Added more documentation on how mixed-mode arithmetic should be implemented. I

also noticed and fixed a bug in Rational's forward operators (they were
claiming all instances of numbers.Rational instead of just the concrete types).

Doc/library/numbers.rst

@@ -99,3 +99,144 @@ The numeric tower
    3-argument form of :func:`pow`, and the bit-string operations: ``<<``,
    ``>>``, ``&``, ``^``, ``|``, ``~``. Provides defaults for :func:`float`,
    :attr:`Rational.numerator`, and :attr:`Rational.denominator`.
+
+
+Notes for type implementors
+---------------------------
+
+Implementors should be careful to make equal numbers equal and hash
+them to the same values. This may be subtle if there are two different
+extensions of the real numbers. For example, :class:`rational.Rational`
+implements :func:`hash` as follows::
+
+    def __hash__(self):
+        if self.denominator == 1:
+            # Get integers right.
+            return hash(self.numerator)
+        # Expensive check, but definitely correct.
+        if self == float(self):
+            return hash(float(self))
+        else:
+            # Use tuple's hash to avoid a high collision rate on
+            # simple fractions.
+            return hash((self.numerator, self.denominator))
+
+
+Adding More Numeric ABCs
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+There are, of course, more possible ABCs for numbers, and this would
+be a poor hierarchy if it precluded the possibility of adding
+those. You can add ``MyFoo`` between :class:`Complex` and
+:class:`Real` with::
+
+    class MyFoo(Complex): ...
+    MyFoo.register(Real)
+
+
+Implementing the arithmetic operations
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+We want to implement the arithmetic operations so that mixed-mode
+operations either call an implementation whose author knew about the
+types of both arguments, or convert both to the nearest built in type
+and do the operation there. For subtypes of :class:`Integral`, this
+means that :meth:`__add__` and :meth:`__radd__` should be defined as::
+
+    class MyIntegral(Integral):
+
+        def __add__(self, other):
+            if isinstance(other, MyIntegral):
+                return do_my_adding_stuff(self, other)
+            elif isinstance(other, OtherTypeIKnowAbout):
+                return do_my_other_adding_stuff(self, other)
+            else:
+                return NotImplemented
+
+        def __radd__(self, other):
+            if isinstance(other, MyIntegral):
+                return do_my_adding_stuff(other, self)
+            elif isinstance(other, OtherTypeIKnowAbout):
+                return do_my_other_adding_stuff(other, self)
+            elif isinstance(other, Integral):
+                return int(other) + int(self)
+            elif isinstance(other, Real):
+                return float(other) + float(self)
+            elif isinstance(other, Complex):
+                return complex(other) + complex(self)
+            else:
+                return NotImplemented
+
+
+There are 5 different cases for a mixed-type operation on subclasses
+of :class:`Complex`. I'll refer to all of the above code that doesn't
+refer to ``MyIntegral`` and ``OtherTypeIKnowAbout`` as
+"boilerplate". ``a`` will be an instance of ``A``, which is a subtype
+of :class:`Complex` (``a : A <: Complex``), and ``b : B <:
+Complex``. I'll consider ``a + b``:
+
+    1. If ``A`` defines an :meth:`__add__` which accepts ``b``, all is
+       well.
+    2. If ``A`` falls back to the boilerplate code, and it were to
+       return a value from :meth:`__add__`, we'd miss the possibility
+       that ``B`` defines a more intelligent :meth:`__radd__`, so the
+       boilerplate should return :const:`NotImplemented` from
+       :meth:`__add__`. (Or ``A`` may not implement :meth:`__add__` at
+       all.)
+    3. Then ``B``'s :meth:`__radd__` gets a chance. If it accepts
+       ``a``, all is well.
+    4. If it falls back to the boilerplate, there are no more possible
+       methods to try, so this is where the default implementation
+       should live.
+    5. If ``B <: A``, Python tries ``B.__radd__`` before
+       ``A.__add__``. This is ok, because it was implemented with
+       knowledge of ``A``, so it can handle those instances before
+       delegating to :class:`Complex`.
+
+If ``A<:Complex`` and ``B<:Real`` without sharing any other knowledge,
+then the appropriate shared operation is the one involving the built
+in :class:`complex`, and both :meth:`__radd__` s land there, so ``a+b
+== b+a``.
+
+Because most of the operations on any given type will be very similar,
+it can be useful to define a helper function which generates the
+forward and reverse instances of any given operator. For example,
+:class:`rational.Rational` uses::
+
+    def _operator_fallbacks(monomorphic_operator, fallback_operator):
+        def forward(a, b):
+            if isinstance(b, (int, long, Rational)):
+                return monomorphic_operator(a, b)
+            elif isinstance(b, float):
+                return fallback_operator(float(a), b)
+            elif isinstance(b, complex):
+                return fallback_operator(complex(a), b)
+            else:
+                return NotImplemented
+        forward.__name__ = '__' + fallback_operator.__name__ + '__'
+        forward.__doc__ = monomorphic_operator.__doc__
+
+        def reverse(b, a):
+            if isinstance(a, RationalAbc):
+                # Includes ints.
+                return monomorphic_operator(a, b)
+            elif isinstance(a, numbers.Real):
+                return fallback_operator(float(a), float(b))
+            elif isinstance(a, numbers.Complex):
+                return fallback_operator(complex(a), complex(b))
+            else:
+                return NotImplemented
+        reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
+        reverse.__doc__ = monomorphic_operator.__doc__
+
+        return forward, reverse
+
+    def _add(a, b):
+        """a + b"""
+        return Rational(a.numerator * b.denominator +
+                        b.numerator * a.denominator,
+                        a.denominator * b.denominator)
+
+    __add__, __radd__ = _operator_fallbacks(_add, operator.add)
+
+    # ...

Lib/numbers.py

@@ -292,7 +292,13 @@ class Rational(Real, Exact):
 
     # Concrete implementation of Real's conversion to float.
     def __float__(self):
-        """float(self) = self.numerator / self.denominator"""
+        """float(self) = self.numerator / self.denominator
+
+        It's important that this conversion use the integer's "true"
+        division rather than casting one side to float before dividing
+        so that ratios of huge integers convert without overflowing.
+
+        """
         return self.numerator / self.denominator
 
 

Lib/rational.py

@@ -179,16 +179,6 @@ class Rational(RationalAbc):
         else:
             return '%s/%s' % (self.numerator, self.denominator)
 
-    """ XXX This section needs a lot more commentary
-
-    * Explain the typical sequence of checks, calls, and fallbacks.
-    * Explain the subtle reasons why this logic was needed.
-    * It is not clear how common cases are handled (for example, how
-      does the ratio of two huge integers get converted to a float
-      without overflowing the long-->float conversion.
-
-    """
-
     def _operator_fallbacks(monomorphic_operator, fallback_operator):
         """Generates forward and reverse operators given a purely-rational
         operator and a function from the operator module.
@@ -196,10 +186,82 @@ class Rational(RationalAbc):
         Use this like:
         __op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
 
+        In general, we want to implement the arithmetic operations so
+        that mixed-mode operations either call an implementation whose
+        author knew about the types of both arguments, or convert both
+        to the nearest built in type and do the operation there. In
+        Rational, that means that we define __add__ and __radd__ as:
+
+            def __add__(self, other):
+                if isinstance(other, (int, long, Rational)):
+                    # Do the real operation.
+                    return Rational(self.numerator * other.denominator +
+                                    other.numerator * self.denominator,
+                                    self.denominator * other.denominator)
+                # float and complex don't follow this protocol, and
+                # Rational knows about them, so special case them.
+                elif isinstance(other, float):
+                    return float(self) + other
+                elif isinstance(other, complex):
+                    return complex(self) + other
+                else:
+                    # Let the other type take over.
+                    return NotImplemented
+
+            def __radd__(self, other):
+                # radd handles more types than add because there's
+                # nothing left to fall back to.
+                if isinstance(other, RationalAbc):
+                    return Rational(self.numerator * other.denominator +
+                                    other.numerator * self.denominator,
+                                    self.denominator * other.denominator)
+                elif isinstance(other, Real):
+                    return float(other) + float(self)
+                elif isinstance(other, Complex):
+                    return complex(other) + complex(self)
+                else:
+                    return NotImplemented
+
+
+        There are 5 different cases for a mixed-type addition on
+        Rational. I'll refer to all of the above code that doesn't
+        refer to Rational, float, or complex as "boilerplate". 'r'
+        will be an instance of Rational, which is a subtype of
+        RationalAbc (r : Rational <: RationalAbc), and b : B <:
+        Complex. The first three involve 'r + b':
+
+            1. If B <: Rational, int, float, or complex, we handle
+               that specially, and all is well.
+            2. If Rational falls back to the boilerplate code, and it
+               were to return a value from __add__, we'd miss the
+               possibility that B defines a more intelligent __radd__,
+               so the boilerplate should return NotImplemented from
+               __add__. In particular, we don't handle RationalAbc
+               here, even though we could get an exact answer, in case
+               the other type wants to do something special.
+            3. If B <: Rational, Python tries B.__radd__ before
+               Rational.__add__. This is ok, because it was
+               implemented with knowledge of Rational, so it can
+               handle those instances before delegating to Real or
+               Complex.
+
+        The next two situations describe 'b + r'. We assume that b
+        didn't know about Rational in its implementation, and that it
+        uses similar boilerplate code:
+
+            4. If B <: RationalAbc, then __radd_ converts both to the
+               builtin rational type (hey look, that's us) and
+               proceeds.
+            5. Otherwise, __radd__ tries to find the nearest common
+               base ABC, and fall back to its builtin type. Since this
+               class doesn't subclass a concrete type, there's no
+               implementation to fall back to, so we need to try as
+               hard as possible to return an actual value, or the user
+               will get a TypeError.
+
         """
         def forward(a, b):
-            if isinstance(b, RationalAbc):
+            if isinstance(b, (int, long, Rational)):
-                # Includes ints.
                 return monomorphic_operator(a, b)
             elif isinstance(b, float):
                 return fallback_operator(float(a), b)