diff --git a/include/fmt/core.h b/include/fmt/core.h
index 50a6fd8d..738b85df 100644
--- a/include/fmt/core.h
+++ b/include/fmt/core.h
@@ -1441,7 +1441,8 @@ template <typename Context> struct arg_mapper {
   template <typename T, typename U = remove_cvref_t<T>>
   struct formattable
       : bool_constant<has_const_formatter<U, Context>() ||
-                      !std::is_const<remove_reference_t<T>>::value> {};
+                      (has_formatter<U, Context>::value &&
+                       !std::is_const<remove_reference_t<T>>::value)> {};
 
   template <typename T, FMT_ENABLE_IF(formattable<T>::value)>
   FMT_CONSTEXPR FMT_INLINE auto do_map(T&& val) -> T& {
@@ -1453,11 +1454,11 @@ template <typename Context> struct arg_mapper {
   }
 
   template <typename T, typename U = remove_cvref_t<T>,
-            FMT_ENABLE_IF(!is_string<U>::value && !is_char<U>::value &&
-                          !std::is_array<U>::value &&
-                          !std::is_pointer<U>::value &&
-                          !std::is_arithmetic<format_as_t<U>>::value &&
-                          has_formatter<U, Context>::value)>
+            FMT_ENABLE_IF((std::is_class<U>::value || std::is_enum<U>::value ||
+                           std::is_union<U>::value) &&
+                          !is_string<U>::value && !is_char<U>::value &&
+                          !is_named_arg<U>::value &&
+                          !std::is_arithmetic<format_as_t<U>>::value)>
   FMT_CONSTEXPR FMT_INLINE auto map(T&& val)
       -> decltype(this->do_map(std::forward<T>(val))) {
     return do_map(std::forward<T>(val));
diff --git a/test/core-test.cc b/test/core-test.cc
index 71686105..22a49def 100644
--- a/test/core-test.cc
+++ b/test/core-test.cc
@@ -672,7 +672,8 @@ TEST(core_test, is_formattable) {
   static_assert(fmt::is_formattable<enabled_formatter>::value, "");
   static_assert(!fmt::is_formattable<enabled_ptr_formatter*>::value, "");
   static_assert(!fmt::is_formattable<disabled_formatter>::value, "");
-  static_assert(fmt::is_formattable<disabled_formatter_convertible>::value, "");
+  static_assert(!fmt::is_formattable<disabled_formatter_convertible>::value,
+                "");
 
   static_assert(fmt::is_formattable<const_formattable&>::value, "");
   static_assert(fmt::is_formattable<const const_formattable&>::value, "");
@@ -798,25 +799,6 @@ TEST(core_test, format_explicitly_convertible_to_std_string_view) {
 #  endif
 #endif
 
-struct convertible_to_long_long {
-  operator long long() const { return 1LL << 32; }
-};
-
-TEST(core_test, format_convertible_to_long_long) {
-  EXPECT_EQ("100000000", fmt::format("{:x}", convertible_to_long_long()));
-}
-
-struct disabled_rvalue_conversion {
-  operator const char*() const& { return "foo"; }
-  operator const char*() & { return "foo"; }
-  operator const char*() const&& = delete;
-  operator const char*() && = delete;
-};
-
-TEST(core_test, disabled_rvalue_conversion) {
-  EXPECT_EQ("foo", fmt::format("{}", disabled_rvalue_conversion()));
-}
-
 namespace adl_test {
 template <typename... T> void make_format_args(const T&...) = delete;
 
diff --git a/test/format-impl-test.cc b/test/format-impl-test.cc
index e39bde74..02101a21 100644
--- a/test/format-impl-test.cc
+++ b/test/format-impl-test.cc
@@ -242,11 +242,11 @@ TEST(fp_test, dragonbox_max_k) {
           floor_log10_pow2(std::numeric_limits<float>::min_exponent -
                            fmt::detail::num_significand_bits<float>() - 1));
   using double_info = fmt::detail::dragonbox::float_info<double>;
-  EXPECT_EQ(
-      fmt::detail::const_check(double_info::max_k),
-      double_info::kappa -
-          floor_log10_pow2(std::numeric_limits<double>::min_exponent -
-                           2 * fmt::detail::num_significand_bits<double>() - 1));
+  EXPECT_EQ(fmt::detail::const_check(double_info::max_k),
+            double_info::kappa -
+                floor_log10_pow2(
+                    std::numeric_limits<double>::min_exponent -
+                    2 * fmt::detail::num_significand_bits<double>() - 1));
 }
 
 TEST(fp_test, get_round_direction) {
@@ -382,6 +382,8 @@ struct double_double {
   auto operator-() const -> double_double { return double_double(-a, -b); }
 };
 
+auto format_as(double_double d) -> double { return d; }
+
 bool operator>=(const double_double& lhs, const double_double& rhs) {
   return lhs.a + lhs.b >= rhs.a + rhs.b;
 }
@@ -394,6 +396,8 @@ struct slow_float {
   auto operator-() const -> slow_float { return slow_float(-value); }
 };
 
+auto format_as(slow_float f) -> float { return f; }
+
 namespace std {
 template <> struct is_floating_point<double_double> : std::true_type {};
 template <> struct numeric_limits<double_double> {
diff --git a/test/format-test.cc b/test/format-test.cc
index 98395353..e798da1d 100644
--- a/test/format-test.cc
+++ b/test/format-test.cc
@@ -1967,15 +1967,10 @@ struct formatter<adl_test::fmt::detail::foo> : formatter<std::string> {
 };
 FMT_END_NAMESPACE
 
-struct convertible_to_int {
-  operator int() const { return 42; }
-};
-
 TEST(format_test, to_string) {
   EXPECT_EQ(fmt::to_string(42), "42");
   EXPECT_EQ(fmt::to_string(reinterpret_cast<void*>(0x1234)), "0x1234");
   EXPECT_EQ(fmt::to_string(adl_test::fmt::detail::foo()), "foo");
-  EXPECT_EQ(fmt::to_string(convertible_to_int()), "42");
   EXPECT_EQ(fmt::to_string(foo), "0");
 
 #if FMT_USE_FLOAT128
diff --git a/test/ostream-test.cc b/test/ostream-test.cc
index b9a6b352..8db9cda4 100644
--- a/test/ostream-test.cc
+++ b/test/ostream-test.cc
@@ -226,30 +226,6 @@ TEST(ostream_test, format_to_n) {
   EXPECT_EQ("xabx", fmt::string_view(buffer, 4));
 }
 
-template <typename T> struct convertible {
-  T value;
-  explicit convertible(const T& val) : value(val) {}
-  operator T() const { return value; }
-};
-
-TEST(ostream_test, disable_builtin_ostream_operators) {
-  EXPECT_EQ("42", fmt::format("{:d}", convertible<unsigned short>(42)));
-  EXPECT_EQ("foo", fmt::format("{}", convertible<const char*>("foo")));
-}
-
-struct streamable_and_convertible_to_bool {
-  operator bool() const { return true; }
-};
-
-std::ostream& operator<<(std::ostream& os, streamable_and_convertible_to_bool) {
-  return os << "foo";
-}
-
-TEST(ostream_test, format_convertible_to_bool) {
-  // operator<< is intentionally not used because of potential ODR violations.
-  EXPECT_EQ(fmt::format("{}", streamable_and_convertible_to_bool()), "true");
-}
-
 struct copyfmt_test {};
 
 std::ostream& operator<<(std::ostream& os, copyfmt_test) {