diff --git a/example/lambert_w_precision_example.cpp b/example/lambert_w_precision_example.cpp
index be4791bc5..e95f3b193 100644
--- a/example/lambert_w_precision_example.cpp
+++ b/example/lambert_w_precision_example.cpp
@@ -63,7 +63,6 @@ int main()
     using boost::math::policies::evaluation_error;
     using boost::math::policies::domain_error;
     using boost::math::policies::overflow_error;
-    using boost::math::policies::domain_error;
     using boost::math::policies::throw_on_error;
 
 //[lambert_w_precision_reference_w
diff --git a/test/test_lambert_w.cpp b/test/test_lambert_w.cpp
index 2d1da1e24..59e591a49 100644
--- a/test/test_lambert_w.cpp
+++ b/test/test_lambert_w.cpp
@@ -1031,16 +1031,24 @@ BOOST_AUTO_TEST_CASE( test_range_of_double_values )
     BOOST_MATH_TEST_VALUE(RealType, -0.99999997649828679),
     5e7 * tolerance);// diff 2.30785e-09 v 2.2204460492503131e-16
 
-  // Compare with previous PB/FK computations at double precision.
-  BOOST_CHECK_CLOSE_FRACTION(  // Check float_next(-exp(-1) )
-    lambert_w0(BOOST_MATH_TEST_VALUE(RealType, -0.36787944117144228)),
-    BOOST_MATH_TEST_VALUE(RealType, -0.99999997892657588),
-    tolerance); // diff 6.03558e-09 v 2.2204460492503131e-16
+   // Compare with previous PB/FK computations at double precision.
+   using std::abs;
+   RealType x = BOOST_MATH_TEST_VALUE(RealType, -0.36787944117144228);
+   RealType w0 = lambert_w0(x);
+   RealType w0_prime = boost::math::lambert_w0_prime(x);
+   RealType mu = std::numeric_limits<RealType>::epsilon()/2;
+   BOOST_CHECK_CLOSE_FRACTION(  // Check float_next(-exp(-1) )
+    w0,
+    BOOST_MATH_TEST_VALUE(RealType, -0.9999999849621573837115797120602890516186071783122773515945338502828025975466699519609633476854139977),
+    2*mu*abs(x*w0_prime/w0)); // diff 6.03558e-09 v 2.2204460492503131e-16
 
-  BOOST_CHECK_CLOSE_FRACTION(  // Check  float_next(float_next(-exp(-1) ))
-    lambert_w0(BOOST_MATH_TEST_VALUE(RealType, -0.36787944117144222)),
+   x = BOOST_MATH_TEST_VALUE(RealType, -0.36787944117144222);
+   w0 = lambert_w0(x);
+   w0_prime = boost::math::lambert_w0_prime(x);
+   BOOST_CHECK_CLOSE_FRACTION(  // Check  float_next(float_next(-exp(-1) ))
+    w0,
     BOOST_MATH_TEST_VALUE(RealType, -0.99999997419043196),
-    tolerance);// diff 2.30785e-09 v 2.2204460492503131e-16
+    2*mu*abs(x*w0_prime/w0));// diff 2.30785e-09 v 2.2204460492503131e-16
 
                // z increasingly close to singularity.
   BOOST_CHECK_CLOSE_FRACTION(lambert_w0(BOOST_MATH_TEST_VALUE(RealType, -0.36)),