491 lines
18 KiB
Python
491 lines
18 KiB
Python
|
from sympy.core.add import Add
|
||
|
|
from sympy.core.function import (Derivative, Function, diff)
|
||
|
|
from sympy.core.mul import Mul
|
||
|
|
from sympy.core.numbers import (I, Rational)
|
||
|
|
from sympy.core.power import Pow
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
from sympy.core.symbol import (Symbol, Wild, symbols)
|
||
|
|
from sympy.functions.elementary.complexes import Abs
|
||
|
|
from sympy.functions.elementary.exponential import (exp, log)
|
||
|
|
from sympy.functions.elementary.miscellaneous import (root, sqrt)
|
||
|
|
from sympy.functions.elementary.trigonometric import (cos, sin)
|
||
|
|
from sympy.polys.polytools import factor
|
||
|
|
from sympy.series.order import O
|
||
|
|
from sympy.simplify.radsimp import (collect, collect_const, fraction, radsimp, rcollect)
|
||
|
|
|
||
|
|
from sympy.core.expr import unchanged
|
||
|
|
from sympy.core.mul import _unevaluated_Mul as umul
|
||
|
|
from sympy.simplify.radsimp import (_unevaluated_Add,
|
||
|
|
collect_sqrt, fraction_expand, collect_abs)
|
||
|
|
from sympy.testing.pytest import raises
|
||
|
|
|
||
|
|
from sympy.abc import x, y, z, a, b, c, d
|
||
|
|
|
||
|
|
|
||
|
|
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(S.Half, -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 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 5994
|
||
|
|
e = S('-(2 + 2*sqrt(2) + 4*2**(1/4))/'
|
||
|
|
'(1 + 2**(3/4) + 3*2**(1/4) + 3*sqrt(2))')
|
||
|
|
assert radsimp(e).expand() == -2*2**Rational(3, 4) - 2*2**Rational(1, 4) + 2 + 2*sqrt(2)
|
||
|
|
|
||
|
|
# issue 5986 (modifications to radimp didn't initially recognize this so
|
||
|
|
# the test is included here)
|
||
|
|
assert radsimp(1/(-sqrt(5)/2 - S.Half + (-sqrt(5)/2 - S.Half)**2)) == 1
|
||
|
|
|
||
|
|
# from issue 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 S.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 7408
|
||
|
|
eq = sqrt(x)/sqrt(y)
|
||
|
|
assert radsimp(eq) == umul(sqrt(x), sqrt(y), 1/y)
|
||
|
|
assert radsimp(eq, symbolic=False) == eq
|
||
|
|
|
||
|
|
# issue 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
|
||
|
|
|
||
|
|
|
||
|
|
def test_radsimp_issue_3214():
|
||
|
|
c, p = symbols('c p', positive=True)
|
||
|
|
s = sqrt(c**2 - p**2)
|
||
|
|
b = (c + I*p - s)/(c + I*p + s)
|
||
|
|
assert radsimp(b) == -I*(c + I*p - sqrt(c**2 - p**2))**2/(2*c*p)
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_1():
|
||
|
|
"""Collect with respect to Symbol"""
|
||
|
|
x, y, z, n = symbols('x,y,z,n')
|
||
|
|
assert collect(1, x) == 1
|
||
|
|
assert collect( x + y*x, x ) == x * (1 + y)
|
||
|
|
assert collect( x + x**2, x ) == x + x**2
|
||
|
|
assert collect( x**2 + y*x**2, x ) == (x**2)*(1 + y)
|
||
|
|
assert collect( x**2 + y*x, x ) == x*y + x**2
|
||
|
|
assert collect( 2*x**2 + y*x**2 + 3*x*y, [x] ) == x**2*(2 + y) + 3*x*y
|
||
|
|
assert collect( 2*x**2 + y*x**2 + 3*x*y, [y] ) == 2*x**2 + y*(x**2 + 3*x)
|
||
|
|
|
||
|
|
assert collect( ((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \
|
||
|
|
x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \
|
||
|
|
x**3*(4*(1 + y)).expand() + x**4
|
||
|
|
# symbols can be given as any iterable
|
||
|
|
expr = x + y
|
||
|
|
assert collect(expr, expr.free_symbols) == expr
|
||
|
|
assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x, x, exact=None
|
||
|
|
) == x*exp(x) + 3*x + (y + 2)*sin(x)
|
||
|
|
assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x + y*x +
|
||
|
|
y*x*exp(x), x, exact=None
|
||
|
|
) == x*exp(x)*(y + 1) + (3 + y)*x + (y + 2)*sin(x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_2():
|
||
|
|
"""Collect with respect to a sum"""
|
||
|
|
a, b, x = symbols('a,b,x')
|
||
|
|
assert collect(a*(cos(x) + sin(x)) + b*(cos(x) + sin(x)),
|
||
|
|
sin(x) + cos(x)) == (a + b)*(cos(x) + sin(x))
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_3():
|
||
|
|
"""Collect with respect to a product"""
|
||
|
|
a, b, c = symbols('a,b,c')
|
||
|
|
f = Function('f')
|
||
|
|
x, y, z, n = symbols('x,y,z,n')
|
||
|
|
|
||
|
|
assert collect(-x/8 + x*y, -x) == x*(y - Rational(1, 8))
|
||
|
|
|
||
|
|
assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2)
|
||
|
|
assert collect( x*y + a*x*y, x*y) == x*y*(1 + a)
|
||
|
|
assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
|
||
|
|
assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
|
||
|
|
|
||
|
|
assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
|
||
|
|
assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
|
||
|
|
x**2*log(x)**2*(a + b)
|
||
|
|
|
||
|
|
# with respect to a product of three symbols
|
||
|
|
assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_4():
|
||
|
|
"""Collect with respect to a power"""
|
||
|
|
a, b, c, x = symbols('a,b,c,x')
|
||
|
|
|
||
|
|
assert collect(a*x**c + b*x**c, x**c) == x**c*(a + b)
|
||
|
|
# issue 6096: 2 stays with c (unless c is integer or x is positive0
|
||
|
|
assert collect(a*x**(2*c) + b*x**(2*c), x**c) == x**(2*c)*(a + b)
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_5():
|
||
|
|
"""Collect with respect to a tuple"""
|
||
|
|
a, x, y, z, n = symbols('a,x,y,z,n')
|
||
|
|
assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [
|
||
|
|
z*(1 + a + x**2*y**4) + x**2*y**4,
|
||
|
|
z*(1 + a) + x**2*y**4*(1 + z) ]
|
||
|
|
assert collect((1 + (x + y) + (x + y)**2).expand(),
|
||
|
|
[x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_pr19431():
|
||
|
|
"""Unevaluated collect with respect to a product"""
|
||
|
|
a = symbols('a')
|
||
|
|
assert collect(a**2*(a**2 + 1), a**2, evaluate=False)[a**2] == (a**2 + 1)
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_D():
|
||
|
|
D = Derivative
|
||
|
|
f = Function('f')
|
||
|
|
x, a, b = symbols('x,a,b')
|
||
|
|
fx = D(f(x), x)
|
||
|
|
fxx = D(f(x), x, x)
|
||
|
|
|
||
|
|
assert collect(a*fx + b*fx, fx) == (a + b)*fx
|
||
|
|
assert collect(a*D(fx, x) + b*D(fx, x), fx) == (a + b)*D(fx, x)
|
||
|
|
assert collect(a*fxx + b*fxx, fx) == (a + b)*D(fx, x)
|
||
|
|
# issue 4784
|
||
|
|
assert collect(5*f(x) + 3*fx, fx) == 5*f(x) + 3*fx
|
||
|
|
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x)) == \
|
||
|
|
(x*f(x) + f(x))*D(f(x), x) + f(x)
|
||
|
|
assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
|
||
|
|
(x*f(x) + f(x))*D(f(x), x) + f(x)
|
||
|
|
assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
|
||
|
|
(1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
|
||
|
|
e = (1 + x*fx + fx)/f(x)
|
||
|
|
assert collect(e.expand(), fx) == fx*(x/f(x) + 1/f(x)) + 1/f(x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_func():
|
||
|
|
f = ((x + a + 1)**3).expand()
|
||
|
|
|
||
|
|
assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \
|
||
|
|
x*(3*a**2 + 6*a + 3) + 1
|
||
|
|
assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \
|
||
|
|
(a + 1)**3
|
||
|
|
|
||
|
|
assert collect(f, x, evaluate=False) == {
|
||
|
|
S.One: a**3 + 3*a**2 + 3*a + 1,
|
||
|
|
x: 3*a**2 + 6*a + 3, x**2: 3*a + 3,
|
||
|
|
x**3: 1
|
||
|
|
}
|
||
|
|
|
||
|
|
assert collect(f, x, factor, evaluate=False) == {
|
||
|
|
S.One: (a + 1)**3, x: 3*(a + 1)**2,
|
||
|
|
x**2: umul(S(3), a + 1), x**3: 1}
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_order():
|
||
|
|
a, b, x, t = symbols('a,b,x,t')
|
||
|
|
|
||
|
|
assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3))
|
||
|
|
assert collect(t + t*x + x**2 + O(x**3), t) == \
|
||
|
|
t*(1 + x + O(x**3)) + x**2 + O(x**3)
|
||
|
|
|
||
|
|
f = a*x + b*x + c*x**2 + d*x**2 + O(x**3)
|
||
|
|
g = x*(a + b) + x**2*(c + d) + O(x**3)
|
||
|
|
|
||
|
|
assert collect(f, x) == g
|
||
|
|
assert collect(f, x, distribute_order_term=False) == g
|
||
|
|
|
||
|
|
f = sin(a + b).series(b, 0, 10)
|
||
|
|
|
||
|
|
assert collect(f, [sin(a), cos(a)]) == \
|
||
|
|
sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10)
|
||
|
|
assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \
|
||
|
|
sin(a)*cos(b).series(b, 0, 10).removeO() + \
|
||
|
|
cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_rcollect():
|
||
|
|
assert rcollect((x**2*y + x*y + x + y)/(x + y), y) == \
|
||
|
|
(x + y*(1 + x + x**2))/(x + y)
|
||
|
|
assert rcollect(sqrt(-((x + 1)*(y + 1))), z) == sqrt(-((x + 1)*(y + 1)))
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_D_0():
|
||
|
|
D = Derivative
|
||
|
|
f = Function('f')
|
||
|
|
x, a, b = symbols('x,a,b')
|
||
|
|
fxx = D(f(x), x, x)
|
||
|
|
|
||
|
|
assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_Wild():
|
||
|
|
"""Collect with respect to functions with Wild argument"""
|
||
|
|
a, b, x, y = symbols('a b x y')
|
||
|
|
f = Function('f')
|
||
|
|
w1 = Wild('.1')
|
||
|
|
w2 = Wild('.2')
|
||
|
|
assert collect(f(x) + a*f(x), f(w1)) == (1 + a)*f(x)
|
||
|
|
assert collect(f(x, y) + a*f(x, y), f(w1)) == f(x, y) + a*f(x, y)
|
||
|
|
assert collect(f(x, y) + a*f(x, y), f(w1, w2)) == (1 + a)*f(x, y)
|
||
|
|
assert collect(f(x, y) + a*f(x, y), f(w1, w1)) == f(x, y) + a*f(x, y)
|
||
|
|
assert collect(f(x, x) + a*f(x, x), f(w1, w1)) == (1 + a)*f(x, x)
|
||
|
|
assert collect(a*(x + 1)**y + (x + 1)**y, w1**y) == (1 + a)*(x + 1)**y
|
||
|
|
assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
|
||
|
|
a*(x + 1)**y + (x + 1)**y
|
||
|
|
assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
|
||
|
|
(1 + a)*(x + 1)**y
|
||
|
|
assert collect(a*(x + 1)**y + (x + 1)**y, w1**w2) == (1 + a)*(x + 1)**y
|
||
|
|
|
||
|
|
|
||
|
|
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 5290
|
||
|
|
assert collect_const(2*x + 2*y + 1, 2) == \
|
||
|
|
collect_const(2*x + 2*y + 1) == \
|
||
|
|
Add(S.One, 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) == \
|
||
|
|
_unevaluated_Add(2*x, Mul(-2, y + z, evaluate=False))
|
||
|
|
|
||
|
|
# this is why the content_primitive is used
|
||
|
|
eq = (sqrt(15 + 5*sqrt(2))*x + sqrt(3 + sqrt(2))*y)*2
|
||
|
|
assert collect_sqrt(eq + 2) == \
|
||
|
|
2*sqrt(sqrt(2) + 3)*(sqrt(5)*x + y) + 2
|
||
|
|
|
||
|
|
# issue 16296
|
||
|
|
assert collect_const(a + b + x/2 + y/2) == a + b + Mul(S.Half, x + y, evaluate=False)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_13143():
|
||
|
|
f = Function('f')
|
||
|
|
fx = f(x).diff(x)
|
||
|
|
e = f(x) + fx + f(x)*fx
|
||
|
|
# collect function before derivative
|
||
|
|
assert collect(e, Wild('w')) == f(x)*(fx + 1) + fx
|
||
|
|
e = f(x) + f(x)*fx + x*fx*f(x)
|
||
|
|
assert collect(e, fx) == (x*f(x) + f(x))*fx + f(x)
|
||
|
|
assert collect(e, f(x)) == (x*fx + fx + 1)*f(x)
|
||
|
|
e = f(x) + fx + f(x)*fx
|
||
|
|
assert collect(e, [f(x), fx]) == f(x)*(1 + fx) + fx
|
||
|
|
assert collect(e, [fx, f(x)]) == fx*(1 + f(x)) + f(x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_6097():
|
||
|
|
assert collect(a*y**(2.0*x) + b*y**(2.0*x), y**x) == (a + b)*(y**x)**2.0
|
||
|
|
assert collect(a*2**(2.0*x) + b*2**(2.0*x), 2**x) == (a + b)*(2**x)**2.0
|
||
|
|
|
||
|
|
|
||
|
|
def test_fraction_expand():
|
||
|
|
eq = (x + y)*y/x
|
||
|
|
assert eq.expand(frac=True) == fraction_expand(eq) == (x*y + y**2)/x
|
||
|
|
assert eq.expand() == y + y**2/x
|
||
|
|
|
||
|
|
|
||
|
|
def test_fraction():
|
||
|
|
x, y, z = map(Symbol, 'xyz')
|
||
|
|
A = Symbol('A', commutative=False)
|
||
|
|
|
||
|
|
assert fraction(S.Half) == (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((1/(x + y))/2, exact=True) == (1, Mul(2,(x + y), evaluate=False))
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
p = symbols('p', positive=True)
|
||
|
|
assert fraction(exp(-p)*log(p), exact=True) == (exp(-p)*log(p), 1)
|
||
|
|
|
||
|
|
m = Mul(1, 1, S.Half, evaluate=False)
|
||
|
|
assert fraction(m) == (1, 2)
|
||
|
|
assert fraction(m, exact=True) == (Mul(1, 1, evaluate=False), 2)
|
||
|
|
|
||
|
|
m = Mul(1, 1, S.Half, S.Half, Pow(1, -1, evaluate=False), evaluate=False)
|
||
|
|
assert fraction(m) == (1, 4)
|
||
|
|
assert fraction(m, exact=True) == \
|
||
|
|
(Mul(1, 1, evaluate=False), Mul(2, 2, 1, evaluate=False))
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5615():
|
||
|
|
aA, Re, a, b, D = symbols('aA Re a b D')
|
||
|
|
e = ((D**3*a + b*aA**3)/Re).expand()
|
||
|
|
assert collect(e, [aA**3/Re, a]) == e
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_5933():
|
||
|
|
from sympy.geometry.polygon import (Polygon, RegularPolygon)
|
||
|
|
from sympy.simplify.radsimp import denom
|
||
|
|
x = Polygon(*RegularPolygon((0, 0), 1, 5).vertices).centroid.x
|
||
|
|
assert abs(denom(x).n()) > 1e-12
|
||
|
|
assert abs(denom(radsimp(x))) > 1e-12 # in case simplify didn't handle it
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_14608():
|
||
|
|
a, b = symbols('a b', commutative=False)
|
||
|
|
x, y = symbols('x y')
|
||
|
|
raises(AttributeError, lambda: collect(a*b + b*a, a))
|
||
|
|
assert collect(x*y + y*(x+1), a) == x*y + y*(x+1)
|
||
|
|
assert collect(x*y + y*(x+1) + a*b + b*a, y) == y*(2*x + 1) + a*b + b*a
|
||
|
|
|
||
|
|
|
||
|
|
def test_collect_abs():
|
||
|
|
s = abs(x) + abs(y)
|
||
|
|
assert collect_abs(s) == s
|
||
|
|
assert unchanged(Mul, abs(x), abs(y))
|
||
|
|
ans = Abs(x*y)
|
||
|
|
assert isinstance(ans, Abs)
|
||
|
|
assert collect_abs(abs(x)*abs(y)) == ans
|
||
|
|
assert collect_abs(1 + exp(abs(x)*abs(y))) == 1 + exp(ans)
|
||
|
|
|
||
|
|
# See https://github.com/sympy/sympy/issues/12910
|
||
|
|
p = Symbol('p', positive=True)
|
||
|
|
assert collect_abs(p/abs(1-p)).is_commutative is True
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_19149():
|
||
|
|
eq = exp(3*x/4)
|
||
|
|
assert collect(eq, exp(x)) == eq
|
||
|
|
|
||
|
|
def test_issue_19719():
|
||
|
|
a, b = symbols('a, b')
|
||
|
|
expr = a**2 * (b + 1) + (7 + 1/b)/a
|
||
|
|
collected = collect(expr, (a**2, 1/a), evaluate=False)
|
||
|
|
# Would return {_Dummy_20**(-2): b + 1, 1/a: 7 + 1/b} without xreplace
|
||
|
|
assert collected == {a**2: b + 1, 1/a: 7 + 1/b}
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_21355():
|
||
|
|
assert radsimp(1/(x + sqrt(x**2))) == 1/(x + sqrt(x**2))
|
||
|
|
assert radsimp(1/(x - sqrt(x**2))) == 1/(x - sqrt(x**2))
|