def test_TR10i(): assert TR10i(cos(1) * cos(3) + sin(1) * sin(3)) == cos(2) assert TR10i(cos(1) * cos(3) - sin(1) * sin(3)) == cos(4) assert TR10i(cos(1) * sin(3) - sin(1) * cos(3)) == sin(2) assert TR10i(cos(1) * sin(3) + sin(1) * cos(3)) == sin(4) assert TR10i(cos(1) * sin(3) + sin(1) * cos(3) + 7) == sin(4) + 7 assert TR10i(cos(1) * sin(3) + sin(1) * cos(3) + cos(3)) == cos(3) + sin(4) assert TR10i(2*cos(1)*sin(3) + 2*sin(1)*cos(3) + cos(3)) == \ 2*sin(4) + cos(3) assert TR10i(cos(2)*cos(3) + sin(2)*(cos(1)*sin(2) + cos(2)*sin(1))) == \ cos(1) eq = (cos(2) * cos(3) + sin(2) * (cos(1) * sin(2) + cos(2) * sin(1))) * cos(5) + sin(1) * sin(5) assert TR10i(eq) == TR10i(eq.expand()) == cos(4) assert TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) == \ 2*sqrt(2)*x*sin(x + pi/6) assert TR10i( cos(x) / sqrt(6) + sin(x) / sqrt(2) + cos(x) / sqrt(6) / 3 + sin(x) / sqrt(2) / 3) == 4 * sqrt(6) * sin(x + pi / 6) / 9 assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) + cos(y)/sqrt(6)/3 + sin(y)/sqrt(2)/3) == \ sqrt(6)*sin(x + pi/6)/3 + sqrt(6)*sin(y + pi/6)/9 assert TR10i(cos(x) + sqrt(3) * sin(x) + 2 * sqrt(3) * cos(x + pi / 6)) == 4 * cos(x) assert TR10i( cos(x) + sqrt(3) * sin(x) + 2 * sqrt(3) * cos(x + pi / 6) + 4 * sin(x)) == 4 * sqrt(2) * sin(x + pi / 4) assert TR10i(cos(2)*sin(3) + sin(2)*cos(4)) == \ sin(2)*cos(4) + sin(3)*cos(2) A = Symbol('A', commutative=False) assert TR10i(sqrt(2)*cos(x)*A + sqrt(6)*sin(x)*A) == \ 2*sqrt(2)*sin(x + pi/6)*A c = cos(x) s = sin(x) h = sin(y) r = cos(y) for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): for argsi in ((c * r, s * h), (c * h, s * r)): # explicit 2-args args = zip(si, argsi) ex = Add(*[Mul(*ai) for ai in args]) t = TR10i(ex) assert not (ex - t.expand(trig=True) or t.is_Add) c = cos(x) s = sin(x) h = sin(pi / 6) r = cos(pi / 6) for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): for argsi in ((c * r, s * h), (c * h, s * r)): # induced args = zip(si, argsi) ex = Add(*[Mul(*ai) for ai in args]) t = TR10i(ex) assert not (ex - t.expand(trig=True) or t.is_Add)
def __trigsimp(expr, deep=False): """recursive helper for trigsimp""" from sympy.simplify.fu import TR10i if _trigpat is None: _trigpats() a, b, c, d, matchers_division, matchers_add, \ matchers_identity, artifacts = _trigpat if expr.is_Mul: # do some simplifications like sin/cos -> tan: if not expr.is_commutative: com, nc = expr.args_cnc() expr = _trigsimp(Mul._from_args(com), deep) * Mul._from_args(nc) else: for i, (pattern, simp, ok1, ok2) in enumerate(matchers_division): if not _dotrig(expr, pattern): continue newexpr = _match_div_rewrite(expr, i) if newexpr is not None: if newexpr != expr: expr = newexpr break else: continue # use SymPy matching instead res = expr.match(pattern) if res and res.get(c, 0): if not res[c].is_integer: ok = ok1.subs(res) if not ok.is_positive: continue ok = ok2.subs(res) if not ok.is_positive: continue # if "a" contains any of trig or hyperbolic funcs with # argument "b" then skip the simplification if any(w.args[0] == res[b] for w in res[a].atoms( TrigonometricFunction, HyperbolicFunction)): continue # simplify and finish: expr = simp.subs(res) break # process below if expr.is_Add: args = [] for term in expr.args: if not term.is_commutative: com, nc = term.args_cnc() nc = Mul._from_args(nc) term = Mul._from_args(com) else: nc = S.One term = _trigsimp(term, deep) for pattern, result in matchers_identity: res = term.match(pattern) if res is not None: term = result.subs(res) break args.append(term * nc) if args != expr.args: expr = Add(*args) expr = min(expr, expand(expr), key=count_ops) if expr.is_Add: for pattern, result in matchers_add: if not _dotrig(expr, pattern): continue expr = TR10i(expr) if expr.has(HyperbolicFunction): res = expr.match(pattern) # if "d" contains any trig or hyperbolic funcs with # argument "a" or "b" then skip the simplification; # this isn't perfect -- see tests if res is None or not (a in res and b in res) or any( w.args[0] in (res[a], res[b]) for w in res[d].atoms(TrigonometricFunction, HyperbolicFunction)): continue expr = result.subs(res) break # Reduce any lingering artifacts, such as sin(x)**2 changing # to 1 - cos(x)**2 when sin(x)**2 was "simpler" for pattern, result, ex in artifacts: if not _dotrig(expr, pattern): continue # Substitute a new wild that excludes some function(s) # to help influence a better match. This is because # sometimes, for example, 'a' would match sec(x)**2 a_t = Wild('a', exclude=[ex]) pattern = pattern.subs(a, a_t) result = result.subs(a, a_t) m = expr.match(pattern) was = None while m and was != expr: was = expr if m[a_t] == 0 or \ -m[a_t] in m[c].args or m[a_t] + m[c] == 0: break if d in m and m[a_t] * m[d] + m[c] == 0: break expr = result.subs(m) m = expr.match(pattern) m.setdefault(c, S.Zero) elif expr.is_Mul or expr.is_Pow or deep and expr.args: expr = expr.func(*[_trigsimp(a, deep) for a in expr.args]) try: if not expr.has(*_trigs): raise TypeError e = expr.atoms(exp) new = expr.rewrite(exp, deep=deep) if new == e: raise TypeError fnew = factor(new) if fnew != new: new = sorted([new, factor(new)], key=count_ops)[0] # if all exp that were introduced disappeared then accept it if not (new.atoms(exp) - e): expr = new except TypeError: pass return expr