diff --git a/src/gf.c b/src/gf.c
index c6df93465cf1a..c61853f0e3218 100644
--- a/src/gf.c
+++ b/src/gf.c
@@ -1397,7 +1397,7 @@ JL_DLLEXPORT void jl_method_table_insert(jl_methtable_t *mt, jl_method_t *method
             size_t ins = 0;
             for (i = 1; i < na; i += 2) {
                 jl_value_t *backedgetyp = backedges[i - 1];
-                if (jl_type_intersection(backedgetyp, (jl_value_t*)type) != (jl_value_t*)jl_bottom_type) {
+                if (!jl_has_empty_intersection(backedgetyp, (jl_value_t*)type)) {
                     jl_method_instance_t *backedge = (jl_method_instance_t*)backedges[i];
                     invalidate_method_instance(backedge, env.max_world, 0);
                     env.invalidated = 1;
@@ -1731,10 +1731,8 @@ JL_DLLEXPORT int jl_has_call_ambiguities(jl_tupletype_t *types, jl_method_t *m)
     if (m->ambig == jl_nothing) return 0;
     for (size_t i = 0; i < jl_array_len(m->ambig); i++) {
         jl_method_t *mambig = (jl_method_t*)jl_array_ptr_ref(m->ambig, i);
-        if (jl_type_intersection((jl_value_t*)mambig->sig,
-                                 (jl_value_t*)types) != (jl_value_t*)jl_bottom_type) {
+        if (!jl_has_empty_intersection((jl_value_t*)mambig->sig, (jl_value_t*)types))
             return 1;
-        }
     }
     return 0;
 }
diff --git a/src/julia.h b/src/julia.h
index f5efda7e378c7..b3188fa2b5905 100644
--- a/src/julia.h
+++ b/src/julia.h
@@ -982,6 +982,7 @@ JL_DLLEXPORT int jl_isa(jl_value_t *a, jl_value_t *t);
 JL_DLLEXPORT int jl_types_equal(jl_value_t *a, jl_value_t *b);
 JL_DLLEXPORT jl_value_t *jl_type_union(jl_value_t **ts, size_t n);
 JL_DLLEXPORT jl_value_t *jl_type_intersection(jl_value_t *a, jl_value_t *b);
+JL_DLLEXPORT int jl_has_empty_intersection(jl_value_t *x, jl_value_t *y);
 JL_DLLEXPORT jl_value_t *jl_type_unionall(jl_tvar_t *v, jl_value_t *body);
 JL_DLLEXPORT const char *jl_typename_str(jl_value_t *v);
 JL_DLLEXPORT const char *jl_typeof_str(jl_value_t *v);
diff --git a/src/subtype.c b/src/subtype.c
index 8a1984c7a285f..113cb1675085e 100644
--- a/src/subtype.c
+++ b/src/subtype.c
@@ -91,6 +91,7 @@ typedef struct {
     int invdepth;             // current number of invariant constructors we're nested in
     int ignore_free;          // treat free vars as black boxes; used during intersection
     int intersection;         // true iff subtype is being called from intersection
+    int emptiness_only;       // true iff intersection only needs to test for emptiness
 } jl_stenv_t;
 
 // state manipulation utilities
@@ -238,7 +239,7 @@ static int obviously_unequal(jl_value_t *a, jl_value_t *b)
     return 0;
 }
 
-static int obviously_disjoint(jl_value_t *a, jl_value_t *b)
+static int obviously_disjoint(jl_value_t *a, jl_value_t *b, int specificity)
 {
     if (a == b || a == (jl_value_t*)jl_any_type || b == (jl_value_t*)jl_any_type)
         return 0;
@@ -280,7 +281,8 @@ static int obviously_disjoint(jl_value_t *a, jl_value_t *b)
             else if (jl_is_va_tuple(bd)) {
                 nb -= 1;
             }
-            else if (na != nb) {
+            else if (!specificity && na != nb) {
+                // note: some disjoint types (e.g. tuples of different lengths) can be more specific
                 return 1;
             }
             np = na < nb ? na : nb;
@@ -298,13 +300,18 @@ static int obviously_disjoint(jl_value_t *a, jl_value_t *b)
                 if (jl_is_type(bi)) {
                     if (istuple && (ai == jl_bottom_type || bi == jl_bottom_type))
                         ; // TODO: this can return 1 if and when Tuple{Union{}} === Union{}
-                    else if (obviously_disjoint(ai, bi))
+                    else if (obviously_disjoint(ai, bi, specificity))
                         return 1;
                 }
-                else {
+                else if (!specificity) {
+                    // Tuple{1} is more specific than Tuple{Any}
                     return 1;
                 }
             }
+            else if (jl_is_type(bi)) {
+                if (!specificity)
+                    return 1;
+            }
             else if (!jl_egal(ai, bi)) {
                 return 1;
             }
@@ -534,8 +541,9 @@ static int subtype_unionall(jl_value_t *t, jl_unionall_t *u, jl_stenv_t *e, int8
     // if the var for this unionall (based on identity) already appears somewhere
     // in the environment, rename to get a fresh var.
     while (btemp != NULL) {
-        if (btemp->var == u->var || jl_has_typevar(btemp->lb, u->var) ||
-            jl_has_typevar(btemp->ub, u->var)) {
+        if (btemp->var == u->var ||
+            (btemp->lb != jl_bottom_type && jl_has_typevar(btemp->lb, u->var)) ||
+            (btemp->ub != (jl_value_t*)jl_any_type && jl_has_typevar(btemp->ub, u->var))) {
             u = rename_unionall(u);
             break;
         }
@@ -619,7 +627,7 @@ static int subtype_unionall(jl_value_t *t, jl_unionall_t *u, jl_stenv_t *e, int8
     btemp = e->vars;
     while (btemp != NULL) {
         jl_value_t *vi = btemp->ub;
-        if (vi != (jl_value_t*)vb.var && jl_has_typevar(vi, vb.var)) {
+        if (vi != (jl_value_t*)vb.var && vi != (jl_value_t*)jl_any_type && jl_has_typevar(vi, vb.var)) {
             btemp->ub = jl_new_struct(jl_unionall_type, vb.var, vi);
             btemp->lb = jl_bottom_type;
         }
@@ -1003,6 +1011,7 @@ static void init_stenv(jl_stenv_t *e, jl_value_t **env, int envsz)
     e->invdepth = 0;
     e->ignore_free = 0;
     e->intersection = 0;
+    e->emptiness_only = 0;
     e->Lunions.depth = 0;      e->Runions.depth = 0;
     e->Lunions.more = 0;       e->Runions.more = 0;
 }
@@ -2012,6 +2021,8 @@ static jl_value_t *intersect_all(jl_value_t *x, jl_value_t *y, jl_stenv_t *e)
     int lastset = 0, niter = 0;
     jl_value_t *ii = intersect(x, y, e, 0);
     while (e->Runions.more) {
+        if (e->emptiness_only && ii != jl_bottom_type)
+            return ii;
         e->Runions.depth = 0;
         int set = e->Runions.more - 1;
         e->Runions.more = 0;
@@ -2042,16 +2053,28 @@ static jl_value_t *intersect_all(jl_value_t *x, jl_value_t *y, jl_stenv_t *e)
 
 // type intersection entry points
 
-JL_DLLEXPORT jl_value_t *jl_intersect_types(jl_value_t *x, jl_value_t *y)
+static jl_value_t *intersect_types(jl_value_t *x, jl_value_t *y, int emptiness_only)
 {
     jl_stenv_t e;
-    if (obviously_disjoint(x, y))
+    if (obviously_disjoint(x, y, 0))
         return jl_bottom_type;
     init_stenv(&e, NULL, 0);
     e.intersection = 1;
+    e.emptiness_only = emptiness_only;
     return intersect_all(x, y, &e);
 }
 
+JL_DLLEXPORT jl_value_t *jl_intersect_types(jl_value_t *x, jl_value_t *y)
+{
+    return intersect_types(x, y, 0);
+}
+
+// TODO: this can probably be done more efficiently
+JL_DLLEXPORT int jl_has_empty_intersection(jl_value_t *x, jl_value_t *y)
+{
+    return intersect_types(x, y, 1) == jl_bottom_type;
+}
+
 // return a SimpleVector of all vars from UnionAlls wrapping a given type
 jl_svec_t *jl_outer_unionall_vars(jl_value_t *u)
 {
@@ -2073,7 +2096,7 @@ jl_svec_t *jl_outer_unionall_vars(jl_value_t *u)
 jl_value_t *jl_type_intersection_env_s(jl_value_t *a, jl_value_t *b, jl_svec_t **penv, int *issubty)
 {
     if (issubty) *issubty = 0;
-    if (obviously_disjoint(a, b)) {
+    if (obviously_disjoint(a, b, 0)) {
         if (issubty && a == jl_bottom_type) *issubty = 1;
         return jl_bottom_type;
     }
@@ -2570,6 +2593,8 @@ static int type_morespecific_(jl_value_t *a, jl_value_t *b, int invariant, jl_ty
 
 JL_DLLEXPORT int jl_type_morespecific(jl_value_t *a, jl_value_t *b)
 {
+    if (obviously_disjoint(a, b, 1))
+        return 0;
     if (jl_subtype(b, a)) return 0;
     if (jl_subtype(a, b)) return 1;
     return type_morespecific_(a, b, 0, NULL);
diff --git a/src/typemap.c b/src/typemap.c
index 0119678b92584..b70accd357b87 100644
--- a/src/typemap.c
+++ b/src/typemap.c
@@ -515,7 +515,7 @@ int jl_typemap_intersection_visitor(union jl_typemap_t map, int offs,
                 else {
                     // else an array scan is required to check subtypes
                     // first, fast-path: optimized pre-intersection test to see if `ty` could intersect with any Type
-                    if (typetype || jl_type_intersection((jl_value_t*)jl_type_type, ty) != jl_bottom_type)
+                    if (typetype || !jl_has_empty_intersection((jl_value_t*)jl_type_type, ty))
                         if (!jl_typemap_intersection_array_visitor(&cache->targ, ty, 1, offs, closure)) return 0;
                 }
             }