numbers.py

import decimal
import fractions
import math
import numbers
import re as regex

import mpmath
import mpmath.libmp as mlib
from mpmath.ctx_mp import mpnumeric
from mpmath.libmp import mpf_e, mpf_pi, mpf_pow, phi_fixed
from mpmath.libmp.libmpf import _normalize as mpf_normalize
from mpmath.libmp.libmpf import finf as _mpf_inf
from mpmath.libmp.libmpf import fnan as _mpf_nan
from mpmath.libmp.libmpf import fninf as _mpf_ninf
from mpmath.libmp.libmpf import fzero as _mpf_zero
from mpmath.libmp.libmpf import prec_to_dps

from ..utilities import filldedent
from .cache import cacheit, clear_cache
from .compatibility import DIOFANT_INTS, HAS_GMPY, as_int
from .containers import Tuple
from .decorators import _sympifyit
from .expr import AtomicExpr, Expr
from .logic import fuzzy_not
from .singleton import SingletonWithManagedProperties as Singleton
from .singleton import S
from .sympify import SympifyError, converter, sympify


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:
            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 sympy/sympy#6639)
        if not bc:
            return _mpf_zero
        else:
            # don't change anything; this should already
            # be a well formed mpf tuple
            return mpf
    rv = mpf_normalize(sign, man, expt, bc, prec, rnd)
    return rv


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


def seterr(divide=False):
    """
    Should diofant 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 _decimal_to_Rational_prec(dec):
    """Convert an ordinary decimal instance to a Rational."""
    assert dec.is_finite()
    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))


@cacheit
def igcd(*args):
    """Computes positive integer greatest common divisor.

    Examples
    ========

    >>> igcd(2, 4)
    2
    >>> igcd(5, 10, 15)
    5
    """
    a = args[0]
    for b in args[1:]:
        a, b = as_int(a), abs(as_int(b))

        a = math.gcd(a, b)
        if a == 1:
            break
    return a


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

    Examples
    ========

    >>> ilcm(5, 10)
    10
    >>> ilcm(7, 3)
    21
    >>> ilcm(5, 10, 15)
    30

    """
    if 0 in args:
        return 0
    a = args[0]
    for b in args[1:]:
        a = a*b // igcd(a, b)
    return a


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

    >>> 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 ) % m == 1 where
    c has the same sign as a. If no such value exists, a
    ValueError is raised.

    Examples
    ========

    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)
    -4

    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(Integer(2)/7, Integer(5)/2)
    7/2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Modular_multiplicative_inverse
    .. [2] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
    """
    c = None
    try:
        a, m = as_int(a), as_int(m)
        if m > 1:
            x, y, g = igcdex(a, m)
            if g == 1:
                c = x % m
            if a < 0:
                c -= 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 any kind of number in diofant.

    Floating point numbers are represented by the Float class.
    Integer numbers (of any size), together with rational numbers (again,
    there is no limit on their size) are represented by the Rational class.

    If you want to represent, for example, ``1+sqrt(2)``, then you need to do::

      Rational(1) + sqrt(Rational(2))
    """

    is_commutative = True
    is_number = True
    is_Number = True

    # 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, DIOFANT_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, str):
            val = sympify(obj)
            if isinstance(val, Number):
                return val
            else:
                raise ValueError('String "%s" does not denote a Number' % obj)
        msg = "expected str|int|float|Decimal|Number object but got %r"
        raise TypeError(msg % type(obj).__name__)

    def invert(self, other, *gens, **args):
        from ..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
        from ..functions.elementary.complexes import sign

        other = Number(other)
        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 = sign(rat)*int(abs(rat))  # = rat.floor()
        r = self - other*w
        return Tuple(w, r)

    def __rdivmod__(self, other):
        other = Number(other)
        return divmod(other, self)

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

    def _as_mpf_val(self, prec):  # pragma: no cover
        """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 _eval_conjugate(self):
        return self

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

    @classmethod
    def class_key(cls):
        """Nice order of classes."""
        return 1, 0, 'Number'

    @cacheit
    def sort_key(self, order=None):
        """Return a sort key."""
        return self.class_key(), (0, ()), (), self

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if self is S.Zero:
            return other
        if other is nan:
            return nan
        elif other is oo:
            return oo
        elif other is -oo:
            return -oo
        else:
            return AtomicExpr.__add__(self, other)

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if other is nan:
            return nan
        elif other is oo:
            return -oo
        elif other is -oo:
            return oo
        else:
            return AtomicExpr.__sub__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if self is S.One:
            return other
        if other is nan:
            return nan
        elif other is oo:
            if self.is_zero:
                return nan
            elif self.is_positive:
                return oo
            else:
                return -oo
        elif other is -oo:
            if self.is_zero:
                return nan
            elif self.is_positive:
                return -oo
            else:
                return oo
        elif isinstance(other, Tuple):
            return NotImplemented
        else:
            return AtomicExpr.__mul__(self, other)

    @_sympifyit('other', NotImplemented)
    def __truediv__(self, other):
        if isinstance(other, Number):
            if other is nan:
                return nan
            elif other in (oo, -oo):
                return S.Zero
        return AtomicExpr.__truediv__(self, other)

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

    def is_constant(self, *wrt, **flags):
        """Return True if self is constant.

        See Also
        ========

        diofant.core.expr.Expr.is_constant
        """
        return True

    def as_coeff_mul(self, *deps, **kwargs):
        """Return the tuple (c, args) where self is written as a Mul.

        See Also
        ========

        diofant.core.expr.Expr.as_coeff_mul
        """
        # a -> c*t
        if self.is_Rational or not kwargs.pop('rational', True):
            return self, ()
        elif self.is_negative:
            return S.NegativeOne, (-self,)
        return S.One, (self,)

    def as_coeff_add(self, *deps):
        """Return the tuple (c, args) where self is written as an Add.

        See Also
        ========

        diofant.core.expr.Expr.as_coeff_add
        """
        # a -> c + t
        if self.is_Rational:
            return self, ()
        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 ..polys import gcd
        return gcd(self, other)

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

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


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

    Examples
    ========

    >>> Float(3.5)
    3.50000000000000
    >>> Float(3)
    3.00000000000000

    Creating Floats from strings (and Python ``int`` type) 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 and ints).

    >>> 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, you can change the precision of Float,
    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:

    >>> Float(approx, 5)
    0.099609

    Please note that you can't increase precision with evalf:

    >>> approx.evalf(5)
    Traceback (most recent call last):
    ...
    PrecisionExhausted: ...

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

    >>> Float(exact, 5)
    0.12500
    >>> Float(exact, 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 diofant 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

    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.
    """

    # A Float represents many real numbers,
    # both rational and irrational.
    is_number = True

    is_extended_real = True

    is_Float = True

    def __new__(cls, num, dps=None):
        if isinstance(num, str):
            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, (DIOFANT_INTS, Integer)):
            num = str(num)  # faster than mlib.from_int
        elif num is oo:
            num = '+inf'
        elif num is -oo:
            num = '-inf'
        elif isinstance(num, mpmath.mpf):
            num = num._mpf_

        if dps is None:
            dps = 15
            if isinstance(num, Float):
                return num
            if isinstance(num, str) 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)
        elif dps == '':
            if not isinstance(num, str):
                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('-')))
                    ok = True
            if ok is None:
                raise ValueError('string-float not recognized: %s' % num)

        prec = mlib.libmpf.dps_to_prec(dps)
        if isinstance(num, float):
            _mpf_ = mlib.from_float(num, prec, rnd)
        elif isinstance(num, str):
            _mpf_ = mlib.from_str(num, prec, rnd)
        elif isinstance(num, decimal.Decimal):
            if num.is_finite():
                _mpf_ = mlib.from_str(str(num), prec, rnd)
            elif num.is_nan():
                _mpf_ = _mpf_nan
            else:
                assert num.is_infinite()
                if num > 0:
                    _mpf_ = _mpf_inf
                else:
                    _mpf_ = _mpf_ninf
        elif isinstance(num, Rational):
            _mpf_ = mlib.from_rational(num.p, num.q, prec, rnd)
        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)
                num[1] = mlib.backend.MPZ(num[1], 16)
                _mpf_ = tuple(num)
            else:
                if not num[1] and len(num) == 4:
                    # handle normalization hack
                    return Float._new(num, prec)
                else:
                    _mpf_ = mpmath.mpf(
                        S.NegativeOne**num[0]*num[1]*2**num[2])._mpf_
        elif isinstance(num, Float):
            _mpf_ = num._mpf_
            if prec < num._prec:
                _mpf_ = mpf_norm(_mpf_, prec)
        else:
            _mpf_ = mpmath.mpf(num)._mpf_

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

        obj = Expr.__new__(cls)
        obj._mpf_ = _mpf_
        obj._prec = prec
        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 nan

        obj = Expr.__new__(cls)
        obj._mpf_ = mpf_norm(_mpf_, _prec)
        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):
        """Compute floor of self."""
        return Integer(int(mlib.to_int(
            mlib.mpf_floor(self._mpf_, self._prec))))

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

    @property
    def num(self):
        """Return mpmath representation of self."""
        return mpmath.mpf(self._mpf_)

    def _as_mpf_val(self, prec):
        return mpf_norm(self._mpf_, prec)

    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 __bool__(self):
        return self._mpf_ != _mpf_zero

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

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

    @_sympifyit('other', NotImplemented)
    def __mod__(self, other):
        if isinstance(other, Rational) and other.q != 1:
            # calculate mod with Rationals, *then* round the result
            return Float(Rational.__mod__(Rational(self), other),
                         prec_to_dps(self._prec))
        if isinstance(other, Float):
            r = self/other
            if r == int(r):
                prec = max(prec_to_dps(i) for i in (self._prec, other._prec))
                return Float(0, prec)
        if isinstance(other, Number):
            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)

    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)*I

    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

    def __eq__(self, other):
        if isinstance(other, float):
            # coerce to Float at same precision
            o = Float(other)
            ompf = o._as_mpf_val(self._prec)
            return bool(mlib.mpf_eq(self._mpf_, ompf))
        try:
            other = sympify(other, strict=True)
        except SympifyError:
            return False    # diofant != other  -->  not ==
        if isinstance(other, NumberSymbol):
            if other.is_irrational:
                return False
            return other.__eq__(self)
        if isinstance(other, Float):
            return bool(mlib.mpf_eq(self._mpf_, other._mpf_))
        if isinstance(other, 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

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__lt__(self)
        if other.is_comparable:
            other = other.evalf(strict=False)
        if isinstance(other, Number) and other is not nan:
            return sympify(bool(mlib.mpf_gt(self._mpf_,
                                            other._as_mpf_val(self._prec))),
                           strict=True)
        return Expr.__gt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__le__(self)
        if other.is_comparable:
            other = other.evalf(strict=False)
        if isinstance(other, Number) and other is not nan:
            return sympify(bool(mlib.mpf_ge(self._mpf_,
                                            other._as_mpf_val(self._prec))),
                           strict=True)
        return Expr.__ge__(self, other)

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__gt__(self)
        if other.is_extended_real and other.is_number:
            other = other.evalf(strict=False)
        if isinstance(other, Number) and other is not nan:
            return sympify(bool(mlib.mpf_lt(self._mpf_,
                                            other._as_mpf_val(self._prec))),
                           strict=True)
        return Expr.__lt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__ge__(self)
        if other.is_extended_real and other.is_number:
            other = other.evalf(strict=False)
        if isinstance(other, Number) and other is not nan:
            return sympify(bool(mlib.mpf_le(self._mpf_,
                                            other._as_mpf_val(self._prec))),
                           strict=True)
        return Expr.__le__(self, other)

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

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

    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 integers and rational numbers (p/q) of any size.

    Examples
    ========

    >>> Rational(3)
    3
    >>> 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

    The conversion of floats to expressions or simple fractions can
    be handled with nsimplify:

    >>> 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 Diofant Integers) so some care
    is needed when using them in expressions:

    >>> r.p/r.q
    0.75

    See Also
    ========

    diofant.core.sympify.sympify
    diofant.simplify.simplify.nsimplify
    """

    is_real = True
    is_integer = False
    is_rational = True
    is_number = True

    is_Rational = True

    @cacheit
    def __new__(cls, p, q=1):
        if q == 1:
            if isinstance(p, Rational):
                return p
            elif isinstance(p, Float):
                p, q = float(p).as_integer_ratio()

        if isinstance(p, Rational):
            p = fractions.Fraction(p.p, p.q)
        if isinstance(q, Rational):
            q = fractions.Fraction(q.p, q.q)

        try:
            f = fractions.Fraction(p)/fractions.Fraction(q)
            p, q = f.numerator, f.denominator
        except ValueError:
            raise TypeError('invalid input: %s, %s' % (p, q))
        except ZeroDivisionError:
            if p == 0:
                if _errdict["divide"]:
                    raise ValueError("Indeterminate 0/0")
                else:
                    return nan
            else:
                return zoo

        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.

        >>> 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 isinstance(other, Rational):
            return Rational(self.p*other.q + self.q*other.p, self.q*other.q)
        elif isinstance(other, (Float, AlgebraicNumber)):
            return other + self
        else:
            return Number.__add__(self, other)

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if isinstance(other, Rational):
            return Rational(self.p*other.q - self.q*other.p, self.q*other.q)
        elif isinstance(other, (Float, AlgebraicNumber)):
            return -other + self
        else:
            return Number.__sub__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if isinstance(other, Rational):
            return Rational(self.p*other.p, self.q*other.q)
        elif isinstance(other, (Float, AlgebraicNumber)):
            return other*self
        else:
            return Number.__mul__(self, other)

    @_sympifyit('other', NotImplemented)
    def __truediv__(self, other):
        if isinstance(other, Rational):
            if self.p and other.p == 0:
                return zoo
            else:
                return Rational(self.p*other.q, self.q*other.p)
        elif isinstance(other, Float):
            return self*(1/other)
        else:
            return Number.__truediv__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mod__(self, other):
        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)
        elif isinstance(other, Float):
            # calculate mod with Rationals, *then* round the answer
            return Float(self.__mod__(Rational(other)),
                         prec_to_dps(other._prec))
        else:
            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 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.p % expt.q) /
                                               Integer(expt.q)) *
                             Rational(self.q, -self.p)**ne)
                else:
                    return Rational(self.q, self.p)**ne
            if expt is oo:  # -oo already caught by test for negative
                if self.p > self.q:
                    # (3/2)**oo -> oo
                    return oo
                if self.p < -self.q:
                    # (-3/2)**oo -> oo + I*oo
                    return oo + oo*I
                return S.Zero
            if isinstance(expt, Float):
                return self._eval_evalf(expt._prec)**expt
            elif isinstance(expt, Integer):
                # (4/3)**2 -> 4**2 / 3**2
                return Rational(self.p**expt.p, self.q**expt.p)
            else:  # 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)

    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 -(-p//q)
        return p//q

    @_sympifyit('other', NotImplemented)
    def __eq__(self, other):
        if isinstance(other, NumberSymbol):
            if other.is_irrational:
                return False
            return other.__eq__(self)
        if isinstance(other, Number):
            if isinstance(other, 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 isinstance(other, Float):
                return mlib.mpf_eq(self._as_mpf_val(other._prec), other._mpf_)
        return False

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__lt__(self)
        expr = self
        if isinstance(other, Number):
            if isinstance(other, Rational):
                return sympify(bool(self.p*other.q > self.q*other.p),
                               strict=True)
            if isinstance(other, Float):
                return sympify(bool(mlib.mpf_gt(self._as_mpf_val(other._prec),
                                                other._mpf_)),
                               strict=True)
        elif other.is_number and other.is_extended_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__gt__(expr, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__le__(self)
        expr = self
        if isinstance(other, Number):
            if isinstance(other, Rational):
                return sympify(bool(self.p*other.q >= self.q*other.p),
                               strict=True)
            if isinstance(other, Float):
                return sympify(bool(mlib.mpf_ge(self._as_mpf_val(other._prec),
                                                other._mpf_)),
                               strict=True)
        elif other.is_number and other.is_extended_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__ge__(expr, other)

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if isinstance(other, NumberSymbol):
            return other.__gt__(self)
        expr = self
        if isinstance(other, Number):
            if isinstance(other, Rational):
                return sympify(bool(self.p*other.q < self.q*other.p),
                               strict=True)
            if isinstance(other, Float):
                return sympify(bool(mlib.mpf_lt(self._as_mpf_val(other._prec),
                                                other._mpf_)),
                               strict=True)
        elif other.is_number and other.is_extended_real:
            expr, other = Integer(self.p), self.q*other
        return Expr.__lt__(expr, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        expr = self
        if isinstance(other, NumberSymbol):
            return other.__ge__(self)
        elif isinstance(other, Number):
            if isinstance(other, Rational):
                return sympify(bool(self.p*other.q <= self.q*other.p),
                               strict=True)
            if isinstance(other, Float):
                return sympify(bool(mlib.mpf_le(self._as_mpf_val(other._prec),
                                                other._mpf_)),
                               strict=True)
        elif other.is_number and other.is_extended_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 ..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):
        """Compute GCD of `self` and `other`. """
        if isinstance(other, Rational):
            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):
        """Compute LCM of `self` and `other`. """
        if isinstance(other, Rational):
            return Rational(
                self.p*other.p//igcd(self.p, other.p),
                igcd(self.q, other.q))
        return Number.lcm(self, other)

    def _eval_as_numer_denom(self):
        """expression -> a/b -> a, b

        See Also
        ========

        diofant.core.expr.Expr.as_numer_denom
        """
        return Integer(self.p), Integer(self.q)

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

        Examples
        ========

        >>> Rational(-3, 2).as_content_primitive()
        (3/2, -1)

        See Also
        ========

        diofant.core.expr.Expr.as_content_primitive
        """

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

    @property
    def numerator(self):
        return self.p

    @property
    def denominator(self):
        return self.q


numbers.Rational.register(Rational)


class Integer(Rational):

    q = 1
    is_integer = True
    is_number = True

    is_Integer = True

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

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

    @cacheit
    def __new__(cls, i):
        if isinstance(i, str):
            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(
                'Integer can only work with integer expressions.')

        # We only work with well-behaved integer types. This converts, for
        # example, numpy.int32 instances.
        if ival == 0:
            obj = S.Zero
        elif ival == 1:
            obj = S.One
        elif ival == -1:
            obj = S.NegativeOne
        else:
            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

    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):
            return Tuple(*(divmod(self.p, other.p)))
        else:
            return Number.__divmod__(self, other)

    def __rdivmod__(self, other):
        from .containers import Tuple
        if isinstance(other, int):
            return Tuple(*(divmod(other, self.p)))
        else:
            other = Number(other)
            return Number.__divmod__(other, self)

    # TODO make it decorator + bytecodehacks?
    def __add__(self, other):
        if isinstance(other, int):
            return Integer(self.p + other)
        elif isinstance(other, Integer):
            return Integer(self.p + other.p)
        return Rational.__add__(self, other)

    def __radd__(self, other):
        if isinstance(other, int):
            return Integer(other + self.p)
        return Rational.__add__(self, other)

    def __sub__(self, other):
        if isinstance(other, int):
            return Integer(self.p - other)
        elif isinstance(other, Integer):
            return Integer(self.p - other.p)
        return Rational.__sub__(self, other)

    def __rsub__(self, other):
        if isinstance(other, int):
            return Integer(other - self.p)
        return Rational.__rsub__(self, other)

    def __mul__(self, other):
        if isinstance(other, int):
            return Integer(self.p*other)
        elif isinstance(other, Integer):
            return Integer(self.p*other.p)
        return Rational.__mul__(self, other)

    def __rmul__(self, other):
        if isinstance(other, int):
            return Integer(other*self.p)
        return Rational.__mul__(self, other)

    def __mod__(self, other):
        if isinstance(other, int):
            return Integer(self.p % other)
        elif isinstance(other, Integer):
            return Integer(self.p % other.p)
        return Rational.__mod__(self, other)

    def __rmod__(self, other):
        if isinstance(other, int):
            return Integer(other % self.p)
        elif isinstance(other, Integer):
            return Integer(other.p % self.p)
        return Rational.__rmod__(self, other)

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

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        if isinstance(other, Integer):
            return sympify(self.p > other.p, strict=True)
        return Rational.__gt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if isinstance(other, Integer):
            return sympify(self.p < other.p, strict=True)
        return Rational.__lt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        if isinstance(other, Integer):
            return sympify(self.p >= other.p, strict=True)
        return Rational.__ge__(self, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        if isinstance(other, Integer):
            return sympify(self.p <= other.p, strict=True)
        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
          - root(2**(3+7)*3**(6+7), 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 ..ntheory import perfect_power

        if expt is oo:
            if self.p > S.One:
                return oo
            # cases -1, 0, 1 are done in their respective classes
            return oo + I*oo
        if expt is -oo:
            return Rational(1, self)**oo
        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 I*Pow(-self, expt)
        if expt.is_negative:
            # invert base and change sign on exponent
            ne = -expt
            if self.is_negative:
                return -((S.NegativeOne)**((expt.p % expt.q) /
                                           Integer(expt.q))*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(self).factors(limit=2**15)

        # now process the dict of factors
        if self.is_negative:
            dict[-1] = 1
        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 == self and out_int == 1 and out_rad == 1:
            result = None
        else:
            result = out_int*out_rad*Pow(sqr_int, Rational(sqr_gcd, expt.q))
        return result

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

        return isprime(self)

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

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

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


numbers.Integral.register(Integer)


# Add sympify converters
converter[int] = Integer


class AlgebraicNumber(Expr):
    r"""Class for algebraic numbers in Diofant.

    Represents the algebraic number in the field `\mathbb Q[\theta]`
    given by

    .. math::
        c_n \theta^n + c_{n-1} \theta^{n-1} + \dots + c_0

    Parameters
    ==========

    expr : Expr
        A generator `\theta` for the algebraic number.

    coeffs : tuple, optional
        A tuple of rational coefficients `(c_n, c_{n-1},\dots,c_0)`.
        The default is ``(1, 0)``.

    alias : Symbol, optional
        Alias to denote the generator `\theta`.

    Examples
    ========

    >>> a = AlgebraicNumber(sqrt(3), alias='a')

    Numbers in the same field are automatically combined by
    arithmetic operations.

    >>> a + 1
    a + 1
    >>> _ + a
    2*a + 1

    Powers with integer exponents also automatically evaluated and
    the coefficient list reduced accordingly to the degree of the minimal
    polynomial of `\theta`.

    >>> _**3
    30*a + 37
    >>> 1/a
    a/3

    The generator `\theta` can be any algebraic number, represented in terms
    of radicals or RootOf objects.

    >>> b = AlgebraicNumber(RootOf(x**7 - x + 1, 1), (1, 2, -1), 'b')
    >>> b
    b**2 + 2*b - 1
    >>> b**7
    490*b**6 - 119*b**5 - 196*b**4 - 203*b**3 - 265*b**2 + 637*b - 198

    See Also
    ========

    diofant.polys.rootoftools.RootOf
    """

    is_AlgebraicNumber = True
    is_algebraic = True
    is_number = True

    def __new__(cls, expr, coeffs=(1, 0), alias=None, **kwargs):
        """Construct a new algebraic number. """
        from ..polys import Poly
        from ..polys.polyclasses import DMP
        from ..polys.numberfields import minimal_polynomial
        from .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:
            if expr.free_symbols:
                raise ValueError("Not a number: %s" % expr)

            minpoly, root = minimal_polynomial(
                expr, kwargs.get('gen'), polys=True), expr

        dom = minpoly.domain.get_field()

        if isinstance(coeffs, DMP):
            rep = coeffs
        else:
            rep = DMP.from_diofant_list(sympify(coeffs), 0, dom)

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

        coeffs = Tuple(*rep.all_coeffs())
        args = root, coeffs

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

        obj = Expr.__new__(cls, *args)

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

        return obj

    def _eval_is_real(self):
        return self.root.is_real

    @property
    def free_symbols(self):
        return set()

    def _eval_power(self, expt):
        if expt.is_Integer:
            A = self.rep.domain.algebraic_field(self.root)
            r = A(self.rep.rep)**int(expt)
            return self.func(self, r.rep, self.alias)

    @_sympifyit('other', NotImplemented)
    def __add__(self, other):
        if other.is_Rational:
            other = self.func(self, (other,), self.alias)

        if other.is_AlgebraicNumber:
            if self.minpoly == other.minpoly and self.root == other.root:
                return self.func(self, self.rep + other.rep, self.alias)
            else:
                return Add(self, other, evaluate=False)
        else:
            return Number.__add__(self, other)

    @_sympifyit('other', NotImplemented)
    def __sub__(self, other):
        if other.is_Rational:
            other = self.func(self, (other,), self.alias)

        if other.is_AlgebraicNumber:
            if self.minpoly == other.minpoly and self.root == other.root:
                return self.func(self, self.rep - other.rep, self.alias)
            else:
                return Add(self, -other, evaluate=False)
        else:
            return Number.__sub__(self, other)

    @_sympifyit('other', NotImplemented)
    def __mul__(self, other):
        if other.is_Rational:
            other = self.func(self, (other,), self.alias)

        if other.is_AlgebraicNumber:
            if self.minpoly == other.minpoly and self.root == other.root:
                return self.func(self, self.rep * other.rep, self.alias)
            else:
                return Mul(self, other, evaluate=False)
        else:
            return Number.__mul__(self, other)

    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 .symbol import Dummy
        from ..polys import 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 Diofant coefficients of an algebraic number. """
        return [self.rep.domain.to_diofant(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 ..polys 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):
        from ..polys import RootOf, minimal_polynomial

        for r in [r for r in self.minpoly.all_roots() if r.func != RootOf]:
            if minimal_polynomial(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.
    """

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


class IntegerConstant(Integer):

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


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

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

    Examples
    ========

    >>> 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
    is_imaginary = True

    def _eval_power(self, expt):
        if expt.is_positive:
            return self
        if expt.is_negative:
            return zoo
        if expt.is_extended_real is False:
            return 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 zoo**terms
        if coeff is not S.One:  # there is a Number to discard
            return self**terms

    def __bool__(self):
        return False


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

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

    Examples
    ========

    >>> Integer(1) is S.One
    True

    References
    ==========

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

    is_number = True

    p = 1
    q = 1

    def factors(self, limit=None, use_trial=True, use_rho=False,
                use_pm1=False, verbose=False, visual=False):
        if visual:
            return self
        else:
            return {}


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

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

    Examples
    ========

    >>> 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

    def _eval_power(self, expt):
        if isinstance(expt, Number):
            if isinstance(expt, Float):
                return Float(-1.0)**expt
            elif expt in (oo, -oo):
                return nan
            elif expt is S.Half:
                return I
            else:
                assert isinstance(expt, Rational)
                if expt.q == 2:
                    return I**Integer(expt.p)
                i, r = divmod(expt.p, expt.q)
                if i:
                    return self**i*self**Rational(r, expt.q)
        if isinstance(expt, Add):
            # Handle (-1)**((-1)**n/2 + m/2)
            e2 = 2*expt
            if e2.is_even and e2.could_extract_minus_sign():
                e2 *= self
            assert e2.is_Add
            i, p = e2.as_two_terms()
            if p.is_Pow and p.base is S.NegativeOne and p.exp.is_integer:
                i = (i + 1)/2
                if i.is_even:
                    return self**p.exp


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

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

    Examples
    ========

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

    References
    ==========

    .. [1] https//en.wikipedia.org/wiki/One_half
    """

    is_number = True

    p = 1
    q = 2


class Infinity(Number, metaclass=Singleton):
    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 ``oo``,
    or can be imported as ``oo``.

    Examples
    ========

    >>> 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

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

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

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

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

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

    @_sympifyit('other', NotImplemented)
    def __truediv__(self, other):
        if isinstance(other, Number):
            if other is oo or other is -oo or other is nan:
                return nan
            elif other.is_Float:
                if other == Float('-inf') or \
                        other == Float('inf'):
                    return nan
                elif other.is_nonnegative:
                    return Float('inf')
                else:
                    return Float('-inf')
            else:
                if other >= 0:
                    return oo
                else:
                    return -oo
        return NotImplemented

    def __abs__(self):
        return oo

    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 ..functions import re

        if expt.is_positive:
            return oo
        if expt.is_negative:
            return S.Zero
        if expt.is_real is False and expt.is_number:
            expt_real = re(expt)
            if expt_real.is_positive:
                return zoo
            elif expt_real.is_negative:
                return S.Zero
            elif expt_real.is_zero:
                return nan

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

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

    def __eq__(self, other):
        return other is oo

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if other.is_extended_real:
            return S.false
        return Expr.__lt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        if other.is_extended_real:
            if other.is_finite or other is -oo:
                return S.false
            elif other.is_nonpositive:
                return S.false
            elif other is oo:
                return S.true
        return Expr.__le__(self, other)

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        if other.is_extended_real:
            if other.is_finite or other is -oo:
                return S.true
            elif other.is_nonpositive:
                return S.true
            elif other is oo:
                return S.false
        return Expr.__gt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        if other.is_extended_real:
            return S.true
        return Expr.__ge__(self, other)

    def __mod__(self, other):
        return nan

    __rmod__ = __mod__


oo = S.Infinity


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

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

    See Also
    ========

    Infinity
    """

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

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

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

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

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

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

    @_sympifyit('other', NotImplemented)
    def __truediv__(self, other):
        if isinstance(other, Number):
            if other is oo or other is -oo or other is nan:
                return nan
            elif other.is_Float:
                if other == Float('-inf') or \
                    other == Float('inf') or \
                        other is nan:
                    return nan
                elif other.is_nonnegative:
                    return Float('-inf')
                else:
                    return Float('inf')
            else:
                if other >= 0:
                    return -oo
                else:
                    return oo
        return NotImplemented

    def __abs__(self):
        return oo

    def __neg__(self):
        return oo

    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 in (oo, -oo, nan):
                return nan

            return S.NegativeOne**expt*oo**expt

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

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

    def __eq__(self, other):
        return other is -oo

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if other.is_extended_real:
            if other.is_finite or other is oo:
                return S.true
            elif other.is_nonnegative:
                return S.true
            elif other is -oo:
                return S.false
        return Expr.__lt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        if other.is_extended_real:
            return S.true
        return Expr.__le__(self, other)

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        if other.is_extended_real:
            return S.false
        return Expr.__gt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        if other.is_extended_real:
            if other.is_finite or other is oo:
                return S.false
            elif other.is_nonnegative:
                return S.false
            elif other is -oo:
                return S.true
        return Expr.__ge__(self, other)

    def __mod__(self, other):
        return nan

    __rmod__ = __mod__


class NaN(Number, metaclass=Singleton):
    """
    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 ``nan``.

    Examples
    ========

    >>> nan is 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_comparable = False
    is_finite = False
    is_number = True

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

    def _latex(self, printer):
        return r"\mathrm{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 __truediv__(self, other):
        return self

    def _as_mpf_val(self, prec):
        return _mpf_nan

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

    def __eq__(self, other):
        # NaN is structurally equal to another NaN
        return other is 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(AtomicExpr, metaclass=Singleton):
    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 as ``zoo``.

    Examples
    ========

    >>> 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_extended_real = False

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

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

    def __abs__(self):
        return oo

    def __neg__(self):
        return self

    def _eval_power(self, expt):
        if expt.is_positive:
            return zoo
        elif expt.is_negative:
            return S.Zero


zoo = S.ComplexInfinity


class NumberSymbol(AtomicExpr):

    is_commutative = True
    is_finite = True
    is_number = True

    is_NumberSymbol = True

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

    def approximation_interval(self, number_cls):
        """ Return an interval with number_cls endpoints that contains the
        value of NumberSymbol.  If not implemented, then return None.
        """
        return  # pragma: no cover

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

    @_sympifyit('other', NotImplemented)
    def __eq__(self, other):
        if self is other:
            return True
        if isinstance(other, Number) and self.is_irrational:
            return False

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

    @_sympifyit('other', NotImplemented)
    def __lt__(self, other):
        if self is other:
            return S.false
        return Expr.__lt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __le__(self, other):
        if self is other:
            return S.true
        return Expr.__le__(self, other)

    @_sympifyit('other', NotImplemented)
    def __gt__(self, other):
        r = sympify((-self) < (-other), strict=True)
        if r in (S.true, S.false):
            return r
        else:
            return Expr.__gt__(self, other)

    @_sympifyit('other', NotImplemented)
    def __ge__(self, other):
        r = sympify((-self) <= (-other), strict=True)
        if r in (S.true, S.false):
            return r
        else:
            return Expr.__ge__(self, other)

    def __int__(self):
        raise NotImplementedError  # pragma: no cover

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


class Exp1(NumberSymbol, metaclass=Singleton):
    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 imported as ``E``.

    Examples
    ========

    >>> 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

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

    def __abs__(self):
        return self

    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)

    def _eval_power(self, arg):
        from ..functions.elementary.exponential import log
        from . import Add, Mul, Pow
        if arg.is_Number:
            if arg is oo:
                return oo
            elif arg is -oo:
                return S.Zero
        elif isinstance(arg, log):
            return arg.args[0]
        elif arg.is_Mul:
            Ioo = I*oo
            if arg in [Ioo, -Ioo]:
                return nan

            coeff = arg.coeff(pi*I)
            if coeff:
                if (2*coeff).is_integer:
                    if coeff.is_even:
                        return S.One
                    elif coeff.is_odd:
                        return S.NegativeOne
                    elif (coeff + S.Half).is_even:
                        return -I
                    elif (coeff + S.Half).is_odd:
                        return I

            # Warning: code in risch.py will be very sensitive to changes
            # in this (see DifferentialExtension).

            # look for a single log factor

            coeff, terms = arg.as_coeff_Mul()

            # but it can't be multiplied by oo
            if coeff in (oo, -oo):
                return None

            coeffs, log_term = [coeff], None
            for term in Mul.make_args(terms):
                if isinstance(term, log):
                    if log_term is None:
                        log_term = term.args[0]
                    else:
                        return None
                elif term.is_comparable:
                    coeffs.append(term)
                else:
                    return None

            return log_term**Mul(*coeffs) if log_term else None
        elif arg.is_Add:
            out = []
            add = []
            for a in arg.args:
                if a is S.One:
                    add.append(a)
                    continue
                newa = self**a
                if newa.is_Pow and newa.base is self:
                    add.append(a)
                else:
                    out.append(newa)
            if out:
                return Mul(*out)*Pow(self, Add(*add), evaluate=False)
        elif arg.is_Matrix:
            return arg.exp()

    def _eval_rewrite_as_sin(self):
        from ..functions import sin
        return sin(I + pi/2) - I*sin(I)

    def _eval_rewrite_as_cos(self):
        from ..functions import cos
        return cos(I) + I*cos(I + pi/2)


E = S.Exp1


class Pi(NumberSymbol, metaclass=Singleton):
    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 imported as ``pi``.

    Examples
    ========

    >>> 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

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

    def __abs__(self):
        return self

    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)


pi = S.Pi


class GoldenRatio(NumberSymbol, metaclass=Singleton):
    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.

    Examples
    ========

    >>> GoldenRatio > 1
    true
    >>> GoldenRatio.expand(func=True)
    1/2 + sqrt(5)/2
    >>> 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

    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 ..functions 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)


class EulerGamma(NumberSymbol, metaclass=Singleton):
    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)

    Examples
    ========

    >>> EulerGamma.is_irrational
    >>> EulerGamma > 0
    true
    >>> 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_number = True

    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)


class Catalan(NumberSymbol, metaclass=Singleton):
    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}

    Examples
    ========

    >>> Catalan.is_irrational
    >>> Catalan > 0
    true
    >>> Catalan > 1
    false

    References
    ==========

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

    is_real = True
    is_positive = True
    is_negative = False
    is_number = True

    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


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

    I is a singleton, and can be imported as ``I``.

    Examples
    ========

    >>> 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_extended_real = False
    is_finite = True
    is_number = True
    is_algebraic = True
    is_transcendental = False
    is_real = False

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

    def __abs__(self):
        return S.One

    def _eval_evalf(self, prec):
        return self

    def _eval_conjugate(self):
        return -I

    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 I
                if expt == 2:
                    return -S.One
                return -I
            return (S.NegativeOne)**(expt*S.Half)
        return

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


I = S.ImaginaryUnit


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


converter[fractions.Fraction] = sympify_fractions


try:
    if HAS_GMPY:
        import gmpy2 as gmpy
    else:
        raise ImportError

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

    def sympify_mpq(x):
        return Rational(int(x.numerator), int(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_complex(a):
    real, imag = list(map(sympify, (a.real, a.imag)))
    return real + I*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()