Coverage for sympy/simplify/trigsimp.py : 15%

from __future__ import print_function, division 

from collections import defaultdict 

from sympy.core.cache import cacheit 

from sympy.core import (sympify, Basic, S, Expr, expand_mul, factor_terms, 

Mul, Dummy, igcd, FunctionClass, Add, symbols, Wild, expand) 

from sympy.core.compatibility import reduce, iterable 

from sympy.core.numbers import I, Integer 

from sympy.core.function import count_ops, _mexpand 

from sympy.functions.elementary.trigonometric import TrigonometricFunction 

from sympy.functions.elementary.hyperbolic import HyperbolicFunction 

from sympy.functions import sin, cos, exp, cosh, tanh, sinh, tan, cot, coth 

from sympy.strategies.core import identity 

from sympy.strategies.tree import greedy 

from sympy.polys import Poly 

from sympy.polys.polyerrors import PolificationFailed 

from sympy.polys.polytools import groebner 

from sympy.polys.domains import ZZ 

from sympy.polys import factor, cancel, parallel_poly_from_expr 

from sympy.utilities.misc import debug 

def trigsimp_groebner(expr, hints=[], quick=False, order="grlex", 

polynomial=False): 

""" 

Simplify trigonometric expressions using a groebner basis algorithm. 

This routine takes a fraction involving trigonometric or hyperbolic 

expressions, and tries to simplify it. The primary metric is the 

total degree. Some attempts are made to choose the simplest possible 

expression of the minimal degree, but this is non-rigorous, and also 

very slow (see the ``quick=True`` option). 

If ``polynomial`` is set to True, instead of simplifying numerator and 

denominator together, this function just brings numerator and denominator 

into a canonical form. This is much faster, but has potentially worse 

results. However, if the input is a polynomial, then the result is 

guaranteed to be an equivalent polynomial of minimal degree. 

The most important option is hints. Its entries can be any of the 

following: 

- a natural number 

- a function 

- an iterable of the form (func, var1, var2, ...) 

- anything else, interpreted as a generator 

A number is used to indicate that the search space should be increased. 

A function is used to indicate that said function is likely to occur in a 

simplified expression. 

An iterable is used indicate that func(var1 + var2 + ...) is likely to 

occur in a simplified . 

An additional generator also indicates that it is likely to occur. 

(See examples below). 

This routine carries out various computationally intensive algorithms. 

The option ``quick=True`` can be used to suppress one particularly slow 

step (at the expense of potentially more complicated results, but never at 

the expense of increased total degree). 

Examples 

======== 

>>> from sympy.abc import x, y 

>>> from sympy import sin, tan, cos, sinh, cosh, tanh 

>>> from sympy.simplify.trigsimp import trigsimp_groebner 

Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens: 

>>> ex = sin(x)*cos(x) 

>>> trigsimp_groebner(ex) 

sin(x)*cos(x) 

This is because ``trigsimp_groebner`` only looks for a simplification 

involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try 

``2*x`` by passing ``hints=[2]``: 

>>> trigsimp_groebner(ex, hints=[2]) 

sin(2*x)/2 

>>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2]) 

-cos(2*x) 

Increasing the search space this way can quickly become expensive. A much 

faster way is to give a specific expression that is likely to occur: 

>>> trigsimp_groebner(ex, hints=[sin(2*x)]) 

sin(2*x)/2 

Hyperbolic expressions are similarly supported: 

>>> trigsimp_groebner(sinh(2*x)/sinh(x)) 

2*cosh(x) 

Note how no hints had to be passed, since the expression already involved 

``2*x``. 

The tangent function is also supported. You can either pass ``tan`` in the 

hints, to indicate that than should be tried whenever cosine or sine are, 

or you can pass a specific generator: 

>>> trigsimp_groebner(sin(x)/cos(x), hints=[tan]) 

tan(x) 

>>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)]) 

tanh(x) 

Finally, you can use the iterable form to suggest that angle sum formulae 

should be tried: 

>>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y)) 

>>> trigsimp_groebner(ex, hints=[(tan, x, y)]) 

tan(x + y) 

""" 

# TODO 

# - preprocess by replacing everything by funcs we can handle 

# - optionally use cot instead of tan 

# - more intelligent hinting. 

# For example, if the ideal is small, and we have sin(x), sin(y), 

# add sin(x + y) automatically... ? 

# - algebraic numbers ... 

# - expressions of lowest degree are not distinguished properly 

# e.g. 1 - sin(x)**2 

# - we could try to order the generators intelligently, so as to influence 

# which monomials appear in the quotient basis 

# THEORY 

# ------ 

# Ratsimpmodprime above can be used to "simplify" a rational function 

# modulo a prime ideal. "Simplify" mainly means finding an equivalent 

# expression of lower total degree. 

# 

# We intend to use this to simplify trigonometric functions. To do that, 

# we need to decide (a) which ring to use, and (b) modulo which ideal to 

# simplify. In practice, (a) means settling on a list of "generators" 

# a, b, c, ..., such that the fraction we want to simplify is a rational 

# function in a, b, c, ..., with coefficients in ZZ (integers). 

# (2) means that we have to decide what relations to impose on the 

# generators. There are two practical problems: 

# (1) The ideal has to be *prime* (a technical term). 

# (2) The relations have to be polynomials in the generators. 

# 

# We typically have two kinds of generators: 

# - trigonometric expressions, like sin(x), cos(5*x), etc 

# - "everything else", like gamma(x), pi, etc. 

# 

# Since this function is trigsimp, we will concentrate on what to do with 

# trigonometric expressions. We can also simplify hyperbolic expressions, 

# but the extensions should be clear. 

# 

# One crucial point is that all *other* generators really should behave 

# like indeterminates. In particular if (say) "I" is one of them, then 

# in fact I**2 + 1 = 0 and we may and will compute non-sensical 

# expressions. However, we can work with a dummy and add the relation 

# I**2 + 1 = 0 to our ideal, then substitute back in the end. 

# 

# Now regarding trigonometric generators. We split them into groups, 

# according to the argument of the trigonometric functions. We want to 

# organise this in such a way that most trigonometric identities apply in 

# the same group. For example, given sin(x), cos(2*x) and cos(y), we would 

# group as [sin(x), cos(2*x)] and [cos(y)]. 

# 

# Our prime ideal will be built in three steps: 

# (1) For each group, compute a "geometrically prime" ideal of relations. 

# Geometrically prime means that it generates a prime ideal in 

# CC[gens], not just ZZ[gens]. 

# (2) Take the union of all the generators of the ideals for all groups. 

# By the geometric primality condition, this is still prime. 

# (3) Add further inter-group relations which preserve primality. 

# 

# Step (1) works as follows. We will isolate common factors in the 

# argument, so that all our generators are of the form sin(n*x), cos(n*x) 

# or tan(n*x), with n an integer. Suppose first there are no tan terms. 

# The ideal [sin(x)**2 + cos(x)**2 - 1] is geometrically prime, since 

# X**2 + Y**2 - 1 is irreducible over CC. 

# Now, if we have a generator sin(n*x), than we can, using trig identities, 

# express sin(n*x) as a polynomial in sin(x) and cos(x). We can add this 

# relation to the ideal, preserving geometric primality, since the quotient 

# ring is unchanged. 

# Thus we have treated all sin and cos terms. 

# For tan(n*x), we add a relation tan(n*x)*cos(n*x) - sin(n*x) = 0. 

# (This requires of course that we already have relations for cos(n*x) and 

# sin(n*x).) It is not obvious, but it seems that this preserves geometric 

# primality. 

# XXX A real proof would be nice. HELP! 

# Sketch that <S**2 + C**2 - 1, C*T - S> is a prime ideal of 

# CC[S, C, T]: 

# - it suffices to show that the projective closure in CP**3 is 

# irreducible 

# - using the half-angle substitutions, we can express sin(x), tan(x), 

# cos(x) as rational functions in tan(x/2) 

# - from this, we get a rational map from CP**1 to our curve 

# - this is a morphism, hence the curve is prime 

# 

# Step (2) is trivial. 

# 

# Step (3) works by adding selected relations of the form 

# sin(x + y) - sin(x)*cos(y) - sin(y)*cos(x), etc. Geometric primality is 

# preserved by the same argument as before. 

def parse_hints(hints): 

"""Split hints into (n, funcs, iterables, gens).""" 

n = 1 

funcs, iterables, gens = [], [], [] 

for e in hints: 

if isinstance(e, (int, Integer)): 

n = e 

elif isinstance(e, FunctionClass): 

funcs.append(e) 

elif iterable(e): 

iterables.append((e[0], e[1:])) 

# XXX sin(x+2y)? 

# Note: we go through polys so e.g. 

# sin(-x) -> -sin(x) -> sin(x) 

gens.extend(parallel_poly_from_expr( 

[e[0](x) for x in e[1:]] + [e[0](Add(*e[1:]))])[1].gens) 

else: 

gens.append(e) 

return n, funcs, iterables, gens 

def build_ideal(x, terms): 

""" 

Build generators for our ideal. Terms is an iterable with elements of 

the form (fn, coeff), indicating that we have a generator fn(coeff*x). 

If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed 

to appear in terms. Similarly for hyperbolic functions. For tan(n*x), 

sin(n*x) and cos(n*x) are guaranteed. 

""" 

gens = [] 

I = [] 

y = Dummy('y') 

for fn, coeff in terms: 

for c, s, t, rel in ( 

[cos, sin, tan, cos(x)**2 + sin(x)**2 - 1], 

[cosh, sinh, tanh, cosh(x)**2 - sinh(x)**2 - 1]): 

if coeff == 1 and fn in [c, s]: 

I.append(rel) 

elif fn == t: 

I.append(t(coeff*x)*c(coeff*x) - s(coeff*x)) 

elif fn in [c, s]: 

cn = fn(coeff*y).expand(trig=True).subs(y, x) 

I.append(fn(coeff*x) - cn) 

return list(set(I)) 

def analyse_gens(gens, hints): 

""" 

Analyse the generators ``gens``, using the hints ``hints``. 

The meaning of ``hints`` is described in the main docstring. 

Return a new list of generators, and also the ideal we should 

work with. 

""" 

# First parse the hints 

n, funcs, iterables, extragens = parse_hints(hints) 

debug('n=%s' % n, 'funcs:', funcs, 'iterables:', 

iterables, 'extragens:', extragens) 

# We just add the extragens to gens and analyse them as before 

gens = list(gens) 

gens.extend(extragens) 

# remove duplicates 

funcs = list(set(funcs)) 

iterables = list(set(iterables)) 

gens = list(set(gens)) 

# all the functions we can do anything with 

allfuncs = {sin, cos, tan, sinh, cosh, tanh} 

# sin(3*x) -> ((3, x), sin) 

trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens 

if g.func in allfuncs] 

# Our list of new generators - start with anything that we cannot 

# work with (i.e. is not a trigonometric term) 

freegens = [g for g in gens if g.func not in allfuncs] 

newgens = [] 

trigdict = {} 

for (coeff, var), fn in trigterms: 

trigdict.setdefault(var, []).append((coeff, fn)) 

res = [] # the ideal 

for key, val in trigdict.items(): 

# We have now assembeled a dictionary. Its keys are common 

# arguments in trigonometric expressions, and values are lists of 

# pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we 

# need to deal with fn(coeff*x0). We take the rational gcd of the 

# coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol", 

# all other arguments are integral multiples thereof. 

# We will build an ideal which works with sin(x), cos(x). 

# If hint tan is provided, also work with tan(x). Moreover, if 

# n > 1, also work with sin(k*x) for k <= n, and similarly for cos 

# (and tan if the hint is provided). Finally, any generators which 

# the ideal does not work with but we need to accomodate (either 

# because it was in expr or because it was provided as a hint) 

# we also build into the ideal. 

# This selection process is expressed in the list ``terms``. 

# build_ideal then generates the actual relations in our ideal, 

# from this list. 

fns = [x[1] for x in val] 

val = [x[0] for x in val] 

gcd = reduce(igcd, val) 

terms = [(fn, v/gcd) for (fn, v) in zip(fns, val)] 

fs = set(funcs + fns) 

for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]): 

if any(x in fs for x in (c, s, t)): 

fs.add(c) 

fs.add(s) 

for fn in fs: 

for k in range(1, n + 1): 

terms.append((fn, k)) 

extra = [] 

for fn, v in terms: 

if fn == tan: 

extra.append((sin, v)) 

extra.append((cos, v)) 

if fn in [sin, cos] and tan in fs: 

extra.append((tan, v)) 

if fn == tanh: 

extra.append((sinh, v)) 

extra.append((cosh, v)) 

if fn in [sinh, cosh] and tanh in fs: 

extra.append((tanh, v)) 

terms.extend(extra) 

x = gcd*Mul(*key) 

r = build_ideal(x, terms) 

res.extend(r) 

newgens.extend(set(fn(v*x) for fn, v in terms)) 

# Add generators for compound expressions from iterables 

for fn, args in iterables: 

if fn == tan: 

# Tan expressions are recovered from sin and cos. 

iterables.extend([(sin, args), (cos, args)]) 

elif fn == tanh: 

# Tanh expressions are recovered from sihn and cosh. 

iterables.extend([(sinh, args), (cosh, args)]) 

else: 

dummys = symbols('d:%i' % len(args), cls=Dummy) 

expr = fn( Add(*dummys)).expand(trig=True).subs(list(zip(dummys, args))) 

res.append(fn(Add(*args)) - expr) 

if myI in gens: 

res.append(myI**2 + 1) 

freegens.remove(myI) 

newgens.append(myI) 

return res, freegens, newgens 

myI = Dummy('I') 

expr = expr.subs(S.ImaginaryUnit, myI) 

subs = [(myI, S.ImaginaryUnit)] 

num, denom = cancel(expr).as_numer_denom() 

try: 

(pnum, pdenom), opt = parallel_poly_from_expr([num, denom]) 

except PolificationFailed: 

return expr 

debug('initial gens:', opt.gens) 

ideal, freegens, gens = analyse_gens(opt.gens, hints) 

debug('ideal:', ideal) 

debug('new gens:', gens, " -- len", len(gens)) 

debug('free gens:', freegens, " -- len", len(gens)) 

# NOTE we force the domain to be ZZ to stop polys from injecting generators 

# (which is usually a sign of a bug in the way we build the ideal) 

if not gens: 

return expr 

G = groebner(ideal, order=order, gens=gens, domain=ZZ) 

debug('groebner basis:', list(G), " -- len", len(G)) 

# If our fraction is a polynomial in the free generators, simplify all 

# coefficients separately: 

from sympy.simplify.ratsimp import ratsimpmodprime 

if freegens and pdenom.has_only_gens(*set(gens).intersection(pdenom.gens)): 

num = Poly(num, gens=gens+freegens).eject(*gens) 

res = [] 

for monom, coeff in num.terms(): 

ourgens = set(parallel_poly_from_expr([coeff, denom])[1].gens) 

# We compute the transitive closure of all generators that can 

# be reached from our generators through relations in the ideal. 

changed = True 

while changed: 

changed = False 

for p in ideal: 

p = Poly(p) 

if not ourgens.issuperset(p.gens) and \ 

not p.has_only_gens(*set(p.gens).difference(ourgens)): 

changed = True 

ourgens.update(p.exclude().gens) 

# NOTE preserve order! 

realgens = [x for x in gens if x in ourgens] 

# The generators of the ideal have now been (implicitely) split 

# into two groups: those involving ourgens and those that don't. 

# Since we took the transitive closure above, these two groups 

# live in subgrings generated by a *disjoint* set of variables. 

# Any sensible groebner basis algorithm will preserve this disjoint 

# structure (i.e. the elements of the groebner basis can be split 

# similarly), and and the two subsets of the groebner basis then 

# form groebner bases by themselves. (For the smaller generating 

# sets, of course.) 

ourG = [g.as_expr() for g in G.polys if 

g.has_only_gens(*ourgens.intersection(g.gens))] 

res.append(Mul(*[a**b for a, b in zip(freegens, monom)]) * \ 

ratsimpmodprime(coeff/denom, ourG, order=order, 

gens=realgens, quick=quick, domain=ZZ, 

polynomial=polynomial).subs(subs)) 

return Add(*res) 

# NOTE The following is simpler and has less assumptions on the 

# groebner basis algorithm. If the above turns out to be broken, 

# use this. 

return Add(*[Mul(*[a**b for a, b in zip(freegens, monom)]) * \ 

ratsimpmodprime(coeff/denom, list(G), order=order, 

gens=gens, quick=quick, domain=ZZ) 

for monom, coeff in num.terms()]) 

else: 

return ratsimpmodprime( 

expr, list(G), order=order, gens=freegens+gens, 

quick=quick, domain=ZZ, polynomial=polynomial).subs(subs) 

_trigs = (TrigonometricFunction, HyperbolicFunction) 

def trigsimp(expr, **opts): 

""" 

reduces expression by using known trig identities 

Notes 

===== 

method: 

- Determine the method to use. Valid choices are 'matching' (default), 

'groebner', 'combined', and 'fu'. If 'matching', simplify the 

expression recursively by targeting common patterns. If 'groebner', apply 

an experimental groebner basis algorithm. In this case further options 

are forwarded to ``trigsimp_groebner``, please refer to its docstring. 

If 'combined', first run the groebner basis algorithm with small 

default parameters, then run the 'matching' algorithm. 'fu' runs the 

collection of trigonometric transformations described by Fu, et al. 

(see the `fu` docstring). 

Examples 

======== 

>>> from sympy import trigsimp, sin, cos, log 

>>> from sympy.abc import x, y 

>>> e = 2*sin(x)**2 + 2*cos(x)**2 

>>> trigsimp(e) 

2 

Simplification occurs wherever trigonometric functions are located. 

>>> trigsimp(log(e)) 

log(2) 

Using `method="groebner"` (or `"combined"`) might lead to greater 

simplification. 

The old trigsimp routine can be accessed as with method 'old'. 

>>> from sympy import coth, tanh 

>>> t = 3*tanh(x)**7 - 2/coth(x)**7 

>>> trigsimp(t, method='old') == t 

True 

>>> trigsimp(t) 

tanh(x)**7 

""" 

from sympy.simplify.fu import fu 

expr = sympify(expr) 

try: 

return expr._eval_trigsimp(**opts) 

except AttributeError: 

pass 

old = opts.pop('old', False) 

if not old: 

opts.pop('deep', None) 

recursive = opts.pop('recursive', None) 

method = opts.pop('method', 'matching') 

else: 

method = 'old' 

def groebnersimp(ex, **opts): 

def traverse(e): 

if e.is_Atom: 

return e 

args = [traverse(x) for x in e.args] 

if e.is_Function or e.is_Pow: 

args = [trigsimp_groebner(x, **opts) for x in args] 

return e.func(*args) 

new = traverse(ex) 

if not isinstance(new, Expr): 

return new 

return trigsimp_groebner(new, **opts) 

trigsimpfunc = { 

'fu': (lambda x: fu(x, **opts)), 

'matching': (lambda x: futrig(x)), 

'groebner': (lambda x: groebnersimp(x, **opts)), 

'combined': (lambda x: futrig(groebnersimp(x, 

polynomial=True, hints=[2, tan]))), 

'old': lambda x: trigsimp_old(x, **opts), 

}[method] 

return trigsimpfunc(expr) 

def exptrigsimp(expr, simplify=True): 

""" 

Simplifies exponential / trigonometric / hyperbolic functions. 

When ``simplify`` is True (default) the expression obtained after the 

simplification step will be then be passed through simplify to 

precondition it so the final transformations will be applied. 

Examples 

======== 

>>> from sympy import exptrigsimp, exp, cosh, sinh 

>>> from sympy.abc import z 

>>> exptrigsimp(exp(z) + exp(-z)) 

2*cosh(z) 

>>> exptrigsimp(cosh(z) - sinh(z)) 

exp(-z) 

""" 

from sympy.simplify.fu import hyper_as_trig, TR2i 

from sympy.simplify.simplify import bottom_up 

def exp_trig(e): 

# select the better of e, and e rewritten in terms of exp or trig 

# functions 

choices = [e] 

if e.has(*_trigs): 

choices.append(e.rewrite(exp)) 

choices.append(e.rewrite(cos)) 

return min(*choices, key=count_ops) 

newexpr = bottom_up(expr, exp_trig) 

if simplify: 

newexpr = newexpr.simplify() 

# conversion from exp to hyperbolic 

ex = newexpr.atoms(exp, S.Exp1) 

ex = [ei for ei in ex if 1/ei not in ex] 

## sinh and cosh 

for ei in ex: 

e2 = ei**-2 

if e2 in ex: 

a = e2.args[0]/2 if not e2 is S.Exp1 else S.Half 

newexpr = newexpr.subs((e2 + 1)*ei, 2*cosh(a)) 

newexpr = newexpr.subs((e2 - 1)*ei, 2*sinh(a)) 

## exp ratios to tan and tanh 

for ei in ex: 

n, d = ei - 1, ei + 1 

et = n/d 

etinv = d/n # not 1/et or else recursion errors arise 

a = ei.args[0] if ei.func is exp else S.One 

if a.is_Mul or a is S.ImaginaryUnit: 

c = a.as_coefficient(I) 

if c: 

t = S.ImaginaryUnit*tan(c/2) 

newexpr = newexpr.subs(etinv, 1/t) 

newexpr = newexpr.subs(et, t) 

continue 

t = tanh(a/2) 

newexpr = newexpr.subs(etinv, 1/t) 

newexpr = newexpr.subs(et, t) 

# sin/cos and sinh/cosh ratios to tan and tanh, respectively 

if newexpr.has(HyperbolicFunction): 

e, f = hyper_as_trig(newexpr) 

newexpr = f(TR2i(e)) 

if newexpr.has(TrigonometricFunction): 

newexpr = TR2i(newexpr) 

# can we ever generate an I where there was none previously? 

if not (newexpr.has(I) and not expr.has(I)): 

expr = newexpr 

return expr 

#-------------------- the old trigsimp routines --------------------- 

def trigsimp_old(expr, **opts): 

""" 

reduces expression by using known trig identities 

Notes 

===== 

deep: 

- Apply trigsimp inside all objects with arguments 

recursive: 

- Use common subexpression elimination (cse()) and apply 

trigsimp recursively (this is quite expensive if the 

expression is large) 

method: 

- Determine the method to use. Valid choices are 'matching' (default), 

'groebner', 'combined', 'fu' and 'futrig'. If 'matching', simplify the 

expression recursively by pattern matching. If 'groebner', apply an 

experimental groebner basis algorithm. In this case further options 

are forwarded to ``trigsimp_groebner``, please refer to its docstring. 

If 'combined', first run the groebner basis algorithm with small 

default parameters, then run the 'matching' algorithm. 'fu' runs the 

collection of trigonometric transformations described by Fu, et al. 

(see the `fu` docstring) while `futrig` runs a subset of Fu-transforms 

that mimic the behavior of `trigsimp`. 

compare: 

- show input and output from `trigsimp` and `futrig` when different, 

but returns the `trigsimp` value. 

Examples 

======== 

>>> from sympy import trigsimp, sin, cos, log, cosh, sinh, tan, cot 

>>> from sympy.abc import x, y 

>>> e = 2*sin(x)**2 + 2*cos(x)**2 

>>> trigsimp(e, old=True) 

2 

>>> trigsimp(log(e), old=True) 

log(2*sin(x)**2 + 2*cos(x)**2) 

>>> trigsimp(log(e), deep=True, old=True) 

log(2) 

Using `method="groebner"` (or `"combined"`) can sometimes lead to a lot 

more simplification: 

>>> e = (-sin(x) + 1)/cos(x) + cos(x)/(-sin(x) + 1) 

>>> trigsimp(e, old=True) 

(-sin(x) + 1)/cos(x) + cos(x)/(-sin(x) + 1) 

>>> trigsimp(e, method="groebner", old=True) 

2/cos(x) 

>>> trigsimp(1/cot(x)**2, compare=True, old=True) 

futrig: tan(x)**2 

cot(x)**(-2) 

""" 

old = expr 

first = opts.pop('first', True) 

if first: 

if not expr.has(*_trigs): 

return expr 

trigsyms = set().union(*[t.free_symbols for t in expr.atoms(*_trigs)]) 

if len(trigsyms) > 1: 

d = separatevars(expr) 

if d.is_Mul: 

d = separatevars(d, dict=True) or d 

if isinstance(d, dict): 

expr = 1 

for k, v in d.items(): 

# remove hollow factoring 

was = v 

v = expand_mul(v) 

opts['first'] = False 

vnew = trigsimp(v, **opts) 

if vnew == v: 

vnew = was 

expr *= vnew 

old = expr 

else: 

if d.is_Add: 

for s in trigsyms: 

r, e = expr.as_independent(s) 

if r: 

opts['first'] = False 

expr = r + trigsimp(e, **opts) 

if not expr.is_Add: 

break 

old = expr 

recursive = opts.pop('recursive', False) 

deep = opts.pop('deep', False) 

method = opts.pop('method', 'matching') 

def groebnersimp(ex, deep, **opts): 

def traverse(e): 

if e.is_Atom: 

return e 

args = [traverse(x) for x in e.args] 

if e.is_Function or e.is_Pow: 

args = [trigsimp_groebner(x, **opts) for x in args] 

return e.func(*args) 

if deep: 

ex = traverse(ex) 

return trigsimp_groebner(ex, **opts) 

trigsimpfunc = { 

'matching': (lambda x, d: _trigsimp(x, d)), 

'groebner': (lambda x, d: groebnersimp(x, d, **opts)), 

'combined': (lambda x, d: _trigsimp(groebnersimp(x, 

d, polynomial=True, hints=[2, tan]), 

d)) 

}[method] 

if recursive: 

w, g = cse(expr) 

g = trigsimpfunc(g[0], deep) 

for sub in reversed(w): 

g = g.subs(sub[0], sub[1]) 

g = trigsimpfunc(g, deep) 

result = g 

else: 

result = trigsimpfunc(expr, deep) 

if opts.get('compare', False): 

f = futrig(old) 

if f != result: 

print('\tfutrig:', f) 

return result 

def _dotrig(a, b): 

"""Helper to tell whether ``a`` and ``b`` have the same sorts 

of symbols in them -- no need to test hyperbolic patterns against 

expressions that have no hyperbolics in them.""" 

return a.func == b.func and ( 

a.has(TrigonometricFunction) and b.has(TrigonometricFunction) or 

a.has(HyperbolicFunction) and b.has(HyperbolicFunction)) 

_trigpat = None 

def _trigpats(): 

global _trigpat 

a, b, c = symbols('a b c', cls=Wild) 

d = Wild('d', commutative=False) 

# for the simplifications like sinh/cosh -> tanh: 

# DO NOT REORDER THE FIRST 14 since these are assumed to be in this 

# order in _match_div_rewrite. 

matchers_division = ( 

(a*sin(b)**c/cos(b)**c, a*tan(b)**c, sin(b), cos(b)), 

(a*tan(b)**c*cos(b)**c, a*sin(b)**c, sin(b), cos(b)), 

(a*cot(b)**c*sin(b)**c, a*cos(b)**c, sin(b), cos(b)), 

(a*tan(b)**c/sin(b)**c, a/cos(b)**c, sin(b), cos(b)), 

(a*cot(b)**c/cos(b)**c, a/sin(b)**c, sin(b), cos(b)), 

(a*cot(b)**c*tan(b)**c, a, sin(b), cos(b)), 

(a*(cos(b) + 1)**c*(cos(b) - 1)**c, 

a*(-sin(b)**2)**c, cos(b) + 1, cos(b) - 1), 

(a*(sin(b) + 1)**c*(sin(b) - 1)**c, 

a*(-cos(b)**2)**c, sin(b) + 1, sin(b) - 1), 

(a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c, S.One, S.One), 

(a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c, S.One, S.One), 

(a*coth(b)**c*sinh(b)**c, a*cosh(b)**c, S.One, S.One), 

(a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c, S.One, S.One), 

(a*coth(b)**c/cosh(b)**c, a/sinh(b)**c, S.One, S.One), 

(a*coth(b)**c*tanh(b)**c, a, S.One, S.One), 

(c*(tanh(a) + tanh(b))/(1 + tanh(a)*tanh(b)), 

tanh(a + b)*c, S.One, S.One), 

) 

matchers_add = ( 

(c*sin(a)*cos(b) + c*cos(a)*sin(b) + d, sin(a + b)*c + d), 

(c*cos(a)*cos(b) - c*sin(a)*sin(b) + d, cos(a + b)*c + d), 

(c*sin(a)*cos(b) - c*cos(a)*sin(b) + d, sin(a - b)*c + d), 

(c*cos(a)*cos(b) + c*sin(a)*sin(b) + d, cos(a - b)*c + d), 

(c*sinh(a)*cosh(b) + c*sinh(b)*cosh(a) + d, sinh(a + b)*c + d), 

(c*cosh(a)*cosh(b) + c*sinh(a)*sinh(b) + d, cosh(a + b)*c + d),