from collections import defaultdict

from .. import DIOFANT_DEBUG
from ..core import (Add, Derivative, I, Integer, Mul, Pow, Rational,
expand_mul, expand_power_base, gcd_terms, symbols, sympify)
from ..core.compatibility import default_sort_key, iterable, ordered
from ..core.exprtools import Factors
from ..core.function import _mexpand
from ..core.mul import _keep_coeff, _unevaluated_Mul
from ..functions import log, sqrt
from ..polys import gcd
from .sqrtdenest import sqrtdenest

[docs]def collect(expr, syms, func=None, evaluate=True, exact=False, distribute_order_term=True):
"""Collect additive terms of an expression.

This function collects additive terms of an expression with respect
to a list of expression up to powers with rational exponents. By the
term symbol here are meant arbitrary expressions, which can contain
powers, products, sums etc. In other words symbol is a pattern which
will be searched for in the expression's terms.

The input expression is not expanded by :func:collect, so user is
expected to provide an expression is an appropriate form (for example,
by using :func:~diofant.core.function.expand prior to calling this
function). This makes :func:collect more predictable as there is no
magic happening behind the scenes. However, it is important to note,
that powers of products are converted to products of powers using the
:func:~diofant.core.function.expand_power_base function.

Parameters
==========

expr : Expr
an expression
syms : iterable of Symbol's
collected symbols
evaluate : boolean
First, if evaluate flag is set (by default), this function will
return an expression with collected terms else it will return a
dictionary with expressions up to rational powers as keys and
collected coefficients as values.

Examples
========

This function can collect symbolic coefficients in polynomials or
rational expressions. It will manage to find all integer or rational
powers of collection variable:

>>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x)
c + x**2*(a + b) + x*(a - b)

The same result can be achieved in dictionary form:

>>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)
>>> d[x**2]
a + b
>>> d[x]
a - b
>>> d
c

You can also work with multivariate polynomials. However, remember that
this function is greedy so it will care only about a single symbol at time,
in specification order:

>>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])
x**2*(y + 1) + x*y + y*(a + 1)

Also more complicated expressions can be used as patterns:

>>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x))
(a + b)*sin(2*x)

>>> collect(a*x*log(x) + b*(x*log(x)), x*log(x))
x*(a + b)*log(x)

You can use wildcards in the pattern:

>>> w = Wild('w1')
>>> collect(a*x**y - b*x**y, w**y)
x**y*(a - b)

It is also possible to work with symbolic powers, although it has more
complicated behavior, because in this case power's base and symbolic part
of the exponent are treated as a single symbol:

>>> collect(a*x**c + b*x**c, x)
a*x**c + b*x**c
>>> collect(a*x**c + b*x**c, x**c)
x**c*(a + b)

However if you incorporate rationals to the exponents, then you will get
well known behavior:

>>> collect(a*x**(2*c) + b*x**(2*c), x**c)
x**(2*c)*(a + b)

Note also that all previously stated facts about :func:collect function
apply to the exponential function, so you can get:

>>> collect(a*exp(2*x) + b*exp(2*x), exp(x))
E**(2*x)*(a + b)

If you are interested only in collecting specific powers of some symbols
then set exact flag in arguments:

>>> collect(a*x**7 + b*x**7, x, exact=True)
a*x**7 + b*x**7
>>> collect(a*x**7 + b*x**7, x**7, exact=True)
x**7*(a + b)

You can also apply this function to differential equations, where
derivatives of arbitrary order can be collected. Note that if you
collect with respect to a function or a derivative of a function, all
derivatives of that function will also be collected. Use
exact=True to prevent this from happening:

>>> f = f(x)

>>> collect(a*Derivative(f, x) + b*Derivative(f, x), Derivative(f, x))
(a + b)*Derivative(f(x), x)

>>> collect(a*Derivative(f, x, 2) + b*Derivative(f, x, 2), f)
(a + b)*Derivative(f(x), x, x)

>>> collect(a*Derivative(f, x, 2) + b*Derivative(f, x, 2), Derivative(f, x), exact=True)
a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x)

>>> collect(a*Derivative(f, x) + b*Derivative(f, x) + a*f + b*f, f)
f(x)*(a + b) + (a + b)*Derivative(f(x), x)

Or you can even match both derivative order and exponent at the same time:

>>> collect(a*Derivative(f, x, 2)**2 + b*Derivative(f, x, 2)**2, Derivative(f, x))
(a + b)*Derivative(f(x), x, x)**2

Finally, you can apply a function to each of the collected coefficients.
For example you can factorize symbolic coefficients of polynomial:

>>> f = expand((x + a + 1)**3)

>>> collect(f, x, factor)
x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3

========

collect_const, collect_sqrt, rcollect

"""
def make_expression(terms):
product = []

for term, rat, sym, deriv in terms:
if deriv is not None:
var, order = deriv

while order > 0:
term, order = Derivative(term, var), order - 1

if sym is None:
if rat == 1:
product.append(term)
else:
product.append(Pow(term, rat))
else:
product.append(Pow(term, rat*sym))

return Mul(*product)

def parse_derivative(deriv):
# scan derivatives tower in the input expression and return
# underlying function and maximal differentiation order
expr, sym, order = deriv.expr, deriv.variables, 1

for s in deriv.variables[1:]:
if s == sym:
order += 1
else:
raise NotImplementedError(
'Improve MV Derivative support in collect')

while isinstance(expr, Derivative):
s0 = expr.variables

for s in expr.variables:
if s != s0:
raise NotImplementedError(
'Improve MV Derivative support in collect')

if s0 == sym:
expr, order = expr.expr, order + len(expr.variables)
else:
break

return expr, (sym, Rational(order))

def parse_term(expr):
"""Parses expression expr and outputs tuple (sexpr, rat_expo,
sym_expo, deriv)
where:
- sexpr is the base expression
- rat_expo is the rational exponent that sexpr is raised to
- sym_expo is the symbolic exponent that sexpr is raised to
- deriv contains the derivatives the the expression

for example, the output of x would be (x, 1, None, None)
the output of 2**x would be (2, 1, x, None)

"""
rat_expo, sym_expo = Integer(1), None
sexpr, deriv = expr, None

if expr.is_Pow:
if isinstance(expr.base, Derivative):
sexpr, deriv = parse_derivative(expr.base)
else:
sexpr = expr.base

if expr.exp.is_Number:
rat_expo = expr.exp
else:
coeff, tail = expr.exp.as_coeff_Mul()

if coeff.is_Number:
rat_expo, sym_expo = coeff, tail
else:
sym_expo = expr.exp
elif isinstance(expr, Derivative):
sexpr, deriv = parse_derivative(expr)

return sexpr, rat_expo, sym_expo, deriv

def parse_expression(terms, pattern):
"""Parse terms searching for a pattern.
terms is a list of tuples as returned by parse_terms;
pattern is an expression treated as a product of factors

"""
pattern = Mul.make_args(pattern)

if len(terms) < len(pattern):
# pattern is longer than matched product
# so no chance for positive parsing result
return
else:
pattern = [parse_term(elem) for elem in pattern]

terms = terms[:]  # need a copy
elems, common_expo, has_deriv = [], None, False

for elem, e_rat, e_sym, e_ord in pattern:

if elem.is_Number and e_rat == 1 and e_sym is None:
# a constant is a match for everything
continue

for j in range(len(terms)):
if terms[j] is None:
continue

term, t_rat, t_sym, t_ord = terms[j]

# keeping track of whether one of the terms had
# a derivative or not as this will require rebuilding
# the expression later
if t_ord is not None:
has_deriv = True

if (term.match(elem) is not None and
(t_sym == e_sym or t_sym is not None and
e_sym is not None and
t_sym.match(e_sym) is not None)):
if exact is False:
# we don't have to be exact so find common exponent
# for both expression's term and pattern's element
expo = t_rat / e_rat

if common_expo is None:
# first time
common_expo = expo
else:
# common exponent was negotiated before so
# there is no chance for a pattern match unless
# common and current exponents are equal
if common_expo != expo:
common_expo = 1
else:
# we ought to be exact so all fields of
# interest must match in every details
if e_rat != t_rat or e_ord != t_ord:
continue

# found common term so remove it from the expression
# and try to match next element in the pattern
elems.append(terms[j])
terms[j] = None

break

else:
return

return [_f for _f in terms if _f], elems, common_expo, has_deriv

if evaluate:
if expr.is_Mul:
return expr.func(*[
collect(term, syms, func, True, exact, distribute_order_term)
for term in expr.args])
elif expr.is_Pow:
b = collect(
expr.base, syms, func, True, exact, distribute_order_term)
return Pow(b, expr.exp)

if iterable(syms):
syms = [expand_power_base(i, deep=False) for i in syms]
else:
syms = [expand_power_base(syms, deep=False)]

expr = sympify(expr)
order_term = None

if distribute_order_term:
order_term = expr.getO()

if order_term is not None:
if order_term.has(*syms):
order_term = None
else:
expr = expr.removeO()

summa = [expand_power_base(i, deep=False) for i in Add.make_args(expr)]

collected, disliked = defaultdict(list), Integer(0)
for product in summa:
terms = [parse_term(i) for i in Mul.make_args(product)]

for symbol in syms:
if DIOFANT_DEBUG:
print("DEBUG: parsing of expression %s with symbol %s " % (
str(terms), str(symbol))
)

result = parse_expression(terms, symbol)

if DIOFANT_DEBUG:
print("DEBUG: returned %s" % str(result))

if result is not None:
terms, elems, common_expo, has_deriv = result

# when there was derivative in current pattern we
# will need to rebuild its expression from scratch
if not has_deriv:
index = 1
for elem in elems:
e = elem
if elem is not None:
e *= elem
index *= Pow(elem, e)
else:
index = make_expression(elems)
terms = expand_power_base(make_expression(terms), deep=False)
index = expand_power_base(index, deep=False)
collected[index].append(terms)
break
else:
# none of the patterns matched
disliked += product
# add terms now for each key
collected = {k: Add(*v) for k, v in collected.items()}

if disliked != 0:
collected[Integer(1)] = disliked

if order_term is not None:
for key, val in collected.items():
collected[key] = val + order_term

if func is not None:
collected = {key: func(val) for key, val in collected.items()}

if evaluate:
return Add(*[key*val for key, val in collected.items()])
else:
return collected

[docs]def rcollect(expr, *vars):
"""
Recursively collect sums in an expression.

Examples
========

>>> expr = (x**2*y + x*y + x + y)/(x + y)

>>> rcollect(expr, y)
(x + y*(x**2 + x + 1))/(x + y)

========
collect, collect_const, collect_sqrt

"""
if expr.is_Atom or not expr.has(*vars):
return expr
else:
expr = expr.__class__(*[rcollect(arg, *vars) for arg in expr.args])

return collect(expr, vars)
else:
return expr

[docs]def collect_sqrt(expr, evaluate=True):
"""Return expr with terms having common square roots collected together.
If evaluate is False a count indicating the number of sqrt-containing
terms will be returned and, if non-zero, the terms of the Add will be
returned, else the expression itself will be returned as a single term.
If evaluate is True, the expression with any collected terms will be
returned.

Note: since I = sqrt(-1), it is collected, too.

Examples
========

>>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]]
>>> collect_sqrt(a*r2 + b*r2)
sqrt(2)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3)
sqrt(2)*(a + b) + sqrt(3)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5)
sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)

If evaluate is False then the arguments will be sorted and
returned as a list and a count of the number of sqrt-containing
terms will be returned:

>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False)
((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3)
>>> collect_sqrt(a*sqrt(2) + b, evaluate=False)
((b, sqrt(2)*a), 1)
>>> collect_sqrt(a + b, evaluate=False)
((a + b,), 0)

========
collect, collect_const, rcollect

"""
# this step will help to standardize any complex arguments
# of sqrts
coeff, expr = expr.as_content_primitive()
vars = set()
for m in a.args_cnc():
if m.is_number and (
m.is_Pow and m.exp.is_Rational and m.exp.denominator == 2 or
m is I):

# we only want radicals, so exclude Number handling; in this case
# d will be evaluated
d = collect_const(expr, *vars, Numbers=False)
hit = expr != d

if not evaluate:
# make the evaluated args canonical
for i, m in enumerate(args):
c, nc = m.args_cnc()
for ci in c:
# XXX should this be restricted to ci.is_number as above?
if ci.is_Pow and ci.exp.is_Rational and ci.exp.denominator == 2 or \
ci is I:
break
args[i] *= coeff

return coeff*d

[docs]def collect_const(expr, *vars, **kwargs):
"""A non-greedy collection of terms with similar number coefficients in
an Add expr. If vars is given then only those constants will be
targeted. Although any Number can also be targeted, if this is not
desired set Numbers=False and no Float or Rational will be collected.

Examples
========

>>> from diofant.abc import s
>>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2)))
sqrt(3)*(sqrt(2) + 2)
>>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7))
(sqrt(3) + sqrt(7))*(s + 1)
>>> s = sqrt(2) + 2
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7))
(sqrt(2) + 3)*(sqrt(3) + sqrt(7))
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3))
sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)

The collection is sign-sensitive, giving higher precedence to the
unsigned values:

>>> collect_const(x - y - z)
x - (y + z)
>>> collect_const(-y - z)
-(y + z)
>>> collect_const(2*x - 2*y - 2*z, 2)
2*(x - y - z)
>>> collect_const(2*x - 2*y - 2*z, -2)
2*x - 2*(y + z)

========
collect, collect_sqrt, rcollect

"""
return expr

recurse = False
Numbers = kwargs.get('Numbers', True)

if not vars:
recurse = True
vars = set()
for a in expr.args:
for m in Mul.make_args(a):
if m.is_number:
else:
vars = sympify(vars)
if not Numbers:
vars = [v for v in vars if not v.is_Number]

vars = list(ordered(vars))
for v in vars:
terms = defaultdict(list)
Fv = Factors(v)
f = Factors(m)
q, r = f.div(Fv)
if r.is_one:
# only accept this as a true factor if
# it didn't change an exponent from an Integer
# to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2)
# -- we aren't looking for this sort of change
fwas = f.factors.copy()
fnow = q.factors
if not any(k in fwas and fwas[k].is_Integer and not
fnow[k].is_Integer for k in fnow):
terms[v].append(q.as_expr())
continue
terms[Integer(1)].append(m)

args = []
hit = False
for k in ordered(terms):
v = terms[k]
if k == 1:
args.extend(v)
continue

if len(v) > 1:
hit = True
if recurse and v != expr:
vars.append(v)
else:
v = v

if Numbers and k.is_Number and v.is_Add:
args.append(_keep_coeff(k, v, sign=True))
else:
args.append(k*v)

if hit:
break

return expr

r"""
Rationalize the denominator by removing square roots.

Note: the expression returned from radsimp must be used with caution
since if the denominator contains symbols, it will be possible to make
substitutions that violate the assumptions of the simplification process:
that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If
there are no symbols, this assumptions is made valid by collecting terms
of sqrt(c) so the match variable a does not contain sqrt(c).) If
you do not want the simplification to occur for symbolic denominators, set
symbolic to False.

If there are more than max_terms radical terms then the expression is
returned unchanged.

Examples
========

(1 - I)/2
(-sqrt(2) + 2)/2
>>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2))
sqrt(2)*(x + y)

No simplification beyond removal of the gcd is done. One might
want to polish the result a little, however, by collecting
square root terms:

>>> r2 = sqrt(2)
>>> r5 = sqrt(5)
>>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5))
>>> pprint(ans, use_unicode=False)
___       ___       ___       ___
\/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y
------------------------------------------
2               2      2              2
5*a  + 10*a*b + 5*b  - 2*x  - 4*x*y - 2*y

>>> n, d = fraction(ans)
___             ___
\/ 5 *(a + b) - \/ 2 *(x + y)
------------------------------------------
2               2      2              2
5*a  + 10*a*b + 5*b  - 2*x  - 4*x*y - 2*y

If radicals in the denominator cannot be removed or there is no denominator,
the original expression will be returned.

sqrt(2)*x + sqrt(2)

Results with symbols will not always be valid for all substitutions:

>>> eq = 1/(a + b*sqrt(c))
>>> eq.subs({a: b*sqrt(c)})
1/(2*b*sqrt(c))
nan

If symbolic=False, symbolic denominators will not be transformed (but
numeric denominators will still be processed):

1/(a + b*sqrt(c))

"""
from .simplify import signsimp

syms = symbols("a:d A:D")

def _num(rterms):
# return the multiplier that will simplify the expression described
# by rterms [(sqrt arg, coeff), ... ]
a, b, c, d, A, B, C, D = syms
if len(rterms) == 2:
reps = dict(zip([A, a, B, b], [j for i in rterms for j in i]))
return (
sqrt(A)*a - sqrt(B)*b).xreplace(reps)
if len(rterms) == 3:
reps = dict(zip([A, a, B, b, C, c], [j for i in rterms for j in i]))
return (
(sqrt(A)*a + sqrt(B)*b - sqrt(C)*c)*(2*sqrt(A)*sqrt(B)*a*b - A*a**2 -
B*b**2 + C*c**2)).xreplace(reps)
elif len(rterms) == 4:
reps = dict(zip([A, a, B, b, C, c, D, d], [j for i in rterms for j in i]))
return ((sqrt(A)*a + sqrt(B)*b - sqrt(C)*c - sqrt(D)*d)*(2*sqrt(A)*sqrt(B)*a*b
- A*a**2 - B*b**2 - 2*sqrt(C)*sqrt(D)*c*d + C*c**2 +
D*d**2)*(-8*sqrt(A)*sqrt(B)*sqrt(C)*sqrt(D)*a*b*c*d + A**2*a**4 -
2*A*B*a**2*b**2 - 2*A*C*a**2*c**2 - 2*A*D*a**2*d**2 + B**2*b**4 -
2*B*C*b**2*c**2 - 2*B*D*b**2*d**2 + C**2*c**4 - 2*C*D*c**2*d**2 +
D**2*d**4)).xreplace(reps)
elif len(rterms) == 1:
return sqrt(rterms)
else:
raise NotImplementedError

def ispow2(d, log2=False):
if not d.is_Pow:
return False
e = d.exp
if e.is_Rational and e.denominator == 2 or symbolic and fraction(e) == 2:
return True
if log2:
q = 1
if e.is_Rational:
q = e.denominator
elif symbolic:
d = fraction(e)
if d.is_Integer:
q = d
if q != 1 and log(q, 2).is_Integer:
return True
return False

def handle(expr):
# Handle first reduces to the case
# expr = 1/d, where d is an add, or d is base**p/2.
# We do this by recursively calling handle on each piece.
from .simplify import nsimplify

n, d = fraction(expr)

if expr.is_Atom or (d.is_Atom and n.is_Atom):
return expr
elif not n.is_Atom:
n = n.func(*[handle(a) for a in n.args])
return _unevaluated_Mul(n, handle(1/d))
elif n != 1:
return _unevaluated_Mul(n, handle(1/d))
elif d.is_Mul:
return _unevaluated_Mul(*[handle(1/d) for d in d.args])

# By this step, expr is 1/d, and d is not a mul.
if not symbolic and d.free_symbols:
return expr

if ispow2(d):
d2 = sqrtdenest(sqrt(d.base))**fraction(d.exp)
if d2 != d:
return handle(1/d2)
elif d.is_Pow and (d.exp.is_integer or d.base.is_positive):
# (1/d**i) = (1/d)**i
return handle(1/d.base)**d.exp

return 1/d.func(*[handle(a) for a in d.args])

# handle 1/d treating d as an Add (though it may not be)

keep = True  # keep changes that are made

# flatten it and collect radicals after checking for special
# conditions
d = _mexpand(d)

# did it change?
if d.is_Atom:
return 1/d

# is it a number that might be handled easily?
if d.is_number:
_d = nsimplify(d)
if _d.is_Number and _d.equals(d):
return 1/_d

while True:
# collect similar terms
collected = defaultdict(list)
p2 = []
other = []
for i in Mul.make_args(m):
if ispow2(i, log2=True):
p2.append(i.base if i.exp == Rational(1, 2) else i.base**(2*i.exp))
elif i is I:
p2.append(Integer(-1))
else:
other.append(i)
collected[tuple(ordered(p2))].append(Mul(*other))
rterms = list(ordered(list(collected.items())))
rterms = [(Mul(*i), Add(*j)) for i, j in rterms]
nrad = len(rterms) - (1 if rterms == 1 else 0)
break
# there may have been invalid operations leading to this point
# so don't keep changes, e.g. this expression is troublesome
# in collecting terms so as not to raise the issue of 2834:
# r = sqrt(sqrt(5) + 5)
# eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r)
keep = False
break
if len(rterms) > 4:
# in general, only 4 terms can be removed with repeated squaring
# but other considerations can guide selection of radical terms
# so that radicals are removed
if all(x.is_Integer and (y**2).is_Rational for x, y in rterms):
[sqrt(x)*y for x, y in rterms]))
n *= nd
else:
# is there anything else that might be attempted?
keep = False
break

from .powsimp import powsimp, powdenest

num = powsimp(_num(rterms))
n *= num
d *= num
d = powdenest(_mexpand(d), force=symbolic)
if d.is_Atom:
break

if not keep:
return expr
return _unevaluated_Mul(n, 1/d)

expr = expr.normal()
old = fraction(expr)
n, d = fraction(handle(expr))
if old != (n, d):
if not d.is_Atom:
was = (n, d)
n = signsimp(n, evaluate=False)
d = signsimp(d, evaluate=False)
u = Factors(_unevaluated_Mul(n, 1/d))
u = _unevaluated_Mul(*[k**v for k, v in u.factors.items()])
n, d = fraction(u)
if old == (n, d):
n, d = was
n = expand_mul(n)
n2, d2 = fraction(gcd_terms(_unevaluated_Mul(n, 1/d)))
if d2.is_Number or (d2.count_ops() <= d.count_ops()):
n, d = [signsimp(i) for i in (n2, d2)]
if n.is_Mul and n.args.is_Number:
n = n.func(*n.args)

return coeff + _unevaluated_Mul(n, 1/d)

"""
Rationalize num/den by removing square roots in the denominator;
num and den are sum of terms whose squares are rationals

Examples
========

(-sqrt(3) + sqrt(6)/3, -7/9)

"""
return num, den
g, a, b = split_surds(den)
a = a*sqrt(g)
num = _mexpand((a - b)*num)
den = _mexpand(a**2 - b**2)

[docs]def fraction(expr, exact=False):
"""Returns a pair with expression's numerator and denominator.

If the given expression is not a fraction then this function
will return the tuple (expr, 1).

This function will not make any attempt to simplify nested
fractions or to do any term rewriting at all.

If only one of the numerator/denominator pair is needed then
use numer(expr) or denom(expr) functions respectively.

>>> fraction(x/y)
(x, y)
>>> fraction(x)
(x, 1)

>>> fraction(1/y**2)
(1, y**2)

>>> fraction(x*y/2)
(x*y, 2)
>>> fraction(Rational(1, 2))
(1, 2)

This function will also work fine with assumptions:

>>> k = Symbol('k', negative=True)
>>> fraction(x * y**k)
(x, y**(-k))

If we know nothing about sign of some exponent and 'exact'
flag is unset, then structure this exponent's structure will
be analyzed and pretty fraction will be returned:

>>> fraction(2*x**(-y))
(2, x**y)

>>> fraction(exp(-x))
(1, E**x)

>>> fraction(exp(-x), exact=True)
(E**(-x), 1)

"""
expr = sympify(expr)

numer, denom = [], []

for term in Mul.make_args(expr):
if term.is_Pow and term.is_commutative:
b, ex = term.as_base_exp()
if ex.is_negative:
if ex == -1:
denom.append(b)
else:
denom.append(Pow(b, -ex))
elif ex.is_positive:
numer.append(term)
elif not exact and ex.is_Mul:
n, d = term.as_numer_denom()
numer.append(n)
denom.append(d)
else:
numer.append(term)
elif term.is_Rational:
n, d = term.as_numer_denom()
numer.append(n)
denom.append(d)
else:
numer.append(term)

return Mul(*numer), Mul(*denom)

def numer(expr):
return fraction(expr)

def denom(expr):
return fraction(expr)

def fraction_expand(expr, **hints):
return expr.expand(frac=True, **hints)

def numer_expand(expr, **hints):
a, b = fraction(expr)
return a.expand(numer=True, **hints) / b

def denom_expand(expr, **hints):
a, b = fraction(expr)
return a / b.expand(denom=True, **hints)

expand_numer = numer_expand
expand_denom = denom_expand
expand_fraction = fraction_expand

def split_surds(expr):
"""
split an expression with terms whose squares are rationals
into a sum of terms whose surds squared have gcd equal to g
and a sum of terms with surds squared prime with g

Examples
========

>>> split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15))
(3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10))

"""
args = sorted(expr.args, key=default_sort_key)
coeff_muls = [x.as_coeff_Mul() for x in args]
surds = [x**2 for x in coeff_muls if x.is_Pow]
surds.sort(key=default_sort_key)
g, b1, b2 = _split_gcd(*surds)
g2 = g
if not b2 and len(b1) >= 2:
b1n = [x/g for x in b1]
b1n = [x for x in b1n if x != 1]
# only a common factor has been factored; split again
g1, b1n, b2 = _split_gcd(*b1n)
g2 = g*g1
a1v, a2v = [], []
for c, s in coeff_muls:
if s.is_Pow and s.exp == Rational(1, 2):
s1 = s.base
if s1 in b1:
a1v.append(c*sqrt(s1/g2))
else:
a2v.append(c*s)
else:
a2v.append(c*s)
return g2, a, b

def _split_gcd(*a):
"""
split the list of integers a into a list of integers, a1 having
g = gcd(a1), and a list a2 whose elements are not divisible by
g.  Returns g, a1, a2

Examples
========

>>> _split_gcd(55, 35, 22, 14, 77, 10)
(5, [55, 35, 10], [22, 14, 77])

"""
g = a
b1 = [g]
b2 = []
for x in a[1:]:
g1 = gcd(g, x)
if g1 == 1:
b2.append(x)
else:
g = g1
b1.append(x)
return g, b1, b2