def test_truediv(self):
x = AD(4)
f = el.log(x, 2) / 3 ** x
assert (np.round(f.val, 6), np.round(f.der[-1], 6)) == (
np.round(2 / 81, 6), np.round((81 / (4 * np.log(2)) - 162 * np.log(3)) / 3 ** 8, 6))
f = el.sin(x) / 4
assert (f.val, f.der) == ((np.sin(4)) / 4, (np.cos(4)) / 4)
with pytest.raises(ZeroDivisionError):
f = el.cos(x) / el.sin(0)
with pytest.raises(ZeroDivisionError):
f = el.cos(x) / 0
f = 3 ** x / el.log(x, 2)
assert (np.round(f.val, 6), np.round(f.der, 6)) == (np.round(81 / 2, 6),
np.round((162 * np.log(3) - 81 / (4 * np.log(2))) / (
np.log(4) / np.log(2)) ** 2, 6))
with pytest.raises(AttributeError):
f = el.cos(x) / "el.sin(0)"
x = AD(0)
f = el.sin(x) / el.cos(x)
assert (f.val, f.der) == (0, 1)
with pytest.raises(ZeroDivisionError):
f = el.cos(x) / el.sin(x)
def test_sin(self):
# N = 1
x = AD(0)
f = el.sin(x)
assert (f.val, f.der) == (0.0, 1.0)
x = 13
f = el.sin(x)
assert (f.val, f.der) == (np.sin(13), 0)
x = 13.0
f = el.sin(x)
assert (f.val, f.der) == (np.sin(13), 0)
# N > 1
x = AD(2, N=4)
f = el.sin(x)
# print(f)
assert (f.val, np.round(f.der[-1], 6)) == (np.sin(2), np.round(np.sin(2), 6))
x = AD(2, N=3)
f = el.sin(x)
# print(f)
assert (f.val, np.round(f.der[-1], 6)) == (np.sin(2), np.round(-np.cos(2), 6))
f = el.sin(x) + 3
assert (f.val, np.round(f.der[-1], 6)) == (np.sin(2) + 3, np.round(-np.cos(2), 6))
f = el.sin(x) - 2
assert (f.val, np.round(f.der[-1], 6)) == (np.sin(2) - 2, np.round(-np.cos(2), 6))
f = el.sin(x) * 4
assert (f.val, np.round(f.der[-1], 6)) == (4 * np.sin(2), np.round(-4 * np.cos(2), 6))
def test_sqrt(self):
x = AD(4)
f = el.sqrt(x)
assert (f.val, f.der) == (2, 0.5 * 4 ** (-0.5))
x = 4
f = el.sqrt(x)
assert (f.val, f.der) == (2, 0)
x = 16.0
f = el.sqrt(x)
assert (f.val, f.der) == (4, 0)
x = AD(-5)
with pytest.raises(ValueError):
f = el.sqrt(x)
x = -5
with pytest.raises(ValueError):
f = el.sqrt(x)
with pytest.raises(AttributeError):
x = "World"
f = el.sqrt(x)
x = AD(4, N=3)
f = el.sqrt(x)
assert (f.val, f.der[-1]) == (2, (3 / 8) * 4 ** (-2.5))
def test_exp(self):
# N = 1
f = el.exp(1)
assert (f.val, f.der) == (np.exp(1), 0)
x = AD(1)
f = el.exp(x)
assert (f.val, f.der) == (np.exp(1), np.exp(1))
x = AD(2)
f = el.power(x, 3)
g = el.exp(f)
print(g)
assert (round(g.val, 6), round(g.der[0], 6)) == (round(np.exp(1) ** 8, 6), round(12 * np.exp(1) ** 8, 6))
with pytest.raises(AttributeError):
x = "hello"
f = el.exp(x)
# N > 1
for N in range(2, 10):
x = AD(3, N=N)
f = el.exp(x)
print("N=%d" % N, f)
x = AD(2, N=5)
f = el.exp(x)
assert (f.val, f.der[-1]) == (np.exp(2), np.exp(2))
def test_log(self):
x = AD(12)
f = el.log(x, 10)
assert (f.val, f.der) == (np.log(12) / np.log(10), 1 / (12 * np.log(10)))
x = 8
f = el.log(x, 2)
assert (np.round(f.val, 6), f.der) == (3, 0)
x = 9.0
f = el.log(x, 3)
assert (f.val, f.der) == (2, 0)
x = AD(0)
with pytest.raises(ValueError):
f = el.log(x, 2)
x = 0
with pytest.raises(ValueError):
f = el.log(x, 2)
with pytest.raises(AttributeError):
x = "s"
f = el.log(x, 3)
with pytest.raises(ValueError):
x = -3.9
f = el.log(x, 3)
def test_arccos(self):
x = AD(0.5)
f = el.arccos(x)
assert (f.val, f.der) == (np.arccos(0.5), -1 / np.sqrt(1 - 0.5 ** 2))
x = 0.5
f = el.arccos(x)
assert (f.val, f.der) == (np.arccos(0.5), 0)
x = -5
with pytest.raises(ValueError):
f = el.arccos(x)
x = 10.0
with pytest.raises(ValueError):
f = el.arccos(x)
x = AD(-5.0)
with pytest.raises(ValueError):
f = el.arccos(x)
x = AD(10)
with pytest.raises(ValueError):
f = el.arccos(x)
with pytest.raises(AttributeError):
x = "--"
f = el.arccos(x)
def test_arcsin(self):
x = AD(0.5)
f = el.arcsin(x)
assert (f.val, f.der) == (np.arcsin(0.5), 1 / np.sqrt(1 - 0.5 ** 2))
x = 0
f = el.arcsin(x)
assert (f.val, f.der) == (np.arcsin(0), 0)
x = 1.0
f = el.arcsin(x)
assert (f.val, f.der) == (np.arcsin(1), 0)
x = -5
with pytest.raises(ValueError):
f = el.arcsin(x)
x = 10.0
with pytest.raises(ValueError):
f = el.arcsin(x)
x = AD(-5.0)
with pytest.raises(ValueError):
f = el.arcsin(x)
x = AD(10)
with pytest.raises(ValueError):
f = el.arcsin(x)
with pytest.raises(AttributeError):
x = "."
f = el.arcsin(x)
def test_compare(self):
n = 20
x = AD(20)
y = 100
z = AD(100)
assert (n < z.val) == n.__lt__(z.val)
assert (x.val < y) == x.val.__lt__(y)
assert (n <= z.val) == n.__le__(z.val)
assert (x.val <= y) == x.val.__le__(y)
assert (y <= z.val) == y.__le__(z.val)
assert (y > x.val) == y.__gt__(x.val)
assert (z.val > n) == z.val.__gt__(n)
assert (y >= x.val) == y.__ge__(x.val)
assert (z.val >= n) == z.val.__ge__(n)
assert (n >= y) == n.__ge__(y)
assert (y == z.val) == y.__eq__(z.val)
assert (z.val == y) == z.val.__eq__(y)
assert (y != x.val) == y.__ne__(x.val)
assert (x.val != y) == x.val.__ne__(y)
assert ("val = 20; der = [1.]") == x.__str__()
def test_inv(self):
x = 10
assert el.inv(x) == 0.1
y = -1 / 30
assert el.inv(y) == -30
with pytest.raises(ZeroDivisionError):
x = 0
f = el.inv(x)
with pytest.raises(AttributeError):
x = "zero"
f = el.inv(x)
x = AD(8)
f = el.inv(x)
assert (f.val, f.der) == (1 / 8, -1 / 64)
x = 92
f = el.inv(x)
assert (f.val, f.der) == (1 / 92, 0)
x = AD(10, N=3)
f = el.inv(x)
assert (f.val, f.der[-1]) == (0.1, -6 * 1 / 10000)
def test_rtruediv(self):
x = AD(4, N=2)
f = 3 / (2 * x)
assert (f.val, f.der[-1]) == (3 / 8, 3 / 64)
x = AD(4, N=3)
f = 3 / (2 * x) # shows rtruediv should work
assert (f.val, f.der[-1]) == (3 / 8, -9 / 4 ** 4)
def test_cosh(self):
# N > 1
x = AD(10, N=4)
f = el.cosh(x)
assert (f.val, f.der[-1]) == (np.cosh(10), np.cosh(10))
x = AD(10, N=7)
f = el.cosh(x)
assert (f.val, f.der[-1]) == (np.cosh(10), np.sinh(10))
def test_truediv(self):
# N = 1 tests all passed in solvers_test
x = AD(3)
f = 5 * x / 3
assert (f.val, f.der[-1]) == (5, 5 / 3)
x = AD(4, N=2)
f = x / x
assert (f.val, f.der[-1]) == (1, 0)
def test_tan(self):
# N > 1
x = AD(10, N=2)
f = el.tan(x)
assert (f.val, np.round(f.der[-1], 6)) == (np.tan(10), np.round(2 * np.tan(10) * (1 / (np.cos(10) ** 2)), 6))
x = AD(10, N=3)
f = el.tan(x)
assert (f.val, np.round(f.der[-1], 6)) == (
np.tan(10), np.round(-2 * (-2 + np.cos(20)) * (1 / (np.cos(10) ** 4)), 6))
def test_ln(self):
x = AD(-10)
with pytest.raises(ValueError):
f = el.ln(x)
x = AD(10, N=3)
f = el.ln(x)
assert (f.val, f.der[-1]) == (np.log(10), 2 * 1 / (1000))
x = AD(10, N=4)
f = el.ln(x)
assert (f.val, f.der[-1]) == (np.log(10), -6 * 1 / (10000))
def test_tanh(self):
# N > 1
x = AD(0.5, N=2)
f = el.tanh(x)
assert (f.val, np.round(f.der[-1], 6)) == (np.tanh(0.5), -0.726862)
x = AD(0.5, N=3)
f = el.tanh(x)
assert (f.val, np.round(f.der[-1], 6)) == (np.tanh(0.5), -0.565209)
x = AD(0, N=3)
f = el.tanh(x)
assert (f.val, f.der[-1]) == (np.tanh(0), -2)
def test_power_k(self):
with pytest.raises(AttributeError):
x = AD(10)
n = "1"
f = el.power_k_order(x, n, 3)
with pytest.raises(AttributeError):
x = AD(10)
n = AD(3)
f = el.power_k_order(x, n, 3)
with pytest.raises(ValueError):
x = AD(-3)
n = 1 / 2
f = el.power_k_order(x, n, 3)
def test_sub(self):
x = AD(5)
f = el.power(x, 2) + -5 * x
assert (f.val, f.der) == (0, 5)
f = el.power(x, 2) - 50
assert (f.val, f.der) == (-25, 10)
f = - 50 + el.power(x, 2)
assert (f.val, f.der) == (-25, 10)
f = 50 - el.power(x, 2)
assert (f.val, f.der) == (25, -10)
f = -5 * x + el.power(x, 2)
assert (f.val, f.der) == (0, 5)
f = -x * 5 + el.power(x, 2)
assert (f.val, f.der) == (0, 5)
f = el.power(x, 2) - 5 * x
assert (f.val, f.der) == (0, 5)
f = x * 5 - el.power(x, 2)
assert (f.val, f.der) == (0, -5)
with pytest.raises(AttributeError):
f = el.sin(x) - "5"
def test_add(self):
x = AD(5)
f = el.power(x, 2) + 5
assert (f.val, f.der) == (30, 10)
f = 5 + el.power(x, 2)
assert (f.val, f.der) == (30, 10)
f = el.power(x, 2) + 5 * x
assert (f.val, f.der) == (50, 15)
f = x * 5 + el.power(x, 2)
assert (f.val, f.der) == (50, 15)
with pytest.raises(AttributeError):
f = el.power(x, 2) + "5"
f = el.power(x, 2)
g = 5
h = f + g
assert (h.val, h.der) == (30, 10)
f = 5
g = el.power(x, 2)
h = f + g
assert (h.val, h.der) == (30, 10)
def test_expn(self):
x = AD(2)
f = el.expn(x, 25)
assert (f.val, f.der) == (625, 625 * np.log(25))
x = 2
f = el.expn(x, 25)
assert (f.val, f.der) == (625, 0)
with pytest.raises(AttributeError):
f = el.expn('25', 0)
with pytest.raises(ValueError):
f = el.expn(x, -10)
x = AD(4)
f = el.expn(x, 2)
def test_exp(self):
f = el.exp(1)
assert (f.val, f.der) == (np.exp(1), 0)
x = AD(1)
f = el.exp(x)
assert (f.val, f.der) == (np.exp(1), np.exp(1))
x = AD(2)
f = el.power(x, 3)
g = el.exp(f)
assert (np.round(g.val, 6), np.round(g.der, 6)) == (
np.round(np.exp(1) ** 8, 6), np.round(12 * np.exp(1) ** 8, 6))
with pytest.raises(AttributeError):
x = "hello"
f = el.exp(x)
def test_inv(self):
# N = 1
f = el.ln(np.e)
assert (f.val, f.der) == (1, 0)
x = AD(np.e)
f = el.ln(x)
assert (f.val, f.der) == (1, 1 / np.e)
# N > 1
# See explicit formula here: https://www.math24.net/higher-order-derivatives/#example2
x = AD(np.e, N=2)
f = el.ln(x)
assert (f.val, f.der[-1]) == (1, -1 / (np.e ** 2))
x = AD(np.e ** 2, N=5)
f = el.ln(x)
assert (f.val, f.der[-1]) == (2, 24 / ((np.e ** 2) ** 5))
def test_cos(self):
# N > 1
x = AD(10, N=5)
f = el.cos(x)
assert (f.val, np.round(f.der[-1], 6)) == (np.cos(10), np.round(-np.sin(10), 6))
x = AD(2, N=2)
f = el.cos(x)
assert (f.val, np.round(f.der[-1], 6)) == (np.cos(2), np.round(-np.cos(2), 6))
f = el.cos(x) + 3
assert (f.val, np.round(f.der[-1], 6)) == (np.cos(2) + 3, np.round(-np.cos(2), 6))
f = el.cos(x) - 2
assert (f.val, np.round(f.der[-1], 6)) == (np.cos(2) - 2, np.round(-np.cos(2), 6))
f = el.cos(x) * 4
assert (f.val, np.round(f.der[-1], 6)) == (4 * np.cos(2), np.round(-4 * np.cos(2), 6))
def test_arctan(self):
x = AD(0.5)
f = el.arctan(x)
assert (f.val, f.der) == (np.arctan(0.5), 1 / (1 + 0.5 ** 2))
x = 0.5
f = el.arctan(x)
assert (f.val, f.der) == (np.arctan(0.5), 0)
with pytest.raises(AttributeError):
x = "AD(0.5)"
f = el.arctan(x)
def test_sub(self):
# N > 1
x = AD(5, N=3)
f = el.sin(x)
g = 10
h = f - g
assert (h.val, np.round(h.der[-1]), 6) == (np.sin(5) - 10, np.round(-np.cos(5)), 6)
f = el.sin(x)
g = x ** 2
h = f - g
assert (h.val, np.round(h.der[-1]), 6) == (np.sin(5) - 25, np.round(-np.cos(5)), 6)
def test_mul(self):
x = AD(4)
f = el.log(x, 2) * 3 ** x
assert (np.round(f.val, 6), np.round(f.der[-1], 6)) == (
162, np.round(81 / (4 * np.log(2)) + 162 * np.log(3), 6))
f = 3 ** x * el.log(x, 2)
assert (np.round(f.val, 6), np.round(f.der[-1], 6)) == (
162, np.round(81 / (4 * np.log(2)) + 162 * np.log(3), 6))
with pytest.raises(AttributeError):
f = x * "5"
def test_cosh(self):
x = AD(0.5)
f = el.cosh(x)
assert (f.val, f.der) == (np.cosh(0.5), np.sinh(0.5))
x = 0.5
f = el.cosh(x)
assert (f.val, f.der) == (np.cosh(0.5), 0)
with pytest.raises(AttributeError):
x = "0.5"
f = el.cosh(x)
def test_tanh(self):
x = AD(0.5)
f = el.tanh(x)
assert (f.val, f.der) == (np.tanh(0.5), 1 - np.tanh(0.5) ** 2)
x = 0.5
f = el.tanh(x)
assert (f.val, f.der) == (np.tanh(0.5), 0)
with pytest.raises(AttributeError):
x = "0.5"
f = el.tanh(x)
def test_logistic(self):
x = AD(100)
f = el.logistic(x, x0=0, L=1, k=1)
assert (f.val, np.round(f.der[-1], 10)) == (1 / (1 + np.exp(-100)),
np.round(np.exp(100) / (1 + np.exp(100)) ** 2, 6))
# assert (f.val, f.der[-1]) == (1/(1+np.exp(-100)), np.exp(100)/(1 + np.exp(100))**2)
f = el.logistic(x, x0=10, L=2, k=3)
assert (f.val, f.der[-1]) == (2 * np.exp(270) / (1 + np.exp(270)), (6 * np.exp(270)) / (1 + np.exp(270)) ** 2)
x = AD(100, N=2)
f = el.logistic(x, x0=0, L=1, k=1)
assert (f.val, np.round(f.der[-1], 6)) == (
1 / (1 + np.exp(-100)), np.round((np.exp(100) - np.exp(200)) / (1 + np.exp(100)) ** 3, 6))
x = 100
f = el.logistic(x, x0=0, L=1, k=1)
assert (f.val, f.der) == (1 / (1 + np.exp(-100)), 0)
with pytest.raises(AttributeError):
x = "AD(-3)"
f = el.logistic(x, x0=0, L=1, k=1)
def test_ln(self):
# N = 1
f = el.ln(np.e)
assert (f.val, f.der) == (1, 0)
x = AD(np.e)
f = el.ln(x)
assert (f.val, f.der) == (1, 1 / np.e)
# N > 1
# See explicit formula here: https://www.math24.net/higher-order-derivatives/#example2
x = AD(np.e, N=2)
f = el.ln(x)
assert (f.val, f.der[-1]) == (1, -1 / (np.e ** 2))
x = AD(np.e ** 2, N=5)
f = el.ln(x)
assert (f.val, f.der[-1]) == (2, 24 / ((np.e ** 2) ** 5))
# composite x
x = AD(np.e, N=4)
f = el.ln(x * 3 + 1)
assert (f.val, f.der[-1]) == (np.log(np.e * 3 + 1), (-6 * 3 ** 4) / ((np.e * 3 + 1) ** 4))
def test_power(self):
x = AD(5)
f = el.power(x, 2)
assert (f.val, f.der) == (25, 10)
x = 10
f = el.power(x, 3)
assert (f.val, f.der) == (1000, 0)
x = 10.0
f = el.power(x, 3)
assert (f.val, f.der) == (1000, 0)
with pytest.raises(AttributeError):
x = "s"
f = el.power(x, 3)