Lib/numbers.py

# Copyright 2007 Google, Inc. All Rights Reserved.
# Licensed to PSF under a Contributor Agreement.

"""Abstract Base Classes (ABCs) for numbers, according to PEP 3141.

TODO: Fill out more detailed documentation on the operators."""

from abc import ABCMeta, abstractmethod, abstractproperty

__all__ = ["Number", "Exact", "Inexact",
           "Complex", "Real", "Rational", "Integral",
           ]


class Number(metaclass=ABCMeta):
    """All numbers inherit from this class.

    If you just want to check if an argument x is a number, without
    caring what kind, use isinstance(x, Number).
    """


class Exact(Number):
    """Operations on instances of this type are exact.

    As long as the result of a homogenous operation is of the same
    type, you can assume that it was computed exactly, and there are
    no round-off errors. Laws like commutativity and associativity
    hold.
    """

Exact.register(int)


class Inexact(Number):
    """Operations on instances of this type are inexact.

    Given X, an instance of Inexact, it is possible that (X + -X) + 3
    == 3, but X + (-X + 3) == 0. The exact form this error takes will
    vary by type, but it's generally unsafe to compare this type for
    equality.
    """

Inexact.register(complex)
Inexact.register(float)
# Inexact.register(decimal.Decimal)


## Notes on Decimal and it how relates to the numeric tower
## --------------------------------------------------------
## Decimal is Real except that it does not support rich comparisons.
##
## Decimal has some of the characteristics of Integrals.  It provides
## logical operations but not as operators.  The logical operations only apply
## to a subset of decimals (those that are non-negative, have a zero exponent,
## and have digits that are only 0 or 1).  It does provide __long__() and
## a three argument form of __pow__ that includes exactness guarantees.
## It does not provide an __index__() method.
##
## Depending on context, decimal operations may be exact or inexact.
##
## When decimal is run in a context with small precision and automatic rounding,
## it is Inexact.  See the "Floating point notes" section of the decimal docs
## for an example of losing the associative and distributive properties of
## addition.
##
## When decimal is used for high precision integer arithmetic, it is Exact.
## When the decimal used as fixed-point, it is Exact.
## When it is run with sufficient precision, it is Exact.
## When the decimal.Inexact trap is set, decimal operations are Exact.
## For an example, see the float_to_decimal() recipe in the "Decimal FAQ"
## section of the docs -- it shows an how traps are used in conjunction
## with variable precision to reliably achieve exact results.

class Complex(Number):
    """Complex defines the operations that work on the builtin complex type.

    In short, those are: a conversion to complex, .real, .imag, +, -,
    *, /, abs(), .conjugate, ==, and !=.

    If it is given heterogenous arguments, and doesn't have special
    knowledge about them, it should fall back to the builtin complex
    type as described below.
    """

    @abstractmethod
    def __complex__(self):
        """Return a builtin complex instance. Called for complex(self)."""

    def __bool__(self):
        """True if self != 0. Called for bool(self)."""
        return self != 0

    @abstractproperty
    def real(self):
        """Retrieve the real component of this number.

        This should subclass Real.
        """
        raise NotImplementedError

    @abstractproperty
    def imag(self):
        """Retrieve the real component of this number.

        This should subclass Real.
        """
        raise NotImplementedError

    @abstractmethod
    def __add__(self, other):
        """self + other"""
        raise NotImplementedError

    @abstractmethod
    def __radd__(self, other):
        """other + self"""
        raise NotImplementedError

    @abstractmethod
    def __neg__(self):
        """-self"""
        raise NotImplementedError

    @abstractmethod
    def __pos__(self):
        """+self"""
        raise NotImplementedError

    def __sub__(self, other):
        """self - other"""
        return self + -other

    def __rsub__(self, other):
        """other - self"""
        return -self + other

    @abstractmethod
    def __mul__(self, other):
        """self * other"""
        raise NotImplementedError

    @abstractmethod
    def __rmul__(self, other):
        """other * self"""
        raise NotImplementedError

    @abstractmethod
    def __truediv__(self, other):
        """self / other: Should promote to float when necessary."""
        raise NotImplementedError

    @abstractmethod
    def __rtruediv__(self, other):
        """other / self"""
        raise NotImplementedError

    @abstractmethod
    def __pow__(self, exponent):
        """self**exponent; should promote to float or complex when necessary."""
        raise NotImplementedError

    @abstractmethod
    def __rpow__(self, base):
        """base ** self"""
        raise NotImplementedError

    @abstractmethod
    def __abs__(self):
        """Returns the Real distance from 0. Called for abs(self)."""
        raise NotImplementedError

    @abstractmethod
    def conjugate(self):
        """(x+y*i).conjugate() returns (x-y*i)."""
        raise NotImplementedError

    @abstractmethod
    def __eq__(self, other):
        """self == other"""
        raise NotImplementedError

    def __ne__(self, other):
        """self != other"""
        # The default __ne__ doesn't negate __eq__ until 3.0.
        return not (self == other)

Complex.register(complex)


class Real(Complex):
    """To Complex, Real adds the operations that work on real numbers.

    In short, those are: a conversion to float, trunc(), divmod,
    %, <, <=, >, and >=.

    Real also provides defaults for the derived operations.
    """

    @abstractmethod
    def __float__(self):
        """Any Real can be converted to a native float object.

        Called for float(self)."""
        raise NotImplementedError

    @abstractmethod
    def __trunc__(self):
        """trunc(self): Truncates self to an Integral.

        Returns an Integral i such that:
          * i>0 iff self>0;
          * abs(i) <= abs(self);
          * for any Integral j satisfying the first two conditions,
            abs(i) >= abs(j) [i.e. i has "maximal" abs among those].
        i.e. "truncate towards 0".
        """
        raise NotImplementedError

    @abstractmethod
    def __floor__(self):
        """Finds the greatest Integral <= self."""
        raise NotImplementedError

    @abstractmethod
    def __ceil__(self):
        """Finds the least Integral >= self."""
        raise NotImplementedError

    @abstractmethod
    def __round__(self, ndigits:"Integral"=None):
        """Rounds self to ndigits decimal places, defaulting to 0.

        If ndigits is omitted or None, returns an Integral, otherwise
        returns a Real. Rounds half toward even.
        """
        raise NotImplementedError

    def __divmod__(self, other):
        """divmod(self, other): The pair (self // other, self % other).

        Sometimes this can be computed faster than the pair of
        operations.
        """
        return (self // other, self % other)

    def __rdivmod__(self, other):
        """divmod(other, self): The pair (self // other, self % other).

        Sometimes this can be computed faster than the pair of
        operations.
        """
        return (other // self, other % self)

    @abstractmethod
    def __floordiv__(self, other):
        """self // other: The floor() of self/other."""
        raise NotImplementedError

    @abstractmethod
    def __rfloordiv__(self, other):
        """other // self: The floor() of other/self."""
        raise NotImplementedError

    @abstractmethod
    def __mod__(self, other):
        """self % other"""
        raise NotImplementedError

    @abstractmethod
    def __rmod__(self, other):
        """other % self"""
        raise NotImplementedError

    @abstractmethod
    def __lt__(self, other):
        """self < other

        < on Reals defines a total ordering, except perhaps for NaN."""
        raise NotImplementedError

    @abstractmethod
    def __le__(self, other):
        """self <= other"""
        raise NotImplementedError

    # Concrete implementations of Complex abstract methods.
    def __complex__(self):
        """complex(self) == complex(float(self), 0)"""
        return complex(float(self))

    @property
    def real(self):
        """Real numbers are their real component."""
        return +self

    @property
    def imag(self):
        """Real numbers have no imaginary component."""
        return 0

    def conjugate(self):
        """Conjugate is a no-op for Reals."""
        return +self

Real.register(float)
# Real.register(decimal.Decimal)


class Rational(Real, Exact):
    """.numerator and .denominator should be in lowest terms."""

    @abstractproperty
    def numerator(self):
        raise NotImplementedError

    @abstractproperty
    def denominator(self):
        raise NotImplementedError

    # Concrete implementation of Real's conversion to float.
    def __float__(self):
        """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


class Integral(Rational):
    """Integral adds a conversion to int and the bit-string operations."""

    @abstractmethod
    def __int__(self):
        """int(self)"""
        raise NotImplementedError

    def __index__(self):
        """index(self)"""
        return int(self)

    @abstractmethod
    def __pow__(self, exponent, modulus=None):
        """self ** exponent % modulus, but maybe faster.

        Accept the modulus argument if you want to support the
        3-argument version of pow(). Raise a TypeError if exponent < 0
        or any argument isn't Integral. Otherwise, just implement the
        2-argument version described in Complex.
        """
        raise NotImplementedError

    @abstractmethod
    def __lshift__(self, other):
        """self << other"""
        raise NotImplementedError

    @abstractmethod
    def __rlshift__(self, other):
        """other << self"""
        raise NotImplementedError

    @abstractmethod
    def __rshift__(self, other):
        """self >> other"""
        raise NotImplementedError

    @abstractmethod
    def __rrshift__(self, other):
        """other >> self"""
        raise NotImplementedError

    @abstractmethod
    def __and__(self, other):
        """self & other"""
        raise NotImplementedError

    @abstractmethod
    def __rand__(self, other):
        """other & self"""
        raise NotImplementedError

    @abstractmethod
    def __xor__(self, other):
        """self ^ other"""
        raise NotImplementedError

    @abstractmethod
    def __rxor__(self, other):
        """other ^ self"""
        raise NotImplementedError

    @abstractmethod
    def __or__(self, other):
        """self | other"""
        raise NotImplementedError

    @abstractmethod
    def __ror__(self, other):
        """other | self"""
        raise NotImplementedError

    @abstractmethod
    def __invert__(self):
        """~self"""
        raise NotImplementedError

    # Concrete implementations of Rational and Real abstract methods.
    def __float__(self):
        """float(self) == float(int(self))"""
        return float(int(self))

    @property
    def numerator(self):
        """Integers are their own numerators."""
        return +self

    @property
    def denominator(self):
        """Integers have a denominator of 1."""
        return 1

Integral.register(int)