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 test_Function():
assert mcode(f(x, y, z)) == "f[x, y, z]"
assert mcode(sin(x)**cos(x)) == "Sin[x]^Cos[x]"
assert mcode(sign(x)) == "Sign[x]"
assert mcode(atanh(x), user_functions={"atanh": "ArcTanh"}) == "ArcTanh[x]"
assert (mcode(meijerg(((1, 1), (3, 4)), ((1, ), ()),
x)) == "MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]")
assert (mcode(hyper((1, 2, 3), (3, 4),
x)) == "HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]")
assert mcode(Min(x, y)) == "Min[x, y]"
assert mcode(Max(x, y)) == "Max[x, y]"
assert mcode(Max(x, 2)) == "Max[2, x]" # issue sympy/sympy#15344
assert mcode(binomial(x, y)) == "Binomial[x, y]"
assert mcode(log(x)) == "Log[x]"
assert mcode(tan(x)) == "Tan[x]"
assert mcode(cot(x)) == "Cot[x]"
assert mcode(asin(x)) == "ArcSin[x]"
assert mcode(acos(x)) == "ArcCos[x]"
assert mcode(atan(x)) == "ArcTan[x]"
assert mcode(sinh(x)) == "Sinh[x]"
assert mcode(cosh(x)) == "Cosh[x]"
assert mcode(tanh(x)) == "Tanh[x]"
assert mcode(coth(x)) == "Coth[x]"
assert mcode(sech(x)) == "Sech[x]"
assert mcode(csch(x)) == "Csch[x]"
assert mcode(erfc(x)) == "Erfc[x]"
assert mcode(conjugate(x)) == "Conjugate[x]"
assert mcode(re(x)) == "Re[x]"
assert mcode(im(x)) == "Im[x]"
assert mcode(polygamma(x, y)) == "PolyGamma[x, y]"
class myfunc1(Function):
@classmethod
def eval(cls, x):
pass
class myfunc2(Function):
@classmethod
def eval(cls, x, y):
pass
pytest.raises(
ValueError,
lambda: mcode(myfunc1(x), user_functions={"myfunc1": ["Myfunc1"]}))
assert mcode(myfunc1(x), user_functions={"myfunc1":
"Myfunc1"}) == "Myfunc1[x]"
assert mcode(myfunc2(x, y),
user_functions={"myfunc2": [(lambda *x: False, "Myfunc2")]
}) == "myfunc2[x, y]"
def test_nocache(clear_imports, monkeypatch):
"""Regression tests with DIOFANT_USE_CACHE=False. """
monkeypatch.setenv('DIOFANT_USE_CACHE', 'False')
from diofant.core.cache import CACHE
from diofant.core.symbol import Symbol
from diofant.functions import sin, sqrt, exp, sinh
# test that we don't use cache
assert CACHE == []
x = Symbol('x')
assert CACHE == []
# issue sympy/sympy#8840
(1 + x)*x # not raises
# issue sympy/sympy#9413
(2*x).is_complex # not raises
# see commit c459d18
sin(x + x) # not raises
# see commit 53dd1eb
mx = -Symbol('x', negative=False)
assert mx.is_positive is not True
px = 2*Symbol('x', positive=False)
assert px.is_positive is not True
# see commit 2eaaba2
s = 1/sqrt(x**2)
y = Symbol('y')
result = s.subs(sqrt(x**2), y)
assert result == 1/y
# problem from https://groups.google.com/forum/#!topic/sympy/LkTMQKC_BOw
# see commit c459d18
a = Symbol('a', positive=True)
f = exp(x*(-a - 1))
g = sinh(x)
f*g # not raises
def test_nocache(clear_imports, monkeypatch):
"""Regression tests with DIOFANT_USE_CACHE=False. """
monkeypatch.setenv('DIOFANT_USE_CACHE', 'False')
from diofant.core.cache import CACHE
from diofant.core.symbol import Symbol
from diofant.functions import sin, sqrt, exp, sinh
# test that we don't use cache
assert CACHE == []
x = Symbol('x')
assert CACHE == []
# issue sympy/sympy#8840
(1 + x)*x # not raises
# issue sympy/sympy#9413
(2*x).is_complex # not raises
# see commit c459d18
sin(x + x) # not raises
# see commit 53dd1eb
mx = -Symbol('x', negative=False)
assert mx.is_positive is not True
px = 2*Symbol('x', positive=False)
assert px.is_positive is not True
# see commit 2eaaba2
s = 1/sqrt(x**2)
y = Symbol('y')
result = s.subs({sqrt(x**2): y})
assert result == 1/y
# problem from https://groups.google.com/forum/#!topic/sympy/LkTMQKC_BOw
# see commit c459d18
a = Symbol('a', positive=True)
f = exp(x*(-a - 1))
g = sinh(x)
f*g # not raises
def _expr_big_minus(cls, a, z, n):
return -1 / sqrt(1 + 1 / z) * sinh(2 * a * asinh(sqrt(z)) +
2 * a * pi * I * n)
def _expr_big(cls, a, z, n):
return -1 / sqrt(1 - 1 / z) * sinh(2 * a * acosh(sqrt(z)) +
a * pi * I * (2 * n - 1))
def _expr_small_minus(cls, a, z):
return -sqrt(z) / sqrt(1 + z) * sinh(2 * a * asinh(sqrt(z)))
def test_Function():
assert mcode(f(x, y, z)) == "f[x, y, z]"
assert mcode(sin(x) ** cos(x)) == "Sin[x]^Cos[x]"
assert mcode(sign(x)) == "Sign[x]"
assert mcode(atanh(x), user_functions={"atanh": "ArcTanh"}) == "ArcTanh[x]"
assert (mcode(meijerg(((1, 1), (3, 4)), ((1,), ()), x)) ==
"MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]")
assert (mcode(hyper((1, 2, 3), (3, 4), x)) ==
"HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]")
assert mcode(Min(x, y)) == "Min[x, y]"
assert mcode(Max(x, y)) == "Max[x, y]"
assert mcode(Max(x, 2)) == "Max[2, x]" # issue sympy/sympy#15344
assert mcode(binomial(x, y)) == "Binomial[x, y]"
assert mcode(log(x)) == "Log[x]"
assert mcode(tan(x)) == "Tan[x]"
assert mcode(cot(x)) == "Cot[x]"
assert mcode(asin(x)) == "ArcSin[x]"
assert mcode(acos(x)) == "ArcCos[x]"
assert mcode(atan(x)) == "ArcTan[x]"
assert mcode(sinh(x)) == "Sinh[x]"
assert mcode(cosh(x)) == "Cosh[x]"
assert mcode(tanh(x)) == "Tanh[x]"
assert mcode(coth(x)) == "Coth[x]"
assert mcode(sech(x)) == "Sech[x]"
assert mcode(csch(x)) == "Csch[x]"
assert mcode(erfc(x)) == "Erfc[x]"
assert mcode(conjugate(x)) == "Conjugate[x]"
assert mcode(re(x)) == "Re[x]"
assert mcode(im(x)) == "Im[x]"
assert mcode(polygamma(x, y)) == "PolyGamma[x, y]"
assert mcode(factorial(x)) == "Factorial[x]"
assert mcode(factorial2(x)) == "Factorial2[x]"
assert mcode(rf(x, y)) == "Pochhammer[x, y]"
assert mcode(gamma(x)) == "Gamma[x]"
assert mcode(zeta(x)) == "Zeta[x]"
assert mcode(asinh(x)) == "ArcSinh[x]"
assert mcode(Heaviside(x)) == "UnitStep[x]"
assert mcode(fibonacci(x)) == "Fibonacci[x]"
assert mcode(polylog(x, y)) == "PolyLog[x, y]"
assert mcode(atanh(x)) == "ArcTanh[x]"
class myfunc1(Function):
@classmethod
def eval(cls, x):
pass
class myfunc2(Function):
@classmethod
def eval(cls, x, y):
pass
pytest.raises(ValueError,
lambda: mcode(myfunc1(x),
user_functions={"myfunc1": ["Myfunc1"]}))
assert mcode(myfunc1(x),
user_functions={"myfunc1": "Myfunc1"}) == "Myfunc1[x]"
assert mcode(myfunc2(x, y),
user_functions={"myfunc2": [(lambda *x: False,
"Myfunc2")]}) == "myfunc2[x, y]"
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),
)
# for cos(x)**2 + sin(x)**2 -> 1
matchers_identity = (
(a * sin(b)**2, a - a * cos(b)**2),
(a * tan(b)**2, a * (1 / cos(b))**2 - a),
(a * cot(b)**2, a * (1 / sin(b))**2 - a),
(a * sin(b + c), a * (sin(b) * cos(c) + sin(c) * cos(b))),
(a * cos(b + c), a * (cos(b) * cos(c) - sin(b) * sin(c))),
(a * tan(b + c), a * ((tan(b) + tan(c)) / (1 - tan(b) * tan(c)))),
(a * sinh(b)**2, a * cosh(b)**2 - a),
(a * tanh(b)**2, a - a * (1 / cosh(b))**2),
(a * coth(b)**2, a + a * (1 / sinh(b))**2),
(a * sinh(b + c), a * (sinh(b) * cosh(c) + sinh(c) * cosh(b))),
(a * cosh(b + c), a * (cosh(b) * cosh(c) + sinh(b) * sinh(c))),
(a * tanh(b + c), a * ((tanh(b) + tanh(c)) / (1 + tanh(b) * tanh(c)))),
)
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a * cos(b)**2 + c, a * sin(b)**2 + c, cos),
(a - a * (1 / cos(b))**2 + c, -a * tan(b)**2 + c, cos),
(a - a * (1 / sin(b))**2 + c, -a * cot(b)**2 + c, sin),
(a - a * cosh(b)**2 + c, -a * sinh(b)**2 + c, cosh),
(a - a * (1 / cosh(b))**2 + c, a * tanh(b)**2 + c, cosh),
(a + a * (1 / sinh(b))**2 + c, a * coth(b)**2 + c, sinh),
# same as above but with noncommutative prefactor
(a * d - a * d * cos(b)**2 + c, a * d * sin(b)**2 + c, cos),
(a * d - a * d * (1 / cos(b))**2 + c, -a * d * tan(b)**2 + c, cos),
(a * d - a * d * (1 / sin(b))**2 + c, -a * d * cot(b)**2 + c, sin),
(a * d - a * d * cosh(b)**2 + c, -a * d * sinh(b)**2 + c, cosh),
(a * d - a * d * (1 / cosh(b))**2 + c, a * d * tanh(b)**2 + c, cosh),
(a * d + a * d * (1 / sinh(b))**2 + c, a * d * coth(b)**2 + c, sinh),
)
_trigpat = (a, b, c, d, matchers_division, matchers_add, matchers_identity,
artifacts)
return _trigpat
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 diofant import exptrigsimp, exp, cosh, sinh
>>> from diofant.abc import z
>>> exptrigsimp(exp(z) + exp(-z))
2*cosh(z)
>>> exptrigsimp(cosh(z) - sinh(z))
E**(-z)
"""
from diofant.simplify.fu import hyper_as_trig, TR2i
from diofant.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 = {a
for a in newexpr.atoms(Pow)
if a.base is S.Exp1} | newexpr.atoms(S.Exp1)
if ex:
ex0 = {list(ex)[0]}
ex = [ei for ei in ex if 1 / ei not in ex]
if not ex:
ex = ex0
# sinh and cosh
for ei in ex:
a = ei.exp if ei is not S.Exp1 else S.One
newexpr = newexpr.subs(ei + 1 / ei, 2 * cosh(a))
newexpr = newexpr.subs(ei - 1 / ei, 2 * sinh(a))
e2 = ei**-2
if e2 in ex:
a = e2.exp / 2 if e2 is not 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.exp if ei.is_Pow and ei.base is S.Exp1 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