Ensure Igamma does not NaN or Inf for large values.

unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h

diff --git a/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h b/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
index fe3f6d7..425231a 100644
--- a/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
+++ b/unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsImpl.h
@@ -713,6 +713,18 @@
 
 enum IgammaComputationMode { VALUE, DERIVATIVE, SAMPLE_DERIVATIVE };
 
+template <typename Scalar>
+static EIGEN_STRONG_INLINE Scalar main_igamma_term(Scalar a, Scalar x) {
+    /* Compute  x**a * exp(-x) / gamma(a)  */
+    Scalar logax = a * numext::log(x) - x - lgamma_impl<Scalar>::run(a);
+    if (logax < -numext::log(NumTraits<Scalar>::highest()) ||
+        // Assuming x and a aren't Nan.
+        (numext::isnan)(logax)) {
+      return Scalar(0);
+    }
+    return numext::exp(logax);
+}
+
 template <typename Scalar, IgammaComputationMode mode>
 EIGEN_DEVICE_FUNC
 int igamma_num_iterations() {
@@ -755,6 +767,15 @@
       return zero;
     }
 
+    Scalar ax = main_igamma_term<Scalar>(a, x);
+    // This is independent of mode. If this value is zero,
+    // then the function value is zero. If the function value is zero,
+    // then we are in a neighborhood where the function value evalutes to zero,
+    // so the derivative is zero.
+    if (ax == zero) {
+      return zero;
+    }
+
     // continued fraction
     Scalar y = one - a;
     Scalar z = x + y + one;
@@ -825,9 +846,7 @@
     }
 
     /* Compute  x**a * exp(-x) / gamma(a)  */
-    Scalar logax = a * numext::log(x) - x - lgamma_impl<Scalar>::run(a);
     Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a);
-    Scalar ax = numext::exp(logax);
     Scalar dax_da = ax * dlogax_da;
 
     switch (mode) {
@@ -858,6 +877,18 @@
     const Scalar one = 1;
     const Scalar machep = cephes_helper<Scalar>::machep();
 
+    Scalar ax = main_igamma_term<Scalar>(a, x);
+
+    // This is independent of mode. If this value is zero,
+    // then the function value is zero. If the function value is zero,
+    // then we are in a neighborhood where the function value evalutes to zero,
+    // so the derivative is zero.
+    if (ax == zero) {
+      return zero;
+    }
+
+    ax /= a;
+
     /* power series */
     Scalar r = a;
     Scalar c = one;
@@ -886,10 +917,7 @@
       }
     }
 
-    /* Compute  x**a * exp(-x) / gamma(a + 1)  */
-    Scalar logax = a * numext::log(x) - x - lgamma_impl<Scalar>::run(a + one);
     Scalar dlogax_da = numext::log(x) - digamma_impl<Scalar>::run(a + one);
-    Scalar ax = numext::exp(logax);
     Scalar dax_da = ax * dlogax_da;
 
     switch (mode) {

unsupported/test/special_functions.cpp

diff --git a/unsupported/test/special_functions.cpp b/unsupported/test/special_functions.cpp
index c104ac3..1027272 100644
--- a/unsupported/test/special_functions.cpp
+++ b/unsupported/test/special_functions.cpp
@@ -7,6 +7,7 @@
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
+#include <limits.h>
 #include "main.h"
 #include "../Eigen/SpecialFunctions"
 
@@ -74,6 +75,7 @@
       ArrayType gamma_a_x = Eigen::igamma(a, x) * a.lgamma().exp();
       ArrayType gamma_a_m1_x = Eigen::igamma(a_m1, x) * a_m1.lgamma().exp();
 
+
       // Gamma(a, 0) == Gamma(a)
       VERIFY_IS_APPROX(Eigen::igammac(a, zero), one);
 
@@ -86,6 +88,19 @@
       // gamma(a, x) == (a - 1) * gamma(a-1, x) - x^(a-1) * exp(-x)
       VERIFY_IS_APPROX(gamma_a_x, (a - 1) * gamma_a_m1_x - x.pow(a-1) * (-x).exp());
     }
+    {
+      // Verify for large a and x that values are between 0 and 1.
+      ArrayType m1 = ArrayType::Random(rows,cols);
+      ArrayType m2 = ArrayType::Random(rows,cols);
+      Scalar max_exponent = std::numeric_limits<Scalar>::max_exponent10;
+      ArrayType a = m1.abs() *  pow(10., max_exponent - 1);
+      ArrayType x = m2.abs() *  pow(10., max_exponent - 1);
+      for (int i = 0; i < a.size(); ++i) {
+        Scalar igam = numext::igamma(a(i), x(i));
+        VERIFY(0 <= igam);
+        VERIFY(igam <= 1);
+      }
+    }
 
     {
       // Check exact values of igamma and igammac against a third party calculation.