp

@@ -350,6 +350,14 @@ void test_spots(RealType)
        BOOST_CHECK_CLOSE_FRACTION(cdf(d, -1), static_cast<RealType>(1.61471461239552e-127), 1e-3);
    }
 
+   // https://github.com/boostorg/math/issues/1410
+   // p=0.99 is below both limiting CDF values:
+   // Phi(2.5) and Phi(3.75).  Previously this could return a tiny,
+   // meaningless degree of freedom instead of reporting that the inverse
+   // cannot be selected uniquely.
+   BOOST_MATH_CHECK_THROW(distro1::find_degrees_of_freedom(static_cast<RealType>(-2.5), static_cast<RealType>(1.25), static_cast<RealType>(0.99)), boost::math::evaluation_error);
+   BOOST_MATH_CHECK_THROW(distro1::find_degrees_of_freedom(boost::math::complement(static_cast<RealType>(-2.5), static_cast<RealType>(1.25), static_cast<RealType>(0.01))), boost::math::evaluation_error);
+
 } // template <class RealType>void test_spots(RealType)
 
 template <class T>