def test_pmint_logexp():
f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x
g = log(x**2 + 2*x*exp(x) + 2*x*log(x) + exp(2*x) + 2*exp(x)*log(x) + log(x)**2)/2 + 1/(x + exp(x) + log(x))
# TODO: Optimal solution is g = 1/(x + log(x) + exp(x)) + log(x + log(x) + exp(x)),
# but Diofant requires a lot of guidance to properly simplify heurisch() output.
assert ratsimp(heurisch(f, x)) == g
def test_pmint_logexp():
f = (1 + x + x * exp(x)) * (x + log(x) + exp(x) - 1) / (x + log(x) +
exp(x))**2 / x
g = log(x**2 + 2 * x * exp(x) + 2 * x * log(x) + exp(2 * x) +
2 * exp(x) * log(x) + log(x)**2) / 2 + 1 / (x + exp(x) + log(x))
# TODO: Optimal solution is g = 1/(x + log(x) + exp(x)) + log(x + log(x) + exp(x)),
# but Diofant requires a lot of guidance to properly simplify heurisch() output.
assert ratsimp(heurisch(f, x)) == g
def test_ratsimp():
f, g = 1/x + 1/y, (x + y)/(x*y)
assert f != g and ratsimp(f) == g
f, g = 1/(1 + 1/x), 1 - 1/(x + 1)
assert f != g and ratsimp(f) == g
f, g = x/(x + y) + y/(x + y), 1
assert f != g and ratsimp(f) == g
f, g = -x - y - y**2/(x + y) + x**2/(x + y), -2*y
assert f != g and ratsimp(f) == g
f = (a*c*x*y + a*c*z - b*d*x*y - b*d*z - b*t*x*y - b*t*x - b*t*z +
e*x)/(x*y + z)
G = [a*c - b*d - b*t + (-b*t*x + e*x)/(x*y + z),
a*c - b*d - b*t - ( b*t*x - e*x)/(x*y + z)]
assert f != g and ratsimp(f) in G
A = sqrt(pi)
B = log(erf(x) - 1)
C = log(erf(x) + 1)
D = 8 - 8*erf(x)
f = A*B/D - A*C/D + A*C*erf(x)/D - A*B*erf(x)/D + 2*A/D
assert ratsimp(f) == A*B/8 - A*C/8 - A/(4*erf(x) - 4)
def test_ratsimp():
f, g = 1/x + 1/y, (x + y)/(x*y)
assert f != g and ratsimp(f) == g
f, g = 1/(1 + 1/x), 1 - 1/(x + 1)
assert f != g and ratsimp(f) == g
f, g = x/(x + y) + y/(x + y), 1
assert f != g and ratsimp(f) == g
f, g = -x - y - y**2/(x + y) + x**2/(x + y), -2*y
assert f != g and ratsimp(f) == g
f = (a*c*x*y + a*c*z - b*d*x*y - b*d*z - b*t*x*y - b*t*x - b*t*z +
e*x)/(x*y + z)
G = [a*c - b*d - b*t + (-b*t*x + e*x)/(x*y + z),
a*c - b*d - b*t - ( b*t*x - e*x)/(x*y + z)]
assert f != g and ratsimp(f) in G
A = sqrt(pi)
B = log(erf(x) - 1)
C = log(erf(x) + 1)
D = 8 - 8*erf(x)
f = A*B/D - A*C/D + A*C*erf(x)/D - A*B*erf(x)/D + 2*A/D
assert ratsimp(f) == A*B/8 - A*C/8 - A/(4*erf(x) - 4)
def test_pmint_rat():
# TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
# would give the optimal result?
def drop_const(expr, x):
if expr.is_Add:
return Add(*[ arg for arg in expr.args if arg.has(x) ])
else:
return expr
f = (x**7 - 24*x**4 - 4*x**2 + 8*x - 8)/(x**8 + 6*x**6 + 12*x**4 + 8*x**2)
g = (4 + 8*x**2 + 6*x + 3*x**3)/(x**5 + 4*x**3 + 4*x) + log(x)
assert drop_const(ratsimp(heurisch(f, x)), x) == g
def test_pmint_rat():
# TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
# would give the optimal result?
def drop_const(expr, x):
if expr.is_Add:
return Add(*[arg for arg in expr.args if arg.has(x)])
else:
return expr
f = (x**7 - 24 * x**4 - 4 * x**2 + 8 * x - 8) / (x**8 + 6 * x**6 +
12 * x**4 + 8 * x**2)
g = (4 + 8 * x**2 + 6 * x + 3 * x**3) / (x**5 + 4 * x**3 + 4 * x) + log(x)
assert drop_const(ratsimp(heurisch(f, x)), x) == g
def test_pmint_erf():
f = exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi) * log(erf(x) - 1) / 8 - sqrt(pi) * log(erf(x) + 1) / 8 - sqrt(
pi) / (4 * erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_pmint_erf():
f = exp(-x**2)*erf(x)/(erf(x)**3 - erf(x)**2 - erf(x) + 1)
g = sqrt(pi)*log(erf(x) - 1)/8 - sqrt(pi)*log(erf(x) + 1)/8 - sqrt(pi)/(4*erf(x) - 4)
assert ratsimp(heurisch(f, x)) == g
def test_ratsimp():
assert ratsimp(A * B - B * A) == A * B - B * A
def test_ratsimp():
assert ratsimp(A*B - B*A) == A*B - B*A
