sympy/simplify/tests/test_simplify.py

   1
   2 from sympy import *
   3 from sympy.utilities.pytest import XFAIL
   4
   5 def test_ratsimp():
   6     x = Symbol("x")
   7     y = Symbol("y")
   8     e = 1/x+1/y
   9     assert e != (x+y)/(x*y)
  10     assert ratsimp(e) == (x+y)/(x*y)
  11
  12     e = 1/(1+1/x)
  13     assert ratsimp(e) == x/(x+1)
  14     assert (x+1)*ratsimp(e)/x == 1
  15     assert ratsimp(exp(e)) == exp(x/(x+1))
  16
  17 @XFAIL
  18 def test_ratsimp_X1():
  19     e = -x-y-(x+y)**(-1)*y**2+(x+y)**(-1)*x**2
  20     assert e != -2*y
  21     assert ratsimp(e) == -2*y
  22
  23 @XFAIL
  24 def test_ratsimp_X2():
  25     e = x/(x+y)+y/(x+y)
  26     assert e != 1
  27     assert ratsimp(e) == 1
  28
  29
  30
  31 def test_trigsimp():
  32     x,y = map(Symbol, 'xy')
  33
  34     assert trigsimp(1 - sin(x)**2) == cos(x)**2
  35     assert trigsimp(1 - cos(x)**2) == sin(x)**2
  36     assert trigsimp(sin(x)**2 + cos(x)**2) == 1
  37     assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
  38     assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
  39     assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
  40     assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
  41     assert trigsimp(1/sin(x)**2 - 1) == cot(x)**2
  42     assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
  43
  44     assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
  45     assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) in \
  46                 [2 + 3*cos(x/2)**2, 5 - 3*sin(x/2)**2]
  47
  48     assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1
  49     e = 2*sin(x)**2 + 2*cos(x)**2
  50     assert trigsimp(log(e), deep=True) == log(2)
  51
  52 @XFAIL
  53 def test_factorial_simplify():
  54     # There are more tests in test_factorials.py. These are just to
  55     # ensure that simplify() calls factorial_simplify correctly
  56     from sympy.specfun.factorials import factorial
  57     x = Symbol('x')
  58     assert simplify(factorial(x)/x) == factorial(x-1)
  59     assert simplify(factorial(factorial(x))) == factorial(factorial(x))
  60
  61 def test_simplify():
  62     x = Symbol('x')
  63     y = Symbol('y')
  64     e = 1/x + 1/y
  65     assert e != (x+y)/(x*y)
  66     assert simplify(e) == (x+y)/(x*y)
  67
  68     e = (4+4*x-2*(2+2*x))/(2+2*x)
  69     assert simplify(e) == 0
  70
  71     e = (-4*x*y**2-2*y**3-2*x**2*y)/(x+y)**2
  72     assert simplify(e) == -2*y
  73
  74     e = (x+y)**2/(-4*x*y**2-2*y**3-2*x**2*y)
  75     assert simplify(e) == 1 / (-2*y)
  76
  77     e = -x-y-(x+y)**(-1)*y**2+(x+y)**(-1)*x**2
  78     assert simplify(e) == -2*y
  79
  80 def test_fraction():
  81     x, y, z = map(Symbol, 'xyz')
  82
  83     assert fraction(Rational(1, 2)) == (1, 2)
  84
  85     assert fraction(x) == (x, 1)
  86     assert fraction(1/x) == (1, x)
  87     assert fraction(x/y) == (x, y)
  88     assert fraction(x/2) == (x, 2)
  89
  90     assert fraction(x*y/z) == (x*y, z)
  91     assert fraction(x/(y*z)) == (x, y*z)
  92
  93     assert fraction(1/y**2) == (1, y**2)
  94     assert fraction(x/y**2) == (x, y**2)
  95
  96     assert fraction((x**2+1)/y) == (x**2+1, y)
  97     assert fraction(x*(y+1)/y**7) == (x*(y+1), y**7)
  98
  99     assert fraction(exp(-x), exact=True) == (exp(-x), 1)
 100
 101 def test_together():
 102     x, y, z = map(Symbol, 'xyz')
 103
 104     assert together(1/x) == 1/x
 105
 106     assert together(1/x + 1) == (x+1)/x
 107     assert together(1/x + x) == (x**2+1)/x
 108
 109     assert together(1/x + Rational(1, 2)) == (x+2)/(2*x)
 110
 111     assert together(1/x + 2/y) == (2*x+y)/(y*x)
 112     assert together(1/(1 + 1/x)) == x/(1+x)
 113     assert together(x/(1 + 1/x)) == x**2/(1+x)
 114
 115     assert together(1/x + 1/y + 1/z) == (x*y + x*z + y*z)/(x*y*z)
 116
 117     assert together(1/(x*y) + 1/(x*y)**2) == y**(-2)*x**(-2)*(1+x*y)
 118     assert together(1/(x*y) + 1/(x*y)**4) == y**(-4)*x**(-4)*(1+x**3*y**3)
 119     assert together(1/(x**7*y) + 1/(x*y)**4) == y**(-4)*x**(-7)*(x**3+y**3)
 120
 121     assert together(sin(1/x+1/y)) == sin(1/x+1/y)
 122     assert together(sin(1/x+1/y), deep=True) == sin((x+y)/(x*y))
 123
 124     assert together(Rational(1,2) + x/2) == (x+1)/2
 125
 126 def test_separate():
 127     x, y, z = map(Symbol, 'xyz')
 128
 129     assert separate((x*y*z)**4) == x**4*y**4*z**4
 130     assert separate((x*y*z)**x) == x**x*y**x*z**x
 131     assert separate((x*(y*z)**2)**3) == x**3*y**6*z**6
 132
 133     assert separate((sin((x*y)**2)*y)**z) == sin((x*y)**2)**z*y**z
 134     assert separate((sin((x*y)**2)*y)**z, deep=True) == sin(x**2*y**2)**z*y**z
 135
 136     assert separate(exp(x)**2) == exp(2*x)
 137     assert separate((exp(x)*exp(y))**2) == exp(2*x)*exp(2*y)
 138
 139     assert separate((exp((x*y)**z)*exp(y))**2) == exp(2*(x*y)**z)*exp(2*y)
 140     assert separate((exp((x*y)**z)*exp(y))**2, deep=True) == exp(2*x**z*y**z)*exp(2*y)
 141
 142 @XFAIL
 143 def test_separate_X1():
 144     assert separate((exp(x)*exp(y))**z) == exp(x*z)*exp(y*z)
 145
 146 def test_powsimp():
 147     x,y,n = symbols('xyn')
 148     assert powsimp( y**n * (y/x)**(-n) ) == x**n
 149     assert powsimp( 4**x * 2**(-x) * 2**(-x) ) == 1
 150     assert powsimp( (-4)**x * (-2)**(-x) * 2**(-x) ) == 1
 151
 152 def test_collect():
 153     x,y,n = symbols('xyn')
 154     assert collect(2*x**2 + y*x**2 + 3*x*y, [x]) == x**2*(2+y) + 3*x*y
 155     assert collect(2*x**2 + y*x**2 + 3*x*y, [y]) == 2*x**2 + y*(x**2+3*x)
 156
 157 def test_hypersimp():
 158     n, k = symbols('nk', integer=True)
 159
 160     assert hypersimp(factorial(k), k) == k + 1
 161     assert hypersimp(factorial(k**2), k) is None
 162
 163     assert hypersimp(2**k/factorial(k)**2, k) == 2/(k+1)**2
 164
 165     assert hypersimp(binomial(n, k), k) == (n-k)/(k+1)
 166     assert hypersimp(binomial(n+1, k), k) == (n-k+1)/(k+1)
 167
 168     term = (4*k+1)*factorial(k)/factorial(2*k+1)
 169     assert hypersimp(term, k) == (4*k + 5)/(6 + 16*k**2 + 28*k)
 170
 171     term = 1/((2*k-1)*factorial(2*k+1))
 172     assert hypersimp(term, k) == (2*k-1)/(2*k+1)/(2*k+2)/(2*k+3)
 173
 174     term = binomial(n, k)*(-1)**k/factorial(k)
 175     assert hypersimp(term, k) == (k - n)/(k+1)**2