Source code for sympy.core.mul

from __future__ import print_function, division

from collections import defaultdict
import operator

from .sympify import sympify
from .basic import Basic, C
from .singleton import S
from .operations import AssocOp
from .cache import cacheit
from .logic import fuzzy_not, _fuzzy_group
from .compatibility import cmp_to_key, reduce, xrange
from .expr import Expr

# internal marker to indicate:
#   "there are still non-commutative objects -- don't forget to process them"

class NC_Marker:
is_Order = False
is_Mul = False
is_Number = False
is_Poly = False

is_commutative = False

# Key for sorting commutative args in canonical order
_args_sortkey = cmp_to_key(Basic.compare)
def _mulsort(args):
# in-place sorting of args
args.sort(key=_args_sortkey)

def _unevaluated_Mul(*args):
"""Return a well-formed unevaluated Mul: Numbers are collected and
put in slot 0, any arguments that are Muls will be flattened, and args
are sorted. Use this when args have changed but you still want to return
an unevaluated Mul.

Examples
========

>>> from sympy.core.mul import _unevaluated_Mul as uMul
>>> from sympy import S, sqrt, Mul
>>> from sympy.abc import x
>>> a = uMul(*[S(3.0), x, S(2)])
>>> a.args[0]
6.00000000000000
>>> a.args[1]
x

Two unevaluated Muls with the same arguments will
always compare as equal during testing:

>>> m = uMul(sqrt(2), sqrt(3))
>>> m == uMul(sqrt(3), sqrt(2))
True
>>> u = Mul(sqrt(3), sqrt(2), evaluate=False)
>>> m == uMul(u)
True
>>> m == Mul(*m.args)
False

"""
args = list(args)
newargs = []
ncargs = []
co = S.One
while args:
a = args.pop()
if a.is_Mul:
c, nc = a.args_cnc()
args.extend(c)
if nc:
ncargs.append(Mul._from_args(nc))
elif a.is_Number:
co *= a
else:
newargs.append(a)
_mulsort(newargs)
if co is not S.One:
newargs.insert(0, co)
if ncargs:
newargs.append(Mul._from_args(ncargs))
return Mul._from_args(newargs)

[docs]class Mul(Expr, AssocOp):

__slots__ = []

is_Mul = True

#identity = S.One
# cyclic import, so defined in numbers.py

@classmethod
[docs]    def flatten(cls, seq):
"""Return commutative, noncommutative and order arguments by
combining related terms.

Notes
=====
* In an expression like a*b*c, python process this through sympy
as Mul(Mul(a, b), c). This can have undesirable consequences.

-  Sometimes terms are not combined as one would like:
{c.f. https://github.com/sympy/sympy/issues/4596}

>>> from sympy import Mul, sqrt
>>> from sympy.abc import x, y, z
>>> 2*(x + 1) # this is the 2-arg Mul behavior
2*x + 2
>>> y*(x + 1)*2
2*y*(x + 1)
>>> 2*(x + 1)*y # 2-arg result will be obtained first
y*(2*x + 2)
>>> Mul(2, x + 1, y) # all 3 args simultaneously processed
2*y*(x + 1)
>>> 2*((x + 1)*y) # parentheses can control this behavior
2*y*(x + 1)

Powers with compound bases may not find a single base to
combine with unless all arguments are processed at once.
Post-processing may be necessary in such cases.
{c.f. https://github.com/sympy/sympy/issues/5728}

>>> a = sqrt(x*sqrt(y))
>>> a**3
(x*sqrt(y))**(3/2)
>>> Mul(a,a,a)
(x*sqrt(y))**(3/2)
>>> a*a*a
x*sqrt(y)*sqrt(x*sqrt(y))
>>> _.subs(a.base, z).subs(z, a.base)
(x*sqrt(y))**(3/2)

-  If more than two terms are being multiplied then all the
previous terms will be re-processed for each new argument.
So if each of a, b and c were :class:Mul
expression, then a*b*c (or building up the product
with *=) will process all the arguments of a and
b twice: once when a*b is computed and again when
c is multiplied.

Using Mul(a, b, c) will process all arguments once.

* The results of Mul are cached according to arguments, so flatten
will only be called once for Mul(a, b, c). If you can
structure a calculation so the arguments are most likely to be
repeats then this can save time in computing the answer. For
example, say you had a Mul, M, that you wished to divide by d[i]
and multiply by n[i] and you suspect there are many repeats
in n. It would be better to compute M*n[i]/d[i] rather
than M/d[i]*n[i] since every time n[i] is a repeat, the
product, M*n[i] will be returned without flattening -- the
cached value will be returned. If you divide by the d[i]
first (and those are more unique than the n[i]) then that will
create a new Mul, M/d[i] the args of which will be traversed
again when it is multiplied by n[i].

{c.f. https://github.com/sympy/sympy/issues/5706}

This consideration is moot if the cache is turned off.

NB
--
The validity of the above notes depends on the implementation
details of Mul and flatten which may change at any time. Therefore,
you should only consider them when your code is highly performance
sensitive.

Removal of 1 from the sequence is already handled by AssocOp.__new__.
"""

rv = None
if len(seq) == 2:
a, b = seq
if b.is_Rational:
a, b = b, a
assert not a is S.One
if not a.is_zero and a.is_Rational:
r, b = b.as_coeff_Mul()
if r is not S.One:  # 2-arg hack
# leave the Mul as a Mul
rv = [cls(a*r, b, evaluate=False)], [], None
elif b.is_commutative:
if a is S.One:
rv = [b], [], None
else:
r, b = b.as_coeff_Add()
bargs = [_keep_coeff(a, bi) for bi in Add.make_args(b)]
ar = a*r
if ar:
bargs.insert(0, ar)
rv = bargs, [], None
if rv:
return rv

# apply associativity, separate commutative part of seq
c_part = []         # out: commutative factors
nc_part = []        # out: non-commutative factors

nc_seq = []

coeff = S.One       # standalone term
# e.g. 3 * ...

c_powers = []       # (base,exp)      n
# e.g. (x,n) for x

num_exp = []        # (num-base, exp)           y
# e.g.  (3, y)  for  ... * 3  * ...

neg1e = S.Zero      # exponent on -1 extracted from Number-based Pow and I

pnum_rat = {}       # (num-base, Rat-exp)          1/2
# e.g.  (3, 1/2)  for  ... * 3     * ...

order_symbols = None

# --- PART 1 ---
#
# "collect powers and coeff":
#
# o coeff
# o c_powers
# o num_exp
# o neg1e
# o pnum_rat
#
# NOTE: this is optimized for all-objects-are-commutative case
for o in seq:
# O(x)
if o.is_Order:
o, order_symbols = o.as_expr_variables(order_symbols)

# Mul([...])
if o.is_Mul:
if o.is_commutative:
seq.extend(o.args)    # XXX zerocopy?

else:
# NCMul can have commutative parts as well
for q in o.args:
if q.is_commutative:
seq.append(q)
else:
nc_seq.append(q)

# append non-commutative marker, so we don't forget to
# process scheduled non-commutative objects
seq.append(NC_Marker)

continue

# 3
elif o.is_Number:
if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero:
# we know for sure the result will be nan
return [S.NaN], [], None
elif coeff.is_Number:  # it could be zoo
coeff *= o
if coeff is S.NaN:
# we know for sure the result will be nan
return [S.NaN], [], None
continue

elif o is S.ComplexInfinity:
if not coeff:
# 0 * zoo = NaN
return [S.NaN], [], None
if coeff is S.ComplexInfinity:
# zoo * zoo = zoo
return [S.ComplexInfinity], [], None
coeff = S.ComplexInfinity
continue

elif o is S.ImaginaryUnit:
neg1e += S.Half
continue

elif o.is_commutative:
#      e
# o = b
b, e = o.as_base_exp()

#  y
# 3
if o.is_Pow:
if b.is_Number:

# get all the factors with numeric base so they can be
# combined below, but don't combine negatives unless
# the exponent is an integer
if e.is_Rational:
if e.is_Integer:
coeff *= Pow(b, e)  # it is an unevaluated power
continue
elif e.is_negative:    # also a sign of an unevaluated power
seq.append(Pow(b, e))
continue
elif b.is_negative:
neg1e += e
b = -b
if b is not S.One:
pnum_rat.setdefault(b, []).append(e)
continue
elif b.is_positive or e.is_integer:
num_exp.append((b, e))
continue

elif b is S.ImaginaryUnit and e.is_Rational:
neg1e += e/2
continue

c_powers.append((b, e))

# NON-COMMUTATIVE
# TODO: Make non-commutative exponents not combine automatically
else:
if o is not NC_Marker:
nc_seq.append(o)

# process nc_seq (if any)
while nc_seq:
o = nc_seq.pop(0)
if not nc_part:
nc_part.append(o)
continue

#                             b    c       b+c
# try to combine last terms: a  * a   ->  a
o1 = nc_part.pop()
b1, e1 = o1.as_base_exp()
b2, e2 = o.as_base_exp()
new_exp = e1 + e2
# Only allow powers to combine if the new exponent is
# not an Add. This allow things like a**2*b**3 == a**5
# if a.is_commutative == False, but prohibits
# a**x*a**y and x**a*x**b from combining (x,y commute).
if b1 == b2 and (not new_exp.is_Add):
o12 = b1 ** new_exp

# now o12 could be a commutative object
if o12.is_commutative:
seq.append(o12)
continue
else:
nc_seq.insert(0, o12)

else:
nc_part.append(o1)
nc_part.append(o)

# We do want a combined exponent if it would not be an Add, such as
#  y    2y     3y
# x  * x   -> x
# We determine if two exponents have the same term by using
# as_coeff_Mul.
#
# Unfortunately, this isn't smart enough to consider combining into
# exponents that might already be adds, so things like:
#  z - y    y
# x      * x  will be left alone.  This is because checking every possible
# combination can slow things down.

# gather exponents of common bases...
def _gather(c_powers):
new_c_powers = []
common_b = {}  # b:e
for b, e in c_powers:
co = e.as_coeff_Mul()
common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
for b, e in common_b.items():
for t, c in e.items():
new_c_powers.append((b, c*t))
return new_c_powers

# in c_powers
c_powers = _gather(c_powers)

# and in num_exp
num_exp = _gather(num_exp)

# --- PART 2 ---
#
# o process collected powers  (x**0 -> 1; x**1 -> x; otherwise Pow)
# o combine collected powers  (2**x * 3**x -> 6**x)
#   with numeric base

# ................................
# now we have:
# - coeff:
# - c_powers:    (b, e)
# - num_exp:     (2, e)
# - pnum_rat:    {(1/3, [1/3, 2/3, 1/4])}

#  0             1
# x  -> 1       x  -> x
for b, e in c_powers:
if e is S.One:
if b.is_Number:
coeff *= b
else:
c_part.append(b)
elif e is not S.Zero:
c_part.append(Pow(b, e))

#  x    x     x
# 2  * 3  -> 6
inv_exp_dict = {}   # exp:Mul(num-bases)     x    x
# e.g.  x:6  for  ... * 2  * 3  * ...
for b, e in num_exp:
inv_exp_dict.setdefault(e, []).append(b)
for e, b in inv_exp_dict.items():
inv_exp_dict[e] = cls(*b)
c_part.extend([Pow(b, e) for e, b in inv_exp_dict.items() if e])

# b, e -> e' = sum(e), b
# {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])}
comb_e = {}
for b, e in pnum_rat.items():
del pnum_rat
# process them, reducing exponents to values less than 1
# and updating coeff if necessary else adding them to
# num_rat for further processing
num_rat = []
for e, b in comb_e.items():
b = cls(*b)
if e.q == 1:
coeff *= Pow(b, e)
continue
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)
coeff *= Pow(b, e_i)
e = Rational(ep, e.q)
num_rat.append((b, e))
del comb_e

# extract gcd of bases in num_rat
# 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4)
pnew = defaultdict(list)
i = 0  # steps through num_rat which may grow
while i < len(num_rat):
bi, ei = num_rat[i]
grow = []
for j in range(i + 1, len(num_rat)):
bj, ej = num_rat[j]
g = bi.gcd(bj)
if g is not S.One:
# 4**r1*6**r2 -> 2**(r1+r2)  *  2**r1 *  3**r2
# this might have a gcd with something else
e = ei + ej
if e.q == 1:
coeff *= Pow(g, e)
else:
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)  # change e in place
coeff *= Pow(g, e_i)
e = Rational(ep, e.q)
grow.append((g, e))
# update the jth item
num_rat[j] = (bj/g, ej)
# update bi that we are checking with
bi = bi/g
if bi is S.One:
break
if bi is not S.One:
obj = Pow(bi, ei)
if obj.is_Number:
coeff *= obj
else:
# changes like sqrt(12) -> 2*sqrt(3)
for obj in Mul.make_args(obj):
if obj.is_Number:
coeff *= obj
else:
assert obj.is_Pow
bi, ei = obj.args
pnew[ei].append(bi)

num_rat.extend(grow)
i += 1

# combine bases of the new powers
for e, b in pnew.items():
pnew[e] = cls(*b)

# handle -1 and I
if neg1e:
# treat I as (-1)**(1/2) and compute -1's total exponent
p, q =  neg1e.as_numer_denom()
# if the integer part is odd, extract -1
n, p = divmod(p, q)
if n % 2:
coeff = -coeff
# if it's a multiple of 1/2 extract I
if q == 2:
c_part.append(S.ImaginaryUnit)
elif p:
# see if there is any positive base this power of
# -1 can join
neg1e = Rational(p, q)
for e, b in pnew.items():
if e == neg1e and b.is_positive:
pnew[e] = -b
break
else:
# keep it separate; we've already evaluated it as
# much as possible so evaluate=False
c_part.append(Pow(S.NegativeOne, neg1e, evaluate=False))

# add all the pnew powers
c_part.extend([Pow(b, e) for e, b in pnew.items()])

# oo, -oo
if (coeff is S.Infinity) or (coeff is S.NegativeInfinity):
def _handle_for_oo(c_part, coeff_sign):
new_c_part = []
for t in c_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_c_part.append(t)
return new_c_part, coeff_sign
c_part, coeff_sign = _handle_for_oo(c_part, 1)
nc_part, coeff_sign = _handle_for_oo(nc_part, coeff_sign)
coeff *= coeff_sign

# zoo
if coeff is S.ComplexInfinity:
# zoo might be
#   infinite_real + bounded_im
#   bounded_real + infinite_im
#   infinite_real + infinite_im
# and non-zero real or imaginary will not change that status.
c_part = [c for c in c_part if not (c.is_nonzero and
c.is_real is not None)]
nc_part = [c for c in nc_part if not (c.is_nonzero and
c.is_real is not None)]

# 0
elif coeff is S.Zero:
# we know for sure the result will be 0
return [coeff], [], order_symbols

# check for straggling Numbers that were produced
_new = []
for i in c_part:
if i.is_Number:
coeff *= i
else:
_new.append(i)
c_part = _new

# order commutative part canonically
_mulsort(c_part)

# current code expects coeff to be always in slot-0
if coeff is not S.One:
c_part.insert(0, coeff)

# we are done
if (not nc_part and len(c_part) == 2 and c_part[0].is_Number and
# 2*(1+a) -> 2 + 2 * a
coeff = c_part[0]
c_part = [Add(*[coeff*f for f in c_part[1].args])]

return c_part, nc_part, order_symbols

def _eval_power(b, e):

# don't break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B
cargs, nc = b.args_cnc(split_1=False)

if e.is_Integer:
return Mul(*[Pow(b, e, evaluate=False) for b in cargs]) * \
Pow(Mul._from_args(nc), e, evaluate=False)

p = Pow(b, e, evaluate=False)

if e.is_Rational or e.is_Float:
return p._eval_expand_power_base()

return p

@classmethod
def class_key(cls):
return 3, 0, cls.__name__

def _eval_evalf(self, prec):
c, m = self.as_coeff_Mul()
if c is S.NegativeOne:
if m.is_Mul:
rv = -AssocOp._eval_evalf(m, prec)
else:
mnew = m._eval_evalf(prec)
if mnew is not None:
m = mnew
rv = -m
else:
rv = AssocOp._eval_evalf(self, prec)
if rv.is_number:
return rv.expand()
return rv

@cacheit
[docs]    def as_two_terms(self):
"""Return head and tail of self.

This is the most efficient way to get the head and tail of an
expression.

- if you want only the head, use self.args[0];
- if you want to process the arguments of the tail then use
self.as_coef_mul() which gives the head and a tuple containing
the arguments of the tail when treated as a Mul.
- if you want the coefficient when self is treated as an Add

>>> from sympy.abc import x, y
>>> (3*x*y).as_two_terms()
(3, x*y)
"""
args = self.args

if len(args) == 1:
return S.One, self
elif len(args) == 2:
return args

else:
return args[0], self._new_rawargs(*args[1:])

@cacheit
def as_coeff_mul(self, *deps):
if deps:
l1 = []
l2 = []
for f in self.args:
if f.has(*deps):
l2.append(f)
else:
l1.append(f)
return self._new_rawargs(*l1), tuple(l2)
args = self.args
if args[0].is_Rational:
return args[0], args[1:]
elif args[0] is S.NegativeInfinity:
return S.NegativeOne, (-args[0],) + args[1:]
return S.One, args

[docs]    def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
coeff, args = self.args[0], self.args[1:]

if coeff.is_Number and not (rational and not coeff.is_Rational):
if len(args) == 1:
return coeff, args[0]
else:
return coeff, self._new_rawargs(*args)
else:
return S.One, self

def as_real_imag(self, deep=True, **hints):
from sympy import expand_mul
other = []
coeffr = []
coeffi = []
for a in self.args:
if a.is_real:
coeffr.append(a)
elif a.is_imaginary:
coeffi.append(a)
elif a.is_commutative:
# search for complex conjugate pairs:
for i, x in enumerate(other):
if x == a.conjugate():
coeffr.append(C.Abs(x)**2)
del other[i]
break
else:
else:
other.append(a)
else:
other.append(a)
m = self.func(*other)
if hints.get('ignore') == m:
return
if len(coeffi) % 2:
imco = C.im(coeffi.pop(0))
# all other pairs make a real factor; they will be
# put into reco below
else:
imco = S.Zero
reco = self.func(*(coeffr + coeffi))
r, i = (reco*C.re(m), reco*C.im(m))
if addterms == 1:
if m == 1:
if imco is S.Zero:
return (reco, S.Zero)
else:
return (S.Zero, reco*imco)
if imco is S.Zero:
return (r, i)
return (-imco*i, imco*r)
if imco is S.Zero:
else:
r, i = -imco*i, imco*r

@staticmethod
def _expandsums(sums):
"""
Helper function for _eval_expand_mul.

sums must be a list of instances of Basic.
"""

L = len(sums)
if L == 1:
return sums[0].args
terms = []
left = Mul._expandsums(sums[:L//2])
right = Mul._expandsums(sums[L//2:])

terms = [Mul(a, b) for a in left for b in right]
return Add.make_args(added)  # it may have collapsed down to one term

def _eval_expand_mul(self, **hints):
from sympy import fraction

# Handle things like 1/(x*(x + 1)), which are automatically converted
# to 1/x*1/(x + 1)
expr = self
n, d = fraction(expr)
if d.is_Mul:
n, d = [i._eval_expand_mul(**hints) if i.is_Mul else i
for i in (n, d)]
expr = n/d
if not expr.is_Mul:
return expr

plain, sums, rewrite = [], [], False
for factor in expr.args:
sums.append(factor)
rewrite = True
else:
if factor.is_commutative:
plain.append(factor)
else:
sums.append(Basic(factor))  # Wrapper

if not rewrite:
return expr
else:
plain = self.func(*plain)
if sums:
terms = self.func._expandsums(sums)
args = []
for term in terms:
t = self.func(plain, term)
if t.is_Mul and any(a.is_Add for a in t.args):
t = t._eval_expand_mul()
args.append(t)
else:
return plain

@cacheit
def _eval_derivative(self, s):
args = list(self.args)
terms = []
for i in xrange(len(args)):
d = args[i].diff(s)
if d:
terms.append(self.func(*(args[:i] + [d] + args[i + 1:])))

def _matches_simple(self, expr, repl_dict):
# handle (w*3).matches('x*5') -> {w: x*5/3}
coeff, terms = self.as_coeff_Mul()
terms = Mul.make_args(terms)
if len(terms) == 1:
newexpr = self.__class__._combine_inverse(expr, coeff)
return terms[0].matches(newexpr, repl_dict)
return

def matches(self, expr, repl_dict={}, old=False):
expr = sympify(expr)
if self.is_commutative and expr.is_commutative:
return AssocOp._matches_commutative(self, expr, repl_dict, old)
elif self.is_commutative is not expr.is_commutative:
return None
c1, nc1 = self.args_cnc()
c2, nc2 = expr.args_cnc()
repl_dict = repl_dict.copy()
if c1:
if not c2:
c2 = [1]
a = self.func(*c1)
if isinstance(a, AssocOp):
repl_dict = a._matches_commutative(self.func(*c2), repl_dict, old)
else:
repl_dict = a.matches(self.func(*c2), repl_dict)
if repl_dict:
a = self.func(*nc1)
if isinstance(a, self.func):
repl_dict = a._matches(self.func(*nc2), repl_dict)
else:
repl_dict = a.matches(self.func(*nc2), repl_dict)
return repl_dict or None

def _matches(self, expr, repl_dict={}):
# weed out negative one prefixes
sign = 1
a, b = self.as_two_terms()
if a is S.NegativeOne:
if b.is_Mul:
sign = -sign
else:
# the remainder, b, is not a Mul anymore
return b.matches(-expr, repl_dict)
expr = sympify(expr)
if expr.is_Mul and expr.args[0] is S.NegativeOne:
expr = -expr
sign = -sign

if not expr.is_Mul:
# expr can only match if it matches b and a matches +/- 1
if len(self.args) == 2:
# quickly test for equality
if b == expr:
return a.matches(Rational(sign), repl_dict)
# do more expensive match
dd = b.matches(expr, repl_dict)
if dd is None:
return None
dd = a.matches(Rational(sign), dd)
return dd
return None

d = repl_dict.copy()

# weed out identical terms
pp = list(self.args)
ee = list(expr.args)
for p in self.args:
if p in expr.args:
ee.remove(p)
pp.remove(p)

# only one symbol left in pattern -> match the remaining expression
if len(pp) == 1 and isinstance(pp[0], C.Wild):
if len(ee) == 1:
d[pp[0]] = sign * ee[0]
else:
d[pp[0]] = sign * expr.func(*ee)
return d

if len(ee) != len(pp):
return None

for p, e in zip(pp, ee):
d = p.xreplace(d).matches(e, d)
if d is None:
return None
return d

@staticmethod
def _combine_inverse(lhs, rhs):
"""
Returns lhs/rhs, but treats arguments like symbols, so things like
oo/oo return 1, instead of a nan.
"""
if lhs == rhs:
return S.One

def check(l, r):
if l.is_Float and r.is_comparable:
# if both objects are added to 0 they will share the same "normalization"
# and are more likely to compare the same. Since Add(foo, 0) will not allow
# the 0 to pass, we use __add__ directly.
return False
if check(lhs, rhs) or check(rhs, lhs):
return S.One
if lhs.is_Mul and rhs.is_Mul:
a = list(lhs.args)
b = [1]
for x in rhs.args:
if x in a:
a.remove(x)
elif -x in a:
a.remove(-x)
b.append(-1)
else:
b.append(x)
return lhs.func(*a)/rhs.func(*b)
return lhs/rhs

def as_powers_dict(self):
d = defaultdict(int)
for term in self.args:
b, e = term.as_base_exp()
d[b] += e
return d

def as_numer_denom(self):
# don't use _from_args to rebuild the numerators and denominators
# as the order is not guaranteed to be the same once they have
# been separated from each other
numers, denoms = list(zip(*[f.as_numer_denom() for f in self.args]))
return self.func(*numers), self.func(*denoms)

def as_base_exp(self):
e1 = None
bases = []
nc = 0
for m in self.args:
b, e = m.as_base_exp()
if not b.is_commutative:
nc += 1
if e1 is None:
e1 = e
elif e != e1 or nc > 1:
return self, S.One
bases.append(b)
return self.func(*bases), e1

def _eval_is_polynomial(self, syms):
return all(term._eval_is_polynomial(syms) for term in self.args)

def _eval_is_rational_function(self, syms):
return all(term._eval_is_rational_function(syms) for term in self.args)

def _eval_is_algebraic_expr(self, syms):
return all(term._eval_is_algebraic_expr(syms) for term in self.args)

_eval_is_finite = lambda self: _fuzzy_group(
a.is_finite for a in self.args)
_eval_is_commutative = lambda self: _fuzzy_group(
a.is_commutative for a in self.args)
_eval_is_complex = lambda self: _fuzzy_group(
(a.is_complex for a in self.args), quick_exit=True)

def _eval_is_infinite(self):
if any(a.is_infinite for a in self.args):
if any(a.is_zero for a in self.args):
return S.NaN.is_infinite
if any(a.is_zero is None for a in self.args):
return None
return True

def _eval_is_rational(self):
r = _fuzzy_group((a.is_rational for a in self.args), quick_exit=True)
if r:
return r
elif r is False:
return self.is_zero

def _eval_is_algebraic(self):
r = _fuzzy_group((a.is_algebraic for a in self.args), quick_exit=True)
if r:
return r
elif r is False:
return self.is_zero

def _eval_is_zero(self):
zero = infinite = False
for a in self.args:
z = a.is_zero
if z:
if infinite:
return  # 0*oo is nan and nan.is_zero is None
zero = True
else:
if not a.is_finite:
if zero:
return  # 0*oo is nan and nan.is_zero is None
infinite = True
if zero is False and z is None:  # trap None
zero = None
return zero

def _eval_is_integer(self):
is_rational = self.is_rational

if is_rational:
n, d = self.as_numer_denom()
if d is S.One:
return True
elif d is S(2):
return n.is_even
elif is_rational is False:
return False

def _eval_is_polar(self):
has_polar = any(arg.is_polar for arg in self.args)
return has_polar and \
all(arg.is_polar or arg.is_positive for arg in self.args)

def _eval_is_real(self):
real = True
zero = one_neither = False

for t in self.args:
if not t.is_complex:
return t.is_complex
elif t.is_imaginary:
real = not real
elif t.is_real:
if not zero:
z = t.is_zero
if not z and zero is False:
zero = z
elif z:
if all(a.is_finite for a in self.args):
return True
return
elif t.is_real is False:
if one_neither:
return  # complex terms might cancel
one_neither = True
else:
return

if one_neither:  # self is a+I*b or I*b
if real:
return zero  # real*self is like self: neither is real
elif zero is False:
return real  # can't be trumped by 0
elif real:
return real  # doesn't matter what zero is

def _eval_is_imaginary(self):
z = self.is_zero
if z:
return False
elif z is False:
return (S.ImaginaryUnit*self).is_real

def _eval_is_hermitian(self):
real = True
one_nc = zero = one_neither = False

for t in self.args:
if not t.is_commutative:
if one_nc:
return
one_nc = True

if t.is_antihermitian:
real = not real
elif t.is_hermitian:
if zero is False:
zero = fuzzy_not(t.is_nonzero)
if zero:
return True
elif t.is_hermitian is False:
if one_neither:
return
one_neither = True
else:
return

if one_neither:
if real:
return zero
elif zero is False or real:
return real

def _eval_is_antihermitian(self):
z = self.is_zero
if z:
return False
elif z is False:
return (S.ImaginaryUnit*self).is_hermitian

def _eval_is_irrational(self):
for t in self.args:
a = t.is_irrational
if a:
others = list(self.args)
others.remove(t)
if all((x.is_rational and x.is_nonzero) is True for x in others):
return True
return
if a is None:
return
return False

def _eval_is_positive(self):
"""Return True if self is positive, False if not, and None if it
cannot be determined.

This algorithm is non-recursive and works by keeping track of the
sign which changes when a negative or nonpositive is encountered.
Whether a nonpositive or nonnegative is seen is also tracked since
the presence of these makes it impossible to return True, but
possible to return False if the end result is nonpositive. e.g.

pos * neg * nonpositive -> pos or zero -> None is returned
pos * neg * nonnegative -> neg or zero -> False is returned
"""

sign = 1
saw_NON = False
for t in self.args:
if t.is_positive:
continue
elif t.is_negative:
sign = -sign
elif t.is_zero:
return False
elif t.is_nonpositive:
sign = -sign
saw_NON = True
elif t.is_nonnegative:
saw_NON = True
else:
return
if sign == 1 and saw_NON is False:
return True
if sign < 0:
return False

def _eval_is_negative(self):
return (-self).is_positive

def _eval_is_odd(self):
is_integer = self.is_integer

if is_integer:
r, acc = True, 1
for t in self.args:
if not t.is_integer:
return None
elif t.is_even:
r = False
elif t.is_integer:
if r is False:
pass
elif acc != 1 and (acc + t).is_odd:
r = False
elif t.is_odd is None:
r = None
acc = t
return r

# !integer -> !odd
elif is_integer is False:
return False

def _eval_is_even(self):
is_integer = self.is_integer

if is_integer:
return fuzzy_not(self.is_odd)

elif is_integer is False:
return False

def _eval_subs(self, old, new):
from sympy.functions.elementary.complexes import sign
from sympy.ntheory.factor_ import multiplicity
from sympy.simplify.simplify import powdenest, fraction

if not old.is_Mul:
return None

# try keep replacement literal so -2*x doesn't replace 4*x
if old.args[0].is_Number and old.args[0] < 0:
if self.args[0].is_Number:
if self.args[0] < 0:
return self._subs(-old, -new)
return None

def base_exp(a):
# if I and -1 are in a Mul, they get both end up with
# a -1 base (see issue 6421); all we want here are the
# true Pow or exp separated into base and exponent
if a.is_Pow or a.func is C.exp:
return a.as_base_exp()
return a, S.One

def breakup(eq):
"""break up powers of eq when treated as a Mul:
b**(Rational*e) -> b**e, Rational
commutatives come back as a dictionary {b**e: Rational}
noncommutatives come back as a list [(b**e, Rational)]
"""

(c, nc) = (defaultdict(int), list())
for a in Mul.make_args(eq):
a = powdenest(a)
(b, e) = base_exp(a)
if e is not S.One:
(co, _) = e.as_coeff_mul()
b = Pow(b, e/co)
e = co
if a.is_commutative:
c[b] += e
else:
nc.append([b, e])
return (c, nc)

def rejoin(b, co):
"""
Put rational back with exponent; in general this is not ok, but
since we took it from the exponent for analysis, it's ok to put
it back.
"""

(b, e) = base_exp(b)
return Pow(b, e*co)

def ndiv(a, b):
"""if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return
the integer number of times it divides, else return 0.
"""
if not b.q % a.q or not a.q % b.q:
return int(a/b)
return 0

# give Muls in the denominator a chance to be changed (see issue 5651)
# rv will be the default return value
rv = None
n, d = fraction(self)
if d is not S.One:
self2 = n._subs(old, new)/d._subs(old, new)
if not self2.is_Mul:
return self2._subs(old, new)
if self2 != self:
self = rv = self2

# Now continue with regular substitution.

# handle the leading coefficient and use it to decide if anything
# should even be started; we always know where to find the Rational
# so it's a quick test

co_self = self.args[0]
co_old = old.args[0]
co_xmul = None
if co_old.is_Rational and co_self.is_Rational:
# if coeffs are the same there will be no updating to do
# below after breakup() step; so skip (and keep co_xmul=None)
if co_old != co_self:
co_xmul = co_self.extract_multiplicatively(co_old)
elif co_old.is_Rational:
return rv

# break self and old into factors

(c, nc) = breakup(self)
(old_c, old_nc) = breakup(old)

# update the coefficients if we had an extraction
# e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5
# then co_self in c is replaced by (3/5)**2 and co_residual
# is 2*(1/7)**2

if co_xmul and co_xmul.is_Rational and abs(co_old) != 1:
mult = S(multiplicity(abs(co_old), co_self))
c.pop(co_self)
if co_old in c:
c[co_old] += mult
else:
c[co_old] = mult
co_residual = co_self/co_old**mult
else:
co_residual = 1

# do quick tests to see if we can't succeed

ok = True
if len(old_nc) > len(nc):
# more non-commutative terms
ok = False
elif len(old_c) > len(c):
# more commutative terms
ok = False
elif set(i[0] for i in old_nc).difference(set(i[0] for i in nc)):
# unmatched non-commutative bases
ok = False
elif set(old_c).difference(set(c)):
# unmatched commutative terms
ok = False
elif any(sign(c[b]) != sign(old_c[b]) for b in old_c):
# differences in sign
ok = False
if not ok:
return rv

if not old_c:
cdid = None
else:
rat = []
for (b, old_e) in old_c.items():
c_e = c[b]
rat.append(ndiv(c_e, old_e))
if not rat[-1]:
return rv
cdid = min(rat)

if not old_nc:
ncdid = None
for i in range(len(nc)):
nc[i] = rejoin(*nc[i])
else:
ncdid = 0  # number of nc replacements we did
take = len(old_nc)  # how much to look at each time
limit = cdid or S.Infinity  # max number that we can take
failed = []  # failed terms will need subs if other terms pass
i = 0
while limit and i + take <= len(nc):
hit = False

# the bases must be equivalent in succession, and
# the powers must be extractively compatible on the
# first and last factor but equal inbetween.

rat = []
for j in range(take):
if nc[i + j][0] != old_nc[j][0]:
break
elif j == 0:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif j == take - 1:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif nc[i + j][1] != old_nc[j][1]:
break
else:
rat.append(1)
j += 1
else:
ndo = min(rat)
if ndo:
if take == 1:
if cdid:
ndo = min(cdid, ndo)
nc[i] = Pow(new, ndo)*rejoin(nc[i][0],
nc[i][1] - ndo*old_nc[0][1])
else:
ndo = 1

# the left residual

l = rejoin(nc[i][0], nc[i][1] - ndo*
old_nc[0][1])

# eliminate all middle terms

mid = new

# the right residual (which may be the same as the middle if take == 2)

ir = i + take - 1
r = (nc[ir][0], nc[ir][1] - ndo*
old_nc[-1][1])
if r[1]:
if i + take < len(nc):
nc[i:i + take] = [l*mid, r]
else:
r = rejoin(*r)
nc[i:i + take] = [l*mid*r]
else:

# there was nothing left on the right

nc[i:i + take] = [l*mid]

limit -= ndo
ncdid += ndo
hit = True
if not hit:

# do the subs on this failing factor

failed.append(i)
i += 1
else:

if not ncdid:
return rv

# although we didn't fail, certain nc terms may have
# failed so we rebuild them after attempting a partial
# subs on them

failed.extend(range(i, len(nc)))
for i in failed:
nc[i] = rejoin(*nc[i]).subs(old, new)

# rebuild the expression

if cdid is None:
do = ncdid
elif ncdid is None:
do = cdid
else:
do = min(ncdid, cdid)

margs = []
for b in c:
if b in old_c:

# calculate the new exponent

e = c[b] - old_c[b]*do
margs.append(rejoin(b, e))
else:
margs.append(rejoin(b.subs(old, new), c[b]))
if cdid and not ncdid:

# in case we are replacing commutative with non-commutative,
# we want the new term to come at the front just like the
# rest of this routine

margs = [Pow(new, cdid)] + margs
return co_residual*self.func(*margs)*self.func(*nc)

def _eval_nseries(self, x, n, logx):
from sympy import powsimp
terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
res = powsimp(self.func(*terms).expand(), combine='exp', deep=True)
if res.has(C.Order):
res += C.Order(x**n, x)
return res

return self.func(*[t.as_leading_term(x) for t in self.args])

def _eval_conjugate(self):
return self.func(*[t.conjugate() for t in self.args])

def _eval_transpose(self):
return self.func(*[t.transpose() for t in self.args[::-1]])

return self.func(*[t.adjoint() for t in self.args[::-1]])

def _sage_(self):
s = 1
for x in self.args:
s *= x._sage_()
return s

[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 sqrt
>>> (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive()
(6, -sqrt(2)*(-sqrt(2) + 1))

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

coef = S.One
args = []
for i, a in enumerate(self.args):
coef *= c
if p is not S.One:
args.append(p)
# don't use self._from_args here to reconstruct args
# since there may be identical args now that should be combined
# e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x))
return coef, self.func(*args)

[docs]    def as_ordered_factors(self, order=None):
"""Transform an expression into an ordered list of factors.

Examples
========

>>> from sympy import sin, cos
>>> from sympy.abc import x, y

>>> (2*x*y*sin(x)*cos(x)).as_ordered_factors()
[2, x, y, sin(x), cos(x)]

"""
cpart, ncpart = self.args_cnc()
cpart.sort(key=lambda expr: expr.sort_key(order=order))
return cpart + ncpart

@property
def _sorted_args(self):
return tuple(self.as_ordered_factors())

[docs]def prod(a, start=1):
"""Return product of elements of a. Start with int 1 so if only
ints are included then an int result is returned.

Examples
========

>>> from sympy import prod, S
>>> prod(range(3))
0
>>> type(_) is int
True
>>> prod([S(2), 3])
6
>>> _.is_Integer
True

You can start the product at something other than 1:
>>> prod([1, 2], 3)
6

"""
return reduce(operator.mul, a, start)

def _keep_coeff(coeff, factors, clear=True, sign=False):
"""Return coeff*factors unevaluated if necessary.

If clear is False, do not keep the coefficient as a factor
if it can be distributed on a single factor such that one or
more terms will still have integer coefficients.

If sign is True, allow a coefficient of -1 to remain factored out.

Examples
========

>>> from sympy.core.mul import _keep_coeff
>>> from sympy.abc import x, y
>>> from sympy import S

>>> _keep_coeff(S.Half, x + 2)
(x + 2)/2
>>> _keep_coeff(S.Half, x + 2, clear=False)
x/2 + 1
>>> _keep_coeff(S.Half, (x + 2)*y, clear=False)
y*(x + 2)/2
>>> _keep_coeff(S(-1), x + y)
-x - y
>>> _keep_coeff(S(-1), x + y, sign=True)
-(x + y)
"""

if not coeff.is_Number:
if factors.is_Number:
factors, coeff = coeff, factors
else:
return coeff*factors
if coeff is S.One:
return factors
elif coeff is S.NegativeOne and not sign:
return -factors
if not clear and coeff.is_Rational and coeff.q != 1:
q = S(coeff.q)
for i in factors.args:
c, t = i.as_coeff_Mul()
r = c/q
if r == int(r):
return coeff*factors
return Mul._from_args((coeff, factors))
elif factors.is_Mul:
margs = list(factors.args)
if margs[0].is_Number:
margs[0] *= coeff
if margs[0] == 1:
margs.pop(0)
else:
margs.insert(0, coeff)
return Mul._from_args(margs)
else:
return coeff*factors

def expand_2arg(e):
from sympy.simplify.simplify import bottom_up
def do(e):
if e.is_Mul:
c, r = e.as_coeff_Mul()
if c.is_Number and r.is_Add:
return _unevaluated_Add(*[c*ri for ri in r.args])
return e
return bottom_up(e, do)

from .numbers import Rational
from .power import Pow