diff --git a/include/boost/math/special_functions/detail/bessel_jn.hpp b/include/boost/math/special_functions/detail/bessel_jn.hpp
index 4e683a7c3..c3a5def87 100644
--- a/include/boost/math/special_functions/detail/bessel_jn.hpp
+++ b/include/boost/math/special_functions/detail/bessel_jn.hpp
@@ -84,7 +84,11 @@ BOOST_MATH_GPU_ENABLED T bessel_jn(int n, T x, const Policy& pol)
             current = value;
         }
     }
-    else if((x < 1) || ((n > x * x / 4) && (x < 5)))
+    else if(x < 1)
+    {
+       return factor * bessel_j_small_z_series(T(n), x, pol);
+    }
+    else if((x < 5) && (n > x * x / 4))
     {
        return factor * bessel_j_small_z_series(T(n), x, pol);
     }
diff --git a/include/boost/math/special_functions/detail/bessel_jy.hpp b/include/boost/math/special_functions/detail/bessel_jy.hpp
index 264b93e05..3b3e20805 100644
--- a/include/boost/math/special_functions/detail/bessel_jy.hpp
+++ b/include/boost/math/special_functions/detail/bessel_jy.hpp
@@ -327,7 +327,12 @@ namespace boost { namespace math {
          // x is positive until reflection
          W = T(2) / (x * pi<T>());               // Wronskian
          T Yv_scale = 1;
-         if(((kind & need_y) == 0) && ((x < 1) || ((v > x * x / 4) && (x < 5))))
+
+         const bool kind_does_not_need_y { ((kind & need_y) == 0) };
+
+         const bool x_is_lt_one { (x < 1) };
+
+         if(kind_does_not_need_y && x_is_lt_one)
          {
             //
             // This series will actually converge rapidly for all small
@@ -337,7 +342,17 @@ namespace boost { namespace math {
             Jv = bessel_j_small_z_series(v, x, pol);
             Yv = boost::math::numeric_limits<T>::quiet_NaN();
          }
-         else if((x < 1) && (u != 0) && (log(policies::get_epsilon<T, Policy>() / 2) > v * log((x/2) * (x/2) / v)))
+         else if(kind_does_not_need_y && ((x < 5) && (v > x * x / 4)))
+         {
+            //
+            // This series will actually converge rapidly for all small
+            // x - say up to x < 20 - but the first few terms are large
+            // and divergent which leads to large errors :-(
+            //
+            Jv = bessel_j_small_z_series(v, x, pol);
+            Yv = boost::math::numeric_limits<T>::quiet_NaN();
+         }
+         else if(x_is_lt_one && (u != 0) && (log(policies::get_epsilon<T, Policy>() / 2) > v * log((x/2) * (x/2) / v)))
          {
             // Evaluate using series representations.
             // This is particularly important for x << v as in this