[Python-checkins] r60471 - in python/trunk: Doc/library/numbers.rst Lib/numbers.py Lib/rational.py
jeffrey.yasskin
python-checkins at python.org
Thu Jan 31 08:44:12 CET 2008
Author: jeffrey.yasskin
Date: Thu Jan 31 08:44:11 2008
New Revision: 60471
Modified:
python/trunk/Doc/library/numbers.rst
python/trunk/Lib/numbers.py
python/trunk/Lib/rational.py
Log:
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).
Modified: python/trunk/Doc/library/numbers.rst
==============================================================================
--- python/trunk/Doc/library/numbers.rst (original)
+++ python/trunk/Doc/library/numbers.rst Thu Jan 31 08:44:11 2008
@@ -99,3 +99,144 @@
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)
+
+ # ...
\ No newline at end of file
Modified: python/trunk/Lib/numbers.py
==============================================================================
--- python/trunk/Lib/numbers.py (original)
+++ python/trunk/Lib/numbers.py Thu Jan 31 08:44:11 2008
@@ -292,7 +292,13 @@
# 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
Modified: python/trunk/Lib/rational.py
==============================================================================
--- python/trunk/Lib/rational.py (original)
+++ python/trunk/Lib/rational.py Thu Jan 31 08:44:11 2008
@@ -179,16 +179,6 @@
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 @@
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):
- # Includes ints.
+ if isinstance(b, (int, long, Rational)):
return monomorphic_operator(a, b)
elif isinstance(b, float):
return fallback_operator(float(a), b)
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