register_diff(np.power, pow_diff) register_diff(np.positive, lambda x: +x[1]) register_diff(np.negative, lambda x: -x[1]) register_diff(np.conj, lambda x: np.conj(x[1])) register_diff(np.conj, lambda x: np.conj(x[1])) register_diff(np.exp, lambda x: x[1] * np.exp(x[0])) register_diff(np.exp2, lambda x: x[1] * np.log(2) * np.exp(x[0])) register_diff(np.log, lambda x: x[1] / x[0]) register_diff(np.log2, lambda x: x[1] / (np.log(2) * x[0])) register_diff(np.log10, lambda x: x[1] / (np.log(10) * x[0])) register_diff(np.sqrt, lambda x: x[1] / (2 * np.sqrt(x[0]))) register_diff(np.square, lambda x: 2 * x[1] * x[0]) register_diff(np.cbrt, lambda x: x[1] / (3 * (x[0]**(2 / 3)))) register_diff(np.reciprocal, lambda x: -x[1] / np.square(x[0])) register_diff(np.broadcast_to, lambda x, shape: np.broadcast_to(x[1], shape)) register_diff(np.sin, lambda x: x[1] * np.cos(x[0])) register_diff(np.cos, lambda x: -x[1] * np.sin(x[0])) register_diff(np.tan, lambda x: x[1] / np.square(np.cos(x[0]))) register_diff(np.arcsin, lambda x: x[1] / np.sqrt(1 - np.square(x[0]))) register_diff(np.arccos, lambda x: -x[1] / np.sqrt(1 - np.square(x[0]))) register_diff(np.arctan, lambda x: x[1] / (1 + np.square(x[0]))) register_diff(np.arctan2, arctan2_diff) register_diff(np.sinh, lambda x: x[1] * np.cosh(x[0])) register_diff(np.cosh, lambda x: x[1] * np.sinh(x[0])) register_diff(np.tanh, lambda x: x[1] / np.square(np.cosh(x[0]))) register_diff(np.arcsinh, lambda x: x[1] / np.sqrt(1 + np.square(x[0]))) register_diff(np.arccosh, lambda x: x[1] / np.sqrt(1 - np.square(x[0]))) register_diff(np.arctanh, lambda x: x[1] / (1 - np.square(x[0])))
# ----- Simple grads -----
defjvp(np.positive, lambda ans, x: lambda g: np.ones_like(x) * g)
defjvp(np.negative, lambda ans, x: lambda g: -np.ones_like(x) * g)
defjvp(np.fabs, lambda ans, x: lambda g: np.sign(x) * g
) # fabs doesn't take complex numbers.
defjvp(np.absolute, lambda ans, x: lambda g: np.real(g * np.conj(x)) / ans)
defjvp(np.reciprocal, lambda ans, x: lambda g: -g / x**2)
defjvp(np.exp, lambda ans, x: lambda g: ans * g)
defjvp(np.exp2, lambda ans, x: lambda g: ans * np.log(2) * g)
defjvp(np.expm1, lambda ans, x: lambda g: (ans + 1) * g)
defjvp(np.log, lambda ans, x: lambda g: g / x)
defjvp(np.log2, lambda ans, x: lambda g: g / x / np.log(2))
defjvp(np.log10, lambda ans, x: lambda g: g / x / np.log(10))
defjvp(np.log1p, lambda ans, x: lambda g: g / (x + 1))
defjvp(np.sin, lambda ans, x: lambda g: g * np.cos(x))
defjvp(np.cos, lambda ans, x: lambda g: -g * np.sin(x))
defjvp(np.tan, lambda ans, x: lambda g: g / np.cos(x)**2)
defjvp(np.arcsin, lambda ans, x: lambda g: g / np.sqrt(1 - x**2))
defjvp(np.arccos, lambda ans, x: lambda g: -g / np.sqrt(1 - x**2))
defjvp(np.arctan, lambda ans, x: lambda g: g / (1 + x**2))
defjvp(np.sinh, lambda ans, x: lambda g: g * np.cosh(x))
defjvp(np.cosh, lambda ans, x: lambda g: g * np.sinh(x))
defjvp(np.tanh, lambda ans, x: lambda g: g / np.cosh(x)**2)
defjvp(np.arcsinh, lambda ans, x: lambda g: g / np.sqrt(x**2 + 1))
defjvp(np.arccosh, lambda ans, x: lambda g: g / np.sqrt(x**2 - 1))
defjvp(np.arctanh, lambda ans, x: lambda g: g / (1 - x**2))
defjvp(np.square, lambda ans, x: lambda g: g * 2 * x)
defjvp(np.sqrt, lambda ans, x: lambda g: g * 0.5 * x**-0.5)
defjvp(
np.sinc,
np.absolute,
lambda ans, x: lambda g: g * replace_zero(np.conj(x), 0.0) / replace_zero(ans, 1.0),
)
defvjp(
np.fabs, lambda ans, x: lambda g: np.sign(x) * g
) # fabs doesn't take complex numbers.
defvjp(np.absolute, lambda ans, x: lambda g: g * np.conj(x) / ans)
defvjp(np.reciprocal, lambda ans, x: lambda g: -g / x ** 2)
defvjp(np.exp, lambda ans, x: lambda g: ans * g)
defvjp(np.exp2, lambda ans, x: lambda g: ans * np.log(2) * g)
defvjp(np.expm1, lambda ans, x: lambda g: (ans + 1) * g)
defvjp(np.log, lambda ans, x: lambda g: g / x)
defvjp(np.log2, lambda ans, x: lambda g: g / x / np.log(2))
defvjp(np.log10, lambda ans, x: lambda g: g / x / np.log(10))
defvjp(np.log1p, lambda ans, x: lambda g: g / (x + 1))
defvjp(np.sin, lambda ans, x: lambda g: g * np.cos(x))
defvjp(np.cos, lambda ans, x: lambda g: -g * np.sin(x))
defvjp(np.tan, lambda ans, x: lambda g: g / np.cos(x) ** 2)
defvjp(np.arcsin, lambda ans, x: lambda g: g / np.sqrt(1 - x ** 2))
defvjp(np.arccos, lambda ans, x: lambda g: -g / np.sqrt(1 - x ** 2))
defvjp(np.arctan, lambda ans, x: lambda g: g / (1 + x ** 2))
defvjp(np.sinh, lambda ans, x: lambda g: g * np.cosh(x))
defvjp(np.cosh, lambda ans, x: lambda g: g * np.sinh(x))
defvjp(np.tanh, lambda ans, x: lambda g: g / np.cosh(x) ** 2)
defvjp(np.arcsinh, lambda ans, x: lambda g: g / np.sqrt(x ** 2 + 1))
defvjp(np.arccosh, lambda ans, x: lambda g: g / np.sqrt(x ** 2 - 1))
defvjp(np.arctanh, lambda ans, x: lambda g: g / (1 - x ** 2))
defvjp(np.rad2deg, lambda ans, x: lambda g: g / np.pi * 180.0)
defvjp(np.degrees, lambda ans, x: lambda g: g / np.pi * 180.0)
defvjp(np.deg2rad, lambda ans, x: lambda g: g * np.pi / 180.0)
defvjp(np.radians, lambda ans, x: lambda g: g * np.pi / 180.0)
if backend in FULLY_TESTED_BACKENDS:
raise
pytest.xfail(
reason="The backend has no implementation for this ufunc.")
if isinstance(ret, da.Array):
ret.compute()
assert_allclose(ret.diffs[x].arr, y_d_arr)
@pytest.mark.parametrize(
"func, y_d",
[(lambda x: np.power(2 * x + 1, 3), lambda x: 6 * np.power(2 * x + 1, 2)),
(lambda x: np.sin(np.power(x, 2)) / np.power(np.sin(x), 2), lambda x:
(2 * x * np.cos(np.power(x, 2)) * np.sin(x) - 2 * np.sin(np.power(x, 2))
* np.cos(x)) / np.power(np.sin(x), 3)),
(lambda x: np.power(np.log(np.power(x, 3)), 1 / 3),
lambda x: 2 * np.power(np.log(np.power(x, 2)), -2 / 3) / (3 * x)),
(lambda x: np.log(
(1 + x) / (1 - x)) / 4 - np.arctan(x) / 2, lambda x: np.power(x, 2) /
(1 - np.power(x, 4)))],
)
def test_arbitrary_function(backend, func, y_d):
x_arr = [0.2, 0.3]
try:
with ua.set_backend(backend), ua.set_backend(udiff, coerce=True):
x = np.asarray(x_arr)
x.var = udiff.Variable('x')
ret = func(x)
y_d_arr = y_d(x)
[
(lambda x: x * x, lambda x: 2, None),
(lambda x: (2 * x + 1)**3, lambda x: 24 * (2 * x + 1), (0.5, None)),
(
lambda x: np.sin(x**2) + np.sin(x)**2,
lambda x: 2 * cos(x**2) - 4 * x**2 * sin(x**2) + 2 * cos(x)**2 - 2
* sin(x)**2,
(0, pi),
),
(
lambda x: np.log(x**2),
lambda x: -2 / x**2,
(1, None),
),
(
lambda x: np.power(np.cos(x), 2) * np.log(x),
lambda x: -2 * cos(2 * x) * log(x) - 2 * sin(2 * x) / x - cos(x)**2
/ x**2,
(0, None),
),
(
lambda x: x / np.sqrt(1 - x**2),
lambda x: 3 * x / (1 - x**2)**(5 / 2),
(-1, 1),
),
],
)
def test_high_order_diff(backend, mode, func, y_d, domain):
if domain is None:
x_arr = generate_test_data()
else:
if isinstance(ret, da.Array):
ret.compute()
assert_allclose(ret.diffs[x].arr, y_d_arr)
@pytest.mark.xfail
@pytest.mark.parametrize(
"func, y_d, domain",
[
(lambda x: (2 * x + 1) ** 3, lambda x: 6 * (2 * x + 1) ** 2, (0.5, None)),
(
lambda x: np.sin(x ** 2) / (np.sin(x)) ** 2,
lambda x: (
2 * x * np.cos(x ** 2) * np.sin(x) - 2 * np.sin(x ** 2) * np.cos(x)
)
/ (np.sin(x)) ** 3,
(0, pi),
),
(
lambda x: (np.log(x ** 2)) ** (1 / 3),
lambda x: 2 * (np.log(x ** 2)) ** (-2 / 3) / (3 * x),
(1, None),
),
(
lambda x: np.log((1 + x) / (1 - x)) / 4 - np.arctan(x) / 2,
lambda x: x ** 2 / (1 - x ** 4),
(-1, 1),
),
(
ret = method(x)
except ua.BackendNotImplementedError:
if backend in FULLY_TESTED_BACKENDS:
raise
pytest.xfail(reason="The backend has no implementation for this ufunc.")
if isinstance(ret, da.Array):
ret.compute()
assert_allclose(ret.diffs[x].arr, y_d_arr)
@pytest.mark.parametrize(
"func, y_d",
[
(lambda x: np.power(2 * x + 1, 3), lambda x: 6 * np.power(2 * x + 1, 2)),
(lambda x: np.sin(np.power(x, 2)) / np.power(np.sin(x), 2), lambda x: (2 * x * np.cos(np.power(x, 2)) * np.sin(x) - 2 * np.sin(np.power(x, 2)) * np.cos(x)) / np.power(np.sin(x), 3)),
(lambda x: np.power(np.log(np.power(x, 3)), 1/3), lambda x: 2 * np.power(np.log(np.power(x, 2)), -2/3) / (3 * x)),
(lambda x: np.log((1 + x) / (1 - x)) / 4 - np.arctan(x) / 2, lambda x: np.power(x, 2) / (1 - np.power(x, 4))),
(lambda x: np.arctanh(3 * x ** 3 + x ** 2 +1), lambda x: (9 * x ** 2 + 2 * x) / (1 - np.power(3 * x ** 3 + x ** 2 + 1 , 2))),
(lambda x: np.sinh(np.cbrt(x)) + np.cosh(4 * x ** 3) , lambda x: np.cosh(np.cbrt(x)) / (3 * x ** (2/3)) + 12 * (x ** 2) * np.sinh(4 * x ** 3)),
(lambda x: np.log(1 + x ** 2) / np.arctanh(x), lambda x: ((2 * x * np.arctanh(x) / (1 + x ** 2)) - (np.log(1 + x ** 2)/(1 - x ** 2))) / np.power(np.arctanh(x) , 2))
],
)
def test_arbitrary_function(backend, func, y_d):
x_arr = [0.2, 0.3]
try:
with ua.set_backend(backend), ua.set_backend(udiff, coerce=True):
x = np.asarray(x_arr)
x.var = udiff.Variable('x')
ret = func(x)
y_d_arr = y_d(x)
if backend in FULLY_TESTED_BACKENDS:
raise
pytest.xfail(
reason="The backend has no implementation for this ufunc.")
if isinstance(ret, da.Array):
ret.compute()
assert_allclose(ret.diffs[x].arr, y_d_arr)
@pytest.mark.parametrize(
"func, y_d, domain",
[(lambda x: (2 * x + 1)**3, lambda x: 6 * (2 * x + 1)**2, (0.5, None)),
(lambda x: np.sin(x**2) / (np.sin(x))**2, lambda x:
(2 * x * np.cos(x**2) * np.sin(x) - 2 * np.sin(x**2) * np.cos(x)) /
(np.sin(x))**3, (0, pi)),
(lambda x: (np.log(x**2))**(1 / 3), lambda x: 2 *
(np.log(x**2))**(-2 / 3) / (3 * x), (1, None)),
(lambda x: np.log((1 + x) /
(1 - x)) / 4 - np.arctan(x) / 2, lambda x: x**2 /
(1 - x**4), (-1, 1)),
(lambda x: np.arctanh(3 * x**3 + x**2 + 1), lambda x: (9 * x**2 + 2 * x) /
(1 - (3 * x**3 + x**2 + 1)**2), (0, None)),
(lambda x: np.sinh(np.cbrt(x)) + np.cosh(4 * x**3),
lambda x: np.cosh(np.cbrt(x)) / (3 * x**(2 / 3)) + 12 *
(x**2) * np.sinh(4 * x**3), (1 / 4, None)),
(lambda x: np.log(1 + x**2) / np.arctanh(x), lambda x:
((2 * x * np.arctanh(x) / (1 + x**2)) -
(np.log(1 + x**2) / (1 - x**2))) / (np.arctanh(x))**2, (0, 1))],
)