def _separate_imaginary_from_complex(cls, complexes):
from diofant.utilities.iterables import sift
def is_imag(c):
'''
return True if all roots are imaginary (ax**2 + b)
return False if no roots are imaginary
return None if 2 roots are imaginary (ax**N'''
u, f, k = c
deg = f.degree()
if f.length() == 2:
if deg == 2:
return True # both imag
elif _ispow2(deg):
if f.LC() * f.TC() < 0:
return # 2 are imag
return False # none are imag
# separate according to the function
sifted = sift(complexes, lambda c: c[1])
del complexes
imag = []
complexes = []
for f in sifted:
isift = sift(sifted[f], lambda c: is_imag(c))
imag.extend(isift.pop(True, []))
complexes.extend(isift.pop(False, []))
mixed = isift.pop(None, [])
assert not isift
if not mixed:
continue
while True:
# the non-imaginary ones will be on one side or the other
# of the y-axis
i = 0
while i < len(mixed):
u, f, k = mixed[i]
if u.ax * u.bx > 0:
complexes.append(mixed.pop(i))
else:
i += 1
if len(mixed) == 2:
imag.extend(mixed)
break
# refine
for i, (u, f, k) in enumerate(mixed):
u = u._inner_refine()
mixed[i] = u, f, k
return imag, complexes
def conglomerate(expr):
""" Conglomerate together identical args x + x -> 2x """
groups = sift(arguments(expr), key)
counts = {k: sum(map(count, args)) for k, args in groups.items()}
newargs = [combine(cnt, mat) for mat, cnt in counts.items()]
if set(newargs) != set(arguments(expr)):
return term(operator(expr), newargs)
else:
return expr
def _eval_factor(self, **hints):
if 1 == len(self.limits):
summand = self.function.factor(**hints)
if summand.is_Mul:
out = sift(
summand.args, lambda w: w.is_commutative and not set(
self.variables) & w.free_symbols)
return Mul(*out[True]) * self.func(Mul(*out[False]), *
self.limits)
else:
summand = self.func(self.function, self.limits[0:-1]).factor()
if not summand.has(self.variables[-1]):
return self.func(1, [self.limits[-1]]).doit() * summand
return self
def cse_separate(r, e):
"""Move expressions that are in the form (symbol, expr) out of the
expressions and sort them into the replacements using the reps_toposort.
Examples
========
>>> from diofant.simplify.cse_main import cse_separate
>>> from diofant.abc import x, y, z
>>> from diofant import cos, exp, cse, Eq, symbols
>>> x0, x1 = symbols('x:2')
>>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
>>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [
... [[(x0, y + 1), (x, z + 1), (x1, x + 1)],
... [x1 + exp(x1/x0) + cos(x0), z - 2]],
... [[(x1, y + 1), (x, z + 1), (x0, x + 1)],
... [x0 + exp(x0/x1) + cos(x1), z - 2]]]
...
True
"""
d = sift(e, lambda w: w.is_Equality and w.lhs.is_Symbol)
r = r + [w.args for w in d[True]]
e = d[False]
return [reps_toposort(r), e]
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([Integer(1)], lambda _: _.has(x)) == {False: [1]}
