def test1(): R, x = ring('x', QQ) r = rs_sin(x, x, 15) * x**(-5) assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \ QQ(1,120) - x**-2/6 + x**-4 p = rs_sin(x, x, 10) r = rs_nth_root(p, 2, x, 10) assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \ x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2) p = rs_sin(x, x, 10) r = rs_nth_root(p, 7, x, 10) r = rs_pow(r, 5, x, 10) assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \ 11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7) r = rs_exp(x**QQ(1, 2), x, 10) assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \ x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \ x**QQ(15,2)/1307674368000 + x**7/87178291200 + \ x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \ x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \ x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \ x**QQ(1,2) + 1
def test1(): R, x = ring('x', QQ) r = rs_sin(x, x, 15)*x**(-5) assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \ QQ(1,120) - x**-2/6 + x**-4 p = rs_sin(x, x, 10) r = rs_nth_root(p, 2, x, 10) assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \ x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2) p = rs_sin(x, x, 10) r = rs_nth_root(p, 7, x, 10) r = rs_pow(r, 5, x, 10) assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \ 11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7) r = rs_exp(x**QQ(1,2), x, 10) assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \ x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \ x**QQ(15,2)/1307674368000 + x**7/87178291200 + \ x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \ x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \ x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \ x**QQ(1,2) + 1
def test_nth_root(): R, x, y = ring("x, y", QQ) assert (rs_nth_root(1 + x**2 * y, 4, x, 10) == -77 * x**8 * y**4 / 2048 + 7 * x**6 * y**3 / 128 - 3 * x**4 * y**2 / 32 + x**2 * y / 4 + 1) assert (rs_nth_root(1 + x * y + x**2 * y**3, 3, x, 5) == -(x**4) * y**6 / 9 + 5 * x**4 * y**5 / 27 - 10 * x**4 * y**4 / 243 - 2 * x**3 * y**4 / 9 + 5 * x**3 * y**3 / 81 + x**2 * y**3 / 3 - x**2 * y**2 / 9 + x * y / 3 + 1) assert rs_nth_root(8 * x, 3, x, 3) == 2 * x**QQ(1, 3) assert rs_nth_root(8 * x + x**2 + x**3, 3, x, 3) == x**QQ(4, 3) / 12 + 2 * x**QQ(1, 3) r = rs_nth_root(8 * x + x**2 * y + x**3, 3, x, 4) assert r == -(x**QQ(7, 3)) * y**2 / 288 + x**QQ(7, 3) / 12 + x**QQ( 4, 3) * y / 12 + 2 * x**QQ(1, 3) # Constant term in series a = symbols("a") R, x, y = ring("x, y", EX) assert rs_nth_root( x + a, 3, x, 4) == EX(5 / (81 * a**QQ(8, 3))) * x**3 - EX( 1 / (9 * a**QQ(5, 3))) * x**2 + EX(1 / (3 * a**QQ(2, 3))) * x + EX( a**QQ(1, 3)) assert rs_nth_root( x**QQ(2, 3) + x**2 * y + 5, 2, x, 3) == -EX(sqrt(5) / 100) * x**QQ(8, 3) * y - EX( sqrt(5) / 16000) * x**QQ(8, 3) + EX(sqrt(5) / 10) * x**2 * y + EX( sqrt(5) / 2000) * x**2 - EX(sqrt(5) / 200) * x**QQ(4, 3) + EX( sqrt(5) / 10) * x**QQ(2, 3) + EX(sqrt(5))
def test_nth_root(): R, x, y = ring('x, y', QQ) r1 = rs_nth_root(1 + x**2*y, 4, x, 10) assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \ 7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1 assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \ 5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \ x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1 # Constant term in series a = symbols('a') R, x, y = ring('x, y', EX) assert rs_nth_root(x + a, 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \ EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3)) assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\ x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \ EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \ EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5))
def test_RR(): rs_funcs = [rs_sin, rs_cos, rs_tan, rs_cot, rs_atan, rs_tanh] sympy_funcs = [sin, cos, tan, cot, atan, tanh] R, x, y = ring('x, y', RR) a = symbols('a') for rs_func, sympy_func in zip(rs_funcs, sympy_funcs): p = rs_func(2 + x, x, 5).compose(x, 5) q = sympy_func(2 + a).series(a, 0, 5).removeO() is_close(p.as_expr(), q.subs(a, 5).n()) p = rs_nth_root(2 + x, 5, x, 5).compose(x, 5) q = ((2 + a)**QQ(1, 5)).series(a, 0, 5).removeO() is_close(p.as_expr(), q.subs(a, 5).n())
def test_puiseux(): R, x, y = ring('x, y', QQ) p = x**QQ(2, 5) + x**QQ(2, 3) + x r = rs_series_inversion(p, x, 1) r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \ 2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \ + x**QQ(-2,5) assert r == r1 r = rs_nth_root(1 + p, 3, x, 1) assert r == -x**QQ(4, 5) / 9 + x**QQ(2, 3) / 3 + x**QQ(2, 5) / 3 + 1 r = rs_log(1 + p, x, 1) assert r == -x**QQ(4, 5) / 2 + x**QQ(2, 3) + x**QQ(2, 5) r = rs_LambertW(p, x, 1) assert r == -x**QQ(4, 5) + x**QQ(2, 3) + x**QQ(2, 5) p1 = x + x**QQ(1, 5) * y r = rs_exp(p1, x, 1) assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \ x**QQ(1,5)*y + 1 r = rs_atan(p, x, 2) assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ x + x**QQ(2,3) + x**QQ(2,5) r = rs_atan(p1, x, 2) assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \ x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y r = rs_asin(p, x, 2) assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_cot(p, x, 1) assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \ 2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \ x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5) r = rs_cos_sin(p, x, 2) assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \ x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1 assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_atanh(p, x, 2) assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \ x**QQ(2,3) + x**QQ(2,5) r = rs_sinh(p, x, 2) assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_cosh(p, x, 2) assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \ x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1 r = rs_tanh(p, x, 2) assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ x + x**QQ(2,3) + x**QQ(2,5)
def test_puiseux(): R, x, y = ring('x, y', QQ) p = x**QQ(2,5) + x**QQ(2,3) + x r = rs_series_inversion(p, x, 1) r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \ 2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \ + x**QQ(-2,5) assert r == r1 r = rs_nth_root(1 + p, 3, x, 1) assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1 r = rs_log(1 + p, x, 1) assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5) r = rs_LambertW(p, x, 1) assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5) p1 = x + x**QQ(1,5)*y r = rs_exp(p1, x, 1) assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \ x**QQ(1,5)*y + 1 r = rs_atan(p, x, 2) assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ x + x**QQ(2,3) + x**QQ(2,5) r = rs_atan(p1, x, 2) assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \ x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y r = rs_asin(p, x, 2) assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_cot(p, x, 1) assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \ 2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \ x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5) r = rs_cos_sin(p, x, 2) assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \ x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1 assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_atanh(p, x, 2) assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \ x**QQ(2,3) + x**QQ(2,5) r = rs_sinh(p, x, 2) assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \ x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5) r = rs_cosh(p, x, 2) assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \ x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1 r = rs_tanh(p, x, 2) assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \ x + x**QQ(2,3) + x**QQ(2,5)