Source code for diofant.simplify.powsimp

from collections import defaultdict
from functools import reduce

from ..core import (Add, Basic, Dummy, E, Integer, Mul, Pow, Rational, cacheit,
count_ops, expand_log, expand_mul, factor_terms, prod,
sympify)
from ..core.compatibility import default_sort_key, ordered
from ..core.mul import _keep_coeff
from ..core.rules import Transform
from ..functions import exp, exp_polar, log, polarify, root, unpolarify
from ..logic import true
from ..ntheory import multiplicity
from ..polys import gcd, lcm_list

[docs]@cacheit
def powsimp(expr, deep=False, combine='all', force=False, measure=count_ops):
"""
reduces expression by combining powers with similar bases and exponents.

Notes
=====

If deep is True then powsimp() will also simplify arguments of
functions. By default deep is set to False.

If force is True then bases will be combined without checking for
assumptions, e.g. sqrt(x)*sqrt(y) -> sqrt(x*y) which is not true
if x and y are both negative.

You can make powsimp() only combine bases or only combine exponents by
changing combine='base' or combine='exp'.  By default, combine='all',
which does both.  combine='base' will only combine::

a   a          a                          2x      x
x * y  =>  (x*y)   as well as things like 2   =>  4

and combine='exp' will only combine
::

a   b      (a + b)
x * x  =>  x

combine='exp' will strictly only combine exponents in the way that used
to be automatic.  Also use deep=True if you need the old behavior.

When combine='all', 'exp' is evaluated first.  Consider the first
example below for when there could be an ambiguity relating to this.
This is done so things like the second example can be completely
combined.  If you want 'base' combined first, do something like
powsimp(powsimp(expr, combine='base'), combine='exp').

Examples
========

>>> powsimp(x**y*x**z*y**z, combine='all')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='exp')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='base', force=True)
x**y*(x*y)**z

>>> powsimp(x**z*x**y*n**z*n**y, combine='all', force=True)
(n*x)**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='exp')
n**(y + z)*x**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='base', force=True)
(n*x)**y*(n*x)**z

>>> x, y = symbols('x y', positive=True)
>>> powsimp(log(exp(x)*exp(y)))
log(E**x*E**y)
>>> powsimp(log(exp(x)*exp(y)), deep=True)
x + y

Radicals with Mul bases will be combined if combine='exp'

>>> x, y = symbols('x y')

Two radicals are automatically joined through Mul:

>>> a = sqrt(x*sqrt(y))
>>> a*a**3 == a**4
True

But if an integer power of that radical has been
autoexpanded then Mul does not join the resulting factors:

>>> a**4  # auto expands to a Mul, no longer a Pow
x**2*y
>>> _*a  # so Mul doesn't combine them
x**2*y*sqrt(x*sqrt(y))
>>> powsimp(_)  # but powsimp will
(x*sqrt(y))**(5/2)
>>> powsimp(x*y*a)  # but won't when doing so would violate assumptions
x*y*sqrt(x*sqrt(y))

"""
from ..matrices import MatrixSymbol

def recurse(arg, **kwargs):
_deep = kwargs.get('deep', deep)
_combine = kwargs.get('combine', combine)
_force = kwargs.get('force', force)
_measure = kwargs.get('measure', measure)
return powsimp(arg, _deep, _combine, _force, _measure)

expr = sympify(expr)

if (not isinstance(expr, Basic) or isinstance(expr, MatrixSymbol) or (
expr.is_Atom or expr in (exp_polar(0), exp_polar(1)))):
return expr

if deep or expr.is_Add or expr.is_Mul and _y not in expr.args:
expr = expr.func(*[recurse(w) for w in expr.args])

if expr.is_Pow:
return recurse(expr*_y, deep=False)/_y

if not expr.is_Mul:
return expr

# handle the Mul
if combine in ('exp', 'all'):
# Collect base/exp data, while maintaining order in the
# non-commutative parts of the product
c_powers = defaultdict(list)
nc_part = []
newexpr = []
coeff = Integer(1)
for term in expr.args:
if term.is_Rational:
coeff *= term
continue
if term.is_Pow:
term = _denest_pow(term)
if term.is_commutative:
b, e = term.as_base_exp()
if deep:
b, e = [recurse(i) for i in [b, e]]
if b.is_Pow:
# don't let smthg like sqrt(x**a) split into x**a, 1/2
# or else it will be joined as x**(a/2) later
b, e = b**e, Integer(1)
c_powers[b].append(e)
else:
# This is the logic that combines exponents for equal,
# but non-commutative bases: A**x*A**y == A**(x+y).
if nc_part:
b1, e1 = nc_part[-1].as_base_exp()
b2, e2 = term.as_base_exp()
if (b1 == b2 and
e1.is_commutative and e2.is_commutative):
continue
nc_part.append(term)

# add up exponents of common bases
for b, e in ordered(c_powers.items()):
# allow 2**x/4 -> 2**(x - 2); don't do this when b and e are
# Numbers since autoevaluation will undo it, e.g.
# 2**(1/3)/4 -> 2**(1/3 - 2) -> 2**(1/3)/4
if (b and b.is_Number and not all(ei.is_Number for ei in e) and
coeff != 1 and b not in (1, -1)):
m = multiplicity(abs(b), abs(coeff))
if m:
e.append(m)
coeff /= b**m
if coeff != 1:
if coeff in c_powers:
c_powers[coeff] += Integer(1)
else:
c_powers[coeff] = Integer(1)

# convert to plain dictionary
c_powers = dict(c_powers)

# check for base and inverted base pairs
be = list(c_powers.items())
skip = set()  # skip if we already saw them
for b, e in be:
if b in skip:
continue
bpos = b.is_positive or b.is_polar
if bpos:
binv = 1/b
if b != binv and binv in c_powers:
if b.as_numer_denom() == 1:
c_powers.pop(b)
c_powers[binv] -= e
else:
e = c_powers.pop(binv)
c_powers[b] -= e

# check for base and negated base pairs
be = list(c_powers.items())
_n = Integer(-1)
for i, (b, e) in enumerate(be):
if ((-b).is_Symbol or b.is_Add) and -b in c_powers:
if (b.is_positive in (0, 1) or e.is_integer):
c_powers[-b] += c_powers.pop(b)
if _n in c_powers:
c_powers[_n] += e
else:
c_powers[_n] = e

# filter c_powers and convert to a list
c_powers = [(b, e) for b, e in c_powers.items() if e]

# ==============================================================
# check for Mul bases of Rational powers that can be combined with
# separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) ->
# (x*sqrt(x*y))**(3/2)
# ---------------- helper functions

def ratq(x):
"""Return Rational part of x's exponent as it appears in the bkey."""
return bkey(x)

def bkey(b, e=None):
"""Return (b**s, c.denominator), c.numerator where e -> c*s. If e is not given then
it will be taken by using as_base_exp() on the input b.
e.g.
x**3/2 -> (x, 2), 3
x**y -> (x**y, 1), 1
x**(2*y/3) -> (x**y, 3), 2
exp(x/2) -> (exp(a), 2), 1

"""
if e is not None:  # coming from c_powers or from below
if e.is_Integer:
return (b, Integer(1)), e
elif e.is_Rational:
return (b, Integer(e.denominator)), Integer(e.numerator)
else:
c, m = e.as_coeff_Mul(rational=True)
if c != 1 and b.is_positive:
return (b**m, Integer(c.denominator)), Integer(c.numerator)
else:
return (b**e, Integer(1)), Integer(1)
else:
return bkey(*b.as_base_exp())

def update(b):
"""Decide what to do with base, b. If its exponent is now an
integer multiple of the Rational denominator, then remove it
and put the factors of its base in the common_b dictionary or
update the existing bases if necessary. If it has been zeroed
out, simply remove the base.

"""
newe, r = divmod(common_b[b], b)
if not r:
common_b.pop(b)
if newe:
for m in Mul.make_args(b**newe):
b, e = bkey(m)
if b not in common_b:
common_b[b] = 0
common_b[b] += e
if b != 1:
bases.append(b)
# ---------------- end of helper functions

# assemble a dictionary of the factors having a Rational power
common_b = {}
done = []
bases = []
for b, e in c_powers:
b, e = bkey(b, e)
if b in common_b:
common_b[b] = common_b[b] + e
else:
common_b[b] = e
if b != 1 and b.is_Mul:
bases.append(b)
c_powers = [(b, e) for b, e in common_b.items() if e]
bases.sort(key=default_sort_key)  # this makes tie-breaking canonical
bases.sort(key=measure, reverse=True)  # handle longest first
for base in bases:
if base not in common_b:  # it may have been removed already
continue
b, exponent = base
last = False  # True when no factor of base is a radical
qlcm = 1  # the lcm of the radical denominators
while True:
bstart = b
qstart = qlcm

bb = []  # list of factors
ee = []  # (factor's expo. and it's current value in common_b)
for bi in Mul.make_args(b):
bib, bie = bkey(bi)
if bib not in common_b or common_b[bib] < bie:
ee = bb = []  # failed
break
ee.append([bie, common_b[bib]])
bb.append(bib)
if ee:
# find the number of extractions possible
# e.g. [(1, 2), (2, 2)] -> min(2/1, 2/2) -> 1
min1 = ee/ee
for i in range(len(ee)):
rat = ee[i]/ee[i]
if rat < 1:
break
min1 = min(min1, rat)
else:
# update base factor counts
# e.g. if ee = [(2, 5), (3, 6)] then min1 = 2
# and the new base counts will be 5-2*2 and 6-2*3
for i in range(len(bb)):
common_b[bb[i]] -= min1*ee[i]
update(bb[i])
# update the count of the base
# e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y)
# will increase by 4 to give bkey (x*sqrt(y), 2, 5)
common_b[base] += min1*qstart*exponent
if (last  # no more radicals in base
or len(common_b) == 1  # nothing left to join with
or all(k == 1 for k in common_b)):  # no rad's in common_b
break
# see what we can exponentiate base by to remove any radicals
# so we know what to search for
# e.g. if base were x**(1/2)*y**(1/3) then we should
# exponentiate by 6 and look for powers of x and y in the ratio
# of 2 to 3
qlcm = lcm_list([ratq(bi) for bi in Mul.make_args(bstart)])
if qlcm == 1:
break  # we are done
b = bstart**qlcm
qlcm *= qstart
if all(ratq(bi) == 1 for bi in Mul.make_args(b)):
last = True  # we are going to be done after this next pass
# this base no longer can find anything to join with and
# since it was longer than any other we are done with it
b, q = base
done.append((b, common_b.pop(base)*Rational(1, q)))

# update c_powers and get ready to continue with powsimp
c_powers = done
# there may be terms still in common_b that were bases that were
# identified as needing processing, so remove those, too
for (b, q), e in common_b.items():
if b.is_Pow and q != 1 and not b.exp.is_Rational:
b, be = b.as_base_exp()
b = b**(be/q)
else:
b = root(b, q)
c_powers.append((b, e))
check = len(c_powers)
c_powers = dict(c_powers)
assert len(c_powers) == check  # there should have been no duplicates
# ==============================================================

# rebuild the expression
newexpr = expr.func(*(newexpr + [Pow(b, e) for b, e in c_powers.items()]))
if combine == 'exp':
return expr.func(newexpr, expr.func(*nc_part))
else:
return recurse(expr.func(*nc_part), combine='base') * \
recurse(newexpr, combine='base')

elif combine == 'base':

# Build c_powers and nc_part.  These must both be lists not
# dicts because exp's are not combined.
c_powers = []
nc_part = []
for term in expr.args:
if term.is_commutative:
c_powers.append(list(term.as_base_exp()))
else:
nc_part.append(term)

# Pull out numerical coefficients from exponent if assumptions allow
# e.g., 2**(2*x) => 4**x
for i in range(len(c_powers)):
b, e = c_powers[i]
if not (all(x.is_nonnegative for x in b.as_numer_denom()) or e.is_integer or force or b.is_polar):
continue
exp_c, exp_t = e.as_coeff_Mul(rational=True)
if exp_c != 1 and exp_t != 1:
c_powers[i] = [Pow(b, exp_c), exp_t]

# Combine bases whenever they have the same exponent and
# assumptions allow
# first gather the potential bases under the common exponent
c_exp = defaultdict(list)
for b, e in c_powers:
if deep:
e = recurse(e)
c_exp[e].append(b)
del c_powers

# Merge back in the results of the above to form a new product
c_powers = defaultdict(list)
for e in c_exp:
bases = c_exp[e]

# calculate the new base for e

if len(bases) == 1:
new_base = bases
elif e.is_integer or force:
new_base = expr.func(*bases)
else:
# see which ones can be joined
unk = []
nonneg = []
neg = []
for bi in bases:
if bi.is_negative:
neg.append(bi)
elif bi.is_nonnegative:
nonneg.append(bi)
elif bi.is_polar:
nonneg.append(
bi)  # polar can be treated like non-negative
else:
unk.append(bi)
if len(unk) == 1 and not neg or len(neg) == 1 and not unk:
# a single neg or a single unk can join the rest
nonneg.extend(unk + neg)
unk = neg = []
elif neg:
# their negative signs cancel in groups of 2*q if we know
# that e = p/q else we have to treat them as unknown
israt = False
if e.is_Rational:
israt = True
else:
p, d = e.as_numer_denom()
if p.is_integer and d.is_integer:
israt = True
if israt:
neg = [-w for w in neg]
unk.extend([Integer(-1)]*len(neg))
else:
unk.extend(neg)
neg = []
del israt

# these shouldn't be joined
for b in unk:
c_powers[b].append(e)
# here is a new joined base
new_base = expr.func(*(nonneg + neg))
# if there are positive parts they will just get separated
# again unless some change is made

def _terms(e):
# return the number of terms of this expression
# when multiplied out -- assuming no joining of terms
return sum(_terms(ai) for ai in e.args)
if e.is_Mul:
return prod([_terms(mi) for mi in e.args])
return 1
xnew_base = expand_mul(new_base, deep=False)
new_base = factor_terms(xnew_base)

c_powers[new_base].append(e)

# break out the powers from c_powers now
c_part = [Pow(b, ei) for b, e in c_powers.items() for ei in e]

# we're done
return expr.func(*(c_part + nc_part))

else:
raise ValueError("combine must be one of ('all', 'exp', 'base').")

[docs]def powdenest(eq, force=False, polar=False):
r"""
Collect exponents on powers as assumptions allow.

Given (bb**be)**e, this can be simplified as follows:
* if bb is positive, or
* e is an integer, or
* |be| < 1 then this simplifies to bb**(be*e)

Given a product of powers raised to a power, (bb1**be1 *
bb2**be2...)**e, simplification can be done as follows:

- if e is positive, the gcd of all bei can be joined with e;
- all non-negative bb can be separated from those that are negative
and their gcd can be joined with e; autosimplification already
handles this separation.
- integer factors from powers that have integers in the denominator
of the exponent can be removed from any term and the gcd of such
integers can be joined with e

Setting force to True will make symbols that are not explicitly
negative behave as though they are positive, resulting in more
denesting.

Setting polar to True will do simplifications on the Riemann surface of
the logarithm, also resulting in more denestings.

When there are sums of logs in exp() then a product of powers may be
obtained e.g. exp(3*(log(a) + 2*log(b))) - > a**3*b**6.

Examples
========

>>> powdenest((x**(2*a/3))**(3*x))
(x**(2*a/3))**(3*x)
>>> powdenest(exp(3*x*log(2)))
2**(3*x)

Assumptions may prevent expansion:

>>> powdenest(sqrt(x**2))
sqrt(x**2)

>>> p = symbols('p', positive=True)
>>> powdenest(sqrt(p**2))
p

No other expansion is done.

>>> i, j = symbols('i j', integer=True)
>>> powdenest((x**x)**(i + j))  # -X-> (x**x)**i*(x**x)**j
x**(x*(i + j))

But exp() will be denested by moving all non-log terms outside of
the function; this may result in the collapsing of the exp to a power
with a different base:

>>> powdenest(exp(3*y*log(x)))
x**(3*y)
>>> powdenest(exp(y*(log(a) + log(b))))
(a*b)**y
>>> powdenest(exp(3*(log(a) + log(b))))
a**3*b**3

If assumptions allow, symbols can also be moved to the outermost exponent:

>>> i = Symbol('i', integer=True)
>>> powdenest(((x**(2*i))**(3*y))**x)
((x**(2*i))**(3*y))**x
>>> powdenest(((x**(2*i))**(3*y))**x, force=True)
x**(6*i*x*y)

>>> powdenest(((x**(2*a/3))**(3*y/i))**x)
((x**(2*a/3))**(3*y/i))**x
>>> powdenest((x**(2*i)*y**(4*i))**z, force=True)
(x*y**2)**(2*i*z)

>>> n = Symbol('n', negative=True)

>>> powdenest((x**i)**y, force=True)
x**(i*y)
>>> powdenest((n**i)**x, force=True)
(n**i)**x

"""
from .simplify import posify

if force:
eq, rep = posify(eq)
return powdenest(eq, force=False).xreplace(rep)

if polar:
eq, rep = polarify(eq)
return unpolarify(powdenest(unpolarify(eq, exponents_only=True)), rep)

new = powsimp(sympify(eq))
return new.xreplace(Transform(_denest_pow, filter=lambda m: m.is_Pow))

_y = Dummy('y')

def _denest_pow(eq):
"""
Denest powers.

This is a helper function for powdenest that performs the actual
transformation.

"""
from .simplify import logcombine

b, e = eq.as_base_exp()
if b.is_Pow and e != 1:
new = b._eval_power(e)
if new is not None:
eq = new
b, e = new.as_base_exp()

# denest exp with log terms in exponent
if b is E and e.is_Mul:
logs = []
other = []
for ei in e.args:
if any(isinstance(ai, log) for ai in Add.make_args(ei)):
logs.append(ei)
else:
other.append(ei)
logs = logcombine(Mul(*logs))
return Pow(exp(logs), Mul(*other))

_, be = b.as_base_exp()
if be == 1 and not (b.is_Mul or
b.is_Rational and b.denominator != 1 or
b.is_positive):
return eq

# denest eq which is either pos**e or Pow**e or Mul**e or
# Mul(b1**e1, b2**e2)

# handle polar numbers specially
polars, nonpolars = [], []
for bb in Mul.make_args(b):
if bb.is_polar:
polars.append(bb.as_base_exp())
else:
nonpolars.append(bb)
if len(polars) == 1 and not polars.is_Mul:
return Pow(polars, polars*e)*powdenest(Mul(*nonpolars)**e)
elif polars:
return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \
* powdenest(Mul(*nonpolars)**e)

if b.is_Integer:
# use log to see if there is a power here
logb = expand_log(log(b))
if logb.is_Mul:
c, logb = logb.args
e *= c
base = logb.args
return Pow(base, e)

# if b is not a Mul or any factor is an atom then there is nothing to do
if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)):
return eq

# let log handle the case of the base of the argument being a Mul, e.g.
# sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we
# will take the log, expand it, and then factor out the common powers that
# now appear as coefficient. We do this manually since terms_gcd pulls out
# fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2;
# gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but
# we want 3*x. Neither work with noncommutatives.

def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a, b).as_numer_denom()
g = Mul(*(a.args_cnc(cset=True) & b.args_cnc(cset=True)))
return _keep_coeff(c, g)

glogb = expand_log(log(b))
args = glogb.args
g = reduce(nc_gcd, args)
if g != 1:
cg, rg = g.as_coeff_Mul()
glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args]))

# now put the log back together again
if isinstance(glogb, log) or not glogb.is_Mul:
if glogb.args.is_Pow:
glogb = _denest_pow(glogb.args)
if (abs(glogb.exp) < 1) == true:
return Pow(glogb.base, glogb.exp*e)
return eq

# the log(b) was a Mul so join any adds with logcombine