Source code for sympy.core.numbers

from __future__ import print_function, division

import decimal
import fractions
import math
import re as regex

from .containers import Tuple
from .sympify import converter, sympify, _sympify, SympifyError, _convert_numpy_types
from .singleton import S, Singleton
from .expr import Expr, AtomicExpr
from .decorators import _sympifyit
from .cache import cacheit, clear_cache
from .logic import fuzzy_not
from sympy.core.compatibility import (
    as_int, integer_types, long, string_types, with_metaclass, HAS_GMPY,
    SYMPY_INTS, int_info)
from sympy.core.cache import lru_cache

import mpmath
import mpmath.libmp as mlib
from mpmath.libmp.backend import MPZ
from mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
from mpmath.ctx_mp import mpnumeric
from mpmath.libmp.libmpf import (
    finf as _mpf_inf, fninf as _mpf_ninf,
    fnan as _mpf_nan, fzero as _mpf_zero, _normalize as mpf_normalize,
    prec_to_dps)
from sympy.utilities.misc import debug, filldedent
from .evaluate import global_evaluate

from sympy.utilities.exceptions import SymPyDeprecationWarning

rnd = mlib.round_nearest

_LOG2 = math.log(2)


def comp(z1, z2, tol=None):
    """Return a bool indicating whether the error between z1 and z2 is <= tol.

    If ``tol`` is None then True will be returned if there is a significant
    difference between the numbers: ``abs(z1 - z2)*10**p <= 1/2`` where ``p``
    is the lower of the precisions of the values. A comparison of strings will
    be made if ``z1`` is a Number and a) ``z2`` is a string or b) ``tol`` is ''
    and ``z2`` is a Number.

    When ``tol`` is a nonzero value, if z2 is non-zero and ``|z1| > 1``
    the error is normalized by ``|z1|``, so if you want to see if the
    absolute error between ``z1`` and ``z2`` is <= ``tol`` then call this
    as ``comp(z1 - z2, 0, tol)``.
    """
    if type(z2) is str:
        if not isinstance(z1, Number):
            raise ValueError('when z2 is a str z1 must be a Number')
        return str(z1) == z2
    if not z1:
        z1, z2 = z2, z1
    if not z1:
        return True
    if not tol:
        if tol is None:
            if type(z2) is str and getattr(z1, 'is_Number', False):
                return str(z1) == z2
            a, b = Float(z1), Float(z2)
            return int(abs(a - b)*10**prec_to_dps(
                min(a._prec, b._prec)))*2 <= 1
        elif all(getattr(i, 'is_Number', False) for i in (z1, z2)):
            return z1._prec == z2._prec and str(z1) == str(z2)
        raise ValueError('exact comparison requires two Numbers')
    diff = abs(z1 - z2)
    az1 = abs(z1)
    if z2 and az1 > 1:
        return diff/az1 <= tol
    else:
        return diff <= tol


def mpf_norm(mpf, prec):
    """Return the mpf tuple normalized appropriately for the indicated
    precision after doing a check to see if zero should be returned or
    not when the mantissa is 0. ``mpf_normlize`` always assumes that this
    is zero, but it may not be since the mantissa for mpf's values "+inf",
    "-inf" and "nan" have a mantissa of zero, too.

    Note: this is not intended to validate a given mpf tuple, so sending
    mpf tuples that were not created by mpmath may produce bad results. This
    is only a wrapper to ``mpf_normalize`` which provides the check for non-
    zero mpfs that have a 0 for the mantissa.
    """
    sign, man, expt, bc = mpf
    if not man:
        # hack for mpf_normalize which does not do this;
        # it assumes that if man is zero the result is 0
        # (see issue 6639)
        if not bc:
            return _mpf_zero
        else:
            # don't change anything; this should already
            # be a well formed mpf tuple
            return mpf

    # Necessary if mpmath is using the gmpy backend
    from mpmath.libmp.backend import MPZ
    rv = mpf_normalize(sign, MPZ(man), expt, bc, prec, rnd)
    return rv

# TODO: we should use the warnings module
_errdict = {"divide": False}


def seterr(divide=False):
    """
    Should sympy raise an exception on 0/0 or return a nan?

    divide == True .... raise an exception
    divide == False ... return nan
    """
    if _errdict["divide"] != divide:
        clear_cache()
        _errdict["divide"] = divide


def _as_integer_ratio(p):
    neg_pow, man, expt, bc = getattr(p, '_mpf_', mpmath.mpf(p)._mpf_)
    p = [1, -1][neg_pow % 2]*man
    if expt < 0:
        q = 2**-expt
    else:
        q = 1
        p *= 2**expt
    return int(p), int(q)


def _decimal_to_Rational_prec(dec):
    """Convert an ordinary decimal instance to a Rational."""
    if not dec.is_finite():
        raise TypeError("dec must be finite, got %s." % dec)
    s, d, e = dec.as_tuple()
    prec = len(d)
    if e >= 0:  # it's an integer
        rv = Integer(int(dec))
    else:
        s = (-1)**s
        d = sum([di*10**i for i, di in enumerate(reversed(d))])
        rv = Rational(s*d, 10**-e)
    return rv, prec


def _literal_float(f):
    """Return True if n can be interpreted as a floating point number."""
    pat = r"[-+]?((\d*\.\d+)|(\d+\.?))(eE[-+]?\d+)?"
    return bool(regex.match(pat, f))

# (a,b) -> gcd(a,b)

# TODO caching with decorator, but not to degrade performance

@lru_cache(1024)
def igcd(*args):
    """Computes nonnegative integer greatest common divisor.

    The algorithm is based on the well known Euclid's algorithm. To
    improve speed, igcd() has its own caching mechanism implemented.

    Examples
    ========

    >>> from sympy.core.numbers import igcd
    >>> igcd(2, 4)
    2
    >>> igcd(5, 10, 15)
    5

    """
    if len(args) < 2:
        raise TypeError(
            'igcd() takes at least 2 arguments (%s given)' % len(args))
    args_temp = [abs(as_int(i)) for i in args]
    if 1 in args_temp:
        return 1
    a = args_temp.pop()
    for b in args_temp:
        a = igcd2(a, b) if b else a
    return a


try:
    from math import gcd as igcd2
except ImportError:
    def igcd2(a, b):
        """Compute gcd of two Python integers a and b."""
        if (a.bit_length() > BIGBITS and
            b.bit_length() > BIGBITS):
            return igcd_lehmer(a, b)

        a, b = abs(a), abs(b)
        while b:
            a, b = b, a % b
        return a


# Use Lehmer's algorithm only for very large numbers.
# The limit could be different on Python 2.7 and 3.x.
# If so, then this could be defined in compatibility.py.
BIGBITS = 5000
def igcd_lehmer(a, b):
    """Computes greatest common divisor of two integers.

    Euclid's algorithm for the computation of the greatest
    common divisor  gcd(a, b)  of two (positive) integers
    a and b is based on the division identity
        a = q*b + r,
    where the quotient  q  and the remainder  r  are integers
    and  0 <= r < b. Then each common divisor of  a  and  b
    divides  r, and it follows that  gcd(a, b) == gcd(b, r).
    The algorithm works by constructing the sequence
    r0, r1, r2, ..., where  r0 = a, r1 = b,  and each  rn
    is the remainder from the division of the two preceding
    elements.

    In Python, q = a // b  and  r = a % b  are obtained by the
    floor division and the remainder operations, respectively.
    These are the most expensive arithmetic operations, especially
    for large  a  and  b.

    Lehmer's algorithm is based on the observation that the quotients
    qn = r(n-1) // rn  are in general small integers even
    when  a  and  b  are very large. Hence the quotients can be
    usually determined from a relatively small number of most
    significant bits.

    The efficiency of the algorithm is further enhanced by not
    computing each long remainder in Euclid's sequence. The remainders
    are linear combinations of  a  and  b  with integer coefficients
    derived from the quotients. The coefficients can be computed
    as far as the quotients can be determined from the chosen
    most significant parts of  a  and  b. Only then a new pair of
    consecutive remainders is computed and the algorithm starts
    anew with this pair.

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm

    """
    a, b = abs(as_int(a)), abs(as_int(b))
    if a < b:
        a, b = b, a

    # The algorithm works by using one or two digit division
    # whenever possible. The outer loop will replace the
    # pair (a, b) with a pair of shorter consecutive elements
    # of the Euclidean gcd sequence until a and b
    # fit into two Python (long) int digits.
    nbits = 2*int_info.bits_per_digit

    while a.bit_length() > nbits and b != 0:
        # Quotients are mostly small integers that can
        # be determined from most significant bits.
        n = a.bit_length() - nbits
        x, y = int(a >> n), int(b >> n)  # most significant bits

        # Elements of the Euclidean gcd sequence are linear
        # combinations of a and b with integer coefficients.
        # Compute the coefficients of consecutive pairs
        #     a' = A*a + B*b, b' = C*a + D*b
        # using small integer arithmetic as far as possible.
        A, B, C, D = 1, 0, 0, 1  # initial values

        while True:
            # The coefficients alternate in sign while looping.
            # The inner loop combines two steps to keep track
            # of the signs.

            # At this point we have
            #   A > 0, B <= 0, C <= 0, D > 0,
            #   x' = x + B <= x < x" = x + A,
            #   y' = y + C <= y < y" = y + D,
            # and
            #   x'*N <= a' < x"*N, y'*N <= b' < y"*N,
            # where N = 2**n.

            # Now, if y' > 0, and x"//y' and x'//y" agree,
            # then their common value is equal to  q = a'//b'.
            # In addition,
            #   x'%y" = x' - q*y" < x" - q*y' = x"%y',
            # and
            #   (x'%y")*N < a'%b' < (x"%y')*N.

            # On the other hand, we also have  x//y == q,
            # and therefore
            #   x'%y" = x + B - q*(y + D) = x%y + B',
            #   x"%y' = x + A - q*(y + C) = x%y + A',
            # where
            #    B' = B - q*D < 0, A' = A - q*C > 0.

            if y + C <= 0:
                break
            q = (x + A) // (y + C)

            # Now  x'//y" <= q, and equality holds if
            #   x' - q*y" = (x - q*y) + (B - q*D) >= 0.
            # This is a minor optimization to avoid division.
            x_qy, B_qD = x - q*y, B - q*D
            if x_qy + B_qD < 0:
                break

            # Next step in the Euclidean sequence.
            x, y = y, x_qy
            A, B, C, D = C, D, A - q*C, B_qD

            # At this point the signs of the coefficients
            # change and their roles are interchanged.
            #   A <= 0, B > 0, C > 0, D < 0,
            #   x' = x + A <= x < x" = x + B,
            #   y' = y + D < y < y" = y + C.

            if y + D <= 0:
                break
            q = (x + B) // (y + D)
            x_qy, A_qC = x - q*y, A - q*C
            if x_qy + A_qC < 0:
                break

            x, y = y, x_qy
            A, B, C, D = C, D, A_qC, B - q*D
            # Now the conditions on top of the loop
            # are again satisfied.
            #   A > 0, B < 0, C < 0, D > 0.

        if B == 0:
            # This can only happen when y == 0 in the beginning
            # and the inner loop does nothing.
            # Long division is forced.
            a, b = b, a % b
            continue

        # Compute new long arguments using the coefficients.
        a, b = A*a + B*b, C*a + D*b

    # Small divisors. Finish with the standard algorithm.
    while b:
        a, b = b, a % b

    return a


def ilcm(*args):
    """Computes integer least common multiple.

    Examples
    ========

    >>> from sympy.core.numbers import ilcm
    >>> ilcm(5, 10)
    10
    >>> ilcm(7, 3)
    21
    >>> ilcm(5, 10, 15)
    30

    """
    if len(args) < 2:
        raise TypeError(
            'ilcm() takes at least 2 arguments (%s given)' % len(args))
    if 0 in args:
        return 0
    a = args[0]
    for b in args[1:]:
        a = a // igcd(a, b) * b # since gcd(a,b) | a
    return a


def igcdex(a, b):
    """Returns x, y, g such that g = x*a + y*b = gcd(a, b).

       >>> from sympy.core.numbers import igcdex
       >>> igcdex(2, 3)
       (-1, 1, 1)
       >>> igcdex(10, 12)
       (-1, 1, 2)

       >>> x, y, g = igcdex(100, 2004)
       >>> x, y, g
       (-20, 1, 4)
       >>> x*100 + y*2004
       4

    """
    if (not a) and (not b):
        return (0, 1, 0)

    if not a:
        return (0, b//abs(b), abs(b))
    if not b:
        return (a//abs(a), 0, abs(a))

    if a < 0:
        a, x_sign = -a, -1
    else:
        x_sign = 1

    if b < 0:
        b, y_sign = -b, -1
    else:
        y_sign = 1

    x, y, r, s = 1, 0, 0, 1

    while b:
        (c, q) = (a % b, a // b)
        (a, b, r, s, x, y) = (b, c, x - q*r, y - q*s, r, s)

    return (x*x_sign, y*y_sign, a)


def mod_inverse(a, m):
    """
    Return the number c such that, (a * c) = 1 (mod m)
    where c has the same sign as m. If no such value exists,
    a ValueError is raised.

    Examples
    ========

    >>> from sympy import S
    >>> from sympy.core.numbers import mod_inverse

    Suppose we wish to find multiplicative inverse x of
    3 modulo 11. This is the same as finding x such
    that 3 * x = 1 (mod 11). One value of x that satisfies
    this congruence is 4. Because 3 * 4 = 12 and 12 = 1 (mod 11).
    This is the value return by mod_inverse:

    >>> mod_inverse(3, 11)
    4
    >>> mod_inverse(-3, 11)
    7

    When there is a common factor between the numerators of
    ``a`` and ``m`` the inverse does not exist:

    >>> mod_inverse(2, 4)
    Traceback (most recent call last):
    ...
    ValueError: inverse of 2 mod 4 does not exist

    >>> mod_inverse(S(2)/7, S(5)/2)
    7/2

    References
    ==========
    - https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
    - https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
    """
    c = None
    try:
        a, m = as_int(a), as_int(m)
        if m != 1 and m != -1:
            x, y, g = igcdex(a, m)
            if g == 1:
                c = x % m
    except ValueError:
        a, m = sympify(a), sympify(m)
        if not (a.is_number and m.is_number):
            raise TypeError(filldedent('''
                Expected numbers for arguments; symbolic `mod_inverse`
                is not implemented
                but symbolic expressions can be handled with the
                similar function,
                sympy.polys.polytools.invert'''))
        big = (m > 1)
        if not (big is S.true or big is S.false):
            raise ValueError('m > 1 did not evaluate; try to simplify %s' % m)
        elif big:
            c = 1/a
    if c is None:
        raise ValueError('inverse of %s (mod %s) does not exist' % (a, m))
    return c


class Number(AtomicExpr):
    """Represents atomic numbers in SymPy.

    Floating point numbers are represented by the Float class.
    Rational numbers (of any size) are represented by the Rational class.
    Integer numbers (of any size) are represented by the Integer class.
    Float and Rational are subclasses of Number; Integer is a subclass
    of Rational.

    For example, ``2/3`` is represented as ``Rational(2, 3)`` which is
    a different object from the floating point number obtained with
    Python division ``2/3``. Even for numbers that are exactly
    represented in binary, there is a difference between how two forms,
    such as ``Rational(1, 2)`` and ``Float(0.5)``, are used in SymPy.
    The rational form is to be preferred in symbolic computations.

    Other kinds of numbers, such as algebraic numbers ``sqrt(2)`` or
    complex numbers ``3 + 4*I``, are not instances of Number class as
    they are not atomic.

    See Also
    ========

    Float, Integer, Rational
    """
    is_commutative = True
    is_number = True
    is_Number = True

    __slots__ = []

    # Used to make max(x._prec, y._prec) return x._prec when only x is a float
    _prec = -1

    def __new__(cls, *obj):
        if len(obj) == 1:
            obj = obj[0]

        if isinstance(obj, Number):
            return obj
        if isinstance(obj, SYMPY_INTS):
            return Integer(obj)
        if isinstance(obj, tuple) and len(obj) == 2:
            return Rational(*obj)
        if isinstance(obj, (float, mpmath.mpf, decimal.Decimal)):
            return Float(obj)
        if isinstance(obj, string_types):
            val = sympify(obj)
            if isinstance(val, Number):
                return val
            else:
                raise ValueError('String "%s" does not denote a Number' % obj)
        msg = "expected str|int|long|float|Decimal|Number object but got %r"
        raise TypeError(msg % type(obj).__name__)

    def invert(self, other, *gens, **args):
        from sympy.polys.polytools import invert
        if getattr(other, 'is_number', True):
            return mod_inverse(self, other)
        return invert(self, other, *gens, **args)

    def __divmod__(self, other):
        from .containers import Tuple

        try:
            other = Number(other)
        except TypeError:
            msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
            raise TypeError(msg % (type(self).__name__, type(other).__name__))
        if not other:
            raise ZeroDivisionError('modulo by zero')
        if self.is_Integer and other.is_Integer:
            return Tuple(*divmod(self.p, other.p))
        else:
            rat = self/other
        w = int(rat) if rat > 0 else int(rat) - 1
        r = self - other*w
        return Tuple(w, r)

    def __rdivmod__(self, other):
        try:
            other = Number(other)
        except TypeError:
            msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
            raise TypeError(msg % (type(other).__name__, type(self).__name__))
        return divmod(other, self)

    def __round__(self, *args):
        return round(float(self), *args)

    def _as_mpf_val(self, prec):
        """Evaluation of mpf tuple accurate to at least prec bits."""
        raise NotImplementedError('%s needs ._as_mpf_val() method' %
            (self.__class__.__name__))

    def _eval_evalf(self, prec):
        return Float._new(self._as_mpf_val(prec), prec)

    def _as_mpf_op(self, prec):
        prec = max(prec, self._prec)
        return self._as_mpf_val(prec), prec

    def __float__(self):
        return mlib.to_float(self._as_mpf_val(53))

    def floor(self):
        raise NotImplementedError('%s needs .floor() method' %
            (self.__class__.__name__))

    def ceiling(self):
        raise NotImplementedError('%s needs .ceiling() method' %
            (self.__class__.__name__))

    def _eval_conjugate(self):
        return self

    def _eval_order(self, *symbols):
        from sympy import Order
        # Order(5, x, y) -> Order(1,x,y)
        return Order(S.One, *symbols)

    def _eval_subs(self, old, new):
        if old == -self:
            return -new
        return self  # there is no other possibility

    def _eval_is_finite(self):
        return True

    @classmethod
    def class_key(cls):
        return 1, 0, 'Number'

    @cacheit
    def sort_key(self, order=None):
        return self.class_key(), (0, ()), (), self

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            if other is S.NaN:
                return S.NaN
            elif other is S.Infinity:
                return S.Infinity
            elif other is S.NegativeInfinity:
                return S.NegativeInfinity
        return AtomicExpr.__add__(self, other)

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            if other is S.NaN:
                return S.NaN
            elif other is S.Infinity:
                return S.NegativeInfinity
            elif other is S.NegativeInfinity:
                return S.Infinity
        return AtomicExpr.__sub__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            if other is S.NaN:
                return S.NaN
            elif other is S.Infinity:
                if self.is_zero:
                    return S.NaN
                elif self.is_positive:
                    return S.Infinity
                else:
                    return S.NegativeInfinity
            elif other is S.NegativeInfinity:
                if self.is_zero:
                    return S.NaN
                elif self.is_positive:
                    return S.NegativeInfinity
                else:
                    return S.Infinity
        elif isinstance(other, Tuple):
            return NotImplemented
        return AtomicExpr.__mul__(self, other)

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            if other is S.NaN:
                return S.NaN
            elif other is S.Infinity or other is S.NegativeInfinity:
                return S.Zero
        return AtomicExpr.__div__(self, other)

    __truediv__ = __div__

    def __eq__(self, other):
        raise NotImplementedError('%s needs .__eq__() method' %
            (self.__class__.__name__))

    def __ne__(self, other):
        raise NotImplementedError('%s needs .__ne__() method' %
            (self.__class__.__name__))

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        raise NotImplementedError('%s needs .__lt__() method' %
            (self.__class__.__name__))

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        raise NotImplementedError('%s needs .__le__() method' %
            (self.__class__.__name__))

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        return _sympify(other).__lt__(self)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        return _sympify(other).__le__(self)

    def __hash__(self):
        return super(Number, self).__hash__()

    def is_constant(self, *wrt, **flags):
        return True

    def as_coeff_mul(self, *deps, **kwargs):
        # a -> c*t
        if self.is_Rational or not kwargs.pop('rational', True):
            return self, tuple()
        elif self.is_negative:
            return S.NegativeOne, (-self,)
        return S.One, (self,)

    def as_coeff_add(self, *deps):
        # a -> c + t
        if self.is_Rational:
            return self, tuple()
        return S.Zero, (self,)

    def as_coeff_Mul(self, rational=False):
        """Efficiently extract the coefficient of a product. """
        if rational and not self.is_Rational:
            return S.One, self
        return (self, S.One) if self else (S.One, self)

    def as_coeff_Add(self, rational=False):
        """Efficiently extract the coefficient of a summation. """
        if not rational:
            return self, S.Zero
        return S.Zero, self

    def gcd(self, other):
        """Compute GCD of `self` and `other`. """
        from sympy.polys import gcd
        return gcd(self, other)

    def lcm(self, other):
        """Compute LCM of `self` and `other`. """
        from sympy.polys import lcm
        return lcm(self, other)

    def cofactors(self, other):
        """Compute GCD and cofactors of `self` and `other`. """
        from sympy.polys import cofactors
        return cofactors(self, other)


class Float(Number):
    """Represent a floating-point number of arbitrary precision.

    Examples
    ========

    >>> from sympy import Float
    >>> Float(3.5)
    3.50000000000000
    >>> Float(3)
    3.00000000000000

    Creating Floats from strings (and Python ``int`` and ``long``
    types) will give a minimum precision of 15 digits, but the
    precision will automatically increase to capture all digits
    entered.

    >>> Float(1)
    1.00000000000000
    >>> Float(10**20)
    100000000000000000000.
    >>> Float('1e20')
    100000000000000000000.

    However, *floating-point* numbers (Python ``float`` types) retain
    only 15 digits of precision:

    >>> Float(1e20)
    1.00000000000000e+20
    >>> Float(1.23456789123456789)
    1.23456789123457

    It may be preferable to enter high-precision decimal numbers
    as strings:

    Float('1.23456789123456789')
    1.23456789123456789

    The desired number of digits can also be specified:

    >>> Float('1e-3', 3)
    0.00100
    >>> Float(100, 4)
    100.0

    Float can automatically count significant figures if a null string
    is sent for the precision; space are also allowed in the string. (Auto-
    counting is only allowed for strings, ints and longs).

    >>> Float('123 456 789 . 123 456', '')
    123456789.123456
    >>> Float('12e-3', '')
    0.012
    >>> Float(3, '')
    3.

    If a number is written in scientific notation, only the digits before the
    exponent are considered significant if a decimal appears, otherwise the
    "e" signifies only how to move the decimal:

    >>> Float('60.e2', '')  # 2 digits significant
    6.0e+3
    >>> Float('60e2', '')  # 4 digits significant
    6000.
    >>> Float('600e-2', '')  # 3 digits significant
    6.00

    Notes
    =====

    Floats are inexact by their nature unless their value is a binary-exact
    value.

    >>> approx, exact = Float(.1, 1), Float(.125, 1)

    For calculation purposes, evalf needs to be able to change the precision
    but this will not increase the accuracy of the inexact value. The
    following is the most accurate 5-digit approximation of a value of 0.1
    that had only 1 digit of precision:

    >>> approx.evalf(5)
    0.099609

    By contrast, 0.125 is exact in binary (as it is in base 10) and so it
    can be passed to Float or evalf to obtain an arbitrary precision with
    matching accuracy:

    >>> Float(exact, 5)
    0.12500
    >>> exact.evalf(20)
    0.12500000000000000000

    Trying to make a high-precision Float from a float is not disallowed,
    but one must keep in mind that the *underlying float* (not the apparent
    decimal value) is being obtained with high precision. For example, 0.3
    does not have a finite binary representation. The closest rational is
    the fraction 5404319552844595/2**54. So if you try to obtain a Float of
    0.3 to 20 digits of precision you will not see the same thing as 0.3
    followed by 19 zeros:

    >>> Float(0.3, 20)
    0.29999999999999998890

    If you want a 20-digit value of the decimal 0.3 (not the floating point
    approximation of 0.3) you should send the 0.3 as a string. The underlying
    representation is still binary but a higher precision than Python's float
    is used:

    >>> Float('0.3', 20)
    0.30000000000000000000

    Although you can increase the precision of an existing Float using Float
    it will not increase the accuracy -- the underlying value is not changed:

    >>> def show(f): # binary rep of Float
    ...     from sympy import Mul, Pow
    ...     s, m, e, b = f._mpf_
    ...     v = Mul(int(m), Pow(2, int(e), evaluate=False), evaluate=False)
    ...     print('%s at prec=%s' % (v, f._prec))
    ...
    >>> t = Float('0.3', 3)
    >>> show(t)
    4915/2**14 at prec=13
    >>> show(Float(t, 20)) # higher prec, not higher accuracy
    4915/2**14 at prec=70
    >>> show(Float(t, 2)) # lower prec
    307/2**10 at prec=10

    The same thing happens when evalf is used on a Float:

    >>> show(t.evalf(20))
    4915/2**14 at prec=70
    >>> show(t.evalf(2))
    307/2**10 at prec=10

    Finally, Floats can be instantiated with an mpf tuple (n, c, p) to
    produce the number (-1)**n*c*2**p:

    >>> n, c, p = 1, 5, 0
    >>> (-1)**n*c*2**p
    -5
    >>> Float((1, 5, 0))
    -5.00000000000000

    An actual mpf tuple also contains the number of bits in c as the last
    element of the tuple:

    >>> _._mpf_
    (1, 5, 0, 3)

    This is not needed for instantiation and is not the same thing as the
    precision. The mpf tuple and the precision are two separate quantities
    that Float tracks.

    """
    __slots__ = ['_mpf_', '_prec']

    # A Float represents many real numbers,
    # both rational and irrational.
    is_rational = None
    is_irrational = None
    is_number = True

    is_real = True

    is_Float = True

    def __new__(cls, num, dps=None, prec=None, precision=None):
        if prec is not None:
            SymPyDeprecationWarning(
                            feature="Using 'prec=XX' to denote decimal precision",
                            useinstead="'dps=XX' for decimal precision and 'precision=XX' "\
                                              "for binary precision",
                            issue=12820,
                            deprecated_since_version="1.1").warn()
            dps = prec
        del prec  # avoid using this deprecated kwarg

        if dps is not None and precision is not None:
            raise ValueError('Both decimal and binary precision supplied. '
                             'Supply only one. ')

        if isinstance(num, string_types):
            num = num.replace(' ', '')
            if num.startswith('.') and len(num) > 1:
                num = '0' + num
            elif num.startswith('-.') and len(num) > 2:
                num = '-0.' + num[2:]
        elif isinstance(num, float) and num == 0:
            num = '0'
        elif isinstance(num, (SYMPY_INTS, Integer)):
            num = str(num)  # faster than mlib.from_int
        elif num is S.Infinity:
            num = '+inf'
        elif num is S.NegativeInfinity:
            num = '-inf'
        elif type(num).__module__ == 'numpy': # support for numpy datatypes
            num = _convert_numpy_types(num)
        elif isinstance(num, mpmath.mpf):
            if precision is None:
                if dps is None:
                    precision = num.context.prec
            num = num._mpf_

        if dps is None and precision is None:
            dps = 15
            if isinstance(num, Float):
                return num
            if isinstance(num, string_types) and _literal_float(num):
                try:
                    Num = decimal.Decimal(num)
                except decimal.InvalidOperation:
                    pass
                else:
                    isint = '.' not in num
                    num, dps = _decimal_to_Rational_prec(Num)
                    if num.is_Integer and isint:
                        dps = max(dps, len(str(num).lstrip('-')))
                    dps = max(15, dps)
                    precision = mlib.libmpf.dps_to_prec(dps)
        elif precision == '' and dps is None or precision is None and dps == '':
            if not isinstance(num, string_types):
                raise ValueError('The null string can only be used when '
                'the number to Float is passed as a string or an integer.')
            ok = None
            if _literal_float(num):
                try:
                    Num = decimal.Decimal(num)
                except decimal.InvalidOperation:
                    pass
                else:
                    isint = '.' not in num
                    num, dps = _decimal_to_Rational_prec(Num)
                    if num.is_Integer and isint:
                        dps = max(dps, len(str(num).lstrip('-')))
                        precision = mlib.libmpf.dps_to_prec(dps)
                    ok = True
            if ok is None:
                raise ValueError('string-float not recognized: %s' % num)

        # decimal precision(dps) is set and maybe binary precision(precision)
        # as well.From here on binary precision is used to compute the Float.
        # Hence, if supplied use binary precision else translate from decimal
        # precision.

        if precision is None or precision == '':
            precision = mlib.libmpf.dps_to_prec(dps)

        precision = int(precision)

        if isinstance(num, float):
            _mpf_ = mlib.from_float(num, precision, rnd)
        elif isinstance(num, string_types):
            _mpf_ = mlib.from_str(num, precision, rnd)
        elif isinstance(num, decimal.Decimal):
            if num.is_finite():
                _mpf_ = mlib.from_str(str(num), precision, rnd)
            elif num.is_nan():
                _mpf_ = _mpf_nan
            elif num.is_infinite():
                if num > 0:
                    _mpf_ = _mpf_inf
                else:
                    _mpf_ = _mpf_ninf
            else:
                raise ValueError("unexpected decimal value %s" % str(num))
        elif isinstance(num, tuple) and len(num) in (3, 4):
            if type(num[1]) is str:
                # it's a hexadecimal (coming from a pickled object)
                # assume that it is in standard form
                num = list(num)
                # If we're loading an object pickled in Python 2 into
                # Python 3, we may need to strip a tailing 'L' because
                # of a shim for int on Python 3, see issue #13470.
                if num[1].endswith('L'):
                    num[1] = num[1][:-1]
                num[1] = MPZ(num[1], 16)
                _mpf_ = tuple(num)
            else:
                if len(num) == 4:
                    # handle normalization hack
                    return Float._new(num, precision)
                else:
                    return (S.NegativeOne**num[0]*num[1]*S(2)**num[2]).evalf(precision)
        else:
            try:
                _mpf_ = num._as_mpf_val(precision)
            except (NotImplementedError, AttributeError):
                _mpf_ = mpmath.mpf(num, prec=precision)._mpf_

        # special cases
        if _mpf_ == _mpf_zero:
            pass  # we want a Float
        elif _mpf_ == _mpf_nan:
            return S.NaN

        obj = Expr.__new__(cls)
        obj._mpf_ = _mpf_
        obj._prec = precision
        return obj

    @classmethod
    def _new(cls, _mpf_, _prec):
        # special cases
        if _mpf_ == _mpf_zero:
            return S.Zero  # XXX this is different from Float which gives 0.0
        elif _mpf_ == _mpf_nan:
            return S.NaN

        obj = Expr.__new__(cls)
        obj._mpf_ = mpf_norm(_mpf_, _prec)
        # XXX: Should this be obj._prec = obj._mpf_[3]?
        obj._prec = _prec
        return obj

    # mpz can't be pickled
    def __getnewargs__(self):
        return (mlib.to_pickable(self._mpf_),)

    def __getstate__(self):
        return {'_prec': self._prec}

    def _hashable_content(self):
        return (self._mpf_, self._prec)

    def floor(self):
        return Integer(int(mlib.to_int(
            mlib.mpf_floor(self._mpf_, self._prec))))

    def ceiling(self):
        return Integer(int(mlib.to_int(
            mlib.mpf_ceil(self._mpf_, self._prec))))

    @property
    def num(self):
        return mpmath.mpf(self._mpf_)

    def _as_mpf_val(self, prec):
        rv = mpf_norm(self._mpf_, prec)
        if rv != self._mpf_ and self._prec == prec:
            debug(self._mpf_, rv)
        return rv

    def _as_mpf_op(self, prec):
        return self._mpf_, max(prec, self._prec)

    def _eval_is_finite(self):
        if self._mpf_ in (_mpf_inf, _mpf_ninf):
            return False
        return True

    def _eval_is_infinite(self):
        if self._mpf_ in (_mpf_inf, _mpf_ninf):
            return True
        return False

    def _eval_is_integer(self):
        return self._mpf_ == _mpf_zero

    def _eval_is_negative(self):
        if self._mpf_ == _mpf_ninf:
            return True
        if self._mpf_ == _mpf_inf:
            return False
        return self.num < 0

    def _eval_is_positive(self):
        if self._mpf_ == _mpf_inf:
            return True
        if self._mpf_ == _mpf_ninf:
            return False
        return self.num > 0

    def _eval_is_zero(self):
        return self._mpf_ == _mpf_zero

    def __nonzero__(self):
        return self._mpf_ != _mpf_zero

    __bool__ = __nonzero__

    def __neg__(self):
        return Float._new(mlib.mpf_neg(self._mpf_), self._prec)

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
        return Number.__add__(self, other)

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_sub(self._mpf_, rhs, prec, rnd), prec)
        return Number.__sub__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if isinstance(other, Number) and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
        return Number.__mul__(self, other)

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        if isinstance(other, Number) and other != 0 and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_div(self._mpf_, rhs, prec, rnd), prec)
        return Number.__div__(self, other)

    __truediv__ = __div__

    @_sympifyit('other', NotImplemented)
    def __mod__(self, other):
        if isinstance(other, Rational) and other.q != 1 and global_evaluate[0]:
            # calculate mod with Rationals, *then* round the result
            return Float(Rational.__mod__(Rational(self), other),
                         precision=self._prec)
        if isinstance(other, Float) and global_evaluate[0]:
            r = self/other
            if r == int(r):
                return Float(0, precision=max(self._prec, other._prec))
        if isinstance(other, Number) and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_mod(self._mpf_, rhs, prec, rnd), prec)
        return Number.__mod__(self, other)

    @_sympifyit('other', NotImplemented)
    def __rmod__(self, other):
        if isinstance(other, Float) and global_evaluate[0]:
            return other.__mod__(self)
        if isinstance(other, Number) and global_evaluate[0]:
            rhs, prec = other._as_mpf_op(self._prec)
            return Float._new(mlib.mpf_mod(rhs, self._mpf_, prec, rnd), prec)
        return Number.__rmod__(self, other)

    def _eval_power(self, expt):
        """
        expt is symbolic object but not equal to 0, 1

        (-p)**r -> exp(r*log(-p)) -> exp(r*(log(p) + I*Pi)) ->
                  -> p**r*(sin(Pi*r) + cos(Pi*r)*I)
        """
        if self == 0:
            if expt.is_positive:
                return S.Zero
            if expt.is_negative:
                return Float('inf')
        if isinstance(expt, Number):
            if isinstance(expt, Integer):
                prec = self._prec
                return Float._new(
                    mlib.mpf_pow_int(self._mpf_, expt.p, prec, rnd), prec)
            elif isinstance(expt, Rational) and \
                    expt.p == 1 and expt.q % 2 and self.is_negative:
                return Pow(S.NegativeOne, expt, evaluate=False)*(
                    -self)._eval_power(expt)
            expt, prec = expt._as_mpf_op(self._prec)
            mpfself = self._mpf_
            try:
                y = mpf_pow(mpfself, expt, prec, rnd)
                return Float._new(y, prec)
            except mlib.ComplexResult:
                re, im = mlib.mpc_pow(
                    (mpfself, _mpf_zero), (expt, _mpf_zero), prec, rnd)
                return Float._new(re, prec) + \
                    Float._new(im, prec)*S.ImaginaryUnit

    def __abs__(self):
        return Float._new(mlib.mpf_abs(self._mpf_), self._prec)

    def __int__(self):
        if self._mpf_ == _mpf_zero:
            return 0
        return int(mlib.to_int(self._mpf_))  # uses round_fast = round_down

    __long__ = __int__

    def __eq__(self, other):
        if isinstance(other, float):
            # coerce to Float at same precision
            o = Float(other)
            try:
                ompf = o._as_mpf_val(self._prec)
            except ValueError:
                return False
            return bool(mlib.mpf_eq(self._mpf_, ompf))
        try:
            other = _sympify(other)
        except SympifyError:
            return NotImplemented
        if other.is_NumberSymbol:
            if other.is_irrational:
                return False
            return other.__eq__(self)
        if other.is_Float:
            return bool(mlib.mpf_eq(self._mpf_, other._mpf_))
        if other.is_Number:
            # numbers should compare at the same precision;
            # all _as_mpf_val routines should be sure to abide
            # by the request to change the prec if necessary; if
            # they don't, the equality test will fail since it compares
            # the mpf tuples
            ompf = other._as_mpf_val(self._prec)
            return bool(mlib.mpf_eq(self._mpf_, ompf))
        return False    # Float != non-Number

    def __ne__(self, other):
        return not self == other

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__lt__(self)
        if other.is_Rational and not other.is_Integer:
            self *= other.q
            other = _sympify(other.p)
        elif other.is_comparable:
            other = other.evalf()
        if other.is_Number and other is not S.NaN:
            return _sympify(bool(
                mlib.mpf_gt(self._mpf_, other._as_mpf_val(self._prec))))
        return Expr.__gt__(self, other)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__le__(self)
        if other.is_Rational and not other.is_Integer:
            self *= other.q
            other = _sympify(other.p)
        elif other.is_comparable:
            other = other.evalf()
        if other.is_Number and other is not S.NaN:
            return _sympify(bool(
                mlib.mpf_ge(self._mpf_, other._as_mpf_val(self._prec))))
        return Expr.__ge__(self, other)

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__gt__(self)
        if other.is_Rational and not other.is_Integer:
            self *= other.q
            other = _sympify(other.p)
        elif other.is_comparable:
            other = other.evalf()
        if other.is_Number and other is not S.NaN:
            return _sympify(bool(
                mlib.mpf_lt(self._mpf_, other._as_mpf_val(self._prec))))
        return Expr.__lt__(self, other)

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__ge__(self)
        if other.is_Rational and not other.is_Integer:
            self *= other.q
            other = _sympify(other.p)
        elif other.is_comparable:
            other = other.evalf()
        if other.is_Number and other is not S.NaN:
            return _sympify(bool(
                mlib.mpf_le(self._mpf_, other._as_mpf_val(self._prec))))
        return Expr.__le__(self, other)

    def __hash__(self):
        return super(Float, self).__hash__()

    def epsilon_eq(self, other, epsilon="1e-15"):
        return abs(self - other) < Float(epsilon)

    def _sage_(self):
        import sage.all as sage
        return sage.RealNumber(str(self))

    def __format__(self, format_spec):
        return format(decimal.Decimal(str(self)), format_spec)


# Add sympify converters
converter[float] = converter[decimal.Decimal] = Float

# this is here to work nicely in Sage
RealNumber = Float


class Rational(Number):
    """Represents rational numbers (p/q) of any size.

    Examples
    ========

    >>> from sympy import Rational, nsimplify, S, pi
    >>> Rational(1, 2)
    1/2

    Rational is unprejudiced in accepting input. If a float is passed, the
    underlying value of the binary representation will be returned:

    >>> Rational(.5)
    1/2
    >>> Rational(.2)
    3602879701896397/18014398509481984

    If the simpler representation of the float is desired then consider
    limiting the denominator to the desired value or convert the float to
    a string (which is roughly equivalent to limiting the denominator to
    10**12):

    >>> Rational(str(.2))
    1/5
    >>> Rational(.2).limit_denominator(10**12)
    1/5

    An arbitrarily precise Rational is obtained when a string literal is
    passed:

    >>> Rational("1.23")
    123/100
    >>> Rational('1e-2')
    1/100
    >>> Rational(".1")
    1/10
    >>> Rational('1e-2/3.2')
    1/320

    The conversion of other types of strings can be handled by
    the sympify() function, and conversion of floats to expressions
    or simple fractions can be handled with nsimplify:

    >>> S('.[3]')  # repeating digits in brackets
    1/3
    >>> S('3**2/10')  # general expressions
    9/10
    >>> nsimplify(.3)  # numbers that have a simple form
    3/10

    But if the input does not reduce to a literal Rational, an error will
    be raised:

    >>> Rational(pi)
    Traceback (most recent call last):
    ...
    TypeError: invalid input: pi


    Low-level
    ---------

    Access numerator and denominator as .p and .q:

    >>> r = Rational(3, 4)
    >>> r
    3/4
    >>> r.p
    3
    >>> r.q
    4

    Note that p and q return integers (not SymPy Integers) so some care
    is needed when using them in expressions:

    >>> r.p/r.q
    0.75

    See Also
    ========
    sympify, sympy.simplify.simplify.nsimplify
    """
    is_real = True
    is_integer = False
    is_rational = True
    is_number = True

    __slots__ = ['p', 'q']

    is_Rational = True

    @cacheit
    def __new__(cls, p, q=None, gcd=None):
        if q is None:
            if isinstance(p, Rational):
                return p

            if isinstance(p, SYMPY_INTS):
                pass
            else:
                if isinstance(p, (float, Float)):
                    return Rational(*_as_integer_ratio(p))

                if not isinstance(p, string_types):
                    try:
                        p = sympify(p)
                    except (SympifyError, SyntaxError):
                        pass  # error will raise below
                else:
                    if p.count('/') > 1:
                        raise TypeError('invalid input: %s' % p)
                    p = p.replace(' ', '')
                    pq = p.rsplit('/', 1)
                    if len(pq) == 2:
                        p, q = pq
                        fp = fractions.Fraction(p)
                        fq = fractions.Fraction(q)
                        p = fp/fq
                    try:
                        p = fractions.Fraction(p)
                    except ValueError:
                        pass  # error will raise below
                    else:
                        return Rational(p.numerator, p.denominator, 1)

                if not isinstance(p, Rational):
                    raise TypeError('invalid input: %s' % p)

            q = 1
            gcd = 1
        else:
            p = Rational(p)
            q = Rational(q)

        if isinstance(q, Rational):
            p *= q.q
            q = q.p
        if isinstance(p, Rational):
            q *= p.q
            p = p.p

        # p and q are now integers
        if q == 0:
            if p == 0:
                if _errdict["divide"]:
                    raise ValueError("Indeterminate 0/0")
                else:
                    return S.NaN
            return S.ComplexInfinity
        if q < 0:
            q = -q
            p = -p
        if not gcd:
            gcd = igcd(abs(p), q)
        if gcd > 1:
            p //= gcd
            q //= gcd
        if q == 1:
            return Integer(p)
        if p == 1 and q == 2:
            return S.Half
        obj = Expr.__new__(cls)
        obj.p = p
        obj.q = q
        return obj

    def limit_denominator(self, max_denominator=1000000):
        """Closest Rational to self with denominator at most max_denominator.

        >>> from sympy import Rational
        >>> Rational('3.141592653589793').limit_denominator(10)
        22/7
        >>> Rational('3.141592653589793').limit_denominator(100)
        311/99

        """
        f = fractions.Fraction(self.p, self.q)
        return Rational(f.limit_denominator(fractions.Fraction(int(max_denominator))))

    def __getnewargs__(self):
        return (self.p, self.q)

    def _hashable_content(self):
        return (self.p, self.q)

    def _eval_is_positive(self):
        return self.p > 0

    def _eval_is_zero(self):
        return self.p == 0

    def __neg__(self):
        return Rational(-self.p, self.q)

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                return Rational(self.p + self.q*other.p, self.q, 1)
            elif isinstance(other, Rational):
                #TODO: this can probably be optimized more
                return Rational(self.p*other.q + self.q*other.p, self.q*other.q)
            elif isinstance(other, Float):
                return other + self
            else:
                return Number.__add__(self, other)
        return Number.__add__(self, other)
    __radd__ = __add__

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                return Rational(self.p - self.q*other.p, self.q, 1)
            elif isinstance(other, Rational):
                return Rational(self.p*other.q - self.q*other.p, self.q*other.q)
            elif isinstance(other, Float):
                return -other + self
            else:
                return Number.__sub__(self, other)
        return Number.__sub__(self, other)
    @_sympifyit('other', NotImplemented)
    def __rsub__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                return Rational(self.q*other.p - self.p, self.q, 1)
            elif isinstance(other, Rational):
                return Rational(self.q*other.p - self.p*other.q, self.q*other.q)
            elif isinstance(other, Float):
                return -self + other
            else:
                return Number.__rsub__(self, other)
        return Number.__rsub__(self, other)
    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                return Rational(self.p*other.p, self.q, igcd(other.p, self.q))
            elif isinstance(other, Rational):
                return Rational(self.p*other.p, self.q*other.q, igcd(self.p, other.q)*igcd(self.q, other.p))
            elif isinstance(other, Float):
                return other*self
            else:
                return Number.__mul__(self, other)
        return Number.__mul__(self, other)
    __rmul__ = __mul__

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                if self.p and other.p == S.Zero:
                    return S.ComplexInfinity
                else:
                    return Rational(self.p, self.q*other.p, igcd(self.p, other.p))
            elif isinstance(other, Rational):
                return Rational(self.p*other.q, self.q*other.p, igcd(self.p, other.p)*igcd(self.q, other.q))
            elif isinstance(other, Float):
                return self*(1/other)
            else:
                return Number.__div__(self, other)
        return Number.__div__(self, other)
    @_sympifyit('other', NotImplemented)
    def __rdiv__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Integer):
                return Rational(other.p*self.q, self.p, igcd(self.p, other.p))
            elif isinstance(other, Rational):
                return Rational(other.p*self.q, other.q*self.p, igcd(self.p, other.p)*igcd(self.q, other.q))
            elif isinstance(other, Float):
                return other*(1/self)
            else:
                return Number.__rdiv__(self, other)
        return Number.__rdiv__(self, other)
    __truediv__ = __div__

    @_sympifyit('other', NotImplemented)
    def __mod__(self, other):
        if global_evaluate[0]:
            if isinstance(other, Rational):
                n = (self.p*other.q) // (other.p*self.q)
                return Rational(self.p*other.q - n*other.p*self.q, self.q*other.q)
            if isinstance(other, Float):
                # calculate mod with Rationals, *then* round the answer
                return Float(self.__mod__(Rational(other)),
                             precision=other._prec)
            return Number.__mod__(self, other)
        return Number.__mod__(self, other)

    @_sympifyit('other', NotImplemented)
    def __rmod__(self, other):
        if isinstance(other, Rational):
            return Rational.__mod__(other, self)
        return Number.__rmod__(self, other)

    def _eval_power(self, expt):
        if isinstance(expt, Number):
            if isinstance(expt, Float):
                return self._eval_evalf(expt._prec)**expt
            if expt.is_negative:
                # (3/4)**-2 -> (4/3)**2
                ne = -expt
                if (ne is S.One):
                    return Rational(self.q, self.p)
                if self.is_negative:
                    return S.NegativeOne**expt*Rational(self.q, -self.p)**ne
                else:
                    return Rational(self.q, self.p)**ne
            if expt is S.Infinity:  # -oo already caught by test for negative
                if self.p > self.q:
                    # (3/2)**oo -> oo
                    return S.Infinity
                if self.p < -self.q:
                    # (-3/2)**oo -> oo + I*oo
                    return S.Infinity + S.Infinity*S.ImaginaryUnit
                return S.Zero
            if isinstance(expt, Integer):
                # (4/3)**2 -> 4**2 / 3**2
                return Rational(self.p**expt.p, self.q**expt.p, 1)
            if isinstance(expt, Rational):
                if self.p != 1:
                    # (4/3)**(5/6) -> 4**(5/6)*3**(-5/6)
                    return Integer(self.p)**expt*Integer(self.q)**(-expt)
                # as the above caught negative self.p, now self is positive
                return Integer(self.q)**Rational(
                expt.p*(expt.q - 1), expt.q) / \
                    Integer(self.q)**Integer(expt.p)

        if self.is_negative and expt.is_even:
            return (-self)**expt

        return

    def _as_mpf_val(self, prec):
        return mlib.from_rational(self.p, self.q, prec, rnd)

    def _mpmath_(self, prec, rnd):
        return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))

    def __abs__(self):
        return Rational(abs(self.p), self.q)

    def __int__(self):
        p, q = self.p, self.q
        if p < 0:
            return -int(-p//q)
        return int(p//q)

    __long__ = __int__

    def floor(self):
        return Integer(self.p // self.q)

    def ceiling(self):
        return -Integer(-self.p // self.q)

    def __eq__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            return NotImplemented
        if other.is_NumberSymbol:
            if other.is_irrational:
                return False
            return other.__eq__(self)
        if other.is_Number:
            if other.is_Rational:
                # a Rational is always in reduced form so will never be 2/4
                # so we can just check equivalence of args
                return self.p == other.p and self.q == other.q
            if other.is_Float:
                return mlib.mpf_eq(self._as_mpf_val(other._prec), other._mpf_)
        return False

    def __ne__(self, other):
        return not self == other

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__lt__(self)
        expr = self
        if other.is_Number:
            if other.is_Rational:
                return _sympify(bool(self.p*other.q > self.q*other.p))
            if other.is_Float:
                return _sympify(bool(mlib.mpf_gt(
                    self._as_mpf_val(other._prec), other._mpf_)))
        elif other.is_number and other.is_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__gt__(expr, other)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__le__(self)
        expr = self
        if other.is_Number:
            if other.is_Rational:
                 return _sympify(bool(self.p*other.q >= self.q*other.p))
            if other.is_Float:
                return _sympify(bool(mlib.mpf_ge(
                    self._as_mpf_val(other._prec), other._mpf_)))
        elif other.is_number and other.is_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__ge__(expr, other)

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        if other.is_NumberSymbol:
            return other.__gt__(self)
        expr = self
        if other.is_Number:
            if other.is_Rational:
                return _sympify(bool(self.p*other.q < self.q*other.p))
            if other.is_Float:
                return _sympify(bool(mlib.mpf_lt(
                    self._as_mpf_val(other._prec), other._mpf_)))
        elif other.is_number and other.is_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__lt__(expr, other)

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        expr = self
        if other.is_NumberSymbol:
            return other.__ge__(self)
        elif other.is_Number:
            if other.is_Rational:
                return _sympify(bool(self.p*other.q <= self.q*other.p))
            if other.is_Float:
                return _sympify(bool(mlib.mpf_le(
                    self._as_mpf_val(other._prec), other._mpf_)))
        elif other.is_number and other.is_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__le__(expr, other)

    def __hash__(self):
        return super(Rational, self).__hash__()

    def factors(self, limit=None, use_trial=True, use_rho=False,
                use_pm1=False, verbose=False, visual=False):
        """A wrapper to factorint which return factors of self that are
        smaller than limit (or cheap to compute). Special methods of
        factoring are disabled by default so that only trial division is used.
        """
        from sympy.ntheory import factorrat

        return factorrat(self, limit=limit, use_trial=use_trial,
                      use_rho=use_rho, use_pm1=use_pm1,
                      verbose=verbose).copy()

    @_sympifyit('other', NotImplemented)
    def gcd(self, other):
        if isinstance(other, Rational):
            if other is S.Zero:
                return other
            return Rational(
                Integer(igcd(self.p, other.p)),
                Integer(ilcm(self.q, other.q)))
        return Number.gcd(self, other)

    @_sympifyit('other', NotImplemented)
    def lcm(self, other):
        if isinstance(other, Rational):
            return Rational(
                self.p // igcd(self.p, other.p) * other.p,
                igcd(self.q, other.q))
        return Number.lcm(self, other)

    def as_numer_denom(self):
        return Integer(self.p), Integer(self.q)

    def _sage_(self):
        import sage.all as sage
        return sage.Integer(self.p)/sage.Integer(self.q)

    def as_content_primitive(self, radical=False, clear=True):
        """Return the tuple (R, self/R) where R is the positive Rational
        extracted from self.

        Examples
        ========

        >>> from sympy import S
        >>> (S(-3)/2).as_content_primitive()
        (3/2, -1)

        See docstring of Expr.as_content_primitive for more examples.
        """

        if self:
            if self.is_positive:
                return self, S.One
            return -self, S.NegativeOne
        return S.One, self

    def as_coeff_Mul(self, rational=False):
        """Efficiently extract the coefficient of a product. """
        return self, S.One

    def as_coeff_Add(self, rational=False):
        """Efficiently extract the coefficient of a summation. """
        return self, S.Zero


class Integer(Rational):
    """Represents integer numbers of any size.

    Examples
    ========

    >>> from sympy import Integer
    >>> Integer(3)
    3

    If a float or a rational is passed to Integer, the fractional part
    will be discarded; the effect is of rounding toward zero.

    >>> Integer(3.8)
    3
    >>> Integer(-3.8)
    -3

    A string is acceptable input if it can be parsed as an integer:

    >>> Integer("9" * 20)
    99999999999999999999

    It is rarely needed to explicitly instantiate an Integer, because
    Python integers are automatically converted to Integer when they
    are used in SymPy expressions.
    """
    q = 1
    is_integer = True
    is_number = True

    is_Integer = True

    __slots__ = ['p']

    def _as_mpf_val(self, prec):
        return mlib.from_int(self.p, prec, rnd)

    def _mpmath_(self, prec, rnd):
        return mpmath.make_mpf(self._as_mpf_val(prec))

    @cacheit
    def __new__(cls, i):
        if isinstance(i, string_types):
            i = i.replace(' ', '')
        # whereas we cannot, in general, make a Rational from an
        # arbitrary expression, we can make an Integer unambiguously
        # (except when a non-integer expression happens to round to
        # an integer). So we proceed by taking int() of the input and
        # let the int routines determine whether the expression can
        # be made into an int or whether an error should be raised.
        try:
            ival = int(i)
        except TypeError:
            raise TypeError(
                "Argument of Integer should be of numeric type, got %s." % i)
        # We only work with well-behaved integer types. This converts, for
        # example, numpy.int32 instances.
        if ival == 1:
            return S.One
        if ival == -1:
            return S.NegativeOne
        if ival == 0:
            return S.Zero
        obj = Expr.__new__(cls)
        obj.p = ival
        return obj

    def __getnewargs__(self):
        return (self.p,)

    # Arithmetic operations are here for efficiency
    def __int__(self):
        return self.p

    __long__ = __int__

    def floor(self):
        return Integer(self.p)

    def ceiling(self):
        return Integer(self.p)

    def __neg__(self):
        return Integer(-self.p)

    def __abs__(self):
        if self.p >= 0:
            return self
        else:
            return Integer(-self.p)

    def __divmod__(self, other):
        from .containers import Tuple
        if isinstance(other, Integer) and global_evaluate[0]:
            return Tuple(*(divmod(self.p, other.p)))
        else:
            return Number.__divmod__(self, other)

    def __rdivmod__(self, other):
        from .containers import Tuple
        if isinstance(other, integer_types) and global_evaluate[0]:
            return Tuple(*(divmod(other, self.p)))
        else:
            try:
                other = Number(other)
            except TypeError:
                msg = "unsupported operand type(s) for divmod(): '%s' and '%s'"
                oname = type(other).__name__
                sname = type(self).__name__
                raise TypeError(msg % (oname, sname))
            return Number.__divmod__(other, self)

    # TODO make it decorator + bytecodehacks?
    def __add__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(self.p + other)
            elif isinstance(other, Integer):
                return Integer(self.p + other.p)
            elif isinstance(other, Rational):
                return Rational(self.p*other.q + other.p, other.q, 1)
            return Rational.__add__(self, other)
        else:
            return Add(self, other)

    def __radd__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(other + self.p)
            elif isinstance(other, Rational):
                return Rational(other.p + self.p*other.q, other.q, 1)
            return Rational.__radd__(self, other)
        return Rational.__radd__(self, other)

    def __sub__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(self.p - other)
            elif isinstance(other, Integer):
                return Integer(self.p - other.p)
            elif isinstance(other, Rational):
                return Rational(self.p*other.q - other.p, other.q, 1)
            return Rational.__sub__(self, other)
        return Rational.__sub__(self, other)

    def __rsub__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(other - self.p)
            elif isinstance(other, Rational):
                return Rational(other.p - self.p*other.q, other.q, 1)
            return Rational.__rsub__(self, other)
        return Rational.__rsub__(self, other)

    def __mul__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(self.p*other)
            elif isinstance(other, Integer):
                return Integer(self.p*other.p)
            elif isinstance(other, Rational):
                return Rational(self.p*other.p, other.q, igcd(self.p, other.q))
            return Rational.__mul__(self, other)
        return Rational.__mul__(self, other)

    def __rmul__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(other*self.p)
            elif isinstance(other, Rational):
                return Rational(other.p*self.p, other.q, igcd(self.p, other.q))
            return Rational.__rmul__(self, other)
        return Rational.__rmul__(self, other)

    def __mod__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(self.p % other)
            elif isinstance(other, Integer):
                return Integer(self.p % other.p)
            return Rational.__mod__(self, other)
        return Rational.__mod__(self, other)

    def __rmod__(self, other):
        if global_evaluate[0]:
            if isinstance(other, integer_types):
                return Integer(other % self.p)
            elif isinstance(other, Integer):
                return Integer(other.p % self.p)
            return Rational.__rmod__(self, other)
        return Rational.__rmod__(self, other)

    def __eq__(self, other):
        if isinstance(other, integer_types):
            return (self.p == other)
        elif isinstance(other, Integer):
            return (self.p == other.p)
        return Rational.__eq__(self, other)

    def __ne__(self, other):
        return not self == other

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        if other.is_Integer:
            return _sympify(self.p > other.p)
        return Rational.__gt__(self, other)

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        if other.is_Integer:
            return _sympify(self.p < other.p)
        return Rational.__lt__(self, other)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        if other.is_Integer:
            return _sympify(self.p >= other.p)
        return Rational.__ge__(self, other)

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        if other.is_Integer:
            return _sympify(self.p <= other.p)
        return Rational.__le__(self, other)

    def __hash__(self):
        return hash(self.p)

    def __index__(self):
        return self.p

    ########################################

    def _eval_is_odd(self):
        return bool(self.p % 2)

    def _eval_power(self, expt):
        """
        Tries to do some simplifications on self**expt

        Returns None if no further simplifications can be done

        When exponent is a fraction (so we have for example a square root),
        we try to find a simpler representation by factoring the argument
        up to factors of 2**15, e.g.

          - sqrt(4) becomes 2
          - sqrt(-4) becomes 2*I
          - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7)

        Further simplification would require a special call to factorint on
        the argument which is not done here for sake of speed.

        """
        from sympy import perfect_power

        if expt is S.Infinity:
            if self.p > S.One:
                return S.Infinity
            # cases -1, 0, 1 are done in their respective classes
            return S.Infinity + S.ImaginaryUnit*S.Infinity
        if expt is S.NegativeInfinity:
            return Rational(1, self)**S.Infinity
        if not isinstance(expt, Number):
            # simplify when expt is even
            # (-2)**k --> 2**k
            if self.is_negative and expt.is_even:
                return (-self)**expt
        if isinstance(expt, Float):
            # Rational knows how to exponentiate by a Float
            return super(Integer, self)._eval_power(expt)
        if not isinstance(expt, Rational):
            return
        if expt is S.Half and self.is_negative:
            # we extract I for this special case since everyone is doing so
            return S.ImaginaryUnit*Pow(-self, expt)
        if expt.is_negative:
            # invert base and change sign on exponent
            ne = -expt
            if self.is_negative:
                    return S.NegativeOne**expt*Rational(1, -self)**ne
            else:
                return Rational(1, self.p)**ne
        # see if base is a perfect root, sqrt(4) --> 2
        x, xexact = integer_nthroot(abs(self.p), expt.q)
        if xexact:
            # if it's a perfect root we've finished
            result = Integer(x**abs(expt.p))
            if self.is_negative:
                result *= S.NegativeOne**expt
            return result

        # The following is an algorithm where we collect perfect roots
        # from the factors of base.

        # if it's not an nth root, it still might be a perfect power
        b_pos = int(abs(self.p))
        p = perfect_power(b_pos)
        if p is not False:
            dict = {p[0]: p[1]}
        else:
            dict = Integer(b_pos).factors(limit=2**15)

        # now process the dict of factors
        out_int = 1  # integer part
        out_rad = 1  # extracted radicals
        sqr_int = 1
        sqr_gcd = 0
        sqr_dict = {}
        for prime, exponent in dict.items():
            exponent *= expt.p
            # remove multiples of expt.q: (2**12)**(1/10) -> 2*(2**2)**(1/10)
            div_e, div_m = divmod(exponent, expt.q)
            if div_e > 0:
                out_int *= prime**div_e
            if div_m > 0:
                # see if the reduced exponent shares a gcd with e.q
                # (2**2)**(1/10) -> 2**(1/5)
                g = igcd(div_m, expt.q)
                if g != 1:
                    out_rad *= Pow(prime, Rational(div_m//g, expt.q//g))
                else:
                    sqr_dict[prime] = div_m
        # identify gcd of remaining powers
        for p, ex in sqr_dict.items():
            if sqr_gcd == 0:
                sqr_gcd = ex
            else:
                sqr_gcd = igcd(sqr_gcd, ex)
                if sqr_gcd == 1:
                    break
        for k, v in sqr_dict.items():
            sqr_int *= k**(v//sqr_gcd)
        if sqr_int == b_pos and out_int == 1 and out_rad == 1:
            result = None
        else:
            result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q))
            if self.is_negative:
                result *= Pow(S.NegativeOne, expt)
        return result

    def _eval_is_prime(self):
        from sympy.ntheory import isprime

        return isprime(self)

    def _eval_is_composite(self):
        if self > 1:
            return fuzzy_not(self.is_prime)
        else:
            return False

    def as_numer_denom(self):
        return self, S.One

    def __floordiv__(self, other):
        return Integer(self.p // Integer(other).p)

    def __rfloordiv__(self, other):
        return Integer(Integer(other).p // self.p)

# Add sympify converters
for i_type in integer_types:
    converter[i_type] = Integer


class AlgebraicNumber(Expr):
    """Class for representing algebraic numbers in SymPy. """

    __slots__ = ['rep', 'root', 'alias', 'minpoly']

    is_AlgebraicNumber = True
    is_algebraic = True
    is_number = True

    def __new__(cls, expr, coeffs=None, alias=None, **args):
        """Construct a new algebraic number. """
        from sympy import Poly
        from sympy.polys.polyclasses import ANP, DMP
        from sympy.polys.numberfields import minimal_polynomial
        from sympy.core.symbol import Symbol

        expr = sympify(expr)

        if isinstance(expr, (tuple, Tuple)):
            minpoly, root = expr

            if not minpoly.is_Poly:
                minpoly = Poly(minpoly)
        elif expr.is_AlgebraicNumber:
            minpoly, root = expr.minpoly, expr.root
        else:
            minpoly, root = minimal_polynomial(
                expr, args.get('gen'), polys=True), expr

        dom = minpoly.get_domain()

        if coeffs is not None:
            if not isinstance(coeffs, ANP):
                rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
                scoeffs = Tuple(*coeffs)
            else:
                rep = DMP.from_list(coeffs.to_list(), 0, dom)
                scoeffs = Tuple(*coeffs.to_list())

            if rep.degree() >= minpoly.degree():
                rep = rep.rem(minpoly.rep)

        else:
            rep = DMP.from_list([1, 0], 0, dom)
            scoeffs = Tuple(1, 0)

        sargs = (root, scoeffs)

        if alias is not None:
            if not isinstance(alias, Symbol):
                alias = Symbol(alias)
            sargs = sargs + (alias,)

        obj = Expr.__new__(cls, *sargs)

        obj.rep = rep
        obj.root = root
        obj.alias = alias
        obj.minpoly = minpoly

        return obj

    def __hash__(self):
        return super(AlgebraicNumber, self).__hash__()

    def _eval_evalf(self, prec):
        return self.as_expr()._evalf(prec)

    @property
    def is_aliased(self):
        """Returns ``True`` if ``alias`` was set. """
        return self.alias is not None

    def as_poly(self, x=None):
        """Create a Poly instance from ``self``. """
        from sympy import Dummy, Poly, PurePoly
        if x is not None:
            return Poly.new(self.rep, x)
        else:
            if self.alias is not None:
                return Poly.new(self.rep, self.alias)
            else:
                return PurePoly.new(self.rep, Dummy('x'))

    def as_expr(self, x=None):
        """Create a Basic expression from ``self``. """
        return self.as_poly(x or self.root).as_expr().expand()

    def coeffs(self):
        """Returns all SymPy coefficients of an algebraic number. """
        return [ self.rep.dom.to_sympy(c) for c in self.rep.all_coeffs() ]

    def native_coeffs(self):
        """Returns all native coefficients of an algebraic number. """
        return self.rep.all_coeffs()

    def to_algebraic_integer(self):
        """Convert ``self`` to an algebraic integer. """
        from sympy import Poly
        f = self.minpoly

        if f.LC() == 1:
            return self

        coeff = f.LC()**(f.degree() - 1)
        poly = f.compose(Poly(f.gen/f.LC()))

        minpoly = poly*coeff
        root = f.LC()*self.root

        return AlgebraicNumber((minpoly, root), self.coeffs())

    def _eval_simplify(self, ratio, measure, rational, inverse):
        from sympy.polys import CRootOf, minpoly

        for r in [r for r in self.minpoly.all_roots() if r.func != CRootOf]:
            if minpoly(self.root - r).is_Symbol:
                # use the matching root if it's simpler
                if measure(r) < ratio*measure(self.root):
                    return AlgebraicNumber(r)
        return self


class RationalConstant(Rational):
    """
    Abstract base class for rationals with specific behaviors

    Derived classes must define class attributes p and q and should probably all
    be singletons.
    """
    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)


class IntegerConstant(Integer):
    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)


class Zero(with_metaclass(Singleton, IntegerConstant)):
    """The number zero.

    Zero is a singleton, and can be accessed by ``S.Zero``

    Examples
    ========

    >>> from sympy import S, Integer, zoo
    >>> Integer(0) is S.Zero
    True
    >>> 1/S.Zero
    zoo

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Zero
    """

    p = 0
    q = 1
    is_positive = False
    is_negative = False
    is_zero = True
    is_number = True

    __slots__ = []

    @staticmethod
    def __abs__():
        return S.Zero

    @staticmethod
    def __neg__():
        return S.Zero

    def _eval_power(self, expt):
        if expt.is_positive:
            return self
        if expt.is_negative:
            return S.ComplexInfinity
        if expt.is_real is False:
            return S.NaN
        # infinities are already handled with pos and neg
        # tests above; now throw away leading numbers on Mul
        # exponent
        coeff, terms = expt.as_coeff_Mul()
        if coeff.is_negative:
            return S.ComplexInfinity**terms
        if coeff is not S.One:  # there is a Number to discard
            return self**terms

    def _eval_order(self, *symbols):
        # Order(0,x) -> 0
        return self

    def __nonzero__(self):
        return False

    __bool__ = __nonzero__

    def as_coeff_Mul(self, rational=False):  # XXX this routine should be deleted
        """Efficiently extract the coefficient of a summation. """
        return S.One, self


class One(with_metaclass(Singleton, IntegerConstant)):
    """The number one.

    One is a singleton, and can be accessed by ``S.One``.

    Examples
    ========

    >>> from sympy import S, Integer
    >>> Integer(1) is S.One
    True

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/1_%28number%29
    """
    is_number = True

    p = 1
    q = 1

    __slots__ = []

    @staticmethod
    def __abs__():
        return S.One

    @staticmethod
    def __neg__():
        return S.NegativeOne

    def _eval_power(self, expt):
        return self

    def _eval_order(self, *symbols):
        return

    @staticmethod
    def factors(limit=None, use_trial=True, use_rho=False, use_pm1=False,
                verbose=False, visual=False):
        if visual:
            return S.One
        else:
            return {}


class NegativeOne(with_metaclass(Singleton, IntegerConstant)):
    """The number negative one.

    NegativeOne is a singleton, and can be accessed by ``S.NegativeOne``.

    Examples
    ========

    >>> from sympy import S, Integer
    >>> Integer(-1) is S.NegativeOne
    True

    See Also
    ========

    One

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/%E2%88%921_%28number%29

    """
    is_number = True

    p = -1
    q = 1

    __slots__ = []

    @staticmethod
    def __abs__():
        return S.One

    @staticmethod
    def __neg__():
        return S.One

    def _eval_power(self, expt):
        if expt.is_odd:
            return S.NegativeOne
        if expt.is_even:
            return S.One
        if isinstance(expt, Number):
            if isinstance(expt, Float):
                return Float(-1.0)**expt
            if expt is S.NaN:
                return S.NaN
            if expt is S.Infinity or expt is S.NegativeInfinity:
                return S.NaN
            if expt is S.Half:
                return S.ImaginaryUnit
            if isinstance(expt, Rational):
                if expt.q == 2:
                    return S.ImaginaryUnit**Integer(expt.p)
                i, r = divmod(expt.p, expt.q)
                if i:
                    return self**i*self**Rational(r, expt.q)
        return


class Half(with_metaclass(Singleton, RationalConstant)):
    """The rational number 1/2.

    Half is a singleton, and can be accessed by ``S.Half``.

    Examples
    ========

    >>> from sympy import S, Rational
    >>> Rational(1, 2) is S.Half
    True

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/One_half
    """
    is_number = True

    p = 1
    q = 2

    __slots__ = []

    @staticmethod
    def __abs__():
        return S.Half


class Infinity(with_metaclass(Singleton, Number)):
    r"""Positive infinite quantity.

    In real analysis the symbol `\infty` denotes an unbounded
    limit: `x\to\infty` means that `x` grows without bound.

    Infinity is often used not only to define a limit but as a value
    in the affinely extended real number system.  Points labeled `+\infty`
    and `-\infty` can be added to the topological space of the real numbers,
    producing the two-point compactification of the real numbers.  Adding
    algebraic properties to this gives us the extended real numbers.

    Infinity is a singleton, and can be accessed by ``S.Infinity``,
    or can be imported as ``oo``.

    Examples
    ========

    >>> from sympy import oo, exp, limit, Symbol
    >>> 1 + oo
    oo
    >>> 42/oo
    0
    >>> x = Symbol('x')
    >>> limit(exp(x), x, oo)
    oo

    See Also
    ========

    NegativeInfinity, NaN

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Infinity
    """

    is_commutative = True
    is_positive = True
    is_infinite = True
    is_number = True
    is_prime = False

    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)

    def _latex(self, printer):
        return r"\infty"

    def _eval_subs(self, old, new):
        if self == old:
            return new

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if isinstance(other, Number):
            if other is S.NegativeInfinity or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('-inf'):
                    return S.NaN
                else:
                    return Float('inf')
            else:
                return S.Infinity
        return NotImplemented
    __radd__ = __add__

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if isinstance(other, Number):
            if other is S.Infinity or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('inf'):
                    return S.NaN
                else:
                    return Float('inf')
            else:
                return S.Infinity
        return NotImplemented

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if isinstance(other, Number):
            if other is S.Zero or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == 0:
                    return S.NaN
                if other > 0:
                    return Float('inf')
                else:
                    return Float('-inf')
            else:
                if other > 0:
                    return S.Infinity
                else:
                    return S.NegativeInfinity
        return NotImplemented
    __rmul__ = __mul__

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        if isinstance(other, Number):
            if other is S.Infinity or \
                other is S.NegativeInfinity or \
                    other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('-inf') or \
                        other == Float('inf'):
                    return S.NaN
                elif other.is_nonnegative:
                    return Float('inf')
                else:
                    return Float('-inf')
            else:
                if other >= 0:
                    return S.Infinity
                else:
                    return S.NegativeInfinity
        return NotImplemented

    __truediv__ = __div__

    def __abs__(self):
        return S.Infinity

    def __neg__(self):
        return S.NegativeInfinity

    def _eval_power(self, expt):
        """
        ``expt`` is symbolic object but not equal to 0 or 1.

        ================ ======= ==============================
        Expression       Result  Notes
        ================ ======= ==============================
        ``oo ** nan``    ``nan``
        ``oo ** -p``     ``0``   ``p`` is number, ``oo``
        ================ ======= ==============================

        See Also
        ========
        Pow
        NaN
        NegativeInfinity

        """
        from sympy.functions import re

        if expt.is_positive:
            return S.Infinity
        if expt.is_negative:
            return S.Zero
        if expt is S.NaN:
            return S.NaN
        if expt is S.ComplexInfinity:
            return S.NaN
        if expt.is_real is False and expt.is_number:
            expt_real = re(expt)
            if expt_real.is_positive:
                return S.ComplexInfinity
            if expt_real.is_negative:
                return S.Zero
            if expt_real.is_zero:
                return S.NaN

            return self**expt.evalf()

    def _as_mpf_val(self, prec):
        return mlib.finf

    def _sage_(self):
        import sage.all as sage
        return sage.oo

    def __hash__(self):
        return super(Infinity, self).__hash__()

    def __eq__(self, other):
        return other is S.Infinity

    def __ne__(self, other):
        return other is not S.Infinity

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        if other.is_real:
            return S.false
        return Expr.__lt__(self, other)

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        if other.is_real:
            if other.is_finite or other is S.NegativeInfinity:
                return S.false
            elif other.is_nonpositive:
                return S.false
            elif other.is_infinite and other.is_positive:
                return S.true
        return Expr.__le__(self, other)

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        if other.is_real:
            if other.is_finite or other is S.NegativeInfinity:
                return S.true
            elif other.is_nonpositive:
                return S.true
            elif other.is_infinite and other.is_positive:
                return S.false
        return Expr.__gt__(self, other)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        if other.is_real:
            return S.true
        return Expr.__ge__(self, other)

    def __mod__(self, other):
        return S.NaN

    __rmod__ = __mod__

    def floor(self):
        return self

    def ceiling(self):
        return self

oo = S.Infinity


class NegativeInfinity(with_metaclass(Singleton, Number)):
    """Negative infinite quantity.

    NegativeInfinity is a singleton, and can be accessed
    by ``S.NegativeInfinity``.

    See Also
    ========

    Infinity
    """

    is_commutative = True
    is_negative = True
    is_infinite = True
    is_number = True

    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)

    def _latex(self, printer):
        return r"-\infty"

    def _eval_subs(self, old, new):
        if self == old:
            return new

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if isinstance(other, Number):
            if other is S.Infinity or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('inf'):
                    return Float('nan')
                else:
                    return Float('-inf')
            else:
                return S.NegativeInfinity
        return NotImplemented
    __radd__ = __add__

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if isinstance(other, Number):
            if other is S.NegativeInfinity or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('-inf'):
                    return Float('nan')
                else:
                    return Float('-inf')
            else:
                return S.NegativeInfinity
        return NotImplemented

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if isinstance(other, Number):
            if other is S.Zero or other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other is S.NaN or other.is_zero:
                    return S.NaN
                elif other.is_positive:
                    return Float('-inf')
                else:
                    return Float('inf')
            else:
                if other.is_positive:
                    return S.NegativeInfinity
                else:
                    return S.Infinity
        return NotImplemented
    __rmul__ = __mul__

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        if isinstance(other, Number):
            if other is S.Infinity or \
                other is S.NegativeInfinity or \
                    other is S.NaN:
                return S.NaN
            elif other.is_Float:
                if other == Float('-inf') or \
                    other == Float('inf') or \
                        other is S.NaN:
                    return S.NaN
                elif other.is_nonnegative:
                    return Float('-inf')
                else:
                    return Float('inf')
            else:
                if other >= 0:
                    return S.NegativeInfinity
                else:
                    return S.Infinity
        return NotImplemented

    __truediv__ = __div__

    def __abs__(self):
        return S.Infinity

    def __neg__(self):
        return S.Infinity

    def _eval_power(self, expt):
        """
        ``expt`` is symbolic object but not equal to 0 or 1.

        ================ ======= ==============================
        Expression       Result  Notes
        ================ ======= ==============================
        ``(-oo) ** nan`` ``nan``
        ``(-oo) ** oo``  ``nan``
        ``(-oo) ** -oo`` ``nan``
        ``(-oo) ** e``   ``oo``  ``e`` is positive even integer
        ``(-oo) ** o``   ``-oo`` ``o`` is positive odd integer
        ================ ======= ==============================

        See Also
        ========

        Infinity
        Pow
        NaN

        """
        if expt.is_number:
            if expt is S.NaN or \
                expt is S.Infinity or \
                    expt is S.NegativeInfinity:
                return S.NaN

            if isinstance(expt, Integer) and expt.is_positive:
                if expt.is_odd:
                    return S.NegativeInfinity
                else:
                    return S.Infinity

            return S.NegativeOne**expt*S.Infinity**expt

    def _as_mpf_val(self, prec):
        return mlib.fninf

    def _sage_(self):
        import sage.all as sage
        return -(sage.oo)

    def __hash__(self):
        return super(NegativeInfinity, self).__hash__()

    def __eq__(self, other):
        return other is S.NegativeInfinity

    def __ne__(self, other):
        return other is not S.NegativeInfinity

    def __lt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s < %s" % (self, other))
        if other.is_real:
            if other.is_finite or other is S.Infinity:
                return S.true
            elif other.is_nonnegative:
                return S.true
            elif other.is_infinite and other.is_negative:
                return S.false
        return Expr.__lt__(self, other)

    def __le__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s <= %s" % (self, other))
        if other.is_real:
            return S.true
        return Expr.__le__(self, other)

    def __gt__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s > %s" % (self, other))
        if other.is_real:
            return S.false
        return Expr.__gt__(self, other)

    def __ge__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            raise TypeError("Invalid comparison %s >= %s" % (self, other))
        if other.is_real:
            if other.is_finite or other is S.Infinity:
                return S.false
            elif other.is_nonnegative:
                return S.false
            elif other.is_infinite and other.is_negative:
                return S.true
        return Expr.__ge__(self, other)

    def __mod__(self, other):
        return S.NaN

    __rmod__ = __mod__

    def floor(self):
        return self

    def ceiling(self):
        return self


class NaN(with_metaclass(Singleton, Number)):
    """
    Not a Number.

    This serves as a place holder for numeric values that are indeterminate.
    Most operations on NaN, produce another NaN.  Most indeterminate forms,
    such as ``0/0`` or ``oo - oo` produce NaN.  Two exceptions are ``0**0``
    and ``oo**0``, which all produce ``1`` (this is consistent with Python's
    float).

    NaN is loosely related to floating point nan, which is defined in the
    IEEE 754 floating point standard, and corresponds to the Python
    ``float('nan')``.  Differences are noted below.

    NaN is mathematically not equal to anything else, even NaN itself.  This
    explains the initially counter-intuitive results with ``Eq`` and ``==`` in
    the examples below.

    NaN is not comparable so inequalities raise a TypeError.  This is in
    constrast with floating point nan where all inequalities are false.

    NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported
    as ``nan``.

    Examples
    ========

    >>> from sympy import nan, S, oo, Eq
    >>> nan is S.NaN
    True
    >>> oo - oo
    nan
    >>> nan + 1
    nan
    >>> Eq(nan, nan)   # mathematical equality
    False
    >>> nan == nan     # structural equality
    True

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/NaN

    """
    is_commutative = True
    is_real = None
    is_rational = None
    is_algebraic = None
    is_transcendental = None
    is_integer = None
    is_comparable = False
    is_finite = None
    is_zero = None
    is_prime = None
    is_positive = None
    is_negative = None
    is_number = True

    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)

    def _latex(self, printer):
        return r"\text{NaN}"

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        return self

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        return self

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        return self

    @_sympifyit('other', NotImplemented)
    def __div__(self, other):
        return self

    __truediv__ = __div__

    def floor(self):
        return self

    def ceiling(self):
        return self

    def _as_mpf_val(self, prec):
        return _mpf_nan

    def _sage_(self):
        import sage.all as sage
        return sage.NaN

    def __hash__(self):
        return super(NaN, self).__hash__()

    def __eq__(self, other):
        # NaN is structurally equal to another NaN
        return other is S.NaN

    def __ne__(self, other):
        return other is not S.NaN

    def _eval_Eq(self, other):
        # NaN is not mathematically equal to anything, even NaN
        return S.false

    # Expr will _sympify and raise TypeError
    __gt__ = Expr.__gt__
    __ge__ = Expr.__ge__
    __lt__ = Expr.__lt__
    __le__ = Expr.__le__

nan = S.NaN


class ComplexInfinity(with_metaclass(Singleton, AtomicExpr)):
    r"""Complex infinity.

    In complex analysis the symbol `\tilde\infty`, called "complex
    infinity", represents a quantity with infinite magnitude, but
    undetermined complex phase.

    ComplexInfinity is a singleton, and can be accessed by
    ``S.ComplexInfinity``, or can be imported as ``zoo``.

    Examples
    ========

    >>> from sympy import zoo, oo
    >>> zoo + 42
    zoo
    >>> 42/zoo
    0
    >>> zoo + zoo
    nan
    >>> zoo*zoo
    zoo

    See Also
    ========

    Infinity
    """

    is_commutative = True
    is_infinite = True
    is_number = True
    is_prime = False
    is_complex = True
    is_real = False

    __slots__ = []

    def __new__(cls):
        return AtomicExpr.__new__(cls)

    def _latex(self, printer):
        return r"\tilde{\infty}"

    @staticmethod
    def __abs__():
        return S.Infinity

    def floor(self):
        return self

    def ceiling(self):
        return self

    @staticmethod
    def __neg__():
        return S.ComplexInfinity

    def _eval_power(self, expt):
        if expt is S.ComplexInfinity:
            return S.NaN

        if isinstance(expt, Number):
            if expt is S.Zero:
                return S.NaN
            else:
                if expt.is_positive:
                    return S.ComplexInfinity
                else:
                    return S.Zero

    def _sage_(self):
        import sage.all as sage
        return sage.UnsignedInfinityRing.gen()


zoo = S.ComplexInfinity


class NumberSymbol(AtomicExpr):

    is_commutative = True
    is_finite = True
    is_number = True

    __slots__ = []

    is_NumberSymbol = True

    def __new__(cls):
        return AtomicExpr.__new__(cls)

    def approximation(self, number_cls):
        """ Return an interval with number_cls endpoints
        that contains the value of NumberSymbol.
        If not implemented, then return None.
        """

    def _eval_evalf(self, prec):
        return Float._new(self._as_mpf_val(prec), prec)

    def __eq__(self, other):
        try:
            other = _sympify(other)
        except SympifyError:
            return NotImplemented
        if self is other:
            return True
        if other.is_Number and self.is_irrational:
            return False

        return False    # NumberSymbol != non-(Number|self)

    def __ne__(self, other):
        return not self == other

    def __le__(self, other):
        if self is other:
            return S.true
        return Expr.__le__(self, other)

    def __ge__(self, other):
        if self is other:
            return S.true
        return Expr.__ge__(self, other)

    def __int__(self):
        # subclass with appropriate return value
        raise NotImplementedError

    def __long__(self):
        return self.__int__()

    def __hash__(self):
        return super(NumberSymbol, self).__hash__()


class Exp1(with_metaclass(Singleton, NumberSymbol)):
    r"""The `e` constant.

    The transcendental number `e = 2.718281828\ldots` is the base of the
    natural logarithm and of the exponential function, `e = \exp(1)`.
    Sometimes called Euler's number or Napier's constant.

    Exp1 is a singleton, and can be accessed by ``S.Exp1``,
    or can be imported as ``E``.

    Examples
    ========

    >>> from sympy import exp, log, E
    >>> E is exp(1)
    True
    >>> log(E)
    1

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/E_%28mathematical_constant%29
    """

    is_real = True
    is_positive = True
    is_negative = False  # XXX Forces is_negative/is_nonnegative
    is_irrational = True
    is_number = True
    is_algebraic = False
    is_transcendental = True

    __slots__ = []

    def _latex(self, printer):
        return r"e"

    @staticmethod
    def __abs__():
        return S.Exp1

    def __int__(self):
        return 2

    def _as_mpf_val(self, prec):
        return mpf_e(prec)

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (Integer(2), Integer(3))
        elif issubclass(number_cls, Rational):
            pass

    def _eval_power(self, expt):
        from sympy import exp
        return exp(expt)

    def _eval_rewrite_as_sin(self, **kwargs):
        from sympy import sin
        I = S.ImaginaryUnit
        return sin(I + S.Pi/2) - I*sin(I)

    def _eval_rewrite_as_cos(self, **kwargs):
        from sympy import cos
        I = S.ImaginaryUnit
        return cos(I) + I*cos(I + S.Pi/2)

    def _sage_(self):
        import sage.all as sage
        return sage.e
E = S.Exp1


class Pi(with_metaclass(Singleton, NumberSymbol)):
    r"""The `\pi` constant.

    The transcendental number `\pi = 3.141592654\ldots` represents the ratio
    of a circle's circumference to its diameter, the area of the unit circle,
    the half-period of trigonometric functions, and many other things
    in mathematics.

    Pi is a singleton, and can be accessed by ``S.Pi``, or can
    be imported as ``pi``.

    Examples
    ========

    >>> from sympy import S, pi, oo, sin, exp, integrate, Symbol
    >>> S.Pi
    pi
    >>> pi > 3
    True
    >>> pi.is_irrational
    True
    >>> x = Symbol('x')
    >>> sin(x + 2*pi)
    sin(x)
    >>> integrate(exp(-x**2), (x, -oo, oo))
    sqrt(pi)

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Pi
    """

    is_real = True
    is_positive = True
    is_negative = False
    is_irrational = True
    is_number = True
    is_algebraic = False
    is_transcendental = True

    __slots__ = []

    def _latex(self, printer):
        return r"\pi"

    @staticmethod
    def __abs__():
        return S.Pi

    def __int__(self):
        return 3

    def _as_mpf_val(self, prec):
        return mpf_pi(prec)

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (Integer(3), Integer(4))
        elif issubclass(number_cls, Rational):
            return (Rational(223, 71), Rational(22, 7))

    def _sage_(self):
        import sage.all as sage
        return sage.pi
pi = S.Pi


class GoldenRatio(with_metaclass(Singleton, NumberSymbol)):
    r"""The golden ratio, `\phi`.

    `\phi = \frac{1 + \sqrt{5}}{2}` is algebraic number.  Two quantities
    are in the golden ratio if their ratio is the same as the ratio of
    their sum to the larger of the two quantities, i.e. their maximum.

    GoldenRatio is a singleton, and can be accessed by ``S.GoldenRatio``.

    Examples
    ========

    >>> from sympy import S
    >>> S.GoldenRatio > 1
    True
    >>> S.GoldenRatio.expand(func=True)
    1/2 + sqrt(5)/2
    >>> S.GoldenRatio.is_irrational
    True

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Golden_ratio
    """

    is_real = True
    is_positive = True
    is_negative = False
    is_irrational = True
    is_number = True
    is_algebraic = True
    is_transcendental = False

    __slots__ = []

    def _latex(self, printer):
        return r"\phi"

    def __int__(self):
        return 1

    def _as_mpf_val(self, prec):
         # XXX track down why this has to be increased
        rv = mlib.from_man_exp(phi_fixed(prec + 10), -prec - 10)
        return mpf_norm(rv, prec)

    def _eval_expand_func(self, **hints):
        from sympy import sqrt
        return S.Half + S.Half*sqrt(5)

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (S.One, Rational(2))
        elif issubclass(number_cls, Rational):
            pass

    def _sage_(self):
        import sage.all as sage
        return sage.golden_ratio

    _eval_rewrite_as_sqrt = _eval_expand_func


class TribonacciConstant(with_metaclass(Singleton, NumberSymbol)):
    r"""The tribonacci constant.

    The tribonacci numbers are like the Fibonacci numbers, but instead
    of starting with two predetermined terms, the sequence starts with
    three predetermined terms and each term afterwards is the sum of the
    preceding three terms.

    The tribonacci constant is the ratio toward which adjacent tribonacci
    numbers tend. It is a root of the polynomial `x^3 - x^2 - x - 1 = 0`,
    and also satisfies the equation `x + x^{-3} = 2`.

    TribonacciConstant is a singleton, and can be accessed
    by ``S.TribonacciConstant``.

    Examples
    ========

    >>> from sympy import S
    >>> S.TribonacciConstant > 1
    True
    >>> S.TribonacciConstant.expand(func=True)
    1/3 + (19 - 3*sqrt(33))**(1/3)/3 + (3*sqrt(33) + 19)**(1/3)/3
    >>> S.TribonacciConstant.is_irrational
    True
    >>> S.TribonacciConstant.n(20)
    1.8392867552141611326

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
    """

    is_real = True
    is_positive = True
    is_negative = False
    is_irrational = True
    is_number = True
    is_algebraic = True
    is_transcendental = False

    __slots__ = []

    def _latex(self, printer):
        return r"\text{TribonacciConstant}"

    def __int__(self):
        return 2

    def _eval_evalf(self, prec):
        rv = self._eval_expand_func(function=True)._eval_evalf(prec + 4)
        return Float(rv, precision=prec)

    def _eval_expand_func(self, **hints):
        from sympy import sqrt, cbrt
        return (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (S.One, Rational(2))
        elif issubclass(number_cls, Rational):
            pass

    _eval_rewrite_as_sqrt = _eval_expand_func


class EulerGamma(with_metaclass(Singleton, NumberSymbol)):
    r"""The Euler-Mascheroni constant.

    `\gamma = 0.5772157\ldots` (also called Euler's constant) is a mathematical
    constant recurring in analysis and number theory.  It is defined as the
    limiting difference between the harmonic series and the
    natural logarithm:

    .. math:: \gamma = \lim\limits_{n\to\infty}
              \left(\sum\limits_{k=1}^n\frac{1}{k} - \ln n\right)

    EulerGamma is a singleton, and can be accessed by ``S.EulerGamma``.

    Examples
    ========

    >>> from sympy import S
    >>> S.EulerGamma.is_irrational
    >>> S.EulerGamma > 0
    True
    >>> S.EulerGamma > 1
    False

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant
    """

    is_real = True
    is_positive = True
    is_negative = False
    is_irrational = None
    is_number = True

    __slots__ = []

    def _latex(self, printer):
        return r"\gamma"

    def __int__(self):
        return 0

    def _as_mpf_val(self, prec):
         # XXX track down why this has to be increased
        v = mlib.libhyper.euler_fixed(prec + 10)
        rv = mlib.from_man_exp(v, -prec - 10)
        return mpf_norm(rv, prec)

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (S.Zero, S.One)
        elif issubclass(number_cls, Rational):
            return (S.Half, Rational(3, 5))

    def _sage_(self):
        import sage.all as sage
        return sage.euler_gamma


class Catalan(with_metaclass(Singleton, NumberSymbol)):
    r"""Catalan's constant.

    `K = 0.91596559\ldots` is given by the infinite series

    .. math:: K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}

    Catalan is a singleton, and can be accessed by ``S.Catalan``.

    Examples
    ========

    >>> from sympy import S
    >>> S.Catalan.is_irrational
    >>> S.Catalan > 0
    True
    >>> S.Catalan > 1
    False

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Catalan%27s_constant
    """

    is_real = True
    is_positive = True
    is_negative = False
    is_irrational = None
    is_number = True

    __slots__ = []

    def __int__(self):
        return 0

    def _as_mpf_val(self, prec):
        # XXX track down why this has to be increased
        v = mlib.catalan_fixed(prec + 10)
        rv = mlib.from_man_exp(v, -prec - 10)
        return mpf_norm(rv, prec)

    def approximation_interval(self, number_cls):
        if issubclass(number_cls, Integer):
            return (S.Zero, S.One)
        elif issubclass(number_cls, Rational):
            return (Rational(9, 10), S.One)

    def _sage_(self):
        import sage.all as sage
        return sage.catalan


class ImaginaryUnit(with_metaclass(Singleton, AtomicExpr)):
    r"""The imaginary unit, `i = \sqrt{-1}`.

    I is a singleton, and can be accessed by ``S.I``, or can be
    imported as ``I``.

    Examples
    ========

    >>> from sympy import I, sqrt
    >>> sqrt(-1)
    I
    >>> I*I
    -1
    >>> 1/I
    -I

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Imaginary_unit
    """

    is_commutative = True
    is_imaginary = True
    is_finite = True
    is_number = True
    is_algebraic = True
    is_transcendental = False

    __slots__ = []

    def _latex(self, printer):
        return printer._settings['imaginary_unit_latex']

    @staticmethod
    def __abs__():
        return S.One

    def _eval_evalf(self, prec):
        return self

    def _eval_conjugate(self):
        return -S.ImaginaryUnit

    def _eval_power(self, expt):
        """
        b is I = sqrt(-1)
        e is symbolic object but not equal to 0, 1

        I**r -> (-1)**(r/2) -> exp(r/2*Pi*I) -> sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal
        I**0 mod 4 -> 1
        I**1 mod 4 -> I
        I**2 mod 4 -> -1
        I**3 mod 4 -> -I
        """

        if isinstance(expt, Number):
            if isinstance(expt, Integer):
                expt = expt.p % 4
                if expt == 0:
                    return S.One
                if expt == 1:
                    return S.ImaginaryUnit
                if expt == 2:
                    return -S.One
                return -S.ImaginaryUnit
        return

    def as_base_exp(self):
        return S.NegativeOne, S.Half

    def _sage_(self):
        import sage.all as sage
        return sage.I

    @property
    def _mpc_(self):
        return (Float(0)._mpf_, Float(1)._mpf_)

I = S.ImaginaryUnit


def sympify_fractions(f):
    return Rational(f.numerator, f.denominator, 1)

converter[fractions.Fraction] = sympify_fractions

try:
    if HAS_GMPY == 2:
        import gmpy2 as gmpy
    elif HAS_GMPY == 1:
        import gmpy
    else:
        raise ImportError

    def sympify_mpz(x):
        return Integer(long(x))

    def sympify_mpq(x):
        return Rational(long(x.numerator), long(x.denominator))

    converter[type(gmpy.mpz(1))] = sympify_mpz
    converter[type(gmpy.mpq(1, 2))] = sympify_mpq
except ImportError:
    pass


def sympify_mpmath(x):
    return Expr._from_mpmath(x, x.context.prec)

converter[mpnumeric] = sympify_mpmath


def sympify_mpq(x):
    p, q = x._mpq_
    return Rational(p, q, 1)

converter[type(mpmath.rational.mpq(1, 2))] = sympify_mpq


def sympify_complex(a):
    real, imag = list(map(sympify, (a.real, a.imag)))
    return real + S.ImaginaryUnit*imag

converter[complex] = sympify_complex

from .power import Pow, integer_nthroot
from .mul import Mul
Mul.identity = One()
from .add import Add
Add.identity = Zero()