# Source code for diofant.simplify.fu

"""
Implementation of the trigsimp algorithm by Fu et al.

The idea behind the fu algorithm is to use a sequence of rules, applied
in what is heuristically known to be a smart order, to select a simpler
expression that is equivalent to the input.

There are transform rules in which a single rule is applied to the
expression tree. The following are just mnemonic in nature; see the
docstrings for examples.

TR0 - simplify expression
TR1 - sec-csc to cos-sin
TR2 - tan-cot to sin-cos ratio
TR2i - sin-cos ratio to tan
TR3 - angle canonicalization
TR4 - functions at special angles
TR5 - powers of sin to powers of cos
TR6 - powers of cos to powers of sin
TR7 - reduce cos power (increase angle)
TR8 - expand products of sin-cos to sums
TR9 - contract sums of sin-cos to products
TR10 - separate sin-cos arguments
TR10i - collect sin-cos arguments
TR11 - reduce double angles
TR12 - separate tan arguments
TR12i - collect tan arguments
TR13 - expand product of tan-cot
TRmorrie - prod(cos(x*2**i), (i, 0, k - 1)) -> sin(2**k*x)/(2**k*sin(x))
TR14 - factored powers of sin or cos to cos or sin power
TR15 - negative powers of sin to cot power
TR16 - negative powers of cos to tan power
TR22 - tan-cot powers to negative powers of sec-csc functions
TR111 - negative sin-cos-tan powers to csc-sec-cot

There are 4 combination transforms (CTR1 - CTR4) in which a sequence of
transformations are applied and the simplest expression is selected from
a few options.

Finally, there are the 2 rule lists (RL1 and RL2), which apply a
sequence of transformations and combined transformations, and the fu
algorithm itself, which applies rules and rule lists and selects the
best expressions. There is also a function L which counts the number
of trigonometric functions that appear in the expression.

Other than TR0, re-writing of expressions is not done by the transformations.
e.g. TR10i finds pairs of terms in a sum that are in the form like
cos(x)*cos(y) + sin(x)*sin(y). Such expression are targeted in a bottom-up
traversal of the expression, but no manipulation to make them appear is
attempted. For example,

Set-up for examples below:

>>> from time import time

>>> eq = cos(x + y)/cos(x)
>>> TR10i(eq.expand(trig=True))
-sin(x)*sin(y)/cos(x) + cos(y)

If the expression is put in "normal" form (with a common denominator) then
the transformation is successful:

>>> TR10i(_.normal())
cos(x + y)/cos(x)

TR11's behavior is similar. It rewrites double angles as smaller angles but
doesn't do any simplification of the result.

>>> TR11(sin(2)**a*cos(1)**(-a), 1)
(2*sin(1)*cos(1))**a*cos(1)**(-a)
>>> powsimp(_)
(2*sin(1))**a

The temptation is to try make these TR rules "smarter" but that should really
be done at a higher level; the TR rules should try maintain the "do one thing
well" principle.  There is one exception, however. In TR10i and TR9 terms are
recognized even when they are each multiplied by a common factor:

>>> fu(a*cos(x)*cos(y) + a*sin(x)*sin(y))
a*cos(x - y)

Factoring with factor_terms is used but it it "JIT"-like, being delayed
until it is deemed necessary. Furthermore, if the factoring does not
help with the simplification, it is not retained, so
a*cos(x)*cos(y) + a*sin(x)*sin(z) does not become the factored
(but unsimplified in the trigonometric sense) expression:

>>> fu(a*cos(x)*cos(y) + a*sin(x)*sin(z))
a*sin(x)*sin(z) + a*cos(x)*cos(y)

In some cases factoring might be a good idea, but the user is left
to make that decision. For example:

>>> expr = ((15*sin(2*x) + 19*sin(x + y) + 17*sin(x + z) + 19*cos(x - z) +
...          25)*(20*sin(2*x) + 15*sin(x + y) + sin(y + z) + 14*cos(x - z) +
...               14*cos(y - z))*(9*sin(2*y) + 12*sin(y + z) +
...                               10*cos(x - y) + 2*cos(y - z) +
...                               18)).expand(trig=True).expand()

In the expanded state, there are nearly 1000 trig functions:

>>> L(expr)
932

If the expression where factored first, this would take time but the
resulting expression would be transformed very quickly:

>>> def clock(f, n=2):
...    t = time(); f(); return round(time() - t, n)
...
>>> clock(lambda: factor(expr))  # doctest: +SKIP
0.86
>>> clock(lambda: TR10i(expr), 3)  # doctest: +SKIP
0.016

If the unexpanded expression is used, the transformation takes longer but
not as long as it took to factor it and then transform it:

>>> clock(lambda: TR10i(expr), 2)  # doctest: +SKIP
0.28

So neither expansion nor factoring is used in TR10i: if the
expression is already factored (or partially factored) then expansion
with trig=True would destroy what is already known and take
longer; if the expression is expanded, factoring may take longer than
simply applying the transformation itself.

Although the algorithms should be canonical, always giving the same
result, they may not yield the best result. This, in general, is
the nature of simplification where searching all possible transformation
paths is very expensive. Here is a simple example. There are 6 terms
in the following sum:

>>> expr = (sin(x)**2*cos(y)*cos(z) + sin(x)*sin(y)*cos(x)*cos(z) +
... sin(x)*sin(z)*cos(x)*cos(y) + sin(y)*sin(z)*cos(x)**2 + sin(y)*sin(z) +
... cos(y)*cos(z))
>>> args = expr.args

Serendipitously, fu gives the best result:

>>> fu(expr)
3*cos(y - z)/2 - cos(2*x + y + z)/2

But if different terms were combined, a less-optimal result might be
obtained, requiring some additional work to get better simplification,
but still less than optimal. The following shows an alternative form
of expr that resists optimal simplification once a given step

>>> TR9(-cos(x)**2*cos(y + z) + 3*cos(y - z)/2 +
...     cos(y + z)/2 + cos(-2*x + y + z)/4 - cos(2*x + y + z)/4)
sin(2*x)*sin(y + z)/2 - cos(x)**2*cos(y + z) + 3*cos(y - z)/2 + cos(y + z)/2

Here is a smaller expression that exhibits the same behavior:

>>> a = sin(x)*sin(z)*cos(x)*cos(y) + sin(x)*sin(y)*cos(x)*cos(z)
>>> TR10i(a)
sin(x)*sin(y + z)*cos(x)
>>> newa = _
>>> TR10i(expr - a)  # this combines two more of the remaining terms
sin(x)**2*cos(y)*cos(z) + sin(y)*sin(z)*cos(x)**2 + cos(y - z)
>>> TR10i(_ + newa) == _ + newa  # but now there is no more simplification
True

Without getting lucky or trying all possible pairings of arguments, the
final result may be less than optimal and impossible to find without
better heuristics or brute force trial of all possibilities.

Notes
=====

This work was started by Dimitar Vlahovski at the Technological School
"Electronic systems" (30.11.2011).

References
==========

Fu, Hongguang, Xiuqin Zhong, and Zhenbing Zeng. "Automated and readable
simplification of trigonometric expressions." Mathematical and computer
modelling 44.11 (2006): 1169-1177.

http://www.sosmath.com/trig/Trig5/trig5/pdf/pdf.html gives a formula sheet.

"""

from collections import defaultdict

from strategies.core import debug, identity
from strategies.tree import greedy

from .. import DIOFANT_DEBUG
from ..core import (Add, Dummy, Expr, I, Integer, Mul, Pow, Rational,
expand_mul, factor_terms, gcd_terms, pi, sympify)
from ..core.compatibility import ordered
from ..core.exprtools import Factors
from ..functions import (binomial, cos, cosh, cot, coth, csc, sec, sin, sinh,
sqrt, tan, tanh)
from ..functions.elementary.hyperbolic import HyperbolicFunction
from ..functions.elementary.trigonometric import TrigonometricFunction
from ..ntheory import perfect_power
from ..polys.polytools import factor
from .simplify import bottom_up

# ================== Fu-like tools ===========================

def TR0(rv):
"""Simplification of rational polynomials, trying to simplify
the expression, e.g. combine things like 3*x + 2*x, etc....

"""
# although it would be nice to use cancel, it doesn't work
# with noncommutatives
return rv.normal().factor().expand()

def TR1(rv):
"""Replace sec, csc with 1/cos, 1/sin

Examples
========

>>> TR1(2*csc(x) + sec(x))
1/cos(x) + 2/sin(x)

"""

def f(rv):
if isinstance(rv, sec):
a = rv.args[0]
return 1/cos(a)
elif isinstance(rv, csc):
a = rv.args[0]
return 1/sin(a)
return rv

return bottom_up(rv, f)

def TR2(rv):
"""Replace tan and cot with sin/cos and cos/sin

Examples
========

>>> TR2(tan(x))
sin(x)/cos(x)
>>> TR2(cot(x))
cos(x)/sin(x)
>>> TR2(tan(tan(x) - sin(x)/cos(x)))
0

"""

def f(rv):
if isinstance(rv, tan):
a = rv.args[0]
return sin(a)/cos(a)
elif isinstance(rv, cot):
a = rv.args[0]
return cos(a)/sin(a)
return rv

return bottom_up(rv, f)

def TR2i(rv, half=False):
"""Converts ratios involving sin and cos as follows::
sin(x)/cos(x) -> tan(x)
sin(x)/(cos(x) + 1) -> tan(x/2) if half=True

Examples
========

>>> TR2i(sin(x)/cos(x))
tan(x)

Powers of the numerator and denominator are also recognized

>>> TR2i(sin(x)**2/(cos(x) + 1)**2, half=True)
tan(x/2)**2

The transformation does not take place unless assumptions allow
(i.e. the base must be positive or the exponent must be an integer
for both numerator and denominator)

>>> TR2i(sin(x)**a/(cos(x) + 1)**a)
(cos(x) + 1)**(-a)*sin(x)**a

"""

def f(rv):
if not rv.is_Mul:
return rv

n, d = rv.as_numer_denom()
if n.is_Atom or d.is_Atom:
return rv

def ok(k, e):
# initial filtering of factors
return (
(e.is_integer or k.is_positive) and (
k.func in (sin, cos) or (half and
len(k.args) >= 2 and
any(any(isinstance(ai, cos) or ai.is_Pow and ai.base is cos
for ai in Mul.make_args(a)) for a in k.args))))

n = n.as_powers_dict()
ndone = [(k, n.pop(k)) for k in list(n) if not ok(k, n[k])]
if not n:
return rv

d = d.as_powers_dict()
ddone = [(k, d.pop(k)) for k in list(d) if not ok(k, d[k])]
if not d:
return rv

# factoring if necessary

def factorize(d, ddone):
newk = []
for k in d:
if k.is_Add and len(k.args) > 1:
knew = factor(k) if half else factor_terms(k)
if knew != k:
newk.append((k, knew))
if newk:
for i, (k, knew) in enumerate(newk):
del d[k]
newk[i] = knew
newk = Mul(*newk).as_powers_dict()
for k in newk:
v = d[k] + newk[k]
if ok(k, v):
d[k] = v
else:
ddone.append((k, v))
factorize(n, ndone)
factorize(d, ddone)

# joining
t = []
for k in n:
if isinstance(k, sin):
a = cos(k.args[0], evaluate=False)
if a in d and d[a] == n[k]:
t.append(tan(k.args[0])**n[k])
n[k] = d[a] = None
elif half:
a1 = 1 + a
if a1 in d and d[a1] == n[k]:
t.append((tan(k.args[0]/2))**n[k])
n[k] = d[a1] = None
elif isinstance(k, cos):
a = sin(k.args[0], evaluate=False)
if a in d and d[a] == n[k]:
t.append(tan(k.args[0])**-n[k])
n[k] = d[a] = None
elif (half and k.is_Add and k.args[0] == 1 and
isinstance(k.args[1], cos)):
a = sin(k.args[1].args[0], evaluate=False)
if a in d and d[a] == n[k] and (d[a].is_integer or
a.is_positive):
t.append(tan(a.args[0]/2)**-n[k])
n[k] = d[a] = None

if t:
rv = Mul(*(t + [b**e for b, e in n.items() if e])) /\
Mul(*[b**e for b, e in d.items() if e])
rv *= Mul(*[b**e for b, e in ndone])/Mul(*[b**e for b, e in ddone])

return rv

return bottom_up(rv, f)

def TR3(rv):
"""Induced formula: example sin(-a) = -sin(a)

Examples
========

>>> TR3(cos(y - x*(y - x)))
cos(x*(x - y) + y)
>>> cos(pi/2 + x)
-sin(x)
>>> cos(30*pi/2 + x)
-cos(x)

"""
from .simplify import signsimp

# Negative argument (already automatic for funcs like sin(-x) -> -sin(x)
# but more complicated expressions can use it, too). Also, trig angles
# between pi/4 and pi/2 are not reduced to an angle between 0 and pi/4.
# The following are automatically handled:
#   Argument of type: pi/2 +/- angle
#   Argument of type: pi +/- angle
#   Argument of type : 2k*pi +/- angle

def f(rv):
if not isinstance(rv, TrigonometricFunction):
return rv
rv = rv.func(signsimp(rv.args[0]))
if (rv.args[0] - pi/4).is_positive is (pi/2 - rv.args[0]).is_positive is True:
fmap = {cos: sin, sin: cos, tan: cot, cot: tan, sec: csc, csc: sec}
rv = fmap[rv.func](pi/2 - rv.args[0])
return rv

return bottom_up(rv, f)

def TR4(rv):
"""Identify values of special angles.

a=  0   pi/6        pi/4        pi/3        pi/2
----------------------------------------------------
cos(a)  0   1/2         sqrt(2)/2   sqrt(3)/2   1
sin(a)  1   sqrt(3)/2   sqrt(2)/2   1/2         0
tan(a)  0   sqt(3)/3    1           sqrt(3)     --

Examples
========

>>> for s in (0, pi/6, pi/4, pi/3, pi/2):
...    print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s)))
...
1 0 0 zoo
sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3)
sqrt(2)/2 sqrt(2)/2 1 1
1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3
0 1 zoo 0

"""
# special values at 0, pi/6, pi/4, pi/3, pi/2 already handled
return rv

def _TR56(rv, f, g, h, max, pow):
"""Helper for TR5 and TR6 to replace f**2 with h(g**2)

Options
=======

max :   controls size of exponent that can appear on f
e.g. if max=4 then f**4 will be changed to h(g**2)**2.
pow :   controls whether the exponent must be a perfect power of 2
e.g. if pow=True (and max >= 6) then f**6 will not be changed
but f**8 will be changed to h(g**2)**4

>>> h = lambda x: 1 - x
>>> _TR56(sin(x)**3, sin, cos, h, 4, False)
sin(x)**3
>>> _TR56(sin(x)**6, sin, cos, h, 6, False)
(-cos(x)**2 + 1)**3
>>> _TR56(sin(x)**6, sin, cos, h, 6, True)
sin(x)**6
>>> _TR56(sin(x)**8, sin, cos, h, 10, True)
(-cos(x)**2 + 1)**4

"""

def _f(rv):
# I'm not sure if this transformation should target all even powers
# or only those expressible as powers of 2. Also, should it only
# make the changes in powers that appear in sums -- making an isolated
# change is not going to allow a simplification as far as I can tell.
if not (rv.is_Pow and rv.base.func == f):
return rv

if rv.exp.is_negative:
return rv
if (rv.exp - max).is_positive:
return rv
if rv.exp == 2:
return h(g(rv.base.args[0])**2)
else:
if rv.exp == 4:
e = 2
elif not pow:
if rv.exp % 2:
return rv
e = rv.exp//2
else:
p = perfect_power(rv.exp)
if not p:
return rv
e = rv.exp//2
return h(g(rv.base.args[0])**2)**e

return bottom_up(rv, _f)

def TR5(rv, max=4, pow=False):
"""Replacement of sin**2 with 1 - cos(x)**2.

See _TR56 docstring for advanced use of max and pow.

Examples
========

>>> TR5(sin(x)**2)
-cos(x)**2 + 1
>>> TR5(sin(x)**-2)  # unchanged
sin(x)**(-2)
>>> TR5(sin(x)**4)
(-cos(x)**2 + 1)**2

"""
return _TR56(rv, sin, cos, lambda x: 1 - x, max=max, pow=pow)

def TR6(rv, max=4, pow=False):
"""Replacement of cos**2 with 1 - sin(x)**2.

See _TR56 docstring for advanced use of max and pow.

Examples
========

>>> TR6(cos(x)**2)
-sin(x)**2 + 1
>>> TR6(cos(x)**-2)  #unchanged
cos(x)**(-2)
>>> TR6(cos(x)**4)
(-sin(x)**2 + 1)**2

"""
return _TR56(rv, cos, sin, lambda x: 1 - x, max=max, pow=pow)

def TR7(rv):
"""Lowering the degree of cos(x)**2

Examples
========

>>> TR7(cos(x)**2)
cos(2*x)/2 + 1/2
>>> TR7(cos(x)**2 + 1)
cos(2*x)/2 + 3/2

"""

def f(rv):
if not (rv.is_Pow and rv.base.func == cos and rv.exp == 2):
return rv
return (1 + cos(2*rv.base.args[0]))/2

return bottom_up(rv, f)

def TR8(rv, first=True):
"""Converting products of cos and/or sin to a sum or
difference of cos and or sin terms.

Examples
========

>>> TR8(cos(2)*cos(3))
cos(5)/2 + cos(1)/2
>>> TR8(cos(2)*sin(3))
sin(5)/2 + sin(1)/2
>>> TR8(sin(2)*sin(3))
-cos(5)/2 + cos(1)/2

"""

def f(rv):
if not (rv.is_Mul or rv.is_Pow and
rv.base.func in (cos, sin) and
(rv.exp.is_integer or rv.base.is_positive)):
return rv

if first:
n, d = [expand_mul(i) for i in rv.as_numer_denom()]
newn = TR8(n, first=False)
newd = TR8(d, first=False)
if newn != n or newd != d:
rv = gcd_terms(newn/newd)
if rv.is_Mul and rv.args[0].is_Rational and \
rv = Mul(*rv.as_coeff_Mul())
return rv

args = {cos: [], sin: [], None: []}
for a in ordered(Mul.make_args(rv)):
if a.func in (cos, sin):
args[a.func].append(a.args[0])
elif (a.is_Pow and a.exp.is_Integer and a.exp > 0 and
a.base.func in (cos, sin)):
# XXX this is ok but pathological expression could be handled
# more efficiently as in TRmorrie
args[a.base.func].extend([a.base.args[0]]*a.exp)
else:
args[None].append(a)
c = args[cos]
s = args[sin]
if not (c and s or len(c) > 1 or len(s) > 1):
return rv

args = args[None]
n = min(len(c), len(s))
for i in range(n):
a1 = s.pop()
a2 = c.pop()
args.append((sin(a1 + a2) + sin(a1 - a2))/2)
while len(c) > 1:
a1 = c.pop()
a2 = c.pop()
args.append((cos(a1 + a2) + cos(a1 - a2))/2)
if c:
args.append(cos(c.pop()))
while len(s) > 1:
a1 = s.pop()
a2 = s.pop()
args.append((-cos(a1 + a2) + cos(a1 - a2))/2)
if s:
args.append(sin(s.pop()))
return TR8(expand_mul(Mul(*args)))

return bottom_up(rv, f)

def TR9(rv):
"""Sum of cos or sin terms as a product of cos or sin.

Examples
========

>>> TR9(cos(1) + cos(2))
2*cos(1/2)*cos(3/2)
>>> TR9(cos(1) + 2*sin(1) + 2*sin(2))
cos(1) + 4*sin(3/2)*cos(1/2)

If no change is made by TR9, no re-arrangement of the
expression will be made. For example, though factoring
of common term is attempted, if the factored expression
wasn't changed, the original expression will be returned:

>>> TR9(cos(3) + cos(3)*cos(2))
cos(3) + cos(2)*cos(3)

"""

def f(rv):
return rv

def do(rv, first=True):
# cos(a)+/-cos(b) can be combined into a product of cosines and
# sin(a)+/-sin(b) can be combined into a product of cosine and
# sine.
#
# If there are more than two args, the pairs which "work" will
# have a gcd extractable and the remaining two terms will have
# the above structure -- all pairs must be checked to find the
# ones that work. args that don't have a common set of symbols
# are skipped since this doesn't lead to a simpler formula and
# also has the arbitrariness of combining, for example, the x
# and y term instead of the y and z term in something like
# cos(x) + cos(y) + cos(z).

return rv

args = list(ordered(rv.args))
if len(args) != 2:
hit = False
for i in range(len(args)):
ai = args[i]
if ai is None:
continue
for j in range(i + 1, len(args)):
aj = args[j]
if aj is None:
continue
was = ai + aj
new = do(was)
if new != was:
args[i] = new  # update in place
args[j] = None
hit = True
break  # go to next i
if hit:
rv = Add(*[_f for _f in args if _f])
rv = do(rv)

return rv

split = trig_split(*args)
if not split:
return rv
gcd, n1, n2, a, b, iscos = split

# application of rule if possible
if iscos:
if n1 == n2:
return gcd*n1*2*cos((a + b)/2)*cos((a - b)/2)
if n1 < 0:
a, b = b, a
return -2*gcd*sin((a + b)/2)*sin((a - b)/2)
else:
if n1 == n2:
return gcd*n1*2*sin((a + b)/2)*cos((a - b)/2)
if n1 < 0:
a, b = b, a
return 2*gcd*cos((a + b)/2)*sin((a - b)/2)

return process_common_addends(rv, do)  # DON'T sift by free symbols

return bottom_up(rv, f)

def TR10(rv, first=True):
"""Separate sums in cos and sin.

Examples
========

>>> TR10(cos(a + b))
-sin(a)*sin(b) + cos(a)*cos(b)
>>> TR10(sin(a + b))
sin(a)*cos(b) + sin(b)*cos(a)
>>> TR10(sin(a + b + c))
(-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) +
(sin(a)*cos(b) + sin(b)*cos(a))*cos(c)

"""

def f(rv):
if rv.func not in (cos, sin):
return rv

f = rv.func
arg = rv.args[0]
if first:
args = list(ordered(arg.args))
else:
args = list(arg.args)
a = args.pop()
if f == sin:
return sin(a)*TR10(cos(b), first=False) + \
cos(a)*TR10(sin(b), first=False)
else:
return cos(a)*TR10(cos(b), first=False) - \
sin(a)*TR10(sin(b), first=False)
else:
if f == sin:
return sin(a)*cos(b) + cos(a)*sin(b)
else:
return cos(a)*cos(b) - sin(a)*sin(b)
return rv

return bottom_up(rv, f)

def TR10i(rv):
"""Sum of products to function of sum.

Examples
========

>>> TR10i(cos(1)*cos(3) + sin(1)*sin(3))
cos(2)
>>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3))
cos(3) + sin(4)
>>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x)
2*sqrt(2)*x*sin(x + pi/6)

"""
global _ROOT2, _ROOT3, _invROOT3
if _ROOT2 is None:
_roots()

def f(rv):
return rv

def do(rv, first=True):
# args which can be expressed as A*(cos(a)*cos(b)+/-sin(a)*sin(b))
# or B*(cos(a)*sin(b)+/-cos(b)*sin(a)) can be combined into
# A*f(a+/-b) where f is either sin or cos.
#
# If there are more than two args, the pairs which "work" will have
# a gcd extractable and the remaining two terms will have the above
# structure -- all pairs must be checked to find the ones that
# work.

return rv

args = list(ordered(rv.args))
if len(args) != 2:
hit = False
for i in range(len(args)):
ai = args[i]
if ai is None:
continue
for j in range(i + 1, len(args)):
aj = args[j]
if aj is None:
continue
was = ai + aj
new = do(was)
if new != was:
args[i] = new  # update in place
args[j] = None
hit = True
break  # go to next i
if hit:
rv = Add(*[_f for _f in args if _f])
rv = do(rv)

return rv

split = trig_split(*args, two=True)
if not split:
return rv
gcd, n1, n2, a, b, same = split

# identify and get c1 to be cos then apply rule if possible
if same:  # coscos, sinsin
gcd = n1*gcd
if n1 == n2:
return gcd*cos(a - b)
return gcd*cos(a + b)
else:  # cossin, cossin
gcd = n1*gcd
if n1 == n2:
return gcd*sin(a + b)
return gcd*sin(b - a)

rv, do, lambda x: tuple(ordered(x.free_symbols)))

# need to check for inducible pairs in ratio of sqrt(3):1 that
# appeared in different lists when sorting by coefficient
for a in rv.args:
hit = 0
if a.is_Mul:
for ai in a.args:
if ai.is_Pow and ai.exp == Rational(1, 2) and \
ai.base.is_Integer:
hit = 1
break
if not hit:

# no need to check all pairs -- just check for the onees
# that have the right ratio
args = []
for b in [_ROOT3*a, _invROOT3]:
continue
continue
new = do(was)
if new != was:
args.append(new)
break
if args:
else:
rv = do(rv)  # final pass to resolve any new inducible pairs
break

return rv

return bottom_up(rv, f)

def TR11(rv, base=None):
"""Function of double angle to product. The base argument can be used
to indicate what is the un-doubled argument, e.g. if 3*pi/7 is the base
then cosine and sine functions with argument 6*pi/7 will be replaced.

Examples
========

>>> TR11(sin(2*x))
2*sin(x)*cos(x)
>>> TR11(cos(2*x))
-sin(x)**2 + cos(x)**2
>>> TR11(sin(4*x))
4*(-sin(x)**2 + cos(x)**2)*sin(x)*cos(x)
>>> TR11(sin(4*x/3))
4*(-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3)

If the arguments are simply integers, no change is made
unless a base is provided:

>>> TR11(cos(2))
cos(2)
>>> TR11(cos(4), 2)
-sin(2)**2 + cos(2)**2

There is a subtle issue here in that autosimplification will convert
some higher angles to lower angles

>>> cos(6*pi/7) + cos(3*pi/7)
-cos(pi/7) + cos(3*pi/7)

The 6*pi/7 angle is now pi/7 but can be targeted with TR11 by supplying
the 3*pi/7 base:

>>> TR11(_, 3*pi/7)
-sin(3*pi/7)**2 + cos(3*pi/7)**2 + cos(3*pi/7)

"""

def f(rv):
if rv.func not in (cos, sin):
return rv

if base:
f = rv.func
t = f(base*2)
co = Integer(1)
if t.is_Mul:
co, t = t.as_coeff_Mul()
if t.func not in (cos, sin):
return rv
if rv.args[0] == t.args[0]:
c = cos(base)
s = sin(base)
if f is cos:
return (c**2 - s**2)/co
else:
return 2*c*s/co
return rv

elif not rv.args[0].is_Number:
# make a change if the leading coefficient's numerator is
# divisible by 2
c, m = rv.args[0].as_coeff_Mul(rational=True)
if c.numerator % 2 == 0:
arg = c.numerator//2*m/c.denominator
c = TR11(cos(arg))
s = TR11(sin(arg))
if rv.func == sin:
rv = 2*s*c
else:
rv = c**2 - s**2
return rv

return bottom_up(rv, f)

def TR12(rv, first=True):
"""Separate sums in tan.

Examples
========

>>> TR12(tan(x + y))
(tan(x) + tan(y))/(-tan(x)*tan(y) + 1)

"""

def f(rv):
if not rv.func == tan:
return rv

arg = rv.args[0]
if first:
args = list(ordered(arg.args))
else:
args = list(arg.args)
a = args.pop()
tb = TR12(tan(b), first=False)
else:
tb = tan(b)
return (tan(a) + tb)/(1 - tan(a)*tb)
return rv

return bottom_up(rv, f)

def TR12i(rv):
"""Combine tan arguments as
(tan(y) + tan(x))/(tan(x)*tan(y) - 1) -> -tan(x + y)

Examples
========

>>> ta, tb, tc = [tan(i) for i in (a, b, c)]
>>> TR12i((ta + tb)/(-ta*tb + 1))
tan(a + b)
>>> TR12i((ta + tb)/(ta*tb - 1))
-tan(a + b)
>>> TR12i((-ta - tb)/(ta*tb - 1))
tan(a + b)
>>> eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1))
>>> TR12i(eq.expand())
-3*tan(a + b)*tan(a + c)/(2*(tan(a) + tan(b) - 1))

"""
from ..polys import factor

def f(rv):
if not (rv.is_Add or rv.is_Mul or rv.is_Pow):
return rv

n, d = rv.as_numer_denom()
if not d.args or not n.args:
return rv

dok = {}

def ok(di):
m = as_f_sign_1(di)
if m:
g, f, s = m
if s == -1 and f.is_Mul and len(f.args) == 2 and \
all(isinstance(fi, tan) for fi in f.args):
return g, f

d_args = list(Mul.make_args(d))
for i, di in enumerate(d_args):
m = ok(di)
if m:
g, t = m
s = Add(*[_.args[0] for _ in t.args])
dok[s] = Integer(1)
d_args[i] = g
continue
di = factor(di)
if di.is_Mul:
d_args.extend(di.args)
d_args[i] = Integer(1)
elif di.is_Pow and (di.exp.is_integer or di.base.is_positive):
m = ok(di.base)
if m:
g, t = m
s = Add(*[_.args[0] for _ in t.args])
dok[s] = di.exp
d_args[i] = g**di.exp
else:
di = factor(di)
if di.is_Mul:
d_args.extend(di.args)
d_args[i] = Integer(1)
if not dok:
return rv

def ok(ni):
if ni.is_Add and len(ni.args) == 2:
a, b = ni.args
if isinstance(a, tan) and isinstance(b, tan):
return a, b
n_args = list(Mul.make_args(factor_terms(n)))
hit = False
for i, ni in enumerate(n_args):
m = ok(ni)
if not m:
m = ok(-ni)
if m:
n_args[i] = Integer(-1)
else:
ni = factor(ni)
if ni.is_Mul:
n_args.extend(ni.args)
n_args[i] = Integer(1)
continue
elif ni.is_Pow and (
ni.exp.is_integer or ni.base.is_positive):
m = ok(ni.base)
if m:
n_args[i] = Integer(1)
else:
ni = factor(ni)
if ni.is_Mul:
n_args.extend(ni.args)
n_args[i] = Integer(1)
continue
else:
continue
else:
n_args[i] = Integer(1)
hit = True
s = Add(*[_.args[0] for _ in m])
ed = dok[s]
if newed is not None:
if newed:
dok[s] = newed
else:
dok.pop(s)
n_args[i] *= -tan(s)

if hit:
tan(a) for a in i.args]) - 1)**e for i, e in dok.items()])

return rv

return bottom_up(rv, f)

def TR13(rv):
"""Change products of tan or cot.

Examples
========

>>> TR13(tan(3)*tan(2))
-tan(2)/tan(5) - tan(3)/tan(5) + 1
>>> TR13(cot(3)*cot(2))
cot(2)*cot(5) + 1 + cot(3)*cot(5)

"""

def f(rv):
if not rv.is_Mul:
return rv

# XXX handle products of powers? or let power-reducing handle it?
args = {tan: [], cot: [], None: []}
for a in ordered(Mul.make_args(rv)):
if a.func in (tan, cot):
args[a.func].append(a.args[0])
else:
args[None].append(a)
t = args[tan]
c = args[cot]
if len(t) < 2 and len(c) < 2:
return rv
args = args[None]
while len(t) > 1:
t1 = t.pop()
t2 = t.pop()
args.append(1 - (tan(t1)/tan(t1 + t2) + tan(t2)/tan(t1 + t2)))
if t:
args.append(tan(t.pop()))
while len(c) > 1:
t1 = c.pop()
t2 = c.pop()
args.append(1 + cot(t1)*cot(t1 + t2) + cot(t2)*cot(t1 + t2))
if c:
args.append(cot(c.pop()))
return Mul(*args)

return bottom_up(rv, f)

def TRmorrie(rv):
"""Returns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x))

Examples
========

>>> TRmorrie(cos(x)*cos(2*x))
sin(4*x)/(4*sin(x))
>>> TRmorrie(7*Mul(*[cos(x) for x in range(10)]))
7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3))

Sometimes autosimplification will cause a power to be
not recognized. e.g. in the following, cos(4*pi/7) automatically
simplifies to -cos(3*pi/7) so only 2 of the 3 terms are
recognized:

>>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7))
-sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7))

A touch by TR8 resolves the expression to a Rational

>>> TR8(_)
-1/8

In this case, if eq is unsimplified, the answer is obtained
directly:

>>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9)
>>> TRmorrie(eq)
1/16

is not simplified without further work:

>>> TR3(eq)
sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2
>>> TRmorrie(_)
sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9))
>>> TR8(_)
cos(7*pi/18)/(16*sin(pi/9))
>>> TR3(_)
1/16

The original expression would have resolve to 1/16 directly with TR8,
however:

>>> TR8(eq)
1/16

References
==========

https://en.wikipedia.org/wiki/Morrie%27s_law

"""

def f(rv):
if not rv.is_Mul:
return rv

args = defaultdict(list)
coss = {}
other = []
for c in rv.args:
b, e = c.as_base_exp()
if e.is_Integer and isinstance(b, cos):
co, a = b.args[0].as_coeff_Mul()
args[a].append(co)
coss[b] = e
else:
other.append(c)

new = []
for a in args:
c = args[a]
c.sort()
no = []
while c:
k = 0
cc = ci = c[0]
while cc in c:
k += 1
cc *= 2
if k > 1:
newarg = sin(2**k*ci*a)/2**k/sin(ci*a)
# see how many times this can be taken
take = None
ccs = []
for i in range(k):
cc /= 2
key = cos(a*cc, evaluate=False)
ccs.append(cc)
take = min(coss[key], take or coss[key])
# update exponent counts
for i in range(k):
cc = ccs.pop()
key = cos(a*cc, evaluate=False)
coss[key] -= take
if not coss[key]:
c.remove(cc)
new.append(newarg**take)
else:
no.append(c.pop(0))
c[:] = no

if new:
rv = Mul(*(new + other + [
cos(k*a, evaluate=False) for a in args for k in args[a]]))

return rv

return bottom_up(rv, f)

def TR14(rv, first=True):
"""Convert factored powers of sin and cos identities into simpler
expressions.

Examples
========

>>> TR14((cos(x) - 1)*(cos(x) + 1))
-sin(x)**2
>>> TR14((sin(x) - 1)*(sin(x) + 1))
-cos(x)**2
>>> p1 = (cos(x) + 1)*(cos(x) - 1)
>>> p2 = (cos(y) - 1)*2*(cos(y) + 1)
>>> p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1))
>>> TR14(p1*p2*p3*(x - 1))
-18*(x - 1)*sin(x)**2*sin(y)**4

"""

def f(rv):
if not rv.is_Mul:
return rv

if first:
# sort them by location in numerator and denominator
# so the code below can just deal with positive exponents
n, d = rv.as_numer_denom()
if d != 1:
newn = TR14(n, first=False)
newd = TR14(d, first=False)
if newn != n or newd != d:
rv = newn/newd
return rv

other = []
process = []
for a in rv.args:
if a.is_Pow:
b, e = a.as_base_exp()
if not (e.is_integer or b.is_positive):
other.append(a)
continue
a = b
else:
e = Integer(1)
m = as_f_sign_1(a)
if not m or m[1].func not in (cos, sin):
if e == 1:
other.append(a)
else:
other.append(a**e)
continue
g, f, si = m
process.append((g, e.is_Number, e, f, si, a))

# sort them to get like terms next to each other
process = list(ordered(process))

# keep track of whether there was any change
nother = len(other)

# access keys
keys = (g, t, e, f, si, a) = list(range(6))

while process:
A = process.pop(0)
if process:
B = process[0]

if A[e].is_Number and B[e].is_Number:
# both exponents are numbers
if A[f] == B[f]:
if A[si] != B[si]:
B = process.pop(0)
take = min(A[e], B[e])

# reinsert any remainder
# the B will likely sort after A so check it first
if B[e] != take:
rem = [B[i] for i in keys]
rem[e] -= take
process.insert(0, rem)
elif A[e] != take:
rem = [A[i] for i in keys]
rem[e] -= take
process.insert(0, rem)

if isinstance(A[f], cos):
t = sin
else:
t = cos
other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take)
continue

elif A[e] == B[e]:
# both exponents are equal symbols
if A[f] == B[f]:
if A[si] != B[si]:
B = process.pop(0)
take = A[e]
if isinstance(A[f], cos):
t = sin
else:
t = cos
other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take)
continue

# either we are done or neither condition above applied
other.append(A[a]**A[e])

if len(other) != nother:
rv = Mul(*other)

return rv

return bottom_up(rv, f)

def TR15(rv, max=4, pow=False):
"""Convert sin(x)*-2 to 1 + cot(x)**2.

See _TR56 docstring for advanced use of max and pow.

Examples
========

>>> TR15(1 - 1/sin(x)**2)
-cot(x)**2

"""

def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, sin)):
return rv

ia = 1/rv
a = _TR56(ia, sin, cot, lambda x: 1 + x, max=max, pow=pow)
if a != ia:
rv = a
return rv

return bottom_up(rv, f)

def TR16(rv, max=4, pow=False):
"""Convert cos(x)*-2 to 1 + tan(x)**2.

See _TR56 docstring for advanced use of max and pow.

Examples
========

>>> TR16(1 - 1/cos(x)**2)
-tan(x)**2

"""

def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, cos)):
return rv

ia = 1/rv
a = _TR56(ia, cos, tan, lambda x: 1 + x, max=max, pow=pow)
if a != ia:
rv = a
return rv

return bottom_up(rv, f)

def TR111(rv):
"""Convert f(x)**-i to g(x)**i where either i is an integer
or the base is positive and f, g are: tan, cot; sin, csc; or cos, sec.

Examples
========

>>> TR111(1 - 1/tan(x)**2)
-cot(x)**2 + 1

"""

def f(rv):
if not (isinstance(rv, Pow) and
(rv.base.is_positive or rv.exp.is_integer and rv.exp.is_negative)):
return rv

if isinstance(rv.base, tan):
return cot(rv.base.args[0])**-rv.exp
elif isinstance(rv.base, sin):
return csc(rv.base.args[0])**-rv.exp
elif isinstance(rv.base, cos):
return sec(rv.base.args[0])**-rv.exp
return rv

return bottom_up(rv, f)

def TR22(rv, max=4, pow=False):
"""Convert tan(x)**2 to sec(x)**2 - 1 and cot(x)**2 to csc(x)**2 - 1.

See _TR56 docstring for advanced use of max and pow.

Examples
========

>>> TR22(1 + tan(x)**2)
sec(x)**2
>>> TR22(1 + cot(x)**2)
csc(x)**2

"""

def f(rv):
if not (isinstance(rv, Pow) and rv.base.func in (cot, tan)):
return rv

rv = _TR56(rv, tan, sec, lambda x: x - 1, max=max, pow=pow)
rv = _TR56(rv, cot, csc, lambda x: x - 1, max=max, pow=pow)
return rv

return bottom_up(rv, f)

def TRpower(rv):
"""Convert sin(x)**n and cos(x)**n with positive n to sums.

Examples
========

>>> TRpower(sin(x)**6)
-15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 + 5/16
>>> TRpower(sin(x)**3*cos(2*x)**4)
(3*sin(x)/4 - sin(3*x)/4)*(cos(4*x)/2 + cos(8*x)/8 + 3/8)

References
==========

* https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae

"""

def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, (sin, cos))):
return rv
b, n = rv.as_base_exp()
x = b.args[0]
if n.is_Integer and n.is_positive:
if n.is_odd and isinstance(b, cos):
rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x)
for k in range((n + 1)/2)])
elif n.is_odd and isinstance(b, sin):
rv = (2**(1-n)*(-1)**((n-1)/2) *
for k in range((n + 1)/2)]))
elif n.is_even and isinstance(b, cos):
rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x)
for k in range(n/2)])
elif n.is_even and isinstance(b, sin):
rv = (2**(1-n)*(-1)**(n/2) *
for k in range(n/2)]))
if n.is_even:
rv += 2**(-n)*binomial(n, n/2)
return rv

return bottom_up(rv, f)

def L(rv):
"""Return count of trigonometric functions in expression.

Examples
========

>>> L(cos(x)+sin(x))
2

"""
return Integer(rv.count(TrigonometricFunction))

# ============== end of basic Fu-like tools =====================

if DIOFANT_DEBUG:  # pragma: no cover
(TR0, TR1, TR2, TR3, TR4, TR5,
TR6, TR7, TR8, TR9, TR10, TR11, TR12, TR13,
TR2i, TRmorrie, TR14, TR15, TR16,
TR12i, TR111, TR22) = list(map(debug, (TR0, TR1, TR2, TR3, TR4, TR5,
TR6, TR7, TR8, TR9, TR10, TR11,
TR12, TR13, TR2i, TRmorrie, TR14,
TR15, TR16, TR12i, TR111, TR22)))

# tuples are chains  --  (f, g) -> lambda x: g(f(x))
# lists are choices  --  [f, g] -> lambda x: min(f(x), g(x), key=objective)

CTR1 = [(TR5, TR0), (TR6, TR0), identity]

CTR2 = (TR11, [(TR5, TR0), (TR6, TR0), TR0])

CTR3 = [(TRmorrie, TR8, TR0), (TRmorrie, TR8, TR10i, TR0), identity]

CTR4 = [(TR4, TR10i), identity]

RL1 = (TR4, TR3, TR4, TR12, TR4, TR13, TR4, TR0)

# XXX it's a little unclear how this one is to be implemented
# see Fu paper of reference, page 7. What is the Union symbol refering to?
# The diagram shows all these as one chain of transformations, but the
# text refers to them being applied independently. Also, a break
# if L starts to increase has not been implemented.
RL2 = [
(TR4, TR3, TR10, TR4, TR3, TR11),
(TR5, TR7, TR11, TR4),
(CTR3, CTR1, TR9, CTR2, TR4, TR9, TR9, CTR4),
identity,
]

[docs]def fu(rv, measure=lambda x: (L(x), x.count_ops())):
"""Attempt to simplify expression by using transformation rules given
in the algorithm by Fu et al.

:func:fu will try to minimize the objective function measure.
By default this first minimizes the number of trig terms and then minimizes
the number of total operations.

Examples
========

>>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6))
3/2
>>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x))
2*sqrt(2)*sin(x + pi/3)

CTR1 example

>>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2
>>> fu(eq)
cos(x)**4 - 2*cos(y)**2 + 2

CTR2 example

>>> fu(Rational(1, 2) - cos(2*x)/2)
sin(x)**2

CTR3 example

>>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b)))
sqrt(2)*sin(a + b + pi/4)

CTR4 example

>>> fu(sqrt(3)*cos(x)/2 + sin(x)/2)
sin(x + pi/3)

Example 1

>>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4)
-cos(x)**2 + cos(y)**2

Example 2

>>> fu(cos(4*pi/9))
sin(pi/18)
>>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9))
1/16

Example 3

>>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18))
-sqrt(3)

Objective function example

>>> fu(sin(x)/cos(x))  # default objective function
tan(x)
>>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count
sin(x)/cos(x)

References
==========
DESTIME2006/DES_contribs/Fu/simplification.pdf

"""
fRL1 = greedy(RL1, measure)
fRL2 = greedy(RL2, measure)

was = rv
rv = sympify(rv)
if not isinstance(rv, Expr):
return rv.func(*[fu(a, measure=measure) for a in rv.args])
rv = TR1(rv)
if rv.has(tan, cot):
rv1 = fRL1(rv)
if (measure(rv1) < measure(rv)):
rv = rv1
if rv.has(tan, cot):
rv = TR2(rv)
if rv.has(sin, cos):
rv1 = fRL2(rv)
rv2 = TR8(TRmorrie(rv1))
rv = min([was, rv, rv1, rv2], key=measure)
return min(TR2i(rv), rv, key=measure)

"""Apply do to addends of rv that (if key1=True) share at least
a common absolute value of their coefficient and the value of key2 when
applied to the argument. If key1 is False key2 must be supplied and
will be the only key applied.

"""

# collect by absolute value of coefficient and key2
absc = defaultdict(list)
if key1:
for a in rv.args:
c, a = a.as_coeff_Mul()
if c < 0:
c = -c
a = -a  # put the sign on a
absc[(c, key2(a) if key2 else 1)].append(a)
elif key2:
for a in rv.args:
absc[(Integer(1), key2(a))].append(a)
else:
raise ValueError('must have at least one key')

args = []
hit = False
for k in absc:
v = absc[k]
c, _ = k
if len(v) > 1:
new = do(e)
if new != e:
e = new
hit = True
args.append(c*e)
else:
args.append(c*v[0])
if hit:

return rv

fufuncs = '''
TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11
TR12 TR13 L TR2i TRmorrie TR12i
TR14 TR15 TR16 TR111 TR22'''.split()
FU = dict(zip(fufuncs, list(map(locals().get, fufuncs))))

def _roots():
global _ROOT2, _ROOT3, _invROOT3
_ROOT2, _ROOT3 = sqrt(2), sqrt(3)
_invROOT3 = 1/_ROOT3

_ROOT2 = None

def trig_split(a, b, two=False):
"""Return the gcd, s1, s2, a1, a2, bool where

If two is False (default) then::
a + b = gcd*(s1*f(a1) + s2*f(a2)) where f = cos if bool else sin
else:
if bool, a + b was +/- cos(a1)*cos(a2) +/- sin(a1)*sin(a2) and equals
n1*gcd*cos(a - b) if n1 == n2 else
n1*gcd*cos(a + b)
else a + b was +/- cos(a1)*sin(a2) +/- sin(a1)*cos(a2) and equals
n1*gcd*sin(a + b) if n1 = n2 else
n1*gcd*sin(b - a)

Examples
========

>>> trig_split(cos(x), cos(y))
(1, 1, 1, x, y, True)
>>> trig_split(2*cos(x), -2*cos(y))
(2, 1, -1, x, y, True)
>>> trig_split(cos(x)*sin(y), cos(y)*sin(y))
(sin(y), 1, 1, x, y, True)

>>> trig_split(cos(x), -sqrt(3)*sin(x), two=True)
(2, 1, -1, x, pi/6, False)
>>> trig_split(cos(x), sin(x), two=True)
(sqrt(2), 1, 1, x, pi/4, False)
>>> trig_split(cos(x), -sin(x), two=True)
(sqrt(2), 1, -1, x, pi/4, False)
>>> trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True)
(2*sqrt(2), 1, -1, x, pi/6, False)
>>> trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True)
(-2*sqrt(2), 1, 1, x, pi/3, False)
>>> trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True)
(sqrt(6)/3, 1, 1, x, pi/6, False)
>>> trig_split(-sqrt(6)*cos(x)*sin(y), -sqrt(2)*sin(x)*sin(y), two=True)
(-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False)

>>> trig_split(cos(x), sin(x))
>>> trig_split(cos(x), sin(z))
>>> trig_split(2*cos(x), -sin(x))
>>> trig_split(cos(x), -sqrt(3)*sin(x))
>>> trig_split(cos(x)*cos(y), sin(x)*sin(z))
>>> trig_split(cos(x)*cos(y), sin(x)*sin(y))
>>> trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True)

"""
global _ROOT2, _ROOT3, _invROOT3
if _ROOT2 is None:
_roots()

a, b = [Factors(i) for i in (a, b)]
ua, ub = a.normal(b)
gcd = a.gcd(b).as_expr()
n1 = n2 = 1
if -1 in ua.factors:
ua = ua.quo(Integer(-1))
n1 = -n1
elif -1 in ub.factors:
ub = ub.quo(Integer(-1))
n2 = -n2
a, b = [i.as_expr() for i in (ua, ub)]

def pow_cos_sin(a, two):
"""Return a as a tuple (r, c, s) such that
a = (r or 1)*(c or 1)*(s or 1).

Three arguments are returned (radical, c-factor, s-factor) as
long as the conditions set by two are met; otherwise None is
returned. If two is True there will be one or two non-None
values in the tuple: c and s or c and r or s and r or s or c with c
being a cosine function (if possible) else a sine, and s being a sine
function (if possible) else oosine. If two is False then there
will only be a c or s term in the tuple.

two also require that either two cos and/or sin be present (with
the condition that if the functions are the same the arguments are
different or vice versa) or that a single cosine or a single sine
be present with an optional radical.

If the above conditions dictated by two are not met then None
is returned.

"""
c = s = None
co = Integer(1)
if a.is_Mul:
co, a = a.as_coeff_Mul()
if len(a.args) > 2 or not two:
return
if a.is_Mul:
args = list(a.args)
else:
args = [a]
a = args.pop(0)
if isinstance(a, cos):
c = a
elif isinstance(a, sin):
s = a
elif a.is_Pow and a.exp == Rational(1, 2):  # autoeval doesn't allow -1/2
co *= a
else:
return
if args:
b = args[0]
if isinstance(b, cos):
if c:
s = b
else:
c = b
elif isinstance(b, sin):
if s:
c = b
else:
s = b
elif b.is_Pow and b.exp == Rational(1, 2):
co *= b
else:
return
return co if co != 1 else None, c, s
elif isinstance(a, cos):
c = a
elif isinstance(a, sin):
s = a
if c is None and s is None:
return
co = co if co != 1 else None
return co, c, s

# get the parts
m = pow_cos_sin(a, two)
if m is None:
return
coa, ca, sa = m
m = pow_cos_sin(b, two)
if m is None:
return
cob, cb, sb = m

# check them
if (not ca) and cb or ca and isinstance(ca, sin):
coa, ca, sa, cob, cb, sb = cob, cb, sb, coa, ca, sa
n1, n2 = n2, n1
if not two:  # need cos(x) and cos(y) or sin(x) and sin(y)
c = ca or sa
s = cb or sb
if not isinstance(c, s.func):
return
return gcd, n1, n2, c.args[0], s.args[0], isinstance(c, cos)
else:
if not coa and not cob:
if (ca and cb and sa and sb):
if not (isinstance(ca, sa.func) is isinstance(cb, sb.func)):
return
args = {j.args for j in (ca, sa)}
if not all(i.args in args for i in (cb, sb)):
return
return gcd, n1, n2, ca.args[0], sa.args[0], isinstance(ca, sa.func)
if ca and sa or cb and sb or \
two and (ca is None and sa is None or cb is None and sb is None):
return
c = ca or sa
s = cb or sb
if c.args != s.args:
return
if not coa:
coa = Integer(1)
if not cob:
cob = Integer(1)
if coa is cob:
gcd *= _ROOT2
return gcd, n1, n2, c.args[0], pi/4, False
elif coa/cob == _ROOT3:
gcd *= 2*cob
return gcd, n1, n2, c.args[0], pi/3, False
elif coa/cob == _invROOT3:
gcd *= 2*coa
return gcd, n1, n2, c.args[0], pi/6, False

def as_f_sign_1(e):
"""If e is a sum that can be written as g*(a + s) where
s is +/-1, return g, a, and s where a does
not have a leading negative coefficient.

Examples
========

>>> as_f_sign_1(x + 1)
(1, x, 1)
>>> as_f_sign_1(x - 1)
(1, x, -1)
>>> as_f_sign_1(-x + 1)
(-1, x, -1)
>>> as_f_sign_1(-x - 1)
(-1, x, 1)
>>> as_f_sign_1(2*x + 2)
(2, x, 1)

"""
if not e.is_Add or len(e.args) != 2:
return
# exact match
a, b = e.args
if a in (-1, 1):
g = Integer(1)
if b.is_Mul and b.args[0].is_Number and b.args[0] < 0:
a, b = -a, -b
g = -g
return g, b, a
# gcd match
a, b = [Factors(i) for i in e.args]
ua, ub = a.normal(b)
gcd = a.gcd(b).as_expr()
if -1 in ua.factors:
ua = ua.quo(Integer(-1))
n1 = -1
n2 = 1
elif -1 in ub.factors:
ub = ub.quo(Integer(-1))
n1 = 1
n2 = -1
else:
n1 = n2 = 1
a, b = [i.as_expr() for i in (ua, ub)]
if a == 1:
a, b = b, a
n1, n2 = n2, n1
if n1 == -1:
gcd = -gcd
n2 = -n2

if b == 1:
return gcd, a, n2

def _osborne(e, d):
"""Replace all hyperbolic functions with trig functions using
the Osborne rule.

Notes
=====

d is a dummy variable to prevent automatic evaluation
of trigonometric/hyperbolic functions.

References
==========

https://en.wikipedia.org/wiki/Hyperbolic_function

"""

def f(rv):
if not isinstance(rv, HyperbolicFunction):
return rv
a = rv.args[0]
a = a*d if not a.is_Add else Add._from_args([i*d for i in a.args])
if isinstance(rv, sinh):
return I*sin(a)
elif isinstance(rv, cosh):
return cos(a)
elif isinstance(rv, tanh):
return I*tan(a)
elif isinstance(rv, coth):
return cot(a)/I
else:
raise NotImplementedError('unhandled %s' % rv.func)

return bottom_up(e, f)

def _osbornei(e, d):
"""Replace all trig functions with hyperbolic functions using
the Osborne rule.

Notes
=====

d is a dummy variable to prevent automatic evaluation
of trigonometric/hyperbolic functions.

References
==========

https://en.wikipedia.org/wiki/Hyperbolic_function

"""

def f(rv):
if not isinstance(rv, TrigonometricFunction):
return rv
a = rv.args[0].xreplace({d: Integer(1)})
if isinstance(rv, sin):
return sinh(a)/I
elif isinstance(rv, cos):
return cosh(a)
elif isinstance(rv, tan):
return tanh(a)/I
elif isinstance(rv, cot):
return coth(a)*I
elif isinstance(rv, sec):
return 1/cosh(a)
elif isinstance(rv, csc):
return I/sinh(a)
else:
raise NotImplementedError('unhandled %s' % rv.func)

return bottom_up(e, f)

def hyper_as_trig(rv):
"""Return an expression containing hyperbolic functions in terms
of trigonometric functions. Any trigonometric functions initially
present are replaced with Dummy symbols and the function to undo
the masking and the conversion back to hyperbolics is also returned. It
should always be true that::

t, f = hyper_as_trig(expr)
expr == f(t)

Examples
========

>>> eq = sinh(x)**2 + cosh(x)**2
>>> t, f = hyper_as_trig(eq)
>>> f(fu(t))
cosh(2*x)

References
==========

https://en.wikipedia.org/wiki/Hyperbolic_function

"""
from .simplify import signsimp

trigs = rv.atoms(TrigonometricFunction)
reps = [(t, Dummy()) for t in trigs]

# get inversion substitutions in place
reps = [(v, k) for k, v in reps]

d = Dummy()

return _osborne(masked, d), lambda x: collect(signsimp(
_osbornei(x, d).xreplace(dict(reps))), I)

def sincos_to_sum(expr):
"""Convert products and powers of sin and cos to sums.

Applied power reduction TRpower first, then expands products, and
converts products to sums with TR8.

Examples
========

>>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2)
7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x)

"""

return TR8(expand_mul(TRpower(expr)))