diff --git a/Doc/library/numbers.rst b/Doc/library/numbers.rst index 505a8af2dc9..6ee8f271866 100644 --- a/Doc/library/numbers.rst +++ b/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) + + # ... \ No newline at end of file diff --git a/Lib/numbers.py b/Lib/numbers.py index 8e02203e308..e391abc387a 100644 --- a/Lib/numbers.py +++ b/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 diff --git a/Lib/rational.py b/Lib/rational.py index 99c5ff6b9f0..f86904dc91f 100755 --- a/Lib/rational.py +++ b/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): - # Includes ints. + if isinstance(b, (int, long, Rational)): return monomorphic_operator(a, b) elif isinstance(b, float): return fallback_operator(float(a), b)