Source code for sympy.core.numbers
from __future__ import print_function, division
import decimal
import fractions
import math
import re as regex
from collections import defaultdict
from .core import C
from .containers import Tuple
from .sympify import converter, sympify, _sympify, SympifyError
from .singleton import S, Singleton
from .expr import Expr, AtomicExpr
from .decorators import _sympifyit, deprecated
from .cache import cacheit, clear_cache
from sympy.core.compatibility import (
as_int, integer_types, long, string_types, with_metaclass, HAS_GMPY,
SYMPY_INTS)
import sympy.mpmath as mpmath
import sympy.mpmath.libmp as mlib
from sympy.mpmath.libmp import mpf_pow, mpf_pi, mpf_e, phi_fixed
from sympy.mpmath.ctx_mp import mpnumeric
from sympy.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)
rnd = mlib.round_nearest
_LOG2 = math.log(2)
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 3540)
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}
[docs]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 _decimal_to_Rational_prec(dec):
"""Convert an ordinary decimal instance to a Rational."""
if not dec.is_finite(): # NOTE: this is_finite is not SymPy's
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)
_gcdcache = {}
# TODO caching with decorator, but not to degrade performance
[docs]def igcd(*args):
"""Computes positive 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
"""
a = args[0]
for b in args[1:]:
try:
a = _gcdcache[(a, b)]
except KeyError:
a, b = as_int(a), as_int(b)
if a and b:
if b < 0:
b = -b
while b:
a, b = b, a % b
else:
a = abs(a or b)
_gcdcache[(a, b)] = a
if a == 1 or b == 1:
return 1
return a
[docs]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 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).
>>> 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)
[docs]class Number(AtomicExpr):
"""
Represents any kind of number in sympy.
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
__slots__ = []
# Used to make max(x._prec, y._prec) return x._prec when only x is a float
_prec = -1
is_Number = True
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)
if isinstance(obj, Number):
return obj
msg = "expected str|int|long|float|Decimal|Number object but got %r"
raise TypeError(msg % type(obj).__name__)
def __divmod__(self, other):
from .containers import Tuple
from sympy.functions.elementary.complexes import sign
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 = sign(rat)*int(abs(rat)) # = rat.floor()
r = self - other*w
#w*other + r == self
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 _eval_conjugate(self):
return self
def _eval_order(self, *symbols):
# Order(5, x, y) -> Order(1,x,y)
return C.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_bounded(self):
return True
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):
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):
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):
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):
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):
raise NotImplementedError('%s needs .__lt__() method' %
(self.__class__.__name__))
def __le__(self, other):
raise NotImplementedError('%s needs .__le__() method' %
(self.__class__.__name__))
def __gt__(self, other):
return _sympify(other).__lt__(self)
def __ge__(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):
# a -> c*t
if self.is_Rational:
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,)
[docs] 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
[docs] def as_coeff_Add(self):
"""Efficiently extract the coefficient of a summation. """
return self, S.Zero
[docs] def gcd(self, other):
"""Compute GCD of `self` and `other`. """
from sympy.polys import gcd
return gcd(self, other)
[docs] def lcm(self, other):
"""Compute LCM of `self` and `other`. """
from sympy.polys import lcm
return lcm(self, other)
[docs] def cofactors(self, other):
"""Compute GCD and cofactors of `self` and `other`. """
from sympy.polys import cofactors
return cofactors(self, other)
[docs]class Float(Number):
"""
Represents a floating point number. It is capable of representing
arbitrary-precision floating-point numbers.
Examples
========
>>> from sympy import Float
>>> Float(3.5)
3.50000000000000
>>> Float(3)
3.00000000000000
Floats can be created from a string representations of Python floats
to force ints to Float or to enter high-precision (> 15 significant
digits) values:
>>> Float('.0010')
0.00100000000000000
>>> Float('1e-3')
0.00100000000000000
>>> Float('1e-3', 3)
0.00100
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']
is_rational = True
is_real = True
is_Float = True
def __new__(cls, num, prec=15):
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 isinstance(num, mpmath.mpf):
num = num._mpf_
if prec == '':
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('-')))
ok = True
if ok is None:
raise ValueError('string-float not recognized: %s' % num)
else:
dps = prec
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):
_mpf_ = mlib.from_str(str(num), prec, rnd)
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] = long(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 S.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 S.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):
return C.Integer(int(mlib.to_int(
mlib.mpf_floor(self._mpf_, self._prec))))
def ceiling(self):
return C.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)
# uncomment to see failures
#if rv != self._mpf_ and self._prec == prec:
# print self._mpf_, rv
return rv
def _as_mpf_op(self, prec):
return self._mpf_, max(prec, self._prec)
def _eval_is_bounded(self):
if self._mpf_ in (_mpf_inf, _mpf_ninf):
return False
return True
def _eval_is_finite(self):
if self._mpf_ in (_mpf_inf, _mpf_ninf, _mpf_zero):
return False
return True
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):
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 __div__(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.__div__(self, other)
__truediv__ = __div__
@_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)
@_sympifyit('other', NotImplemented)
def __rmod__(self, other):
if isinstance(other, Float):
return other.__mod__(self)
if isinstance(other, Number):
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)
expt, prec = expt._as_mpf_op(self._prec)
self = self._mpf_
try:
y = mpf_pow(self, expt, prec, rnd)
return Float._new(y, prec)
except mlib.ComplexResult:
re, im = mlib.mpc_pow(
(self, _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 False # sympy != 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
def __ne__(self, other):
return not self.__eq__(other)
def __gt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other
if isinstance(other, NumberSymbol):
return other.__le__(self)
if other.is_comparable:
other = other.evalf()
if isinstance(other, Number):
return 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:
return False # sympy > other --> ! <=
if isinstance(other, NumberSymbol):
return other.__lt__(self)
if other.is_comparable:
other = other.evalf()
if isinstance(other, Number):
return 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:
return False # sympy > other
if isinstance(other, NumberSymbol):
return other.__ge__(self)
if other.is_real and other.is_number:
other = other.evalf()
if isinstance(other, Number):
return 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:
return False # sympy > other --> ! <=
if isinstance(other, NumberSymbol):
return other.__gt__(self)
if other.is_real and other.is_number:
other = other.evalf()
if isinstance(other, Number):
return 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))
# Add sympify converters
converter[float] = converter[decimal.Decimal] = Float
# this is here to work nicely in Sage
RealNumber = Float
[docs]class Rational(Number):
"""Represents integers and rational numbers (p/q) of any size.
Examples
========
>>> from sympy import Rational, nsimplify, S, pi
>>> 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
>>> 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
__slots__ = ['p', 'q']
is_Rational = True
@cacheit
def __new__(cls, p, q=None):
if q is None:
if isinstance(p, Rational):
return p
if isinstance(p, string_types):
p = p.replace(' ', '')
try:
# we might have a Float
neg_pow, digits, expt = decimal.Decimal(p).as_tuple()
p = [1, -1][neg_pow]*int("".join(str(x) for x in digits))
if expt > 0:
# TODO: this branch needs a test
return Rational(p*Pow(10, expt), 1)
return Rational(p, Pow(10, -expt))
except decimal.InvalidOperation:
f = regex.match('^([-+]?[0-9]+)/([0-9]+)$', p)
if f:
n, d = f.groups()
return Rational(int(n), int(d))
elif p.count('/') == 1:
p, q = p.split('/')
return Rational(Rational(p), Rational(q))
else:
pass # error will raise below
else:
try:
if isinstance(p, fractions.Fraction):
return Rational(p.numerator, p.denominator)
except NameError:
pass # error will raise below
if isinstance(p, (float, Float)):
return Rational(*float(p).as_integer_ratio())
if not isinstance(p, SYMPY_INTS + (Rational,)):
raise TypeError('invalid input: %s' % p)
q = S.One
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
if p < 0:
return S.NegativeInfinity
return S.Infinity
if q < 0:
q = -q
p = -p
n = igcd(abs(p), q)
if n > 1:
p //= n
q //= n
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
#obj._args = (p, q)
return obj
[docs] 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
"""
# Algorithm notes: For any real number x, define a *best upper
# approximation* to x to be a rational number p/q such that:
#
# (1) p/q >= x, and
# (2) if p/q > r/s >= x then s > q, for any rational r/s.
#
# Define *best lower approximation* similarly. Then it can be
# proved that a rational number is a best upper or lower
# approximation to x if, and only if, it is a convergent or
# semiconvergent of the (unique shortest) continued fraction
# associated to x.
#
# To find a best rational approximation with denominator <= M,
# we find the best upper and lower approximations with
# denominator <= M and take whichever of these is closer to x.
# In the event of a tie, the bound with smaller denominator is
# chosen. If both denominators are equal (which can happen
# only when max_denominator == 1 and self is midway between
# two integers) the lower bound---i.e., the floor of self, is
# taken.
if max_denominator < 1:
raise ValueError("max_denominator should be at least 1")
if self.q <= max_denominator:
return self
p0, q0, p1, q1 = 0, 1, 1, 0
n, d = self.p, self.q
while True:
a = n//d
q2 = q0 + a*q1
if q2 > max_denominator:
break
p0, q0, p1, q1 = p1, q1, p0 + a*p1, q2
n, d = d, n - a*d
k = (max_denominator - q0)//q1
bound1 = Rational(p0 + k*p1, q0 + k*q1)
bound2 = Rational(p1, q1)
if abs(bound2 - self) <= abs(bound1 - self):
return bound2
else:
return bound1
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):
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):
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):
return other*self
else:
return Number.__mul__(self, other)
@_sympifyit('other', NotImplemented)
def __div__(self, other):
if isinstance(other, Rational):
if self.p and other.p == S.Zero:
return S.ComplexInfinity
else:
return Rational(self.p*other.q, self.q*other.p)
elif isinstance(other, Float):
return self*(1/other)
else:
return Number.__div__(self, other)
__truediv__ = __div__
@_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)
if isinstance(other, Float):
# calculate mod with Rationals, *then* round the answer
return Float(self.__mod__(Rational(other)),
prec_to_dps(other._prec))
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:
if expt.q != 1:
return -(S.NegativeOne)**((expt.p % expt.q) /
S(expt.q))*Rational(self.q, -self.p)**ne
else:
return S.NegativeOne**ne*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)
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 -(-p//q)
return p//q
__long__ = __int__
def __eq__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy != other --> not ==
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
def __ne__(self, other):
return not self.__eq__(other)
def __gt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <
if isinstance(other, NumberSymbol):
return other.__le__(self)
if other.is_real and other.is_number and not isinstance(other, Rational):
other = other.evalf()
if isinstance(other, Number):
if isinstance(other, Rational):
return bool(self.p*other.q > self.q*other.p)
if isinstance(other, Float):
return bool(mlib.mpf_gt(
self._as_mpf_val(other._prec), other._mpf_))
if other is S.NaN:
return other.__le__(self)
return Expr.__gt__(self, other)
def __ge__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <=
if isinstance(other, NumberSymbol):
return other.__lt__(self)
if other.is_real and other.is_number and not isinstance(other, Rational):
other = other.evalf()
if isinstance(other, Number):
if isinstance(other, Rational):
return bool(self.p*other.q >= self.q*other.p)
if isinstance(other, Float):
return bool(mlib.mpf_ge(
self._as_mpf_val(other._prec), other._mpf_))
if other is S.NaN:
return other.__lt__(self)
return Expr.__ge__(self, other)
def __lt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <
if isinstance(other, NumberSymbol):
return other.__ge__(self)
if other.is_real and other.is_number and not isinstance(other, Rational):
other = other.evalf()
if isinstance(other, Number):
if isinstance(other, Rational):
return bool(self.p*other.q < self.q*other.p)
if isinstance(other, Float):
return bool(mlib.mpf_lt(
self._as_mpf_val(other._prec), other._mpf_))
if other is S.NaN:
return other.__ge__(self)
return Expr.__lt__(self, other)
def __le__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <=
if isinstance(other, NumberSymbol):
return other.__gt__(self)
if other.is_real and other.is_number and not isinstance(other, Rational):
other = other.evalf()
if isinstance(other, Number):
if isinstance(other, Rational):
return bool(self.p*other.q <= self.q*other.p)
if isinstance(other, Float):
return bool(mlib.mpf_le(
self._as_mpf_val(other._prec), other._mpf_))
if other is S.NaN:
return other.__gt__(self)
return Expr.__le__(self, other)
def __hash__(self):
return super(Rational, self).__hash__()
[docs] 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 factorint
f = factorint(self.p, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose).copy()
f = defaultdict(int, f)
for p, e in factorint(self.q, limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose).items():
f[p] += -e
if len(f) > 1 and 1 in f:
del f[1]
if not f:
f = {1: 1}
if not visual:
return dict(f)
else:
if -1 in f:
f.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend([Pow(*i, evaluate=False)
for i in sorted(f.items())])
return Mul(*args, evaluate=False)
@_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*other.p//igcd(self.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)
[docs] def as_content_primitive(self, radical=False):
"""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
# int -> Integer
_intcache = {}
# TODO move this tracing facility to sympy/core/trace.py ?
def _intcache_printinfo():
ints = sorted(_intcache.keys())
nhit = _intcache_hits
nmiss = _intcache_misses
if nhit == 0 and nmiss == 0:
print()
print('Integer cache statistic was not collected')
return
miss_ratio = float(nmiss) / (nhit + nmiss)
print()
print('Integer cache statistic')
print('-----------------------')
print()
print('#items: %i' % len(ints))
print()
print(' #hit #miss #total')
print()
print('%5i %5i (%7.5f %%) %5i' % (
nhit, nmiss, miss_ratio*100, nhit + nmiss)
)
print()
print(ints)
_intcache_hits = 0
_intcache_misses = 0
def int_trace(f):
import os
if os.getenv('SYMPY_TRACE_INT', 'no').lower() != 'yes':
return f
def Integer_tracer(cls, i):
global _intcache_hits, _intcache_misses
try:
_intcache_hits += 1
return _intcache[i]
except KeyError:
_intcache_hits -= 1
_intcache_misses += 1
return f(cls, i)
# also we want to hook our _intcache_printinfo into sys.atexit
import atexit
atexit.register(_intcache_printinfo)
return Integer_tracer
[docs]class Integer(Rational):
q = 1
is_integer = True
is_Integer = True
__slots__ = ['p']
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))
# TODO caching with decorator, but not to degrade performance
@int_trace
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(
'Integer can only work with integer expressions.')
try:
return _intcache[ival]
except KeyError:
# We only work with well-behaved integer types. This converts, for
# example, numpy.int32 instances.
obj = Expr.__new__(cls)
obj.p = ival
_intcache[ival] = obj
return obj
def __getnewargs__(self):
return (self.p,)
# Arithmetic operations are here for efficiency
def __int__(self):
return self.p
__long__ = __int__
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, integer_types):
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 isinstance(other, integer_types):
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, integer_types):
return Integer(other + self.p)
return Rational.__add__(self, other)
def __sub__(self, other):
if isinstance(other, integer_types):
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, integer_types):
return Integer(other - self.p)
return Rational.__rsub__(self, other)
def __mul__(self, other):
if isinstance(other, integer_types):
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, integer_types):
return Integer(other*self.p)
return Rational.__mul__(self, other)
def __mod__(self, other):
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)
def __rmod__(self, other):
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)
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.__eq__(other)
def __gt__(self, other):
if isinstance(other, integer_types):
return (self.p > other)
elif isinstance(other, Integer):
return (self.p > other.p)
return Rational.__gt__(self, other)
def __lt__(self, other):
if isinstance(other, integer_types):
return (self.p < other)
elif isinstance(other, Integer):
return (self.p < other.p)
return Rational.__lt__(self, other)
def __ge__(self, other):
if isinstance(other, integer_types):
return (self.p >= other)
elif isinstance(other, Integer):
return (self.p >= other.p)
return Rational.__ge__(self, other)
def __le__(self, other):
if isinstance(other, integer_types):
return (self.p <= other)
elif isinstance(other, Integer):
return (self.p <= other.p)
return Rational.__le__(self, other)
def __hash__(self):
return super(Integer, self).__hash__()
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 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:
if expt.q != 1:
return -(S.NegativeOne)**((expt.p % expt.q) /
S(expt.q))*Rational(1, -self)**ne
else:
return (S.NegativeOne)**ne*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 sympy.ntheory import isprime
return isprime(self)
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 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)
[docs]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] http://en.wikipedia.org/wiki/Zero
"""
p = 0
q = 1
is_positive = False
is_negative = False
is_finite = False
is_zero = True
is_composite = False
__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__
[docs]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] http://en.wikipedia.org/wiki/1_%28number%29
"""
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
return {1: 1}
[docs]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] http://en.wikipedia.org/wiki/%E2%88%921_%28number%29
"""
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
[docs]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] http://en.wikipedia.org/wiki/One_half
"""
p = 1
q = 2
__slots__ = []
@staticmethod
def __abs__():
return S.Half
[docs]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] http://en.wikipedia.org/wiki/Infinity
"""
is_commutative = True
is_positive = True
is_bounded = False
is_finite = False
is_infinitesimal = False
is_integer = None
is_rational = None
is_odd = None
__slots__ = []
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 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
"""
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_number:
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
@_sympifyit('other', NotImplemented)
def __lt__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return False
@_sympifyit('other', NotImplemented)
def __le__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return other is S.Infinity
@_sympifyit('other', NotImplemented)
def __gt__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return other is not S.Infinity
@_sympifyit('other', NotImplemented)
def __ge__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return True
def __mod__(self, other):
return S.NaN
__rmod__ = __mod__
oo = S.Infinity
[docs]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_real = True
is_positive = False
is_bounded = False
is_finite = False
is_infinitesimal = False
is_integer = None
is_rational = None
__slots__ = []
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 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 isinstance(expt, 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
@_sympifyit('other', NotImplemented)
def __lt__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return other is not S.NegativeInfinity
@_sympifyit('other', NotImplemented)
def __le__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return True
@_sympifyit('other', NotImplemented)
def __gt__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return False
@_sympifyit('other', NotImplemented)
def __ge__(self, other):
if other.is_number and other.is_real is False:
raise TypeError("Invalid comparison of %s and %s" % (self, other))
return other is S.NegativeInfinity
[docs]class NaN(with_metaclass(Singleton, Number)):
"""
Not a Number.
This represents the corresponding data type to floating point nan, which
is defined in the IEEE 754 floating point standard, and corresponds to the
Python ``float('nan')``.
NaN 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 a singleton, and can be accessed by ``S.NaN``, or can be imported
as ``nan``.
Examples
========
>>> from sympy import nan, S, oo
>>> nan is S.NaN
True
>>> oo - oo
nan
>>> nan + 1
nan
References
==========
.. [1] http://en.wikipedia.org/wiki/NaN
"""
is_commutative = True
is_real = None
is_rational = None
is_integer = None
is_comparable = False
is_finite = None
is_bounded = None
is_zero = None
is_prime = None
is_positive = None
is_negative = None
__slots__ = []
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 __div__(self, other):
return self
__truediv__ = __div__
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):
return other is S.NaN
def __ne__(self, other):
return other is not S.NaN
def __gt__(self, other):
return False
def __ge__(self, other):
return False
def __lt__(self, other):
return False
def __le__(self, other):
return False
nan = S.NaN
[docs]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_bounded = False
is_real = None
is_number = True
__slots__ = []
def __new__(cls):
return AtomicExpr.__new__(cls)
def _latex(self, printer):
return r"\tilde{\infty}"
@staticmethod
def __abs__():
return S.Infinity
@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
zoo = S.ComplexInfinity
[docs]class NumberSymbol(AtomicExpr):
is_commutative = True
is_bounded = True
is_finite = True
is_number = True
__slots__ = []
is_NumberSymbol = True
def __new__(cls):
return AtomicExpr.__new__(cls)
[docs] 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 False # sympy != other --> not ==
if self is other:
return True
if isinstance(other, Number) and self.is_irrational:
return False
return False # NumberSymbol != non-(Number|self)
def __ne__(self, other):
return not self.__eq__(other)
def __lt__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <
if self is other:
return False
if isinstance(other, Number):
approx = self.approximation_interval(other.__class__)
if approx is not None:
l, u = approx
if other < l:
return False
if other > u:
return True
return self.evalf() < other
if other.is_real and other.is_number:
other = other.evalf()
return self.evalf() < other
return Expr.__lt__(self, other)
def __le__(self, other):
try:
other = _sympify(other)
except SympifyError:
return False # sympy > other --> not <=
if self is other:
return True
if other.is_real and other.is_number:
other = other.evalf()
if isinstance(other, Number):
return self.evalf() <= other
return Expr.__le__(self, other)
def __gt__(self, other):
return (-self) < (-other)
def __ge__(self, other):
return (-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__()
[docs]class Exp1(with_metaclass(Singleton, NumberSymbol)):
r"""The `e` constant.
The transcendental number `e = 2.718281828\dots` 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] http://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
__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):
return C.exp(expt)
def _eval_rewrite_as_sin(self):
I = S.ImaginaryUnit
return C.sin(I + S.Pi/2) - I*C.sin(I)
def _eval_rewrite_as_cos(self):
I = S.ImaginaryUnit
return C.cos(I) + I*C.cos(I + S.Pi/2)
def _sage_(self):
import sage.all as sage
return sage.e
E = S.Exp1
[docs]class Pi(with_metaclass(Singleton, NumberSymbol)):
r"""The `\pi` constant.
The transcendental number `\pi = 3.141592654\dots` 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] http://en.wikipedia.org/wiki/Pi
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = 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
[docs]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] http://en.wikipedia.org/wiki/Golden_ratio
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = True
__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
[docs]class EulerGamma(with_metaclass(Singleton, NumberSymbol)):
r"""The Euler-Mascheroni constant.
`\gamma = 0.5772157\dots` (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] http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = None
__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
[docs]class Catalan(with_metaclass(Singleton, NumberSymbol)):
r"""Catalan's constant.
`K = 0.91596559\dots` 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] http://en.wikipedia.org/wiki/Catalan%27s_constant
"""
is_real = True
is_positive = True
is_negative = False
is_irrational = None
__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
[docs]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] http://en.wikipedia.org/wiki/Imaginary_unit
"""
is_commutative = True
is_imaginary = True
is_bounded = True
is_finite = True
is_number = True
__slots__ = []
def _latex(self, printer):
return r"i"
@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 (S.NegativeOne)**(expt*S.Half)
return
def as_base_exp(self):
return S.NegativeOne, S.Half
def _sage_(self):
import sage.all as sage
return sage.I
I = S.ImaginaryUnit
def sympify_fractions(f):
return Rational(f.numerator, f.denominator)
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_complex(a):
real, imag = list(map(sympify, (a.real, a.imag)))
return real + S.ImaginaryUnit*imag
converter[complex] = sympify_complex
_intcache[0] = S.Zero
_intcache[1] = S.One
_intcache[-1] = S.NegativeOne
from .power import Pow, integer_nthroot
from .mul import Mul
Mul.identity = One()
from .add import Add
Add.identity = Zero()
