def test_combine_inverse():
assert Mul._combine_inverse(x * I * y, x * I) == y
assert Mul._combine_inverse(x * I * y, y * I) == x
assert Mul._combine_inverse(oo * I * y, y * I) == oo
assert Mul._combine_inverse(oo * I * y, oo * I) == y
assert Add._combine_inverse(oo, oo) == Integer(0)
assert Add._combine_inverse(oo * I, oo * I) == Integer(0)
def test_lookup_table():
table = defaultdict(list)
_create_lookup_table(table)
for _, l in sorted(table.items(), key=default_sort_key):
for formula, terms, cond, hint in sorted(l, key=default_sort_key):
subs = {}
for a in list(formula.free_symbols) + [z_dummy]:
if hasattr(a, 'properties') and a.properties:
# these Wilds match positive integers
subs[a] = randrange(1, 10)
else:
subs[a] = uniform(1.5, 2.0)
if not isinstance(terms, list):
terms = terms(subs)
# First test that hyperexpand can do this.
expanded = [hyperexpand(g) for (_, g) in terms]
assert all(x.is_Piecewise or not x.has(meijerg) for x in expanded)
# Now test that the meijer g-function is indeed as advertised.
expanded = Add(*[f * x for (f, x) in terms])
a, b = formula.evalf(subs=subs,
strict=False), expanded.evalf(subs=subs,
strict=False)
r = min(abs(a), abs(b))
if r < 1:
assert abs(a - b).evalf(strict=False) <= 1e-10
else:
assert (abs(a - b) / r).evalf(strict=False) <= 1e-10
def test_lookup_table():
table = defaultdict(list)
_create_lookup_table(table)
for _, l in sorted(table.items(), key=default_sort_key):
for formula, terms, cond, hint in sorted(l, key=default_sort_key):
subs = {}
for a in list(formula.free_symbols) + [z_dummy]:
if hasattr(a, 'properties') and a.properties:
# these Wilds match positive integers
subs[a] = randrange(1, 10)
else:
subs[a] = uniform(1.5, 2.0)
if not isinstance(terms, list):
terms = terms(subs)
# First test that hyperexpand can do this.
expanded = [hyperexpand(g) for (_, g) in terms]
assert all(x.is_Piecewise or not x.has(meijerg) for x in expanded)
# Now test that the meijer g-function is indeed as advertised.
expanded = Add(*[f*x for (f, x) in terms])
a, b = formula.evalf(subs=subs, strict=False), expanded.evalf(subs=subs, strict=False)
r = min(abs(a), abs(b))
if r < 1:
assert abs(a - b).evalf(strict=False) <= 1e-10
else:
assert (abs(a - b)/r).evalf(strict=False) <= 1e-10
def test_combine_inverse():
assert Mul._combine_inverse(x*I*y, x*I) == y
assert Mul._combine_inverse(x*I*y, y*I) == x
assert Mul._combine_inverse(oo*I*y, y*I) == oo
assert Mul._combine_inverse(oo*I*y, oo*I) == y
assert Add._combine_inverse(oo, oo) == Integer(0)
assert Add._combine_inverse(oo*I, oo*I) == Integer(0)
def test_TR10i():
assert TR10i(cos(1) * cos(3) + sin(1) * sin(3)) == cos(2)
assert TR10i(cos(1) * cos(3) - sin(1) * sin(3)) == cos(4)
assert TR10i(cos(1) * sin(3) - sin(1) * cos(3)) == sin(2)
assert TR10i(cos(1) * sin(3) + sin(1) * cos(3)) == sin(4)
assert TR10i(cos(1) * sin(3) + sin(1) * cos(3) + 7) == sin(4) + 7
assert TR10i(cos(1) * sin(3) + sin(1) * cos(3) + cos(3)) == cos(3) + sin(4)
assert TR10i(2*cos(1)*sin(3) + 2*sin(1)*cos(3) + cos(3)) == \
2*sin(4) + cos(3)
assert TR10i(cos(2)*cos(3) + sin(2)*(cos(1)*sin(2) + cos(2)*sin(1))) == \
cos(1)
eq = (cos(2) * cos(3) + sin(2) *
(cos(1) * sin(2) + cos(2) * sin(1))) * cos(5) + sin(1) * sin(5)
assert TR10i(eq) == TR10i(eq.expand()) == cos(4)
assert TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) == \
2*sqrt(2)*x*sin(x + pi/6)
assert TR10i(
cos(x) / sqrt(6) + sin(x) / sqrt(2) + cos(x) / sqrt(6) / 3 +
sin(x) / sqrt(2) / 3) == 4 * sqrt(6) * sin(x + pi / 6) / 9
assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) +
cos(y)/sqrt(6)/3 + sin(y)/sqrt(2)/3) == \
sqrt(6)*sin(x + pi/6)/3 + sqrt(6)*sin(y + pi/6)/9
assert TR10i(cos(x) + sqrt(3) * sin(x) +
2 * sqrt(3) * cos(x + pi / 6)) == 4 * cos(x)
assert TR10i(
cos(x) + sqrt(3) * sin(x) + 2 * sqrt(3) * cos(x + pi / 6) +
4 * sin(x)) == 4 * sqrt(2) * sin(x + pi / 4)
assert TR10i(cos(2)*sin(3) + sin(2)*cos(4)) == \
sin(2)*cos(4) + sin(3)*cos(2)
A = Symbol('A', commutative=False)
assert TR10i(sqrt(2)*cos(x)*A + sqrt(6)*sin(x)*A) == \
2*sqrt(2)*sin(x + pi/6)*A
c = cos(x)
s = sin(x)
h = sin(y)
r = cos(y)
for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
for a in ((c * r, s * h), (c * h, s * r)): # explicit 2-args
args = zip(si, a)
ex = Add(*[Mul(*ai) for ai in args])
t = TR10i(ex)
assert not (ex - t.expand(trig=True) or t.is_Add)
c = cos(x)
s = sin(x)
h = sin(pi / 6)
r = cos(pi / 6)
for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
for a in ((c * r, s * h), (c * h, s * r)): # induced
args = zip(si, a)
ex = Add(*[Mul(*ai) for ai in args])
t = TR10i(ex)
assert not (ex - t.expand(trig=True) or t.is_Add)
def test_evaluate_false():
cases = {
'2 + 3': Add(2, 3, evaluate=False),
'2**2 / 3': Mul(Pow(2, 2, evaluate=False), Pow(3, -1, evaluate=False), evaluate=False),
'2 + 3 * 5': Add(2, Mul(3, 5, evaluate=False), evaluate=False),
'2 - 3 * 5': Add(2, -Mul(3, 5, evaluate=False), evaluate=False),
'1 / 3': Mul(1, Pow(3, -1, evaluate=False), evaluate=False),
'True | False': Or(True, False, evaluate=False),
'1 + 2 + 3 + 5*3 + integrate(x)': Add(1, 2, 3, Mul(5, 3, evaluate=False), x**2/2, evaluate=False),
'2 * 4 * 6 + 8': Add(Mul(2, 4, 6, evaluate=False), 8, evaluate=False),
}
for case, result in cases.items():
assert sympify(case, evaluate=False) == result
def test_gcd_terms():
f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + \
(2*x + 2)*(x + 6)/(5*x**2 + 5)
assert _gcd_terms(f) == (Rational(6, 5) * ((1 + x) / (1 + x**2)), 5 + x, 1)
assert _gcd_terms(Add.make_args(f)) == \
(Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
newf = Rational(6, 5) * ((1 + x) * (5 + x) / (1 + x**2))
assert gcd_terms(f) == newf
args = Add.make_args(f)
# non-Basic sequences of terms treated as terms of Add
assert gcd_terms(list(args)) == newf
assert gcd_terms(tuple(args)) == newf
assert gcd_terms(set(args)) == newf
# but a Basic sequence is treated as a container
assert gcd_terms(Tuple(*args)) != newf
assert gcd_terms(Basic(Tuple(1, 3*y + 3*x*y), Tuple(1, 3))) == \
Basic((1, 3*y*(x + 1)), (1, 3))
# but we shouldn't change keys of a dictionary or some may be lost
assert gcd_terms(Dict((x*(1 + y), 2), (x + x*y, y + x*y))) == \
Dict({x*(y + 1): 2, x + x*y: y*(1 + x)})
assert gcd_terms((2 * x + 2)**3 +
(2 * x + 2)**2) == 4 * (x + 1)**2 * (2 * x + 3)
assert gcd_terms(0) == 0
assert gcd_terms(1) == 1
assert gcd_terms(x) == x
assert gcd_terms(2 + 2 * x) == Mul(2, 1 + x, evaluate=False)
arg = x * (2 * x + 4 * y)
garg = 2 * x * (x + 2 * y)
assert gcd_terms(arg) == garg
assert gcd_terms(sin(arg)) == sin(garg)
# issue sympy/sympy#6139-like
alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
a = alpha**2 - alpha * x**2 + alpha + x**3 - x * (alpha + 1)
rep = {
alpha: (1 + sqrt(5)) / 2 + alpha1 * x + alpha2 * x**2 + alpha3 * x**3
}
s = (a / (x - alpha)).subs(rep).series(x, 0, 1)
assert simplify(collect(s, x)) == -sqrt(5) / 2 - Rational(3, 2) + O(x)
# issue sympy/sympy#5917
assert _gcd_terms([Integer(0), Integer(0)]) == (0, 0, 1)
assert _gcd_terms([2 * x + 4]) == (2, x + 2, 1)
eq = x / (x + 1 / x)
assert gcd_terms(eq, fraction=False) == eq
def _minpoly_cos(ex, x):
"""
Returns the minimal polynomial of ``cos(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
from diofant import sqrt
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
if c.p == 1:
if c.q == 7:
return 8 * x**3 - 4 * x**2 - 4 * x + 1
if c.q == 9:
return 8 * x**3 - 6 * x + 1
elif c.p == 2:
q = sympify(c.q)
if q.is_prime:
s = _minpoly_sin(ex, x)
return _mexpand(s.subs({x: sqrt((1 - x) / 2)}))
# for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
n = int(c.q)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i) * a[i] for i in range(n + 1)]
r = Add(*a) - (-1)**c.p
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def test_Add():
assert str(x + y) == "x + y"
assert str(x + 1) == "x + 1"
assert str(x + x**2) == "x**2 + x"
assert str(5 + x + y + x*y + x**2 + y**2) == "x**2 + x*y + x + y**2 + y + 5"
assert str(1 + x + x**2/2 + x**3/3) == "x**3/3 + x**2/2 + x + 1"
assert str(2*x - 7*x**2 + 2 + 3*y) == "-7*x**2 + 2*x + 3*y + 2"
assert str(x - y) == "x - y"
assert str(2 - x) == "-x + 2"
assert str(x - 2) == "x - 2"
assert str(x - y - z - w) == "-w + x - y - z"
assert str(x - z*y**2*z*w) == "-w*y**2*z**2 + x"
assert str(x - 1*y*x*y) == "-x*y**2 + x"
assert str(sin(x).series(x, 0, 15)) == "x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 - x**11/39916800 + x**13/6227020800 + O(x**15)"
assert str(Add(0, 0, evaluate=False)) == "0 + 0"
assert str(Add(And(x, y), z)) == 'z + (x & y)'
def test_diop_general_sum_of_squares_quick():
for i in range(3, 10):
assert check_solutions(sum(i**2 for i in symbols(f':{i:d}')) - i)
pytest.raises(ValueError, lambda: _diop_general_sum_of_squares((x, y), 2))
assert _diop_general_sum_of_squares((x, y, z), -2) == set()
eq = x**2 + y**2 + z**2 - (1 + 4 + 9)
assert diop_general_sum_of_squares(eq) == {(1, 2, 3)}
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313
assert len(diop_general_sum_of_squares(eq, 3)) == 3
# issue sympy/sympy#11016
var = symbols(':5') + (symbols('6', negative=True), )
eq = Add(*[i**2 for i in var]) - 112
assert diophantine(eq) == {(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8),
(2, 3, 3, 4, 5, -7), (0, 1, 1, 1, 3, -10),
(0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8),
(0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6),
(1, 1, 3, 4, 6, -7), (0, 2, 3, 3, 3, -9),
(0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9),
(0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6),
(1, 3, 4, 5, 5, -6), (0, 2, 2, 2, 6, -8),
(0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7),
(0, 1, 5, 5, 5, -6)}
# handle negated squares with signsimp
assert diophantine(12 - x**2 - y**2 - z**2) == {(2, 2, 2)}
# diophantine handles simplification, so classify_diop should
# not have to look for additional patterns that are removed
# by diophantine
eq = a**2 + b**2 + c**2 + d**2 - 4
pytest.raises(NotImplementedError, lambda: classify_diop(-eq))
def test_process_common_addends():
# this tests that the args are not evaluated as they are given to do
# and that key2 works when key1 is False
def do(x):
return Add(*[i**(i % 2) for i in x.args])
process_common_addends(Add(*[1, 2, 3, 4], evaluate=False), do,
key2=lambda x: x % 2, key1=False) == 1**1 + 3**1 + 2**0 + 4**0
def test_add():
assert (a**2 - b - c).subs({a**2 - b: d}) in [d - c, a**2 - b - c]
assert (a**2 - c).subs({a**2 - c: d}) == d
assert (a**2 - b - c).subs({a**2 - c: d}) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs({a**2 - c: d}) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs({a**2 - sqrt(a): c}) == c - b
assert (a + b + exp(a + b)).subs({a + b: c}) == c + exp(c)
assert (c + b + exp(c + b)).subs({c + b: a}) == a + exp(a)
assert (a + b + c + d).subs({b + c: x}) == a + d + x
assert (a + b + c + d).subs({-b - c: x}) == a + d - x
assert ((x + 1) * y).subs({x + 1: t}) == t * y
assert ((-x - 1) * y).subs({x + 1: t}) == -t * y
assert ((x - 1) * y).subs({x + 1: t}) == y * (t - 2)
assert ((-x + 1) * y).subs({x + 1: t}) == y * (-t + 2)
# this should work everytime:
e = a**2 - b - c
assert e.subs({Add(*e.args[:2]): d}) == d + e.args[2]
assert e.subs({a**2 - c: d}) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs({0.1: Rational(1, 10)}) == Rational(1, 10) + a
e = (-x * (-y + 1) - y * (y - 1))
ans = (-x * x - y * (-x)).expand()
assert e.subs({-y + 1: x}) == ans
def test_add():
a, b, c, d, x, y, t = symbols('a b c d x y t')
assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
assert (a**2 - c).subs(a**2 - c, d) == d
assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
assert (a + b + c + d).subs(b + c, x) == a + d + x
assert (a + b + c + d).subs(-b - c, x) == a + d - x
assert ((x + 1) * y).subs(x + 1, t) == t * y
assert ((-x - 1) * y).subs(x + 1, t) == -t * y
assert ((x - 1) * y).subs(x + 1, t) == y * (t - 2)
assert ((-x + 1) * y).subs(x + 1, t) == y * (-t + 2)
# this should work everytime:
e = a**2 - b - c
assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
assert e.subs(a**2 - c, d) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a
e = (-x * (-y + 1) - y * (y - 1))
ans = (-x * (x) - y * (-x)).expand()
assert e.subs(-y + 1, x) == ans
def test_as_content_primitive():
# although the _as_content_primitive methods do not alter the underlying structure,
# the as_content_primitive function will touch up the expression and join
# bases that would otherwise have not been joined.
assert ((x*(2 + 2*x)*(3*x + 3)**2)).as_content_primitive() == \
(18, x*(x + 1)**3)
assert (2 + 2*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(2, x + 3*y*(y + 1) + 1)
assert ((2 + 6*x)**2).as_content_primitive() == (4, (3*x + 1)**2)
assert ((2 + 6*x)**(2*y)).as_content_primitive() == \
(1, (_keep_coeff(Integer(2), (3*x + 1)))**(2*y))
assert (5 + 10*x + 2*y*(3 + 3*y)).as_content_primitive() == \
(1, 10*x + 6*y*(y + 1) + 5)
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive() == \
(11, x*(y + 1))
assert ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive() == \
(121, x**2*(y + 1)**2)
assert (y**2).as_content_primitive() == (1, y**2)
assert oo.as_content_primitive() == (1, oo)
eq = x**(2 + y)
assert (eq).as_content_primitive() == (1, eq)
assert (Rational(1, 2)**(2 + x)).as_content_primitive() == (Rational(1, 4), 2**-x)
assert (Rational(-1, 2)**(2 + x)).as_content_primitive() == \
(Rational(1, 4), Rational(-1, 2)**x)
assert (Rational(-1, 2)**(2 + x)).as_content_primitive() == \
(Rational(1, 4), Rational(-1, 2)**x)
assert (4**((1 + y)/2)).as_content_primitive() == (2, 4**(y/2))
assert (3**((1 + y)/2)).as_content_primitive() == \
(1, 3**(Mul(Rational(1, 2), 1 + y, evaluate=False)))
assert (5**Rational(3, 4)).as_content_primitive() == (1, 5**Rational(3, 4))
assert (5**Rational(7, 4)).as_content_primitive() == (5, 5**Rational(3, 4))
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).as_content_primitive() == \
(Rational(1, 14), 7.0*x + 21*y + 10*z)
assert (2**Rational(3, 4) + root(2, 4)*sqrt(3)).as_content_primitive(radical=True) == \
(1, root(2, 4)*(sqrt(2) + sqrt(3)))
def test_TR9():
a = Rational(1, 2)
b = 3 * a
assert TR9(a) == a
assert TR9(cos(1) + cos(2)) == 2 * cos(a) * cos(b)
assert TR9(cos(1) - cos(2)) == 2 * sin(a) * sin(b)
assert TR9(sin(1) - sin(2)) == -2 * sin(a) * cos(b)
assert TR9(sin(1) + sin(2)) == 2 * sin(b) * cos(a)
assert TR9(cos(1) + 2 * sin(1) +
2 * sin(2)) == cos(1) + 4 * sin(b) * cos(a)
assert TR9(cos(4) + cos(2) + 2 * cos(1) * cos(3)) == 4 * cos(1) * cos(3)
assert TR9((cos(4) + cos(2)) / cos(3) / 2 + cos(3)) == 2 * cos(1) * cos(2)
assert TR9(cos(3) + cos(4) + cos(5) + cos(6)) == \
4*cos(Rational(1, 2))*cos(1)*cos(Rational(9, 2))
assert TR9(cos(3) + cos(3) * cos(2)) == cos(3) + cos(2) * cos(3)
assert TR9(-cos(y) +
cos(x *
y)) == -2 * sin(x * y / 2 - y / 2) * sin(x * y / 2 + y / 2)
assert TR9(-sin(y) + sin(x * y)) == 2 * sin(x * y / 2 -
y / 2) * cos(x * y / 2 + y / 2)
c = cos(x)
s = sin(x)
for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
for a in ((c, s), (s, c), (cos(x), cos(x * y)), (sin(x), sin(x * y))):
args = zip(si, a)
ex = Add(*[Mul(*ai) for ai in args])
t = TR9(ex)
assert not (a[0].func == a[1].func and
(not verify_numerically(ex, t.expand(trig=True))
or t.is_Add) or a[1].func != a[0].func and ex != t)
def test_cse_single():
# Simple substitution.
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse([e])
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
assert cse([e], order='none') == cse([e])
def test_gcd_terms():
f = 2*(x + 1)*(x + 4)/(5*x**2 + 5) + (2*x + 2)*(x + 5)/(x**2 + 1)/5 + \
(2*x + 2)*(x + 6)/(5*x**2 + 5)
assert _gcd_terms(f) == (Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
assert _gcd_terms(Add.make_args(f)) == \
(Rational(6, 5)*((1 + x)/(1 + x**2)), 5 + x, 1)
newf = Rational(6, 5)*((1 + x)*(5 + x)/(1 + x**2))
assert gcd_terms(f) == newf
args = Add.make_args(f)
# non-Basic sequences of terms treated as terms of Add
assert gcd_terms(list(args)) == newf
assert gcd_terms(tuple(args)) == newf
assert gcd_terms(set(args)) == newf
# but a Basic sequence is treated as a container
assert gcd_terms(Tuple(*args)) != newf
assert gcd_terms(Basic(Tuple(1, 3*y + 3*x*y), Tuple(1, 3))) == \
Basic((1, 3*y*(x + 1)), (1, 3))
# but we shouldn't change keys of a dictionary or some may be lost
assert gcd_terms(Dict((x*(1 + y), 2), (x + x*y, y + x*y))) == \
Dict({x*(y + 1): 2, x + x*y: y*(1 + x)})
assert gcd_terms((2*x + 2)**3 + (2*x + 2)**2) == 4*(x + 1)**2*(2*x + 3)
assert gcd_terms(0) == 0
assert gcd_terms(1) == 1
assert gcd_terms(x) == x
assert gcd_terms(2 + 2*x) == Mul(2, 1 + x, evaluate=False)
arg = x*(2*x + 4*y)
garg = 2*x*(x + 2*y)
assert gcd_terms(arg) == garg
assert gcd_terms(sin(arg)) == sin(garg)
# issue sympy/sympy#6139-like
alpha, alpha1, alpha2, alpha3 = symbols('alpha:4')
a = alpha**2 - alpha*x**2 + alpha + x**3 - x*(alpha + 1)
rep = {alpha: (1 + sqrt(5))/2 + alpha1*x + alpha2*x**2 + alpha3*x**3}
s = (a/(x - alpha)).subs(rep).series(x, 0, 1)
assert simplify(collect(s, x)) == -sqrt(5)/2 - Rational(3, 2) + O(x)
# issue sympy/sympy#5917
assert _gcd_terms([Integer(0), Integer(0)]) == (0, 0, 1)
assert _gcd_terms([2*x + 4]) == (2, x + 2, 1)
eq = x/(x + 1/x)
assert gcd_terms(eq, fraction=False) == eq
def test_cse_single2():
# Simple substitution, test for being able to pass the expression directly
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse(e)
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
substs, reduced = cse(Matrix([[1]]))
assert isinstance(reduced[0], Matrix)
def _muly(p, x, y):
"""
Returns ``_mexpand(y**deg*p.subs({x:x / y}))``
"""
p1 = poly_from_expr(p, x)[0]
n = degree(p1)
a = [c * x**i * y**(n - i) for (i, ), c in p1.terms()]
return Add(*a)
def _invertx(p, x):
"""
Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))``
"""
p1 = poly_from_expr(p, x)[0]
n = degree(p1)
a = [c * x**(n - i) for (i, ), c in p1.terms()]
return Add(*a)
def test_glom():
def key(x):
return x.as_coeff_Mul()[1]
def count(x):
return x.as_coeff_Mul()[0]
def newargs(cnt, arg):
return cnt * arg
rl = glom(key, count, newargs)
result = rl(Add(x, -x, 3 * x, 2, 3, evaluate=False))
expected = Add(3 * x, 5)
assert set(result.args) == set(expected.args)
result = rl(Add(*expected.args, evaluate=False))
assert set(result.args) == set(expected.args)
def test_line_wrapping():
assert fcode(((x + y)**10).expand(), assign_to='var') == (
' var = x**10 + 10*x**9*y + 45*x**8*y**2 + 120*x**7*y**3 + 210*x**6*\n'
' @ y**4 + 252*x**5*y**5 + 210*x**4*y**6 + 120*x**3*y**7 + 45*x**2*y\n'
' @ **8 + 10*x*y**9 + y**10')
e = [x**i for i in range(11)]
assert fcode(Add(*e)) == (
' x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x\n'
' @ + 1')
def test_collect_const():
# coverage not provided by above tests
assert collect_const(2*sqrt(3) + 4*a*sqrt(5)) == \
2*(2*sqrt(5)*a + sqrt(3)) # let the primitive reabsorb
assert collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3)) == \
2*sqrt(3) + 4*a*sqrt(5)
assert collect_const(sqrt(2)*(1 + sqrt(2)) + sqrt(3) + x*sqrt(2)) == \
sqrt(2)*(x + 1 + sqrt(2)) + sqrt(3)
# issue sympy/sympy#5290
assert collect_const(2*x + 2*y + 1, 2) == \
collect_const(2*x + 2*y + 1) == \
Add(Integer(1), Mul(2, x + y, evaluate=False), evaluate=False)
assert collect_const(-y - z) == Mul(-1, y + z, evaluate=False)
assert collect_const(2*x - 2*y - 2*z, 2) == \
Mul(2, x - y - z, evaluate=False)
assert collect_const(2*x - 2*y - 2*z, -2) == \
Add(2*x, Mul(-2, y + z, evaluate=False), evaluate=False)
def test_cse_not_possible():
# No substitution possible.
e = Add(x, y)
substs, reduced = cse([e])
assert substs == []
assert reduced == [x + y]
# issue sympy/sympy#6329
eq = (meijerg((1, 2), (y, 4), (5, ), [], x) + meijerg((1, 3), (y, 4),
(5, ), [], x))
assert cse(eq) == ([], [eq])
def test_line_wrapping():
x, y = symbols('x,y')
assert fcode(((x + y)**10).expand(), assign_to="var") == (
" var = x**10 + 10*x**9*y + 45*x**8*y**2 + 120*x**7*y**3 + 210*x**6*\n"
" @ y**4 + 252*x**5*y**5 + 210*x**4*y**6 + 120*x**3*y**7 + 45*x**2*y\n"
" @ **8 + 10*x*y**9 + y**10")
e = [x**i for i in range(11)]
assert fcode(Add(*e)) == (
" x**10 + x**9 + x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + x\n"
" @ + 1")
def test_minimal_polynomial_sq():
p = expand_multinomial((1 + 5 * sqrt(2) + 2 * sqrt(3))**3)
mp = minimal_polynomial(cbrt(p))(x)
assert mp == x**4 - 4 * x**3 - 118 * x**2 + 244 * x + 1321
p = expand_multinomial((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(cbrt(p))(x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = Add(*[sqrt(i) for i in range(1, 12)])
mp = minimal_polynomial(p)(x)
assert mp.subs({x: 0}) == -71965773323122507776
def sampling_E(expr, given_condition=None, numsamples=1,
evalf=True, **kwargs):
"""
Sampling version of E
See Also
========
diofant.stats.P
diofant.stats.rv.sampling_P
diofant.stats.rv.sampling_density
"""
samples = sample_iter(expr, given_condition,
numsamples=numsamples, **kwargs)
result = Add(*list(samples)) / numsamples
if evalf:
return result.evalf()
else:
return result
def test_rewrite():
assert rewrite(Integer(1), x, m) == (1, None)
e = exp(x)
assert rewrite(e, x, m) == (1/m, -x)
e = exp(x**2)
assert rewrite(e, x, m) == (1/m, -x**2)
e = exp(x + 1/x)
assert rewrite(e, x, m) == (1/m, -x - 1/x)
e = 1/exp(-x + exp(-x)) - exp(x)
assert rewrite(e, x, m) == (Add(-1/m, 1/(m*exp(m)), evaluate=False), -x)
e = exp(x)*log(log(exp(x)))
assert mrv(e, x) == {exp(x)}
assert rewrite(e, x, m) == (1/m*log(x), -x)
e = exp(-x + 1/x**2) - exp(x + 1/x)
assert rewrite(e, x, m) == (Add(m, Mul(-1, exp(1/x + x**(-2))/m,
evaluate=False),
evaluate=False), -x + 1/x**2)
def expectation(expr, condition=None, numsamples=None, evaluate=True, **kwargs):
"""
Returns the expected value of a random expression
Parameters
==========
expr : Expr containing RandomSymbols
The expression of which you want to compute the expectation value
given : Expr containing RandomSymbols
A conditional expression. E(X, X>0) is expectation of X given X > 0
numsamples : int
Enables sampling and approximates the expectation with this many samples
evalf : Bool (defaults to True)
If sampling return a number rather than a complex expression
evaluate : Bool (defaults to True)
In case of continuous systems return unevaluated integral
Examples
========
>>> from diofant.stats import E, Die
>>> X = Die('X', 6)
>>> E(X)
7/2
>>> E(2*X + 1)
8
>>> E(X, X > 3) # Expectation of X given that it is above 3
5
"""
if not random_symbols(expr): # expr isn't random?
return expr
if numsamples: # Computing by monte carlo sampling?
return sampling_E(expr, condition, numsamples=numsamples)
# Create new expr and recompute E
if condition is not None: # If there is a condition
return expectation(given(expr, condition), evaluate=evaluate)
# A few known statements for efficiency
if expr.is_Add: # We know that E is Linear
return Add(*[expectation(arg, evaluate=evaluate)
for arg in expr.args])
# Otherwise case is simple, pass work off to the ProbabilitySpace
result = pspace(expr).integrate(expr)
if evaluate and hasattr(result, 'doit'):
return result.doit(**kwargs)
else:
return result
def _minpoly_sin(ex, x):
"""
Returns the minimal polynomial of ``sin(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
n = c.q
q = sympify(n)
if q.is_prime:
# for a = pi*p/q with q odd prime, using chebyshevt
# write sin(q*a) = mp(sin(a))*sin(a);
# the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
a = dup_chebyshevt(n, ZZ)
return Add(*[x**(n - i - 1) * a[i] for i in range(n)])
if c.p == 1:
if q == 9:
return 64 * x**6 - 96 * x**4 + 36 * x**2 - 3
if n % 2 == 1:
# for a = pi*p/q with q odd, use
# sin(q*a) = 0 to see that the minimal polynomial must be
# a factor of dup_chebyshevt(n, ZZ)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i) * a[i] for i in range(n + 1)]
r = Add(*a)
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
expr = ((1 - cos(2 * c * pi)) / 2)**S.Half
res = _minpoly_compose(expr, x, QQ)
return res
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def bottom_up_scan(ex):
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Mul:
return Mul(*[bottom_up_scan(g) for g in ex.args])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0 and ex.base.is_Add:
coeff, terms = ex.base.as_coeff_add()
elt, _ = primitive_element(terms, polys=True)
alg = ex.base - coeff
# XXX: turn this into eval()
inverse = invert(elt.gen + coeff, elt).as_expr()
base = inverse.subs(elt.gen, alg).expand()
if ex.exp == -1:
return bottom_up_scan(base)
else:
ex = base**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (ex.base**ex.exp.p).expand(), Rational(
1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex.root not in mapping:
return update_mapping(ex.root, ex.minpoly)
else:
return symbols[ex.root]
raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)
def _eval_aseries(self, n, args0, x, logx):
from diofant import Order, Add, combsimp
point = args0[0]
if point is S.Infinity:
z = self.args[0]
l = [
gamma(k + Rational(5, 6)) * gamma(k + Rational(1, 6)) *
Rational(3, 4)**k /
(2 * S.Pi**2 * factorial(k) * z**Rational(3 * k + 1, 2))
for k in range(n)
]
l = [combsimp(t) for t in l]
o = Order(1 / z**Rational(3 * n + 1, 2), x)
return (Add(*l))._eval_nseries(x, n, logx) + o
# All other points are not handled
return super(_airybis, self)._eval_aseries(n, args0, x, logx)
def field_isomorphism_factor(a, b):
"""Construct field isomorphism via factorization. """
_, factors = factor_list(a.minpoly, extension=b)
for f, _ in factors:
if f.degree() == 1:
coeffs = f.rep.TC().to_diofant_list()
d, terms = len(coeffs) - 1, []
for i, coeff in enumerate(coeffs):
terms.append(coeff * b.root**(d - i))
root = Add(*terms)
if (a.root - root).evalf(chop=True) == 0:
return coeffs
if (a.root + root).evalf(chop=True) == 0:
return [-c for c in coeffs]
else:
return
