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 every time: 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 #Test issue 18747 assert (exp(x) + cos(x)).subs(x, oo) == oo assert Add(*[AccumBounds(-1, 1), oo]) == oo assert Add(*[oo, AccumBounds(-1, 1)]) == oo
def test_frac(): assert isinstance(frac(x), frac) assert frac(oo) == AccumBounds(0, 1) assert frac(-oo) == AccumBounds(0, 1) assert frac(n) == 0 assert frac(nan) == nan assert frac(Rational(4, 3)) == Rational(1, 3) assert frac(-Rational(4, 3)) == Rational(2, 3) r = Symbol('r', real=True) assert frac(I * r) == I * frac(r) assert frac(1 + I * r) == I * frac(r) assert frac(0.5 + I * r) == 0.5 + I * frac(r) assert frac(n + I * r) == I * frac(r) assert frac(n + I * k) == 0 assert unchanged(frac, x + I * x) assert frac(x + I * n) == frac(x) assert frac(x).rewrite(floor) == x - floor(x) assert frac(x).rewrite(ceiling) == x + ceiling(-x) assert frac(y).rewrite(floor).subs(y, pi) == frac(pi) assert frac(y).rewrite(floor).subs(y, -E) == frac(-E) assert frac(y).rewrite(ceiling).subs(y, -pi) == frac(-pi) assert frac(y).rewrite(ceiling).subs(y, E) == frac(E) assert Eq(frac(y), y - floor(y)) assert Eq(frac(y), y + ceiling(-y))
def test_atanh(): x = Symbol('x') #at specific points assert atanh(0) == 0 assert atanh(I) == I * pi / 4 assert atanh(-I) == -I * pi / 4 assert atanh(1) == oo assert atanh(-1) == -oo # at infinites assert atanh(oo) == -I * pi / 2 assert atanh(-oo) == I * pi / 2 assert atanh(I * oo) == I * pi / 2 assert atanh(-I * oo) == -I * pi / 2 assert atanh(zoo) == I * AccumBounds(-pi / 2, pi / 2) #properties assert atanh(-x) == -atanh(x) assert atanh(I / sqrt(3)) == I * pi / 6 assert atanh(-I / sqrt(3)) == -I * pi / 6 assert atanh(I * sqrt(3)) == I * pi / 3 assert atanh(-I * sqrt(3)) == -I * pi / 3 assert atanh(I * (1 + sqrt(2))) == 3 * pi * I / 8 assert atanh(I * (sqrt(2) - 1)) == pi * I / 8 assert atanh(I * (1 - sqrt(2))) == -pi * I / 8 assert atanh(-I * (1 + sqrt(2))) == -3 * pi * I / 8 assert atanh(I * sqrt(5 + 2 * sqrt(5))) == 2 * I * pi / 5 assert atanh(-I * sqrt(5 + 2 * sqrt(5))) == -2 * I * pi / 5 assert atanh(I * (2 - sqrt(3))) == pi * I / 12 assert atanh(I * (sqrt(3) - 2)) == -pi * I / 12 assert atanh(oo) == -I * pi / 2
def test_atanh(): x = Symbol('x') #at specific points assert atanh(0) == 0 assert atanh(I) == I*pi/4 assert atanh(-I) == -I*pi/4 assert atanh(1) is oo assert atanh(-1) is -oo assert atanh(nan) is nan # at infinites assert atanh(oo) == -I*pi/2 assert atanh(-oo) == I*pi/2 assert atanh(I*oo) == I*pi/2 assert atanh(-I*oo) == -I*pi/2 assert atanh(zoo) == I*AccumBounds(-pi/2, pi/2) #properties assert atanh(-x) == -atanh(x) assert atanh(I/sqrt(3)) == I*pi/6 assert atanh(-I/sqrt(3)) == -I*pi/6 assert atanh(I*sqrt(3)) == I*pi/3 assert atanh(-I*sqrt(3)) == -I*pi/3 assert atanh(I*(1 + sqrt(2))) == pi*I*Rational(3, 8) assert atanh(I*(sqrt(2) - 1)) == pi*I/8 assert atanh(I*(1 - sqrt(2))) == -pi*I/8 assert atanh(-I*(1 + sqrt(2))) == pi*I*Rational(-3, 8) assert atanh(I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(2, 5) assert atanh(-I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(-2, 5) assert atanh(I*(2 - sqrt(3))) == pi*I/12 assert atanh(I*(sqrt(3) - 2)) == -pi*I/12 assert atanh(oo) == -I*pi/2 # Symmetry assert atanh(Rational(-1, 2)) == -atanh(S.Half) # inverse composition assert unchanged(atanh, tanh(Symbol('v1'))) assert atanh(tanh(-5, evaluate=False)) == -5 assert atanh(tanh(0, evaluate=False)) == 0 assert atanh(tanh(7, evaluate=False)) == 7 assert atanh(tanh(I, evaluate=False)) == I assert atanh(tanh(-I, evaluate=False)) == -I assert atanh(tanh(-11*I, evaluate=False)) == -11*I + 4*I*pi assert atanh(tanh(3 + I)) == 3 + I assert atanh(tanh(4 + 5*I)) == 4 - 2*I*pi + 5*I assert atanh(tanh(pi/2)) == pi/2 assert atanh(tanh(pi)) == pi assert atanh(tanh(-3 + 7*I)) == -3 - 2*I*pi + 7*I assert atanh(tanh(9 - I*Rational(2, 3))) == 9 - I*Rational(2, 3) assert atanh(tanh(-32 - 123*I)) == -32 - 123*I + 39*I*pi
def test_asech(): x = Symbol('x') assert unchanged(asech, -x) # values at fixed points assert asech(1) == 0 assert asech(-1) == pi*I assert asech(0) is oo assert asech(2) == I*pi/3 assert asech(-2) == 2*I*pi / 3 assert asech(nan) is nan # at infinites assert asech(oo) == I*pi/2 assert asech(-oo) == I*pi/2 assert asech(zoo) == I*AccumBounds(-pi/2, pi/2) assert asech(I) == log(1 + sqrt(2)) - I*pi/2 assert asech(-I) == log(1 + sqrt(2)) + I*pi/2 assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12 assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10 assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10 assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8 assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8 assert asech(sqrt(5) - 1) == I*pi / 5 assert asech(1 - sqrt(5)) == 4*I*pi / 5 assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8 # properties # asech(x) == acosh(1/x) assert asech(sqrt(2)) == acosh(1/sqrt(2)) assert asech(2/sqrt(3)) == acosh(sqrt(3)/2) assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2) assert asech(2) == acosh(S.Half) # asech(x) == I*acos(1/x) # (Note: the exact formula is asech(x) == +/- I*acos(1/x)) assert asech(-sqrt(2)) == I*acos(-1/sqrt(2)) assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2) assert asech(-S(2)) == I*acos(Rational(-1, 2)) assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2) # sech(asech(x)) / x == 1 assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1 assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1 assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1 assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1 assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1 assert expand_mul(sech(asech((1 + sqrt(5)))) / ((1 + sqrt(5)))) == 1 assert expand_mul(sech(asech((-1 - sqrt(5)))) / ((-1 - sqrt(5)))) == 1 assert expand_mul(sech(asech((-sqrt(6) - sqrt(2)))) / ((-sqrt(6) - sqrt(2)))) == 1 # numerical evaluation assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I' assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'
def test_frac(): assert isinstance(frac(x), frac) assert frac(oo) == AccumBounds(0, 1) assert frac(-oo) == AccumBounds(0, 1) assert frac(n) == 0 assert frac(nan) == nan assert frac(Rational(4, 3)) == Rational(1, 3) assert frac(-Rational(4, 3)) == Rational(2, 3) r = Symbol('r', real=True) assert frac(I * r) == I * frac(r) assert frac(1 + I * r) == I * frac(r) assert frac(0.5 + I * r) == 0.5 + I * frac(r) assert frac(n + I * r) == I * frac(r) assert frac(n + I * k) == 0 assert frac(x + I * x) == frac(x + I * x) assert frac(x + I * n) == frac(x) assert frac(x).rewrite(floor) == x - floor(x)
def _eval(arg): if arg is S.Infinity or arg is S.NegativeInfinity: return AccumBounds(0, 1) if arg.is_integer: return S.Zero if arg.is_number: if arg is S.NaN: return S.NaN elif arg is S.ComplexInfinity: return None else: return arg - floor(arg) return cls(arg, evaluate=False)
def test_subs_AccumBounds(): e = x e = e.subs(x, AccumBounds(1, 3)) assert e == AccumBounds(1, 3) e = 2 * x e = e.subs(x, AccumBounds(1, 3)) assert e == AccumBounds(2, 6) e = x + x**2 e = e.subs(x, AccumBounds(-1, 1)) assert e == AccumBounds(-1, 2)
def test_log_AccumBounds(): assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
def test_exp_AccumBounds(): assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2)
def test_AccumBounds(): a = Symbol('a', real=True) assert str(AccumBounds(0, a)) == "AccumBounds(0, a)" assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)"
def test_frac(): assert isinstance(frac(x), frac) assert frac(oo) == AccumBounds(0, 1) assert frac(-oo) == AccumBounds(0, 1) assert frac(zoo) is nan assert frac(n) == 0 assert frac(nan) is nan assert frac(Rational(4, 3)) == Rational(1, 3) assert frac(-Rational(4, 3)) == Rational(2, 3) assert frac(Rational(-4, 3)) == Rational(2, 3) r = Symbol('r', real=True) assert frac(I * r) == I * frac(r) assert frac(1 + I * r) == I * frac(r) assert frac(0.5 + I * r) == 0.5 + I * frac(r) assert frac(n + I * r) == I * frac(r) assert frac(n + I * k) == 0 assert unchanged(frac, x + I * x) assert frac(x + I * n) == frac(x) assert frac(x).rewrite(floor) == x - floor(x) assert frac(x).rewrite(ceiling) == x + ceiling(-x) assert frac(y).rewrite(floor).subs(y, pi) == frac(pi) assert frac(y).rewrite(floor).subs(y, -E) == frac(-E) assert frac(y).rewrite(ceiling).subs(y, -pi) == frac(-pi) assert frac(y).rewrite(ceiling).subs(y, E) == frac(E) assert Eq(frac(y), y - floor(y)) assert Eq(frac(y), y + ceiling(-y)) r = Symbol('r', real=True) p_i = Symbol('p_i', integer=True, positive=True) n_i = Symbol('p_i', integer=True, negative=True) np_i = Symbol('np_i', integer=True, nonpositive=True) nn_i = Symbol('nn_i', integer=True, nonnegative=True) p_r = Symbol('p_r', real=True, positive=True) n_r = Symbol('n_r', real=True, negative=True) np_r = Symbol('np_r', real=True, nonpositive=True) nn_r = Symbol('nn_r', real=True, nonnegative=True) # Real frac argument, integer rhs assert frac(r) <= p_i assert not frac(r) <= n_i assert (frac(r) <= np_i).has(Le) assert (frac(r) <= nn_i).has(Le) assert frac(r) < p_i assert not frac(r) < n_i assert not frac(r) < np_i assert (frac(r) < nn_i).has(Lt) assert not frac(r) >= p_i assert frac(r) >= n_i assert frac(r) >= np_i assert (frac(r) >= nn_i).has(Ge) assert not frac(r) > p_i assert frac(r) > n_i assert (frac(r) > np_i).has(Gt) assert (frac(r) > nn_i).has(Gt) assert not Eq(frac(r), p_i) assert not Eq(frac(r), n_i) assert Eq(frac(r), np_i).has(Eq) assert Eq(frac(r), nn_i).has(Eq) assert Ne(frac(r), p_i) assert Ne(frac(r), n_i) assert Ne(frac(r), np_i).has(Ne) assert Ne(frac(r), nn_i).has(Ne) # Real frac argument, real rhs assert (frac(r) <= p_r).has(Le) assert not frac(r) <= n_r assert (frac(r) <= np_r).has(Le) assert (frac(r) <= nn_r).has(Le) assert (frac(r) < p_r).has(Lt) assert not frac(r) < n_r assert not frac(r) < np_r assert (frac(r) < nn_r).has(Lt) assert (frac(r) >= p_r).has(Ge) assert frac(r) >= n_r assert frac(r) >= np_r assert (frac(r) >= nn_r).has(Ge) assert (frac(r) > p_r).has(Gt) assert frac(r) > n_r assert (frac(r) > np_r).has(Gt) assert (frac(r) > nn_r).has(Gt) assert not Eq(frac(r), n_r) assert Eq(frac(r), p_r).has(Eq) assert Eq(frac(r), np_r).has(Eq) assert Eq(frac(r), nn_r).has(Eq) assert Ne(frac(r), p_r).has(Ne) assert Ne(frac(r), n_r) assert Ne(frac(r), np_r).has(Ne) assert Ne(frac(r), nn_r).has(Ne) # Real frac argument, +/- oo rhs assert frac(r) < oo assert frac(r) <= oo assert not frac(r) > oo assert not frac(r) >= oo assert not frac(r) < -oo assert not frac(r) <= -oo assert frac(r) > -oo assert frac(r) >= -oo assert frac(r) < 1 assert frac(r) <= 1 assert not frac(r) > 1 assert not frac(r) >= 1 assert not frac(r) < 0 assert (frac(r) <= 0).has(Le) assert (frac(r) > 0).has(Gt) assert frac(r) >= 0 # Some test for numbers assert frac(r) <= sqrt(2) assert (frac(r) <= sqrt(3) - sqrt(2)).has(Le) assert not frac(r) <= sqrt(2) - sqrt(3) assert not frac(r) >= sqrt(2) assert (frac(r) >= sqrt(3) - sqrt(2)).has(Ge) assert frac(r) >= sqrt(2) - sqrt(3) assert not Eq(frac(r), sqrt(2)) assert Eq(frac(r), sqrt(3) - sqrt(2)).has(Eq) assert not Eq(frac(r), sqrt(2) - sqrt(3)) assert Ne(frac(r), sqrt(2)) assert Ne(frac(r), sqrt(3) - sqrt(2)).has(Ne) assert Ne(frac(r), sqrt(2) - sqrt(3)) assert frac(p_i, evaluate=False).is_zero assert frac(p_i, evaluate=False).is_finite assert frac(p_i, evaluate=False).is_integer assert frac(p_i, evaluate=False).is_real assert frac(r).is_finite assert frac(r).is_real assert frac(r).is_zero is None assert frac(r).is_integer is None assert frac(oo).is_finite assert frac(oo).is_real
def test_issue_19558(): e = (7*x*cos(x) - 12*log(x)**3)*(-log(x)**4 + 2*sin(x) + 1)**2/ \ (2*(x*cos(x) - 2*log(x)**3)*(3*log(x)**4 - 7*sin(x) + 3)**2) assert e.subs(x, oo) == AccumBounds(-oo, oo) assert (sin(x) + cos(x)).subs(x, oo) == AccumBounds(-2, 2)