diff --git a/include/boost/math/tools/precision.hpp b/include/boost/math/tools/precision.hpp
index 94c22b55ed..fac022f53a 100644
--- a/include/boost/math/tools/precision.hpp
+++ b/include/boost/math/tools/precision.hpp
@@ -182,9 +182,13 @@ struct log_limit_traits
 {
    typedef typename boost::math::conditional<
       (boost::math::numeric_limits<T>::radix == 2) &&
-      (boost::math::numeric_limits<T>::max_exponent == 128
-         || boost::math::numeric_limits<T>::max_exponent == 1024
-         || boost::math::numeric_limits<T>::max_exponent == 16384),
+      (
+         (     boost::math::numeric_limits<T>::max_exponent == 128
+            || boost::math::numeric_limits<T>::max_exponent == 1024
+            || boost::math::numeric_limits<T>::max_exponent == 16384
+         )
+         && (-boost::math::numeric_limits<T>::min_exponent + 1 == boost::math::numeric_limits<T>::max_exponent)
+      ),
       boost::math::integral_constant<int, (boost::math::numeric_limits<T>::max_exponent > (boost::math::numeric_limits<int>::max)() ? (boost::math::numeric_limits<int>::max)() : static_cast<int>(boost::math::numeric_limits<T>::max_exponent))>,
       boost::math::integral_constant<int, 0>
    >::type tag_type;
diff --git a/test/test_bessel_k.hpp b/test/test_bessel_k.hpp
index 6a2a8179d9..66e648b459 100644
--- a/test/test_bessel_k.hpp
+++ b/test/test_bessel_k.hpp
@@ -133,7 +133,9 @@ void test_bessel(T, const char* name)
         {{ SC_(-10.0), SC_(1.0), SC_(1.80713289901029454691597861302340015908245782948536080022119e8) }},
         {{ SC_(100.0), SC_(5.0), SC_(7.03986019306167654653386616796116726248616158936088056952477e115) }},
         {{ SC_(100.0), SC_(80.0), SC_(8.39287107246490782848985384895907681748152272748337807033319e-12) }},
-        {{ SC_(-1000.0), SC_(700.0), SC_(6.51561979144735818903553852606383312984409361984128221539405e-31) }},
+        ((std::numeric_limits<T>::is_specialized) && (std::numeric_limits<T>::min_exponent10 > -293))
+           ? std::array<typename table_type<T>::type, 3> {{ SC_(-1000.0), SC_(664.0), SC_(0.00372564418019942154373086796021082031912651177619335584767976) }}
+           : std::array<typename table_type<T>::type, 3> {{ SC_(-1000.0), SC_(700.0), SC_(6.51561979144735818903553852606383312984409361984128221539405e-31) }},
     }};
     static const std::array<std::array<typename table_type<T>::type, 3>, 11> kv_data = {{
         {{ SC_(0.5), SC_(0.875), SC_(0.558532231646608646115729767013630967055657943463362504577189) }},