sympy/simplify/tests/test_rewrite.py

   1 from sympy import sin, cos, exp, cot, sqrt, S, I, E, pi, symbols, Function, Matrix, Eq, RootSum, Lambda
   2 from sympy.simplify import cancel, trim, apart, together
   3 from sympy.integrals import integrate
   4 from sympy.utilities.pytest import XFAIL
   5
   6 x,y,z,n = symbols('xyzn')
   7
   8 def test_has():
   9     assert cot(x).has(x)
  10     assert cot(x).has(cot)
  11     assert not cot(x).has(sin)
  12     assert sin(x).has(x)
  13     assert sin(x).has(sin)
  14     assert not sin(x).has(cot)
  15
  16 def test_sin_exp_rewrite():
  17     assert sin(x).rewrite(sin, exp) == -I/2*(exp(I*x)-exp(-I*x))
  18     assert sin(x).rewrite(sin, exp).rewrite(exp, sin) == sin(x)
  19     assert cos(x).rewrite(cos, exp).rewrite(exp, cos) == cos(x)
  20     assert (sin(5*y) - sin(2*x)).rewrite(sin, exp).rewrite(exp, sin) == sin(5*y) - sin(2*x)
  21     assert sin(x+y).rewrite(sin, exp).rewrite(exp, sin) == sin(x+y)
  22     assert cos(x+y).rewrite(cos, exp).rewrite(exp, cos) == cos(x+y)
  23     # This next test currently passes... not clear whether it should or not?
  24     assert cos(x).rewrite(cos, exp).rewrite(exp, sin) == cos(x)
  25
  26 def test_cancel():
  27     assert cancel((x**2-2)/(x+sqrt(2))) == x - sqrt(2)
  28     assert cancel((x**2-y**2)/(x-y), x, y) == x + y
  29     assert cancel((x**2-y)/(x-y)) == 1/(x - y)*(x**2 - y)
  30     assert cancel((x**2-y**2)/(x-y), x) == x + y
  31     assert cancel((x**2-y**2)/(x-y), y) == x + y
  32     assert cancel((x**2-y**2)/(x-y)) == x + y
  33     assert cancel((E*x+2)/(x-pi)*(x-1)) == (-2 + x*(2 - E) + E*x**2)/(x - pi)
  34     assert cancel(Eq((x**3-1)/(x-1), sin(x))) == Eq(1 + x + x**2, sin(x))
  35     assert cancel((x**2-1)/(x-1) == (x**2+1)/(x-I), x) == (1 + x == I + x)
  36     assert cancel((x**2-1)/(x-1) + (x**2+1)/(x-I), x) == 1 + I + 2*x
  37     assert cancel((x**2-1)/(x-1) + (x**2+1)/(x-I), y) in [
  38             1/(1 - x)*(1 - x**2) + 1/(x - I)*(1 + x**2),
  39             -1/(1 - x)*(-1 + x**2) + 1/(x - I)*(1 + x**2),
  40             1/(-1 + x)*(-1 + x**2) + 1/(x - I)*(1 + x**2),
  41             ]
  42
  43 def test_trim():
  44     f = Function('f')
  45
  46     assert trim((f(x)**2+f(x))/f(x)) == 1 + f(x)
  47     assert trim((sin(x)**2+sin(x))/sin(x)) == 1 + sin(x)
  48
  49     assert trim((f(x)+y*f(x))/f(x)) == 1 + y
  50
  51     expr = integrate(1/(x**3+1), x)
  52
  53     assert trim(together(expr.diff(x))) == 1/(x**3+1)
  54     assert cancel(together(expr.diff(x))) == 1/(x**3+1)
  55
  56     expr = together(expr.subs(x, sin(x)).diff(x))
  57
  58     assert trim(expr) == cos(x)/(1 + sin(x)**3)
  59
  60     assert trim((2 * (1/n - cos(n * pi)/n))/pi) == \
  61         1/pi/n*(2 - 2*cos(pi*n))
  62
  63     assert trim(sin((f(x)**2+f(x))/f(x))) == sin(1 + f(x))
  64
  65     assert trim(exp(x)*sin(x)/2 + cos(x)*exp(x)) == \
  66         exp(x)*(sin(x) + 2*cos(x))/2
  67
  68 @XFAIL  # because of #666
  69 def test_trim_xfail():
  70     assert trim((exp(3*x)+exp(2*x*y) + y*exp(x))/exp(x)) == \
  71         y + exp(2*x) + exp(-x + 2*x*y)
  72
  73 def test_apart():
  74     assert apart(1/(x+2)/(x+1), x) == 1/(1 + x) - 1/(2 + x)
  75     assert apart(1/(x+1)/(x+5), x) == -1/(5 + x)/4 + 1/(1 + x)/4
  76
  77     f = apart(1/(x-y)/(x-z), x)
  78
  79     assert f.subs({y:1,z:2}) == apart(1/(x-1)/(x-2), x)
  80
  81     assert apart((E*x+2)/(x-pi)*(x-1), x) in [
  82         2 - E + E*pi + E*x - 1/(x - pi)*( 2 - 2*pi + E*pi - E*pi**2),
  83         2 - E + E*pi + E*x + 1/(x - pi)*(-2 + 2*pi - E*pi + E*pi**2),
  84     ]
  85
  86     M = Matrix(2, 2, lambda i, j: 1/(x-(i+1))/(x-(1-j)))
  87
  88     assert apart(M, x) in [
  89             Matrix([
  90                 [(x-1)**(-2),     -1/x-1/(1-x)          ],
  91                 [1/(1-x)-1/(2-x), -S.Half/x-S.Half/(2-x)],
  92             ]),
  93             Matrix([
  94                 [(-1+x)**(-2),     -1/x+1/(-1+x)          ],
  95                 [-1/(-1+x)+1/(-2+x), -S.Half/x+S.Half/(-2+x)],
  96             ]),
  97             ]
  98
  99     assert apart(Eq((x**2+1)/(x+1), sin(x)), x) == \
 100         Eq(x - 1 + 2/(x+1), sin(x))
 101
 102     assert str(apart(1/(1+x**5), x, evaluate=False)) in [
 103         "RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)",
 104         "RootSum(Lambda(_a, -_a/(5*(x - _a))), x**5 + 1, x)"]