diff -r 909099686e6e Lib/test/test_float.py
--- a/Lib/test/test_float.py	Tue May 10 12:01:56 2016 +0300
+++ b/Lib/test/test_float.py	Tue May 10 22:07:12 2016 +0300
@@ -161,11 +161,12 @@ class GeneralFloatCases(unittest.TestCas
             def __float__(self):
                 return float(str(self)) + 1
 
-        self.assertAlmostEqual(float(Foo1()), 42.)
-        self.assertAlmostEqual(float(Foo2()), 42.)
-        self.assertAlmostEqual(float(Foo3(21)), 42.)
+        self.assertEqual(float(Foo1()), 42.)
+        self.assertEqual(float(Foo2()), 42.)
+        with self.assertWarns(DeprecationWarning):
+            self.assertEqual(float(Foo3(21)), 42.)
         self.assertRaises(TypeError, float, Foo4(42))
-        self.assertAlmostEqual(float(FooStr('8')), 9.)
+        self.assertEqual(float(FooStr('8')), 9.)
 
         class Foo5:
             def __float__(self):
@@ -176,10 +177,14 @@ class GeneralFloatCases(unittest.TestCas
         class F:
             def __float__(self):
                 return OtherFloatSubclass(42.)
-        self.assertAlmostEqual(float(F()), 42.)
-        self.assertIs(type(float(F())), OtherFloatSubclass)
-        self.assertAlmostEqual(FloatSubclass(F()), 42.)
-        self.assertIs(type(FloatSubclass(F())), FloatSubclass)
+        with self.assertWarns(DeprecationWarning):
+            self.assertEqual(float(F()), 42.)
+        with self.assertWarns(DeprecationWarning):
+            self.assertIs(type(float(F())), float)
+        with self.assertWarns(DeprecationWarning):
+            self.assertEqual(FloatSubclass(F()), 42.)
+        with self.assertWarns(DeprecationWarning):
+            self.assertIs(type(FloatSubclass(F())), FloatSubclass)
 
     def test_is_integer(self):
         self.assertFalse((1.1).is_integer())
diff -r 909099686e6e Objects/abstract.c
--- a/Objects/abstract.c	Tue May 10 12:01:56 2016 +0300
+++ b/Objects/abstract.c	Tue May 10 22:07:12 2016 +0300
@@ -1353,15 +1353,29 @@ PyNumber_Float(PyObject *o)
         return null_error();
     m = o->ob_type->tp_as_number;
     if (m && m->nb_float) { /* This should include subclasses of float */
-        PyObject *res = m->nb_float(o);
-        if (res && !PyFloat_Check(res)) {
+        PyObject *res = m->nb_float(o), *res2;
+        if (!res || PyFloat_CheckExact(res)) {
+            return res;
+        }
+        if (!PyFloat_Check(res)) {
             PyErr_Format(PyExc_TypeError,
               "__float__ returned non-float (type %.200s)",
               res->ob_type->tp_name);
             Py_DECREF(res);
             return NULL;
         }
-        return res;
+        /* Issue #26983: warn if 'res' not of exact type float. */
+        if (PyErr_WarnFormat(PyExc_DeprecationWarning, 1,
+                "__float__ returned non-float (type %.200s).  "
+                "The ability to return an instance of a strict subclass of float "
+                "is deprecated, and may be removed in a future version of Python.",
+                res->ob_type->tp_name)) {
+            Py_DECREF(res);
+            return NULL;
+        }
+        res2 = PyFloat_FromDouble(((PyFloatObject *)res)->ob_fval);
+        Py_DECREF(res);
+        return res2;
     }
     if (PyFloat_Check(o)) { /* A float subclass with nb_float == NULL */
         PyFloatObject *po = (PyFloatObject *)o;