def test_nested_sin_cos():
x = Forward("x", 2)
y = Forward("y", 5)
f = sin(cos(x * y))
assert f.value == approx(np.sin(np.cos(2 * 5)))
assert f.get_gradient("x") == approx(-5 * np.sin(10) * np.cos(np.cos(10)))
assert f.get_gradient("y") == approx(-2 * np.sin(10) * np.cos(np.cos(10)))
def test_vector():
x = Forward("x", 2)
y = Forward("y", -3)
f = fVector([x, y])
assert f.value == approx([2, -3])
assert f.get_gradient("x") == approx([1, 0])
assert f.get_gradient("y") == approx([0, 1])
assert f.get_gradient("z") == approx([0, 0])
def test_var():
f = Forward("x", 0)
assert f.value == approx(0)
assert f.get_gradient("x") == approx(1)
f = Forward("x", -10)
assert f.get_gradient("x") == approx(1)
assert f.get_gradient("y") == approx(0)
def test_log_weird():
x = Forward("x", 2)
y = Forward("y", 5)
f = log(x**y)
assert f.value == approx(np.log(2**5))
assert f.get_gradient("x") == approx(5 / 2)
assert f.get_gradient("y") == approx(np.log(2))
assert f.get_gradient("z") == approx(0)
def test_multiplication():
f = 4 * Forward("x", 1)
assert f.value == approx(4)
assert f.get_gradient("x") == approx(4)
f = Forward("x", 2) * 4
assert f.value == approx(8)
assert f.get_gradient("x") == approx(4)
def test_multi_dim_var():
f = Forward("x", 0) * Forward("y", 0)
assert f.value == approx(0)
assert f.get_gradient("x") == approx(0)
assert f.get_gradient("y") == approx(0)
f = Forward("x", 2) * Forward("y", -2)
assert f.value == approx(-4)
assert f.get_gradient("x") == approx(-2)
assert f.get_gradient("y") == approx(2)
def test_eq():
x = Forward("x", 23)
f = x == x
assert f
f = Forward("x", 1) == Forward("y", 2)
assert not f
f = Forward("x", 1) == Forward("y", 1)
assert f
def test_weirdness():
x = Forward("x", 2)
f = x**x
assert f.value == approx(4)
assert f.get_gradient("x") == approx(4 + np.log(16))
x = Forward("x", -1.2)
f = x**x
assert f.value == approx((-1.2)**(-1.2))
with raises(ValueError):
f.get_gradient("x")
def test_invalid_creation():
with raises(ValueError):
_x = Forward("x")
with raises(ValueError):
Forward("x", "y")
with raises(ValueError):
Forward(2, 3)
with raises(ValueError):
Forward(2, 3, 4)
with raises(ValueError):
Forward()
def test_arccot():
x = Forward("x", 0.5)
f = arccot(x)
assert f.value == approx(np.arctan(1 / 0.5))
assert f.get_gradient("x") == approx(-1 / (1 + 0.5**2))
def test_sin_chain():
val = np.pi + 1
x = Forward("x", val)
f = sin(2 * x)
assert f.value == approx(np.sin(2 * val))
assert f.get_gradient("x") == approx(2 * np.cos(2 * val))
def test_sondak():
x = Forward("x", 10)
f = cos(x)**2
assert f.value == approx(np.cos(10)**2)
assert f.get_gradient("x") == approx(2 * np.cos(10) * -np.sin(10))
def test_log_negative():
x = Forward("x", -2)
f = log(x)
assert np.isnan(f.value)
assert f.get_gradient("x") == approx(1 / -2)
def test_arcsec():
x = Forward("x", 3)
f = arcsec(x**2)
assert f.value == approx(np.arccos(1 / 9))
assert f.get_gradient("x") == approx(1 / (abs(9) * np.sqrt(81 - 1)) * 2 *
3)
def test_arccsc():
x = Forward("x", -4)
f = arccsc(2 * x)
assert f.value == approx(np.arcsin(1 / (2 * -4)))
y = 2 * -4
assert f.get_gradient("x") == approx(-1 / (abs(y) * np.sqrt(y**2 - 1)) * 2)
def test_other_trig_funcs():
x = Forward("x", 2)
f = sinh(x) + sinh(2)
assert f.value == approx(2 * np.sinh(2))
assert f.get_gradient("x") == approx(np.cosh(2))
f = cosh(x) + cosh(2)
assert f.value == approx(2 * np.cosh(2))
assert f.get_gradient("x") == approx(np.sinh(2))
f = tanh(x) + tanh(2)
assert f.value == approx(2 * np.tanh(2))
assert f.get_gradient("x") == approx(
(np.cosh(2)**2 - np.sinh(2)**2) / np.cosh(2)**2)
f = sech(x) + sech(2)
assert f.value == approx(2 * 1 / np.cosh(2))
assert f.get_gradient("x") == approx(
(-1 / np.cosh(2)) * (np.sinh(2) / np.cosh(2)))
f = csch(x) + csch(2)
assert f.value == approx(2 * 1 / np.sinh(2))
assert f.get_gradient("x") == approx(
(-1 / np.sinh(2)) * (np.cosh(2) / np.sinh(2)))
f = coth(x) + coth(2)
assert f.value == approx(2 * (np.cosh(2) / np.sinh(2)))
assert f.get_gradient("x") == approx(-1 / (np.sinh(2)**2))
def test_log2():
x = Forward("x", 3)
f = log2(x)**3
assert f.value == approx(np.log2(3)**3)
assert f.get_gradient("x") == approx(1 / np.log(2)**3 / 3 * np.log(3)**2 *
3)
def test_sin_of_const():
x = Forward(2)
f = sin(x)
assert f.value == approx(np.sin(2))
assert f.get_gradient("x") == approx(0)
assert f.get_gradient("asdf") == approx(0)
def test_vector_more():
x = Forward("x", -3.5)
f = fVector([x, 2 * x**x])
assert f.value == approx([-3.5, 2 * (-3.5)**(-3.5)])
with raises(ValueError):
f.get_gradient("x")
def test_power():
x = Forward("x", 5)
f = x**5
assert f.value == approx(5**5)
assert f.get_gradient("x") == approx(5 * 5**4)
x = Forward("x", 0)
f = x**5
assert f.value == approx(0)
assert f.get_gradient("x") == approx(0)
f = Forward("x", -5)**2
assert f.value == approx((-5)**2)
assert f.get_gradient("x") == approx(2 * -5)
def test_comparison_with_math():
x = Forward("x", 2)
f = x**2 < x**3
assert f
f = x**x**2 >= x**3
assert f
f = x**x**3 <= x**0.5
assert not f
def test_logistic():
l = lambda x: 1 / (1 + np.e**-x)
x = Forward("x", 0)
f = logistic(x)
assert f.value == approx(0.5)
assert f.get_gradient("x") == approx(l(0) * (1 - l(0)))
x = Forward("x", -5)
f = logistic(x)
assert f.value == approx(l(-5))
assert f.get_gradient("x") == approx(l(-5) * (1 - l(-5)))
x = Forward("x", 20)
f = logistic(x)
assert f.value == approx(l(20))
assert f.get_gradient("x") == approx(l(20) * (1 - l(20)))
def test_division():
x = Forward("x", -2)
f = 4 / x
assert f.value == approx(4 / -2)
assert f.get_gradient("x") == approx(-1 * 4 / (-2)**2)
f = x / 4
assert f.value == approx(-2 / 4)
assert f.get_gradient("x") == approx(1 / 4)
def newtons_method(f, initial_guess=0):
while True:
x = Forward("x", initial_guess)
updated = f(x)
update = updated.value / updated.get_gradient("x")
if abs(update) < 10 ** -12:
break
initial_guess -= update
return initial_guess
def test_ne():
f = Forward("x", 48) != Forward("y", 1)
assert f
f = Forward("x", 24) != Forward("y", 24)
assert not f
f = Forward("x", -1) != Forward("y", 23)
assert f
def test_lt():
f = Forward("x", 1) < Forward("y", 2)
assert f
f = Forward("x", 2) < Forward("y", 1)
assert not f
f = Forward("x", 1) < Forward("y", 1)
assert not f
def test_gt():
f = Forward("x", 24) > Forward("y", 1)
assert f
f = Forward("x", 1) > Forward("y", 24)
assert not f
f = Forward("x", 2) > Forward("y", 2)
assert not f
def test_lte():
f = Forward("x", -24) <= Forward("y", 45)
assert f
f = Forward("x", 36) <= Forward("y", -2)
assert not f
f = Forward("x", 37) <= Forward("x", 37)
assert f
def test_gte():
f = Forward("x", 24) >= Forward("y", 2)
assert f
f = Forward("x", -2) >= Forward("y", 45)
assert not f
f = Forward("x", 3) >= Forward("y", 3)
assert f
def test_multiplication_by_variable():
x = Forward("x", 2)
f = x * x
assert f.value == approx(4)
assert f.get_gradient("x") == approx(4)