def test_fraction():
A = Symbol('A', commutative=False)
assert fraction(Rational(1, 2)) == (1, 2)
assert fraction(x) == (x, 1)
assert fraction(1 / x) == (1, x)
assert fraction(x / y) == (x, y)
assert fraction(x / 2) == (x, 2)
assert fraction(x * y / z) == (x * y, z)
assert fraction(x / (y * z)) == (x, y * z)
assert fraction(1 / y**2) == (1, y**2)
assert fraction(x / y**2) == (x, y**2)
assert fraction((x**2 + 1) / y) == (x**2 + 1, y)
assert fraction(x * (y + 1) / y**7) == (x * (y + 1), y**7)
assert fraction(exp(-x), exact=True) == (exp(-x), 1)
assert fraction(x * A / y) == (x * A, y)
assert fraction(x * A**-1 / y) == (x * A**-1, y)
n = symbols('n', negative=True)
assert fraction(exp(n)) == (1, exp(-n))
assert fraction(exp(-n)) == (exp(-n), 1)
def test_fraction():
A = Symbol('A', commutative=False)
assert fraction(Rational(1, 2)) == (1, 2)
assert fraction(x) == (x, 1)
assert fraction(1/x) == (1, x)
assert fraction(x/y) == (x, y)
assert fraction(x/2) == (x, 2)
assert fraction(x*y/z) == (x*y, z)
assert fraction(x/(y*z)) == (x, y*z)
assert fraction(1/y**2) == (1, y**2)
assert fraction(x/y**2) == (x, y**2)
assert fraction((x**2 + 1)/y) == (x**2 + 1, y)
assert fraction(x*(y + 1)/y**7) == (x*(y + 1), y**7)
assert fraction(exp(-x), exact=True) == (exp(-x), 1)
assert fraction(x*A/y) == (x*A, y)
assert fraction(x*A**-1/y) == (x*A**-1, y)
n = symbols('n', negative=True)
assert fraction(exp(n)) == (1, exp(-n))
assert fraction(exp(-n)) == (exp(-n), 1)
def test_radsimp():
r2 = sqrt(2)
r3 = sqrt(3)
r5 = sqrt(5)
r7 = sqrt(7)
assert fraction(radsimp(1 / r2)) == (sqrt(2), 2)
assert radsimp(1/(1 + r2)) == \
-1 + sqrt(2)
assert radsimp(1/(r2 + r3)) == \
-sqrt(2) + sqrt(3)
assert fraction(radsimp(1/(1 + r2 + r3))) == \
(-sqrt(6) + sqrt(2) + 2, 4)
assert fraction(radsimp(1/(r2 + r3 + r5))) == \
(-sqrt(30) + 2*sqrt(3) + 3*sqrt(2), 12)
assert fraction(radsimp(
1 /
(1 + r2 + r3 + r5))) == ((-34 * sqrt(10) - 26 * sqrt(15) -
55 * sqrt(3) - 61 * sqrt(2) + 14 * sqrt(30) +
93 + 46 * sqrt(6) + 53 * sqrt(5), 71))
assert fraction(radsimp(
1 / (r2 + r3 + r5 + r7))) == ((-50 * sqrt(42) - 133 * sqrt(5) -
34 * sqrt(70) - 145 * sqrt(3) +
22 * sqrt(105) + 185 * sqrt(2) +
62 * sqrt(30) + 135 * sqrt(7), 215))
z = radsimp(1 / (1 + r2 / 3 + r3 / 5 + r5 + r7))
assert len((3616791619821680643598 * z).args) == 16
assert radsimp(1 / z) == 1 / z
assert radsimp(1 / z,
max_terms=20).expand() == 1 + r2 / 3 + r3 / 5 + r5 + r7
assert radsimp(1/(r2*3)) == \
sqrt(2)/6
assert radsimp(1 / (r2 * a + r3 + r5 + r7)) == (
(8 * sqrt(2) * a**7 - 8 * sqrt(7) * a**6 - 8 * sqrt(5) * a**6 -
8 * sqrt(3) * a**6 - 180 * sqrt(2) * a**5 + 8 * sqrt(30) * a**5 +
8 * sqrt(42) * a**5 + 8 * sqrt(70) * a**5 - 24 * sqrt(105) * a**4 +
84 * sqrt(3) * a**4 + 100 * sqrt(5) * a**4 + 116 * sqrt(7) * a**4 -
72 * sqrt(70) * a**3 - 40 * sqrt(42) * a**3 - 8 * sqrt(30) * a**3 +
782 * sqrt(2) * a**3 - 462 * sqrt(3) * a**2 - 302 * sqrt(7) * a**2 -
254 * sqrt(5) * a**2 + 120 * sqrt(105) * a**2 - 795 * sqrt(2) * a -
62 * sqrt(30) * a + 82 * sqrt(42) * a + 98 * sqrt(70) * a -
118 * sqrt(105) + 59 * sqrt(7) + 295 * sqrt(5) + 531 * sqrt(3)) /
(16 * a**8 - 480 * a**6 + 3128 * a**4 - 6360 * a**2 + 3481))
assert radsimp(1 / (r2 * a + r2 * b + r3 + r7)) == (
(sqrt(2) * a *
(a + b)**2 - 5 * sqrt(2) * a + sqrt(42) * a + sqrt(2) * b *
(a + b)**2 - 5 * sqrt(2) * b + sqrt(42) * b - sqrt(7) *
(a + b)**2 - sqrt(3) * (a + b)**2 - 2 * sqrt(3) + 2 * sqrt(7)) /
(2 * a**4 + 8 * a**3 * b + 12 * a**2 * b**2 - 20 * a**2 +
8 * a * b**3 - 40 * a * b + 2 * b**4 - 20 * b**2 + 8))
assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \
sqrt(2)/(2*a + 2*b + 2*c + 2*d)
assert radsimp(1 / (1 + r2 * a + r2 * b + r2 * c + r2 * d)) == (
(sqrt(2) * a + sqrt(2) * b + sqrt(2) * c + sqrt(2) * d - 1) /
(2 * a**2 + 4 * a * b + 4 * a * c + 4 * a * d + 2 * b**2 + 4 * b * c +
4 * b * d + 2 * c**2 + 4 * c * d + 2 * d**2 - 1))
assert radsimp((y**2 - x)/(y - sqrt(x))) == \
sqrt(x) + y
assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \
-(sqrt(x) + y)
assert radsimp(1/(1 - I + a*I)) == \
(-I*a + 1 + I)/(a**2 - 2*a + 2)
assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \
(-x - sqrt(y))/((x - y)*(x**2 - y))
e = (3 + 3 * sqrt(2)) * x * (3 * x - 3 * sqrt(y))
assert radsimp(e) == x * (3 + 3 * sqrt(2)) * (3 * x - 3 * sqrt(y))
assert radsimp(1 / e) == (
(-9 * x + 9 * sqrt(2) * x - 9 * sqrt(y) + 9 * sqrt(2) * sqrt(y)) /
(9 * x * (9 * x**2 - 9 * y)))
assert radsimp(1 + 1/(1 + sqrt(3))) == \
Mul(Rational(1, 2), -1 + sqrt(3), evaluate=False) + 1
A = symbols('A', commutative=False)
assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \
x**2 + sqrt(2)*x**2 - sqrt(2)*x*A
assert radsimp(1 / sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3)
assert radsimp(1 / sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3
# issue sympy/sympy#6532
assert fraction(radsimp(1 / sqrt(x))) == (sqrt(x), x)
assert fraction(radsimp(1 / sqrt(2 * x + 3))) == (sqrt(2 * x + 3),
2 * x + 3)
assert fraction(radsimp(1 / sqrt(2 * (x + 3)))) == (sqrt(2 * x + 6),
2 * x + 6)
# issue sympy/sympy#5994
e = -(2 + 2 * sqrt(2) + 4 * root(2, 4)) / (1 + 2**Rational(3, 4) +
3 * root(2, 4) + 3 * sqrt(2))
assert radsimp(e).expand(
) == -2 * 2**Rational(3, 4) - 2 * root(2, 4) + 2 + 2 * sqrt(2)
# issue sympy/sympy#5986 (modifications to radimp didn't initially recognize this so
# the test is included here)
assert radsimp(1 / (-sqrt(5) / 2 - Rational(1, 2) +
(-sqrt(5) / 2 - Rational(1, 2))**2)) == 1
# from issue sympy/sympy#5934
eq = ((-240 * sqrt(2) * sqrt(sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40) -
360 * sqrt(2) * sqrt(-8 * sqrt(5) + 40) * sqrt(-sqrt(5) + 5) -
120 * sqrt(10) * sqrt(-8 * sqrt(5) + 40) * sqrt(-sqrt(5) + 5) +
120 * sqrt(2) * sqrt(-sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40) +
120 * sqrt(2) * sqrt(-8 * sqrt(5) + 40) * sqrt(sqrt(5) + 5) +
120 * sqrt(10) * sqrt(-sqrt(5) + 5) * sqrt(8 * sqrt(5) + 40) +
120 * sqrt(10) * sqrt(-8 * sqrt(5) + 40) * sqrt(sqrt(5) + 5)) /
(-36000 - 7200 * sqrt(5) + (12 * sqrt(10) * sqrt(sqrt(5) + 5) +
24 * sqrt(10) * sqrt(-sqrt(5) + 5))**2))
assert radsimp(eq) is nan # it's 0/0
# work with normal form
e = 1 / sqrt(sqrt(7) / 7 + 2 * sqrt(2) + 3 * sqrt(3) + 5 * sqrt(5)) + 3
assert radsimp(e) == (
-sqrt(sqrt(7) + 14 * sqrt(2) + 21 * sqrt(3) + 35 * sqrt(5)) *
(-11654899 * sqrt(35) - 1577436 * sqrt(210) - 1278438 * sqrt(15) -
1346996 * sqrt(10) + 1635060 * sqrt(6) + 5709765 +
7539830 * sqrt(14) + 8291415 * sqrt(21)) / 1300423175 + 3)
# obey power rules
base = sqrt(3) - sqrt(2)
assert radsimp(1 / base**3) == (sqrt(3) + sqrt(2))**3
assert radsimp(1 / (-base)**3) == -(sqrt(2) + sqrt(3))**3
assert radsimp(1 / (-base)**x) == (-base)**(-x)
assert radsimp(1 / base**x) == (sqrt(2) + sqrt(3))**x
assert radsimp(root(1 / (-1 - sqrt(2)),
-x)) == (-1)**(-1 / x) * (1 + sqrt(2))**(1 / x)
# recurse
e = cos(1 / (1 + sqrt(2)))
assert radsimp(e) == cos(-sqrt(2) + 1)
assert radsimp(e / 2) == cos(-sqrt(2) + 1) / 2
assert radsimp(1 / e) == 1 / cos(-sqrt(2) + 1)
assert radsimp(2 / e) == 2 / cos(-sqrt(2) + 1)
assert fraction(radsimp(e / sqrt(x))) == (sqrt(x) * cos(-sqrt(2) + 1), x)
# test that symbolic denominators are not processed
r = 1 + sqrt(2)
assert radsimp(x / r, symbolic=False) == -x * (-sqrt(2) + 1)
assert radsimp(x / (y + r), symbolic=False) == x / (y + 1 + sqrt(2))
assert radsimp(x/(y + r)/r, symbolic=False) == \
-x*(-sqrt(2) + 1)/(y + 1 + sqrt(2))
# issue sympy/sympy#7408
eq = sqrt(x) / sqrt(y)
assert radsimp(eq) == UMul(sqrt(x), sqrt(y), 1 / y)
assert radsimp(eq, symbolic=False) == eq
# issue sympy/sympy#7498
assert radsimp(sqrt(x) / sqrt(y)**3) == UMul(sqrt(x), sqrt(y**3), 1 / y**3)
# for coverage
eq = sqrt(x) / y**2
assert radsimp(eq) == eq
assert (radsimp(1 / (1 + r2 / 3 + r3 / 5 + r5 + sqrt(r7))) == 15 /
(3 * sqrt(3) + 5 * sqrt(2) + 15 + 15 * root(7, 4) + 15 * sqrt(5)))
def test_radsimp():
r2 = sqrt(2)
r3 = sqrt(3)
r5 = sqrt(5)
r7 = sqrt(7)
assert fraction(radsimp(1/r2)) == (sqrt(2), 2)
assert radsimp(1/(1 + r2)) == \
-1 + sqrt(2)
assert radsimp(1/(r2 + r3)) == \
-sqrt(2) + sqrt(3)
assert fraction(radsimp(1/(1 + r2 + r3))) == \
(-sqrt(6) + sqrt(2) + 2, 4)
assert fraction(radsimp(1/(r2 + r3 + r5))) == \
(-sqrt(30) + 2*sqrt(3) + 3*sqrt(2), 12)
assert fraction(radsimp(1/(1 + r2 + r3 + r5))) == (
(-34*sqrt(10) - 26*sqrt(15) - 55*sqrt(3) - 61*sqrt(2) + 14*sqrt(30) +
93 + 46*sqrt(6) + 53*sqrt(5), 71))
assert fraction(radsimp(1/(r2 + r3 + r5 + r7))) == (
(-50*sqrt(42) - 133*sqrt(5) - 34*sqrt(70) - 145*sqrt(3) + 22*sqrt(105)
+ 185*sqrt(2) + 62*sqrt(30) + 135*sqrt(7), 215))
z = radsimp(1/(1 + r2/3 + r3/5 + r5 + r7))
assert len((3616791619821680643598*z).args) == 16
assert radsimp(1/z) == 1/z
assert radsimp(1/z, max_terms=20).expand() == 1 + r2/3 + r3/5 + r5 + r7
assert radsimp(1/(r2*3)) == \
sqrt(2)/6
assert radsimp(1/(r2*a + r3 + r5 + r7)) == (
(8*sqrt(2)*a**7 - 8*sqrt(7)*a**6 - 8*sqrt(5)*a**6 - 8*sqrt(3)*a**6 -
180*sqrt(2)*a**5 + 8*sqrt(30)*a**5 + 8*sqrt(42)*a**5 + 8*sqrt(70)*a**5
- 24*sqrt(105)*a**4 + 84*sqrt(3)*a**4 + 100*sqrt(5)*a**4 +
116*sqrt(7)*a**4 - 72*sqrt(70)*a**3 - 40*sqrt(42)*a**3 -
8*sqrt(30)*a**3 + 782*sqrt(2)*a**3 - 462*sqrt(3)*a**2 -
302*sqrt(7)*a**2 - 254*sqrt(5)*a**2 + 120*sqrt(105)*a**2 -
795*sqrt(2)*a - 62*sqrt(30)*a + 82*sqrt(42)*a + 98*sqrt(70)*a -
118*sqrt(105) + 59*sqrt(7) + 295*sqrt(5) + 531*sqrt(3))/(16*a**8 -
480*a**6 + 3128*a**4 - 6360*a**2 + 3481))
assert radsimp(1/(r2*a + r2*b + r3 + r7)) == (
(sqrt(2)*a*(a + b)**2 - 5*sqrt(2)*a + sqrt(42)*a + sqrt(2)*b*(a +
b)**2 - 5*sqrt(2)*b + sqrt(42)*b - sqrt(7)*(a + b)**2 - sqrt(3)*(a +
b)**2 - 2*sqrt(3) + 2*sqrt(7))/(2*a**4 + 8*a**3*b + 12*a**2*b**2 -
20*a**2 + 8*a*b**3 - 40*a*b + 2*b**4 - 20*b**2 + 8))
assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \
sqrt(2)/(2*a + 2*b + 2*c + 2*d)
assert radsimp(1/(1 + r2*a + r2*b + r2*c + r2*d)) == (
(sqrt(2)*a + sqrt(2)*b + sqrt(2)*c + sqrt(2)*d - 1)/(2*a**2 + 4*a*b +
4*a*c + 4*a*d + 2*b**2 + 4*b*c + 4*b*d + 2*c**2 + 4*c*d + 2*d**2 - 1))
assert radsimp((y**2 - x)/(y - sqrt(x))) == \
sqrt(x) + y
assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \
-(sqrt(x) + y)
assert radsimp(1/(1 - I + a*I)) == \
(-I*a + 1 + I)/(a**2 - 2*a + 2)
assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \
(-x - sqrt(y))/((x - y)*(x**2 - y))
e = (3 + 3*sqrt(2))*x*(3*x - 3*sqrt(y))
assert radsimp(e) == x*(3 + 3*sqrt(2))*(3*x - 3*sqrt(y))
assert radsimp(1/e) == (
(-9*x + 9*sqrt(2)*x - 9*sqrt(y) + 9*sqrt(2)*sqrt(y))/(9*x*(9*x**2 -
9*y)))
assert radsimp(1 + 1/(1 + sqrt(3))) == \
Mul(Rational(1, 2), -1 + sqrt(3), evaluate=False) + 1
A = symbols("A", commutative=False)
assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \
x**2 + sqrt(2)*x**2 - sqrt(2)*x*A
assert radsimp(1/sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3)
assert radsimp(1/sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3
# issue sympy/sympy#6532
assert fraction(radsimp(1/sqrt(x))) == (sqrt(x), x)
assert fraction(radsimp(1/sqrt(2*x + 3))) == (sqrt(2*x + 3), 2*x + 3)
assert fraction(radsimp(1/sqrt(2*(x + 3)))) == (sqrt(2*x + 6), 2*x + 6)
# issue sympy/sympy#5994
e = -(2 + 2*sqrt(2) + 4*root(2, 4))/(1 + 2**Rational(3, 4) + 3*root(2, 4) + 3*sqrt(2))
assert radsimp(e).expand() == -2*2**Rational(3, 4) - 2*root(2, 4) + 2 + 2*sqrt(2)
# issue sympy/sympy#5986 (modifications to radimp didn't initially recognize this so
# the test is included here)
assert radsimp(1/(-sqrt(5)/2 - Rational(1, 2) + (-sqrt(5)/2 - Rational(1, 2))**2)) == 1
# from issue sympy/sympy#5934
eq = (
(-240*sqrt(2)*sqrt(sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) -
360*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) -
120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) +
120*sqrt(2)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
120*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5) +
120*sqrt(10)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5))/(-36000 -
7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
24*sqrt(10)*sqrt(-sqrt(5) + 5))**2))
assert radsimp(eq) is nan # it's 0/0
# work with normal form
e = 1/sqrt(sqrt(7)/7 + 2*sqrt(2) + 3*sqrt(3) + 5*sqrt(5)) + 3
assert radsimp(e) == (
-sqrt(sqrt(7) + 14*sqrt(2) + 21*sqrt(3) +
35*sqrt(5))*(-11654899*sqrt(35) - 1577436*sqrt(210) - 1278438*sqrt(15)
- 1346996*sqrt(10) + 1635060*sqrt(6) + 5709765 + 7539830*sqrt(14) +
8291415*sqrt(21))/1300423175 + 3)
# obey power rules
base = sqrt(3) - sqrt(2)
assert radsimp(1/base**3) == (sqrt(3) + sqrt(2))**3
assert radsimp(1/(-base)**3) == -(sqrt(2) + sqrt(3))**3
assert radsimp(1/(-base)**x) == (-base)**(-x)
assert radsimp(1/base**x) == (sqrt(2) + sqrt(3))**x
assert radsimp(root(1/(-1 - sqrt(2)), -x)) == (-1)**(-1/x)*(1 + sqrt(2))**(1/x)
# recurse
e = cos(1/(1 + sqrt(2)))
assert radsimp(e) == cos(-sqrt(2) + 1)
assert radsimp(e/2) == cos(-sqrt(2) + 1)/2
assert radsimp(1/e) == 1/cos(-sqrt(2) + 1)
assert radsimp(2/e) == 2/cos(-sqrt(2) + 1)
assert fraction(radsimp(e/sqrt(x))) == (sqrt(x)*cos(-sqrt(2)+1), x)
# test that symbolic denominators are not processed
r = 1 + sqrt(2)
assert radsimp(x/r, symbolic=False) == -x*(-sqrt(2) + 1)
assert radsimp(x/(y + r), symbolic=False) == x/(y + 1 + sqrt(2))
assert radsimp(x/(y + r)/r, symbolic=False) == \
-x*(-sqrt(2) + 1)/(y + 1 + sqrt(2))
# issue sympy/sympy#7408
eq = sqrt(x)/sqrt(y)
assert radsimp(eq) == UMul(sqrt(x), sqrt(y), 1/y)
assert radsimp(eq, symbolic=False) == eq
# issue sympy/sympy#7498
assert radsimp(sqrt(x)/sqrt(y)**3) == UMul(sqrt(x), sqrt(y**3), 1/y**3)
# for coverage
eq = sqrt(x)/y**2
assert radsimp(eq) == eq
assert (radsimp(1/(1 + r2/3 + r3/5 + r5 + sqrt(r7))) ==
15/(3*sqrt(3) + 5*sqrt(2) + 15 + 15*root(7, 4) + 15*sqrt(5)))