diff --git a/math/gen-tgmath-tests.py b/math/gen-tgmath-tests.py
index c841db6000..be10c47877 100755
--- a/math/gen-tgmath-tests.py
+++ b/math/gen-tgmath-tests.py
@@ -19,14 +19,13 @@
 
 # As glibc does not support decimal floating point, the types to
 # consider for generic parameters are standard and binary
-# floating-point types, and integer types which are treated as double.
-# The corresponding complex types may also be used (including complex
-# integer types, which are a GNU extension, but are currently disabled
-# here because they do not work properly with tgmath.h).
-
-# The proposed resolution to TS 18661-1 DR#9
-# <http://www.open-std.org/jtc1/sc22/wg14/www/docs/n2149.htm#dr_9>
-# makes the <tgmath.h> rules for selecting a function to call
+# floating-point types, and integer types which are treated as
+# _Float32x if any argument has a _FloatNx type and otherwise as
+# double.  The corresponding complex types may also be used (including
+# complex integer types, which are a GNU extension, but are currently
+# disabled here because they do not work properly with tgmath.h).
+
+# C2x makes the <tgmath.h> rules for selecting a function to call
 # correspond to the usual arithmetic conversions (applied successively
 # to the arguments for generic parameters in order), which choose the
 # type whose set of values contains that of the other type (undefined
@@ -69,10 +68,6 @@ class Type(object):
     # Real argument types that correspond to a standard floating type
     # (float, double or long double; not _FloatN or _FloatNx).
     standard_real_argument_types_list = []
-    # Real argument types other than float, double and long double
-    # (i.e., those that are valid as arguments to narrowing macros
-    # returning _FloatN or _FloatNx).
-    non_standard_real_argument_types_list = []
     # The real floating types by their order properties (which are
     # tuples giving the positions in both the possible orders above).
     real_types_order = {}
@@ -86,13 +81,16 @@ class Type(object):
     float64_type = None
     # The type _Complex _Float64.
     complex_float64_type = None
+    # The type _Float32x.
+    float32x_type = None
+    # The type _Complex _Float32x.
+    complex_float32x_type = None
     # The type _Float64x.
     float64x_type = None
-    # The type _Float64x if available, otherwise _Float64.
-    float32x_ext_type = None
 
     def __init__(self, name, suffix=None, mant_dig=None, condition='1',
-                 order=None, integer=False, complex=False, real_type=None):
+                 order=None, integer=False, complex=False, real_type=None,
+                 floatnx=False):
         """Initialize a Type object, creating any corresponding complex type
         in the process."""
         self.name = name
@@ -102,6 +100,7 @@ class Type(object):
         self.order = order
         self.integer = integer
         self.complex = complex
+        self.floatnx = floatnx
         if complex:
             self.complex_type = self
             self.real_type = real_type
@@ -119,8 +118,6 @@ class Type(object):
                 Type.real_argument_types_list.append(self)
                 if not self.name.startswith('_Float'):
                     Type.standard_real_argument_types_list.append(self)
-                if self.name not in ('float', 'double', 'long double'):
-                    Type.non_standard_real_argument_types_list.append(self)
         if self.order is not None:
             Type.real_types_order[self.order] = self
         if self.name == 'double':
@@ -133,26 +130,28 @@ class Type(object):
             Type.float64_type = self
         if self.name == '_Complex _Float64':
             Type.complex_float64_type = self
+        if self.name == '_Float32x':
+            Type.float32x_type = self
+        if self.name == '_Complex _Float32x':
+            Type.complex_float32x_type = self
         if self.name == '_Float64x':
             Type.float64x_type = self
-        if self.name == 'Float32x_ext':
-            Type.float32x_ext_type = self
 
     @staticmethod
     def create_type(name, suffix=None, mant_dig=None, condition='1', order=None,
                     integer=False, complex_name=None, complex_ok=True,
-                    internal=False):
+                    floatnx=False, internal=False):
         """Create and register a Type object for a real type, creating any
         corresponding complex type in the process."""
         real_type = Type(name, suffix=suffix, mant_dig=mant_dig,
                          condition=condition, order=order, integer=integer,
-                         complex=False)
+                         complex=False, floatnx=floatnx)
         if complex_ok:
             if complex_name is None:
                 complex_name = '_Complex %s' % name
             complex_type = Type(complex_name, condition=condition,
                                 integer=integer, complex=True,
-                                real_type=real_type)
+                                real_type=real_type, floatnx=floatnx)
         else:
             complex_type = None
         real_type.complex_type = complex_type
@@ -160,13 +159,13 @@ class Type(object):
         if complex_type is not None:
             complex_type.register_type(internal)
 
-    def floating_type(self, floatn):
+    def floating_type(self, integer_float32x):
         """Return the corresponding floating type."""
         if self.integer:
-            if floatn:
-                return (Type.complex_float64_type
+            if integer_float32x:
+                return (Type.complex_float32x_type
                         if self.complex
-                        else Type.float64_type)
+                        else Type.float32x_type)
             else:
                 return (Type.complex_double_type
                         if self.complex
@@ -174,9 +173,9 @@ class Type(object):
         else:
             return self
 
-    def real_floating_type(self, floatn):
+    def real_floating_type(self, integer_float32x):
         """Return the corresponding real floating type."""
-        return self.real_type.floating_type(floatn)
+        return self.real_type.floating_type(integer_float32x)
 
     def __str__(self):
         """Return string representation of a type."""
@@ -194,7 +193,8 @@ class Type(object):
                          condition='defined HUGE_VAL_F32', order=(2, 2))
         Type.create_type('_Float32x', 'f32x', 'FLT32X_MANT_DIG',
                          complex_name='__CFLOAT32X',
-                         condition='defined HUGE_VAL_F32X', order=(3, 3))
+                         condition='defined HUGE_VAL_F32X', order=(3, 3),
+                         floatnx=True)
         Type.create_type('double', '', 'DBL_MANT_DIG', order=(4, 4))
         Type.create_type('long double', 'l', 'LDBL_MANT_DIG', order=(5, 7))
         Type.create_type('_Float64', 'f64', 'FLT64_MANT_DIG',
@@ -202,7 +202,8 @@ class Type(object):
                          condition='defined HUGE_VAL_F64', order=(6, 5))
         Type.create_type('_Float64x', 'f64x', 'FLT64X_MANT_DIG',
                          complex_name='__CFLOAT64X',
-                         condition='defined HUGE_VAL_F64X', order=(7, 6))
+                         condition='defined HUGE_VAL_F64X', order=(7, 6),
+                         floatnx=True)
         Type.create_type('_Float128', 'f128', 'FLT128_MANT_DIG',
                          complex_name='__CFLOAT128',
                          condition='defined HUGE_VAL_F128', order=(8, 8))
@@ -235,21 +236,16 @@ class Type(object):
                          complex_name='complex_long_double_Float64x',
                          condition='defined HUGE_VAL_F64X', order=(7, 7),
                          internal=True)
-        # An internal type for the argument type used by f32x*
-        # narrowing macros (_Float64x if available, otherwise
-        # _Float64).
-        Type.create_type('Float32x_ext', None, 'FLT32X_EXT_MANT_DIG',
-                         complex_name='complex_Float32x_ext',
-                         condition='1', internal=True)
 
     @staticmethod
-    def can_combine_types(types, floatn):
+    def can_combine_types(types):
         """Return a C preprocessor conditional for whether the given list of
         types can be used together as type-generic macro arguments."""
         have_long_double = False
         have_float128 = False
+        integer_float32x = any(t.floatnx for t in types)
         for t in types:
-            t = t.real_floating_type(floatn)
+            t = t.real_floating_type(integer_float32x)
             if t.name == 'long double':
                 have_long_double = True
             if t.name == '_Float128' or t.name == '_Float64x':
@@ -262,14 +258,15 @@ class Type(object):
         return '1'
 
     @staticmethod
-    def combine_types(types, floatn):
+    def combine_types(types):
         """Return the result of combining a set of types."""
         have_complex = False
         combined = None
+        integer_float32x = any(t.floatnx for t in types)
         for t in types:
             if t.complex:
                 have_complex = True
-            t = t.real_floating_type(floatn)
+            t = t.real_floating_type(integer_float32x)
             if combined is None:
                 combined = t
             else:
@@ -375,18 +372,8 @@ class Tests(object):
                          '# endif\n')
         float64x_text = if_cond_text([Type.float64x_type.condition],
                                      float64x_text)
-        float32x_ext_text = ('#ifdef HUGE_VAL_F64X\n'
-                             'typedef _Float64x Float32x_ext;\n'
-                             'typedef __CFLOAT64X complex_Float32x_ext;\n'
-                             '# define FLT32X_EXT_MANT_DIG FLT64X_MANT_DIG\n'
-                             '#else\n'
-                             'typedef _Float64 Float32x_ext;\n'
-                             'typedef __CFLOAT64 complex_Float32x_ext;\n'
-                             '# define FLT32X_EXT_MANT_DIG FLT64_MANT_DIG\n'
-                             '#endif\n')
         self.header_list.append(float64_text)
         self.header_list.append(float64x_text)
-        self.header_list.append(float32x_ext_text)
         self.types_seen = set()
         for t in Type.all_types_list:
             self.add_type_var(t.name, t.condition)
@@ -439,39 +426,33 @@ class Tests(object):
             narrowing_std = True
             narrow_cond = '1'
             narrow_args = [Type.double_type, Type.long_double_type]
-            narrow_fallback = Type.double_type
         elif ret == 'double':
             narrowing = True
             narrowing_std = True
             narrow_cond = '1'
             narrow_args = [Type.long_double_type]
-            narrow_fallback = Type.long_double_type
         elif ret.startswith('_Float'):
             narrowing = True
-            narrow_args = []
+            narrow_args_1 = []
+            narrow_args_2 = []
             nret_type = None
-            narrow_fallback = None
             for order, real_type in sorted(Type.real_types_order.items()):
                 if real_type.name == ret:
                     nret_type = real_type
                 elif nret_type and real_type.name.startswith('_Float'):
-                    narrow_args.append(real_type)
-                    if (narrow_fallback is None
-                        and ret.endswith('x') == real_type.name.endswith('x')):
-                        narrow_fallback = real_type
+                    if ret.endswith('x') == real_type.name.endswith('x'):
+                        narrow_args_1.append(real_type)
+                    else:
+                        narrow_args_2.append(real_type)
+            narrow_args = narrow_args_1 + narrow_args_2
             if narrow_args:
                 narrow_cond = ('(%s && (%s))'
                                % (nret_type.condition,
                                   ' || '.join(t.condition
                                               for t in narrow_args)))
-                if narrow_fallback is None:
-                    narrow_fallback = narrow_args[0]
-                if ret == '_Float32x':
-                    narrow_fallback = Type.float32x_ext_type
             else:
                 # No possible argument types, even conditionally.
                 narrow_cond = '0'
-        narrowing_nonstd = narrowing and not narrowing_std
         types = [ret] + args
         for t in types:
             if t != 'c' and t != 'g' and t != 'r' and t != 's':
@@ -530,19 +511,13 @@ class Tests(object):
             if t == 'g' or t == 'c':
                 arg_types.append(Type.argument_types_list)
             elif t == 'r':
-                if narrowing_std:
-                    arg_types.append(Type.standard_real_argument_types_list)
-                elif narrowing:
-                    arg_types.append(
-                        Type.non_standard_real_argument_types_list)
-                else:
-                    arg_types.append(Type.real_argument_types_list)
+                arg_types.append(Type.real_argument_types_list)
             elif t == 's':
                 arg_types.append(Type.standard_real_argument_types_list)
         arg_types_product = list_product(arg_types)
         test_num = 0
         for this_args in arg_types_product:
-            comb_type = Type.combine_types(this_args, narrowing_nonstd)
+            comb_type = Type.combine_types(this_args)
             if narrowing:
                 # As long as there are no integer arguments, and as
                 # long as the chosen argument type is as wide as all
@@ -550,22 +525,22 @@ class Tests(object):
                 # of the macro call do not depend on the exact
                 # function chosen.  In particular, for f32x functions
                 # when _Float64x exists, the chosen type should differ
-                # for _Float32x and _Float64 arguments, but it is not
-                # always possible to distinguish those types before
-                # GCC 7 and the implementation does not attempt to do
-                # so before GCC 8.
+                # for double / _Float32x and _Float64 arguments, but
+                # it is not always possible to distinguish those types
+                # before GCC 7 (resulting in some cases - only real
+                # arguments - where a wider argument type is used,
+                # which is semantically OK, and others - integer
+                # arguments present - where it may not be OK, but is
+                # unavoidable).
                 narrow_mant_dig = comb_type.real_type.mant_dig
                 for arg_type in this_args:
                     if arg_type.integer:
                         narrow_mant_dig = 0
             else:
                 narrow_mant_dig = 0
-            if (narrowing
-                and comb_type not in narrow_args
-                and narrow_fallback is not None):
-                comb_type = narrow_fallback
-            can_comb = Type.can_combine_types(this_args, narrowing_nonstd)
+            can_comb = Type.can_combine_types(this_args)
             all_conds = [t.condition for t in this_args]
+            narrow_args_cond = '(%s)' % ' && '.join(sorted(set(all_conds)))
             all_conds.append(can_comb)
             if narrowing:
                 all_conds.append(narrow_cond)
@@ -579,10 +554,69 @@ class Tests(object):
             test_func_name = 'test_%s_%d' % (macro, test_num)
             test_num += 1
             mant_dig = comb_type.real_type.mant_dig
+            test_mant_dig_comp = ''
+            if (narrowing
+                and comb_type not in narrow_args):
+                # The expected argument type is the first in
+                # narrow_args that can represent all the values of
+                # comb_type (which, for the supported cases, means the
+                # first with mant_dig at least as large as that for
+                # comb_type, provided this isn't the case of an IBM
+                # long double argument with binary128 type from
+                # narrow_args).
+                narrow_extra_conds = []
+                test_mant_dig_list = ['#undef NARROW_MANT_DIG\n#if 0\n']
+                for t in narrow_args:
+                    t_cond = '(%s && %s && %s <= %s && %s)' % (
+                        narrow_args_cond, t.condition, mant_dig, t.mant_dig,
+                        Type.can_combine_types(this_args + [t]))
+                    narrow_extra_conds.append(t_cond)
+                    test_mant_dig_list.append('#elif %s\n'
+                                              '#define NARROW_MANT_DIG %s\n'
+                                              % (t_cond, t.mant_dig))
+                test_mant_dig_list.append('#endif\n')
+                test_mant_dig_comp = ''.join(test_mant_dig_list)
+                all_conds.append('(%s)' % ' || '.join(narrow_extra_conds))
+                # A special case where this logic isn't correct is
+                # where comb_type is the internal long_double_Float64
+                # or long_double_Float64x, which will be detected as
+                # not in narrow_args even if the actual type chosen in
+                # a particular configuration would have been in
+                # narrow_args, so check for that case and handle it
+                # appropriately.  In particular, if long double has
+                # the same format as double and there are long double
+                # and _Float64 arguments, and the macro returns
+                # _Float32x, the function called should be one for
+                # _Float64 arguments, not one for _Float64x arguments
+                # that would arise from this logic.
+                if comb_type.real_type.name == 'long_double_Float64':
+                    comb_type_1 = Type.long_double_type
+                    comb_type_2 = Type.float64_type
+                    comb_type_is_2_cond = 'LDBL_MANT_DIG <= FLT64_MANT_DIG'
+                elif comb_type.real_type.name == 'long_double_Float64x':
+                    comb_type_1 = Type.long_double_type
+                    comb_type_2 = Type.float64x_type
+                    comb_type_is_2_cond = 'LDBL_MANT_DIG < FLT64X_MANT_DIG'
+                else:
+                    comb_type_1 = None
+                    comb_type_2 = None
+                if comb_type_1 is None:
+                    mant_dig = 'NARROW_MANT_DIG'
+                else:
+                    mant_dig = ''
+                    if comb_type_1 in narrow_args:
+                        mant_dig += '!(%s) ? %s : ' % (comb_type_is_2_cond,
+                                                       comb_type_1.mant_dig)
+                    if comb_type_2 in narrow_args:
+                        mant_dig += '%s ? %s : ' % (comb_type_is_2_cond,
+                                                    comb_type_2.mant_dig)
+                    mant_dig += 'NARROW_MANT_DIG'
+                if narrow_mant_dig != 0:
+                    narrow_mant_dig = mant_dig
             test_text = '%s, "%s", "%s", %s, %s' % (test_func_name, func_name,
                                                     test_name, mant_dig,
                                                     narrow_mant_dig)
-            test_text = '    { %s },\n' % test_text
+            test_text = '%s    { %s },\n' % (test_mant_dig_comp, test_text)
             test_text = if_cond_text(all_conds, test_text)
             self.test_array_list.append(test_text)
             call_args = []
@@ -742,7 +776,7 @@ class Tests(object):
                        '          && strcmp (called_func_name,\n'
                        '                     tests[i].func_name) == 0)\n'
                        '        num_pass++;\n'
-                       '#if !__GNUC_PREREQ (8, 0)\n'
+                       '#if !__GNUC_PREREQ (7, 0)\n'
                        '      else if (tests[i].narrow_mant_dig > 0\n'
                        '               && (called_mant_dig\n'
                        '                   >= tests[i].narrow_mant_dig)\n'
@@ -759,6 +793,21 @@ class Tests(object):
                        '                  tests[i].mant_dig,\n'
                        '                  called_func_name, called_mant_dig);\n'
                        '        }\n'
+                       '      else if (tests[i].narrow_mant_dig == 0\n'
+                       '               && strcmp (called_func_name,\n'
+                       '                          tests[i].func_name) == 0)\n'
+                       '        {\n'
+                       '          num_pass++;\n'
+                       '          printf ("Test %zu (%s):\\n"\n'
+                       '                  "  Expected: %s precision %d\\n"\n'
+                       '                  "  Actual: %s precision %d\\n"\n'
+                       '                  "  (unavoidable with old GCC)'
+                       '\\n\\n",\n'
+                       '                  i, tests[i].test_name,\n'
+                       '                  tests[i].func_name,\n'
+                       '                  tests[i].mant_dig,\n'
+                       '                  called_func_name, called_mant_dig);\n'
+                       '        }\n'
                        '#endif\n'
                        '      else\n'
                        '        {\n'
diff --git a/math/tgmath.h b/math/tgmath.h
index a50c218a52..7a1e77f393 100644
--- a/math/tgmath.h
+++ b/math/tgmath.h
@@ -37,9 +37,17 @@
    for older GCC, using other compiler extensions but with macros
    expanding their arguments many times (so resulting in exponential
    blowup of the size of expansions when calls to such macros are
-   nested inside arguments to such macros).  */
+   nested inside arguments to such macros).  Because of a long series
+   of defect fixes made after the initial release of TS 18661-1, GCC
+   versions before GCC 13 have __builtin_tgmath semantics that, when
+   integer arguments are passed to narrowing macros returning
+   _Float32x, or non-narrowing macros with at least two generic
+   arguments, do not always correspond to the C2X semantics, so more
+   complicated macro definitions are also used in some cases for
+   versions from GCC 8 to GCC 12.  */
 
 #define __HAVE_BUILTIN_TGMATH __GNUC_PREREQ (8, 0)
+#define __HAVE_BUILTIN_TGMATH_C2X __GNUC_PREREQ (13, 0)
 
 #if __GNUC_PREREQ (2, 7)
 
@@ -163,7 +171,7 @@
   __builtin_tgmath (__TGMATH_NARROW_FUNCS_F64 (F) (X), (Y))
 #  define __TGMATH_3_NARROW_F64(F, X, Y, Z)				\
   __builtin_tgmath (__TGMATH_NARROW_FUNCS_F64 (F) (X), (Y), (Z))
-#  if __HAVE_FLOAT128
+#  if __HAVE_FLOAT128 && __HAVE_BUILTIN_TGMATH_C2X
 #   define __TGMATH_1_NARROW_F32X(F, X)				\
   __builtin_tgmath (__TGMATH_NARROW_FUNCS_F32X (F) (X))
 #   define __TGMATH_2_NARROW_F32X(F, X, Y)			\
@@ -172,8 +180,9 @@
   __builtin_tgmath (__TGMATH_NARROW_FUNCS_F32X (F) (X), (Y), (Z))
 #  endif
 
-# else /* !__HAVE_BUILTIN_TGMATH.  */
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH_C2X
 #  ifdef __NO_LONG_DOUBLE_MATH
 #   define __tgml(fct) fct
 #  else
@@ -213,13 +222,17 @@
 /* Whether an expression (of arithmetic type) has a real type.  */
 #  define __expr_is_real(E) (__builtin_classify_type (E) != 9)
 
+/* Type T1 if E is 1, type T2 is E is 0.  */
+#  define __tgmath_type_if(T1, T2, E)					\
+  __typeof__ (*(0 ? (__typeof__ (0 ? (T2 *) 0 : (void *) (E))) 0	\
+		: (__typeof__ (0 ? (T1 *) 0 : (void *) (!(E)))) 0))
+
 /* The tgmath real type for T, where E is 0 if T is an integer type
    and 1 for a floating type.  If T has a complex type, it is
    unspecified whether the return type is real or complex (but it has
    the correct corresponding real type).  */
 #  define __tgmath_real_type_sub(T, E) \
-  __typeof__ (*(0 ? (__typeof__ (0 ? (double *) 0 : (void *) (E))) 0	      \
-		  : (__typeof__ (0 ? (T *) 0 : (void *) (!(E)))) 0))
+  __tgmath_type_if (T, double, E)
 
 /* The tgmath real type of EXPR.  */
 #  define __tgmath_real_type(expr) \
@@ -247,6 +260,56 @@
 			     __real_integer_type (__typeof__ (+(expr))), \
 			     __complex_integer_type (__typeof__ (+(expr))))
 
+/* The tgmath real type of EXPR1 combined with EXPR2, without handling
+   the C2X rule of interpreting integer arguments as _Float32x if any
+   argument is _FloatNx.  */
+#  define __tgmath_real_type2_base(expr1, expr2)			\
+  __typeof ((__tgmath_real_type (expr1)) 0 + (__tgmath_real_type (expr2)) 0)
+
+/* The tgmath complex type of EXPR1 combined with EXPR2, without
+   handling the C2X rule of interpreting integer arguments as
+   _Float32x if any argument is _FloatNx.  */
+#  define __tgmath_complex_type2_base(expr1, expr2)	\
+  __typeof ((__tgmath_complex_type (expr1)) 0		\
+	    + (__tgmath_complex_type (expr2)) 0)
+
+/* The tgmath real type of EXPR1 combined with EXPR2 and EXPR3,
+   without handling the C2X rule of interpreting integer arguments as
+   _Float32x if any argument is _FloatNx.  */
+#  define __tgmath_real_type3_base(expr1, expr2, expr3)	\
+  __typeof ((__tgmath_real_type (expr1)) 0		\
+	    + (__tgmath_real_type (expr2)) 0		\
+	    + (__tgmath_real_type (expr3)) 0)
+
+/* The tgmath real or complex type of EXPR1 combined with EXPR2 (and
+   EXPR3 if applicable).  */
+#  if __HAVE_FLOATN_NOT_TYPEDEF
+#   define __tgmath_real_type2(expr1, expr2)				\
+  __tgmath_type_if (_Float32x, __tgmath_real_type2_base (expr1, expr2), \
+		    _Generic ((expr1) + (expr2), _Float32x: 1, default: 0))
+#   define __tgmath_complex_type2(expr1, expr2)				\
+  __tgmath_type_if (_Float32x,						\
+		    __tgmath_type_if (_Complex _Float32x,		\
+				      __tgmath_complex_type2_base (expr1, \
+								   expr2), \
+				      _Generic ((expr1) + (expr2),	\
+						_Complex _Float32x: 1,	\
+						default: 0)),		\
+		    _Generic ((expr1) + (expr2), _Float32x: 1, default: 0))
+#   define __tgmath_real_type3(expr1, expr2, expr3)			\
+  __tgmath_type_if (_Float32x,						\
+		    __tgmath_real_type3_base (expr1, expr2, expr3),	\
+		    _Generic ((expr1) + (expr2) + (expr3),		\
+			      _Float32x: 1, default: 0))
+#  else
+#   define __tgmath_real_type2(expr1, expr2)	\
+  __tgmath_real_type2_base (expr1, expr2)
+#   define __tgmath_complex_type2(expr1, expr2)	\
+  __tgmath_complex_type2_base (expr1, expr2)
+#   define __tgmath_real_type3(expr1, expr2, expr3)	\
+  __tgmath_real_type3_base (expr1, expr2, expr3)
+#  endif
+
 #  if (__HAVE_DISTINCT_FLOAT16			\
       || __HAVE_DISTINCT_FLOAT32		\
       || __HAVE_DISTINCT_FLOAT64		\
@@ -258,7 +321,10 @@
 
 /* Expand to text that checks if ARG_COMB has type _Float128, and if
    so calls the appropriately suffixed FCT (which may include a cast),
-   or FCT and CFCT for complex functions, with arguments ARG_CALL.  */
+   or FCT and CFCT for complex functions, with arguments ARG_CALL.
+   __TGMATH_F128LD (only used in the __HAVE_FLOAT64X_LONG_DOUBLE case,
+   for narrowing macros) handles long double the same as
+   _Float128.  */
 #  if __HAVE_DISTINCT_FLOAT128 && __GLIBC_USE (IEC_60559_TYPES_EXT)
 #   if (!__HAVE_FLOAT64X			\
        || __HAVE_FLOAT64X_LONG_DOUBLE		\
@@ -266,6 +332,10 @@
 #    define __TGMATH_F128(arg_comb, fct, arg_call)			\
   __builtin_types_compatible_p (__typeof (+(arg_comb)), _Float128)	\
   ? fct ## f128 arg_call :
+#    define __TGMATH_F128LD(arg_comb, fct, arg_call)			\
+  (__builtin_types_compatible_p (__typeof (+(arg_comb)), _Float128)	\
+   || __builtin_types_compatible_p (__typeof (+(arg_comb)), long double)) \
+  ? fct ## f128 arg_call :
 #    define __TGMATH_CF128(arg_comb, fct, cfct, arg_call)		\
   __builtin_types_compatible_p (__typeof (+__real__ (arg_comb)), _Float128) \
   ? (__expr_is_real (arg_comb)						\
@@ -291,7 +361,7 @@
 #   define __TGMATH_CF128(arg_comb, fct, cfct, arg_call) /* Nothing.  */
 #  endif
 
-# endif /* !__HAVE_BUILTIN_TGMATH.  */
+# endif /* !__HAVE_BUILTIN_TGMATH_C2X.  */
 
 /* We have two kinds of generic macros: to support functions which are
    only defined on real valued parameters and those which are defined
@@ -304,14 +374,18 @@
   __TGMATH_2 (Fct, (Val1), (Val2))
 #  define __TGMATH_BINARY_FIRST_REAL_STD_ONLY(Val1, Val2, Fct)	\
   __TGMATH_2STD (Fct, (Val1), (Val2))
-#  define __TGMATH_BINARY_REAL_ONLY(Val1, Val2, Fct)	\
+#  if __HAVE_BUILTIN_TGMATH_C2X
+#   define __TGMATH_BINARY_REAL_ONLY(Val1, Val2, Fct)	\
   __TGMATH_2 (Fct, (Val1), (Val2))
+#  endif
 #  define __TGMATH_BINARY_REAL_STD_ONLY(Val1, Val2, Fct)	\
   __TGMATH_2STD (Fct, (Val1), (Val2))
-#  define __TGMATH_TERNARY_FIRST_SECOND_REAL_ONLY(Val1, Val2, Val3, Fct) \
+#  if __HAVE_BUILTIN_TGMATH_C2X
+#   define __TGMATH_TERNARY_FIRST_SECOND_REAL_ONLY(Val1, Val2, Val3, Fct) \
   __TGMATH_3 (Fct, (Val1), (Val2), (Val3))
-#  define __TGMATH_TERNARY_REAL_ONLY(Val1, Val2, Val3, Fct)	\
+#   define __TGMATH_TERNARY_REAL_ONLY(Val1, Val2, Val3, Fct)	\
   __TGMATH_3 (Fct, (Val1), (Val2), (Val3))
+#  endif
 #  define __TGMATH_TERNARY_FIRST_REAL_RET_ONLY(Val1, Val2, Val3, Fct)	\
   __TGMATH_3 (Fct, (Val1), (Val2), (Val3))
 #  define __TGMATH_UNARY_REAL_IMAG(Val, Fct, Cfct)	\
@@ -321,11 +395,14 @@
   __TGMATH_1C (Fct, Cfct, (Val))
 #  define __TGMATH_UNARY_REAL_IMAG_RET_REAL_SAME(Val, Cfct)	\
   __TGMATH_1 (Cfct, (Val))
-#  define __TGMATH_BINARY_REAL_IMAG(Val1, Val2, Fct, Cfct)	\
+#  if __HAVE_BUILTIN_TGMATH_C2X
+#   define __TGMATH_BINARY_REAL_IMAG(Val1, Val2, Fct, Cfct)	\
   __TGMATH_2C (Fct, Cfct, (Val1), (Val2))
+#  endif
 
-# else /* !__HAVE_BUILTIN_TGMATH.  */
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH
 #  define __TGMATH_UNARY_REAL_ONLY(Val, Fct)				\
   (__extension__ ((sizeof (+(Val)) == sizeof (double)			      \
 		      || __builtin_classify_type (Val) != 8)		      \
@@ -362,29 +439,28 @@
 		     : (sizeof (+(Val1)) == sizeof (float))		      \
 		     ? (__tgmath_real_type (Val1)) Fct##f (Val1, Val2)	      \
 		     : (__tgmath_real_type (Val1)) __tgml(Fct) (Val1, Val2)))
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH_C2X
 #  define __TGMATH_BINARY_REAL_ONLY(Val1, Val2, Fct) \
      (__extension__ ((sizeof ((Val1) + (Val2)) > sizeof (double)	      \
 		      && __builtin_classify_type ((Val1) + (Val2)) == 8)      \
 		     ? __TGMATH_F128 ((Val1) + (Val2),			      \
-				      (__typeof				      \
-				       ((__tgmath_real_type (Val1)) 0	      \
-					+ (__tgmath_real_type (Val2)) 0)) Fct, \
+				      (__tgmath_real_type2 (Val1, Val2)) Fct, \
 				      (Val1, Val2))			      \
-		     (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				+ (__tgmath_real_type (Val2)) 0))	      \
+		     (__tgmath_real_type2 (Val1, Val2))			      \
 		     __tgml(Fct) (Val1, Val2)				      \
 		     : (sizeof (+(Val1)) == sizeof (double)		      \
 			|| sizeof (+(Val2)) == sizeof (double)		      \
 			|| __builtin_classify_type (Val1) != 8		      \
 			|| __builtin_classify_type (Val2) != 8)		      \
-		     ? (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0))	      \
+		     ? (__tgmath_real_type2 (Val1, Val2))		      \
 		       Fct (Val1, Val2)					      \
-		     : (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0))	      \
+		     : (__tgmath_real_type2 (Val1, Val2))		      \
 		       Fct##f (Val1, Val2)))
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH
 #  define __TGMATH_BINARY_REAL_STD_ONLY(Val1, Val2, Fct) \
      (__extension__ ((sizeof ((Val1) + (Val2)) > sizeof (double)	      \
 		      && __builtin_classify_type ((Val1) + (Val2)) == 8)      \
@@ -401,27 +477,24 @@
 		     : (__typeof ((__tgmath_real_type (Val1)) 0		      \
 				   + (__tgmath_real_type (Val2)) 0))	      \
 		       Fct##f (Val1, Val2)))
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH_C2X
 #  define __TGMATH_TERNARY_FIRST_SECOND_REAL_ONLY(Val1, Val2, Val3, Fct) \
      (__extension__ ((sizeof ((Val1) + (Val2)) > sizeof (double)	      \
 		      && __builtin_classify_type ((Val1) + (Val2)) == 8)      \
 		     ? __TGMATH_F128 ((Val1) + (Val2),			      \
-				      (__typeof				      \
-				       ((__tgmath_real_type (Val1)) 0	      \
-					+ (__tgmath_real_type (Val2)) 0)) Fct, \
+				      (__tgmath_real_type2 (Val1, Val2)) Fct, \
 				      (Val1, Val2, Val3))		      \
-		     (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				+ (__tgmath_real_type (Val2)) 0))	      \
+		     (__tgmath_real_type2 (Val1, Val2))			      \
 		     __tgml(Fct) (Val1, Val2, Val3)			      \
 		     : (sizeof (+(Val1)) == sizeof (double)		      \
 			|| sizeof (+(Val2)) == sizeof (double)		      \
 			|| __builtin_classify_type (Val1) != 8		      \
 			|| __builtin_classify_type (Val2) != 8)		      \
-		     ? (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0))	      \
+		     ? (__tgmath_real_type2 (Val1, Val2))		      \
 		       Fct (Val1, Val2, Val3)				      \
-		     : (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0))	      \
+		     : (__tgmath_real_type2 (Val1, Val2))		      \
 		       Fct##f (Val1, Val2, Val3)))
 
 #  define __TGMATH_TERNARY_REAL_ONLY(Val1, Val2, Val3, Fct) \
@@ -429,14 +502,10 @@
 		      && __builtin_classify_type ((Val1) + (Val2) + (Val3))   \
 			 == 8)						      \
 		     ? __TGMATH_F128 ((Val1) + (Val2) + (Val3),		      \
-				      (__typeof				      \
-				       ((__tgmath_real_type (Val1)) 0	      \
-					+ (__tgmath_real_type (Val2)) 0	      \
-					+ (__tgmath_real_type (Val3)) 0)) Fct, \
+				      (__tgmath_real_type3 (Val1, Val2,	      \
+							    Val3)) Fct,	      \
 				      (Val1, Val2, Val3))		      \
-		     (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				+ (__tgmath_real_type (Val2)) 0		      \
-				+ (__tgmath_real_type (Val3)) 0))	      \
+		     (__tgmath_real_type3 (Val1, Val2, Val3))		      \
 		       __tgml(Fct) (Val1, Val2, Val3)			      \
 		     : (sizeof (+(Val1)) == sizeof (double)		      \
 			|| sizeof (+(Val2)) == sizeof (double)		      \
@@ -444,15 +513,13 @@
 			|| __builtin_classify_type (Val1) != 8		      \
 			|| __builtin_classify_type (Val2) != 8		      \
 			|| __builtin_classify_type (Val3) != 8)		      \
-		     ? (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0	      \
-				   + (__tgmath_real_type (Val3)) 0))	      \
+		     ? (__tgmath_real_type3 (Val1, Val2, Val3))		      \
 		       Fct (Val1, Val2, Val3)				      \
-		     : (__typeof ((__tgmath_real_type (Val1)) 0		      \
-				   + (__tgmath_real_type (Val2)) 0	      \
-				   + (__tgmath_real_type (Val3)) 0))	      \
+		     : (__tgmath_real_type3 (Val1, Val2, Val3))		      \
 		       Fct##f (Val1, Val2, Val3)))
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH
 #  define __TGMATH_TERNARY_FIRST_REAL_RET_ONLY(Val1, Val2, Val3, Fct) \
      (__extension__ ((sizeof (+(Val1)) == sizeof (double)		\
 		      || __builtin_classify_type (Val1) != 8)		\
@@ -528,7 +595,9 @@
 		      __tgml(Cfct) (Val))))
 #  define __TGMATH_UNARY_REAL_IMAG_RET_REAL_SAME(Val, Cfct)	\
   __TGMATH_UNARY_REAL_IMAG_RET_REAL ((Val), Cfct, Cfct)
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH_C2X
 /* XXX This definition has to be changed as soon as the compiler understands
    the imaginary keyword.  */
 #  define __TGMATH_BINARY_REAL_IMAG(Val1, Val2, Fct, Cfct) \
@@ -537,41 +606,33 @@
 		      && __builtin_classify_type (__real__ (Val1)	      \
 						  + __real__ (Val2)) == 8)    \
 		     ? __TGMATH_CF128 ((Val1) + (Val2),			      \
-				       (__typeof			      \
-					((__tgmath_complex_type (Val1)) 0     \
-					 + (__tgmath_complex_type (Val2)) 0)) \
+				       (__tgmath_complex_type2 (Val1, Val2))  \
 				       Fct,				      \
-				       (__typeof			      \
-					((__tgmath_complex_type (Val1)) 0     \
-					 + (__tgmath_complex_type (Val2)) 0)) \
+				       (__tgmath_complex_type2 (Val1, Val2))  \
 				       Cfct,				      \
 				       (Val1, Val2))			      \
 		     (__expr_is_real ((Val1) + (Val2))			      \
-		      ? (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+		      ? (__tgmath_complex_type2 (Val1, Val2))		      \
 		      __tgml(Fct) (Val1, Val2)				      \
-		      : (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+		      : (__tgmath_complex_type2 (Val1, Val2))		      \
 		      __tgml(Cfct) (Val1, Val2))			      \
 		     : (sizeof (+__real__ (Val1)) == sizeof (double)	      \
 			|| sizeof (+__real__ (Val2)) == sizeof (double)	      \
 			|| __builtin_classify_type (__real__ (Val1)) != 8     \
 			|| __builtin_classify_type (__real__ (Val2)) != 8)    \
 		     ? (__expr_is_real ((Val1) + (Val2))		      \
-			? (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+			? (__tgmath_complex_type2 (Val1, Val2))		      \
 			  Fct (Val1, Val2)				      \
-			: (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+			: (__tgmath_complex_type2 (Val1, Val2))		      \
 			  Cfct (Val1, Val2))				      \
 		     : (__expr_is_real ((Val1) + (Val2))		      \
-			? (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+			? (__tgmath_complex_type2 (Val1, Val2))		      \
 			  Fct##f (Val1, Val2)				      \
-			: (__typeof ((__tgmath_complex_type (Val1)) 0	      \
-				   + (__tgmath_complex_type (Val2)) 0))	      \
+			: (__tgmath_complex_type2 (Val1, Val2))		      \
 			  Cfct##f (Val1, Val2))))
+# endif
 
+# if !__HAVE_BUILTIN_TGMATH
 #  define __TGMATH_1_NARROW_F(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (double) \
 		  ? F ## l (X)						\
@@ -587,6 +648,7 @@
 			  + (__tgmath_real_type (Z)) 0) > sizeof (double) \
 		  ? F ## l (X, Y, Z)					\
 		  : F (X, Y, Z)))
+# endif
 /* In most cases, these narrowing macro definitions based on sizeof
    ensure that the function called has the right argument format, as
    for other <tgmath.h> macros for compilers before GCC 8, but may not
@@ -595,60 +657,74 @@
 
    In the case of macros for _Float32x return type, when _Float64x
    exists, _Float64 arguments should result in the *f64 function being
-   called while _Float32x arguments should result in the *f64x
-   function being called.  These cases cannot be distinguished using
-   sizeof (or at all if the types are typedefs rather than different
-   types).  However, for these functions it is OK (does not affect the
-   final result) to call a function with any argument format at least
-   as wide as all the floating-point arguments, unless that affects
-   rounding of integer arguments.  Integer arguments are considered to
-   have type _Float64, so the *f64 functions are preferred for f32x*
-   macros when no argument has a wider floating-point type.  */
-#  if __HAVE_FLOAT64X_LONG_DOUBLE && __HAVE_DISTINCT_FLOAT128
+   called while _Float32x, float and double arguments should result in
+   the *f64x function being called (and integer arguments are
+   considered to have type _Float32x if any argument has type
+   _FloatNx, or double otherwise).  These cases cannot be
+   distinguished using sizeof (or at all if the types are typedefs
+   rather than different types, in which case we err on the side of
+   using the wider type if unsure).  */
+# if !__HAVE_BUILTIN_TGMATH_C2X
+#  if __HAVE_FLOATN_NOT_TYPEDEF
+#   define __TGMATH_NARROW_F32X_USE_F64X(X)			\
+  !__builtin_types_compatible_p (__typeof (+(X)), _Float64)
+#  else
+#   define __TGMATH_NARROW_F32X_USE_F64X(X)			\
+  (__builtin_types_compatible_p (__typeof (+(X)), double)	\
+   || __builtin_types_compatible_p (__typeof (+(X)), float)	\
+   || !__floating_type (__typeof (+(X))))
+#  endif
+# endif
+# if __HAVE_FLOAT64X_LONG_DOUBLE && __HAVE_DISTINCT_FLOAT128
+#  if !__HAVE_BUILTIN_TGMATH
 #   define __TGMATH_1_NARROW_F32(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X), F, (X))				\
+		  ? __TGMATH_F128LD ((X), F, (X))			\
 		  F ## f64x (X)						\
 		  : F ## f64 (X)))
 #   define __TGMATH_2_NARROW_F32(F, X, Y)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X) + (Y), F, (X, Y))		\
+		  ? __TGMATH_F128LD ((X) + (Y), F, (X, Y))		\
 		  F ## f64x (X, Y)					\
 		  : F ## f64 (X, Y)))
 #   define __TGMATH_3_NARROW_F32(F, X, Y, Z)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0			\
 			  + (__tgmath_real_type (Z)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X) + (Y) + (Z), F, (X, Y, Z))	\
+		  ? __TGMATH_F128LD ((X) + (Y) + (Z), F, (X, Y, Z))	\
 		  F ## f64x (X, Y, Z)					\
 		  : F ## f64 (X, Y, Z)))
 #   define __TGMATH_1_NARROW_F64(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X), F, (X))				\
+		  ? __TGMATH_F128LD ((X), F, (X))			\
 		  F ## f64x (X)						\
 		  : F ## f128 (X)))
 #   define __TGMATH_2_NARROW_F64(F, X, Y)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X) + (Y), F, (X, Y))		\
+		  ? __TGMATH_F128LD ((X) + (Y), F, (X, Y))		\
 		  F ## f64x (X, Y)					\
 		  : F ## f128 (X, Y)))
 #   define __TGMATH_3_NARROW_F64(F, X, Y, Z)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0			\
 			  + (__tgmath_real_type (Z)) 0) > sizeof (_Float64) \
-		  ? __TGMATH_F128 ((X) + (Y) + (Z), F, (X, Y, Z))	\
+		  ? __TGMATH_F128LD ((X) + (Y) + (Z), F, (X, Y, Z))	\
 		  F ## f64x (X, Y, Z)					\
 		  : F ## f128 (X, Y, Z)))
+#  endif
+#  if !__HAVE_BUILTIN_TGMATH_C2X
 #   define __TGMATH_1_NARROW_F32X(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (_Float64) \
+		  || __TGMATH_NARROW_F32X_USE_F64X (X)			\
 		  ? __TGMATH_F128 ((X), F, (X))				\
 		  F ## f64x (X)						\
 		  : F ## f64 (X)))
 #   define __TGMATH_2_NARROW_F32X(F, X, Y)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0) > sizeof (_Float64) \
+		  || __TGMATH_NARROW_F32X_USE_F64X ((X) + (Y))		\
 		  ? __TGMATH_F128 ((X) + (Y), F, (X, Y))		\
 		  F ## f64x (X, Y)					\
 		  : F ## f64 (X, Y)))
@@ -656,10 +732,13 @@
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0			\
 			  + (__tgmath_real_type (Z)) 0) > sizeof (_Float64) \
+		  || __TGMATH_NARROW_F32X_USE_F64X ((X) + (Y) + (Z))	\
 		  ? __TGMATH_F128 ((X) + (Y) + (Z), F, (X, Y, Z))	\
 		  F ## f64x (X, Y, Z)					\
 		  : F ## f64 (X, Y, Z)))
-#  elif __HAVE_FLOAT128
+#  endif
+# elif __HAVE_FLOAT128
+#  if !__HAVE_BUILTIN_TGMATH
 #   define __TGMATH_1_NARROW_F32(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (_Float64) \
 		  ? F ## f128 (X)					\
@@ -681,22 +760,29 @@
   (F ## f128 (X, Y))
 #   define __TGMATH_3_NARROW_F64(F, X, Y, Z)	\
   (F ## f128 (X, Y, Z))
+#  endif
+#  if !__HAVE_BUILTIN_TGMATH_C2X
 #   define __TGMATH_1_NARROW_F32X(F, X)					\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0) > sizeof (_Float32x) \
+		  || __TGMATH_NARROW_F32X_USE_F64X (X)			\
 		  ? F ## f64x (X)					\
 		  : F ## f64 (X)))
 #   define __TGMATH_2_NARROW_F32X(F, X, Y)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0) > sizeof (_Float32x) \
+		  || __TGMATH_NARROW_F32X_USE_F64X ((X) + (Y))		\
 		  ? F ## f64x (X, Y)					\
 		  : F ## f64 (X, Y)))
 #   define __TGMATH_3_NARROW_F32X(F, X, Y, Z)				\
   (__extension__ (sizeof ((__tgmath_real_type (X)) 0			\
 			  + (__tgmath_real_type (Y)) 0			\
 			  + (__tgmath_real_type (Z)) 0) > sizeof (_Float32x) \
+		  || __TGMATH_NARROW_F32X_USE_F64X ((X) + (Y) + (Z))	\
 		  ? F ## f64x (X, Y, Z)					\
 		  : F ## f64 (X, Y, Z)))
-#  else
+#  endif
+# else
+#  if !__HAVE_BUILTIN_TGMATH
 #   define __TGMATH_1_NARROW_F32(F, X)		\
   (F ## f64 (X))
 #   define __TGMATH_2_NARROW_F32(F, X, Y)	\
@@ -704,7 +790,7 @@
 #   define __TGMATH_3_NARROW_F32(F, X, Y, Z)	\
   (F ## f64 (X, Y, Z))
 #  endif
-# endif /* !__HAVE_BUILTIN_TGMATH.  */
+# endif
 #else
 # error "Unsupported compiler; you cannot use <tgmath.h>"
 #endif
