from sympy.core.expr import unchanged from sympy.core.mul import Mul from sympy.core.numbers import (I, Rational as R, pi) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.functions.combinatorial.factorials import factorial from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin) from sympy.series.order import O from sympy.simplify.radsimp import expand_numer from sympy.core.function import (expand, expand_multinomial, expand_power_base, expand_log) from sympy.testing.pytest import raises from sympy.core.random import verify_numerically from sympy.abc import x, y, z def test_expand_no_log(): assert ( (1 + log(x**4))**2).expand(log=False) == 1 + 2*log(x**4) + log(x**4)**2 assert ((1 + log(x**4))*(1 + log(x**3))).expand( log=False) == 1 + log(x**4) + log(x**3) + log(x**4)*log(x**3) def test_expand_no_multinomial(): assert ((1 + x)*(1 + (1 + x)**4)).expand(multinomial=False) == \ 1 + x + (1 + x)**4 + x*(1 + x)**4 def test_expand_negative_integer_powers(): expr = (x + y)**(-2) assert expr.expand() == 1 / (2*x*y + x**2 + y**2) assert expr.expand(multinomial=False) == (x + y)**(-2) expr = (x + y)**(-3) assert expr.expand() == 1 / (3*x*x*y + 3*x*y*y + x**3 + y**3) assert expr.expand(multinomial=False) == (x + y)**(-3) expr = (x + y)**(2) * (x + y)**(-4) assert expr.expand() == 1 / (2*x*y + x**2 + y**2) assert expr.expand(multinomial=False) == (x + y)**(-2) def test_expand_non_commutative(): A = Symbol('A', commutative=False) B = Symbol('B', commutative=False) C = Symbol('C', commutative=False) a = Symbol('a') b = Symbol('b') i = Symbol('i', integer=True) n = Symbol('n', negative=True) m = Symbol('m', negative=True) p = Symbol('p', polar=True) np = Symbol('p', polar=False) assert (C*(A + B)).expand() == C*A + C*B assert (C*(A + B)).expand() != A*C + B*C assert ((A + B)**2).expand() == A**2 + A*B + B*A + B**2 assert ((A + B)**3).expand() == (A**2*B + B**2*A + A*B**2 + B*A**2 + A**3 + B**3 + A*B*A + B*A*B) # issue 6219 assert ((a*A*B*A**-1)**2).expand() == a**2*A*B**2/A # Note that (a*A*B*A**-1)**2 is automatically converted to a**2*(A*B*A**-1)**2 assert ((a*A*B*A**-1)**2).expand(deep=False) == a**2*(A*B*A**-1)**2 assert ((a*A*B*A**-1)**2).expand() == a**2*(A*B**2*A**-1) assert ((a*A*B*A**-1)**2).expand(force=True) == a**2*A*B**2*A**(-1) assert ((a*A*B)**2).expand() == a**2*A*B*A*B assert ((a*A)**2).expand() == a**2*A**2 assert ((a*A*B)**i).expand() == a**i*(A*B)**i assert ((a*A*(B*(A*B/A)**2))**i).expand() == a**i*(A*B*A*B**2/A)**i # issue 6558 assert (A*B*(A*B)**-1).expand() == 1 assert ((a*A)**i).expand() == a**i*A**i assert ((a*A*B*A**-1)**3).expand() == a**3*A*B**3/A assert ((a*A*B*A*B/A)**3).expand() == \ a**3*A*B*(A*B**2)*(A*B**2)*A*B*A**(-1) assert ((a*A*B*A*B/A)**-2).expand() == \ A*B**-1*A**-1*B**-2*A**-1*B**-1*A**-1/a**2 assert ((a*b*A*B*A**-1)**i).expand() == a**i*b**i*(A*B/A)**i assert ((a*(a*b)**i)**i).expand() == a**i*a**(i**2)*b**(i**2) e = Pow(Mul(a, 1/a, A, B, evaluate=False), S(2), evaluate=False) assert e.expand() == A*B*A*B assert sqrt(a*(A*b)**i).expand() == sqrt(a*b**i*A**i) assert (sqrt(-a)**a).expand() == sqrt(-a)**a assert expand((-2*n)**(i/3)) == 2**(i/3)*(-n)**(i/3) assert expand((-2*n*m)**(i/a)) == (-2)**(i/a)*(-n)**(i/a)*(-m)**(i/a) assert expand((-2*a*p)**b) == 2**b*p**b*(-a)**b assert expand((-2*a*np)**b) == 2**b*(-a*np)**b assert expand(sqrt(A*B)) == sqrt(A*B) assert expand(sqrt(-2*a*b)) == sqrt(2)*sqrt(-a*b) def test_expand_radicals(): a = (x + y)**R(1, 2) assert (a**1).expand() == a assert (a**3).expand() == x*a + y*a assert (a**5).expand() == x**2*a + 2*x*y*a + y**2*a assert (1/a**1).expand() == 1/a assert (1/a**3).expand() == 1/(x*a + y*a) assert (1/a**5).expand() == 1/(x**2*a + 2*x*y*a + y**2*a) a = (x + y)**R(1, 3) assert (a**1).expand() == a assert (a**2).expand() == a**2 assert (a**4).expand() == x*a + y*a assert (a**5).expand() == x*a**2 + y*a**2 assert (a**7).expand() == x**2*a + 2*x*y*a + y**2*a def test_expand_modulus(): assert ((x + y)**11).expand(modulus=11) == x**11 + y**11 assert ((x + sqrt(2)*y)**11).expand(modulus=11) == x**11 + 10*sqrt(2)*y**11 assert (x + y/2).expand(modulus=1) == y/2 raises(ValueError, lambda: ((x + y)**11).expand(modulus=0)) raises(ValueError, lambda: ((x + y)**11).expand(modulus=x)) def test_issue_5743(): assert (x*sqrt( x + y)*(1 + sqrt(x + y))).expand() == x**2 + x*y + x*sqrt(x + y) assert (x*sqrt( x + y)*(1 + x*sqrt(x + y))).expand() == x**3 + x**2*y + x*sqrt(x + y) def test_expand_frac(): assert expand((x + y)*y/x/(x + 1), frac=True) == \ (x*y + y**2)/(x**2 + x) assert expand((x + y)*y/x/(x + 1), numer=True) == \ (x*y + y**2)/(x*(x + 1)) assert expand((x + y)*y/x/(x + 1), denom=True) == \ y*(x + y)/(x**2 + x) eq = (x + 1)**2/y assert expand_numer(eq, multinomial=False) == eq # issue 26329 eq = (exp(x*z) - exp(y*z))/exp(z*(x + y)) ans = exp(-y*z) - exp(-x*z) assert eq.expand(numer=True) != ans assert eq.expand(numer=True, exact=True) == ans assert expand_numer(eq) != ans assert expand_numer(eq, exact=True) == ans def test_issue_6121(): eq = -I*exp(-3*I*pi/4)/(4*pi**(S(3)/2)*sqrt(x)) assert eq.expand(complex=True) # does not give oo recursion eq = -I*exp(-3*I*pi/4)/(4*pi**(R(3, 2))*sqrt(x)) assert eq.expand(complex=True) # does not give oo recursion def test_expand_power_base(): assert expand_power_base((x*y*z)**4) == x**4*y**4*z**4 assert expand_power_base((x*y*z)**x).is_Pow assert expand_power_base((x*y*z)**x, force=True) == x**x*y**x*z**x assert expand_power_base((x*(y*z)**2)**3) == x**3*y**6*z**6 assert expand_power_base((sin((x*y)**2)*y)**z).is_Pow assert expand_power_base( (sin((x*y)**2)*y)**z, force=True) == sin((x*y)**2)**z*y**z assert expand_power_base( (sin((x*y)**2)*y)**z, deep=True) == (sin(x**2*y**2)*y)**z assert expand_power_base(exp(x)**2) == exp(2*x) assert expand_power_base((exp(x)*exp(y))**2) == exp(2*x)*exp(2*y) assert expand_power_base( (exp((x*y)**z)*exp(y))**2) == exp(2*(x*y)**z)*exp(2*y) assert expand_power_base((exp((x*y)**z)*exp( y))**2, deep=True, force=True) == exp(2*x**z*y**z)*exp(2*y) assert expand_power_base((exp(x)*exp(y))**z).is_Pow assert expand_power_base( (exp(x)*exp(y))**z, force=True) == exp(x)**z*exp(y)**z def test_expand_arit(): a = Symbol("a") b = Symbol("b", positive=True) c = Symbol("c") p = R(5) e = (a + b)*c assert e == c*(a + b) assert (e.expand() - a*c - b*c) == R(0) e = (a + b)*(a + b) assert e == (a + b)**2 assert e.expand() == 2*a*b + a**2 + b**2 e = (a + b)*(a + b)**R(2) assert e == (a + b)**3 assert e.expand() == 3*b*a**2 + 3*a*b**2 + a**3 + b**3 assert e.expand() == 3*b*a**2 + 3*a*b**2 + a**3 + b**3 e = (a + b)*(a + c)*(b + c) assert e == (a + c)*(a + b)*(b + c) assert e.expand() == 2*a*b*c + b*a**2 + c*a**2 + b*c**2 + a*c**2 + c*b**2 + a*b**2 e = (a + R(1))**p assert e == (1 + a)**5 assert e.expand() == 1 + 5*a + 10*a**2 + 10*a**3 + 5*a**4 + a**5 e = (a + b + c)*(a + c + p) assert e == (5 + a + c)*(a + b + c) assert e.expand() == 5*a + 5*b + 5*c + 2*a*c + b*c + a*b + a**2 + c**2 x = Symbol("x") s = exp(x*x) - 1 e = s.nseries(x, 0, 6)/x**2 assert e.expand() == 1 + x**2/2 + O(x**4) e = (x*(y + z))**(x*(y + z))*(x + y) assert e.expand(power_exp=False, power_base=False) == x*(x*y + x* z)**(x*y + x*z) + y*(x*y + x*z)**(x*y + x*z) assert e.expand(power_exp=False, power_base=False, deep=False) == x* \ (x*(y + z))**(x*(y + z)) + y*(x*(y + z))**(x*(y + z)) e = x * (x + (y + 1)**2) assert e.expand(deep=False) == x**2 + x*(y + 1)**2 e = (x*(y + z))**z assert e.expand(power_base=True, mul=True, deep=True) in [x**z*(y + z)**z, (x*y + x*z)**z] assert ((2*y)**z).expand() == 2**z*y**z p = Symbol('p', positive=True) assert sqrt(-x).expand().is_Pow assert sqrt(-x).expand(force=True) == I*sqrt(x) assert ((2*y*p)**z).expand() == 2**z*p**z*y**z assert ((2*y*p*x)**z).expand() == 2**z*p**z*(x*y)**z assert ((2*y*p*x)**z).expand(force=True) == 2**z*p**z*x**z*y**z assert ((2*y*p*-pi)**z).expand() == 2**z*pi**z*p**z*(-y)**z assert ((2*y*p*-pi*x)**z).expand() == 2**z*pi**z*p**z*(-x*y)**z n = Symbol('n', negative=True) m = Symbol('m', negative=True) assert ((-2*x*y*n)**z).expand() == 2**z*(-n)**z*(x*y)**z assert ((-2*x*y*n*m)**z).expand() == 2**z*(-m)**z*(-n)**z*(-x*y)**z # issue 5482 assert sqrt(-2*x*n) == sqrt(2)*sqrt(-n)*sqrt(x) # issue 5605 (2) assert (cos(x + y)**2).expand(trig=True) in [ (-sin(x)*sin(y) + cos(x)*cos(y))**2, sin(x)**2*sin(y)**2 - 2*sin(x)*sin(y)*cos(x)*cos(y) + cos(x)**2*cos(y)**2 ] # Check that this isn't too slow x = Symbol('x') W = 1 for i in range(1, 21): W = W * (x - i) W = W.expand() assert W.has(-1672280820*x**15) def test_expand_mul(): # part of issue 20597 e = Mul(2, 3, evaluate=False) assert e.expand() == 6 e = Mul(2, 3, 1/x, evaluate=False) assert e.expand() == 6/x e = Mul(2, R(1, 3), evaluate=False) assert e.expand() == R(2, 3) def test_power_expand(): """Test for Pow.expand()""" a = Symbol('a') b = Symbol('b') p = (a + b)**2 assert p.expand() == a**2 + b**2 + 2*a*b p = (1 + 2*(1 + a))**2 assert p.expand() == 9 + 4*(a**2) + 12*a p = 2**(a + b) assert p.expand() == 2**a*2**b A = Symbol('A', commutative=False) B = Symbol('B', commutative=False) assert (2**(A + B)).expand() == 2**(A + B) assert (A**(a + b)).expand() != A**(a + b) def test_issues_5919_6830(): # issue 5919 n = -1 + 1/x z = n/x/(-n)**2 - 1/n/x assert expand(z) == 1/(x**2 - 2*x + 1) - 1/(x - 2 + 1/x) - 1/(-x + 1) # issue 6830 p = (1 + x)**2 assert expand_multinomial((1 + x*p)**2) == ( x**2*(x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 2*x*(x**2 + 2*x + 1) + 1) assert expand_multinomial((1 + (y + x)*p)**2) == ( 2*((x + y)*(x**2 + 2*x + 1)) + (x**2 + 2*x*y + y**2)* (x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 1) A = Symbol('A', commutative=False) p = (1 + A)**2 assert expand_multinomial((1 + x*p)**2) == ( x**2*(1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 2*x*(1 + 2*A + A**2) + 1) assert expand_multinomial((1 + (y + x)*p)**2) == ( (x + y)*(1 + 2*A + A**2)*2 + (x**2 + 2*x*y + y**2)* (1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 1) assert expand_multinomial((1 + (y + x)*p)**3) == ( (x + y)*(1 + 2*A + A**2)*3 + (x**2 + 2*x*y + y**2)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4)*3 + (x**3 + 3*x**2*y + 3*x*y**2 + y**3)*(1 + 6*A + 15*A**2 + 20*A**3 + 15*A**4 + 6*A**5 + A**6) + 1) # unevaluate powers eq = (Pow((x + 1)*((A + 1)**2), 2, evaluate=False)) # - in this case the base is not an Add so no further # expansion is done assert expand_multinomial(eq) == \ (x**2 + 2*x + 1)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4) # - but here, the expanded base *is* an Add so it gets expanded eq = (Pow(((A + 1)**2), 2, evaluate=False)) assert expand_multinomial(eq) == 1 + 4*A + 6*A**2 + 4*A**3 + A**4 # coverage def ok(a, b, n): e = (a + I*b)**n return verify_numerically(e, expand_multinomial(e)) for a in [2, S.Half]: for b in [3, R(1, 3)]: for n in range(2, 6): assert ok(a, b, n) assert expand_multinomial((x + 1 + O(z))**2) == \ 1 + 2*x + x**2 + O(z) assert expand_multinomial((x + 1 + O(z))**3) == \ 1 + 3*x + 3*x**2 + x**3 + O(z) assert expand_multinomial(3**(x + y + 3)) == 27*3**(x + y) def test_expand_log(): t = Symbol('t', positive=True) # after first expansion, -2*log(2) + log(4); then 0 after second assert expand(log(t**2) - log(t**2/4) - 2*log(2)) == 0 assert expand_log(log(7*6)/log(6)) == 1 + log(7)/log(6) b = factorial(10) assert expand_log(log(7*b**4)/log(b) ) == 4 + log(7)/log(b) def test_issue_23952(): assert (x**(y + z)).expand(force=True) == x**y*x**z one = Symbol('1', integer=True, prime=True, odd=True, positive=True) two = Symbol('2', integer=True, prime=True, even=True) e = two - one for b in (0, x): # 0**e = 0, 0**-e = zoo; but if expanded then nan assert unchanged(Pow, b, e) # power_exp assert unchanged(Pow, b, -e) # power_exp assert unchanged(Pow, b, y - x) # power_exp assert unchanged(Pow, b, 3 - x) # multinomial assert (b**e).expand().is_Pow # power_exp assert (b**-e).expand().is_Pow # power_exp assert (b**(y - x)).expand().is_Pow # power_exp assert (b**(3 - x)).expand().is_Pow # multinomial nn1 = Symbol('nn1', nonnegative=True) nn2 = Symbol('nn2', nonnegative=True) nn3 = Symbol('nn3', nonnegative=True) assert (x**(nn1 + nn2)).expand() == x**nn1*x**nn2 assert (x**(-nn1 - nn2)).expand() == x**-nn1*x**-nn2 assert unchanged(Pow, x, nn1 + nn2 - nn3) assert unchanged(Pow, x, 1 + nn2 - nn3) assert unchanged(Pow, x, nn1 - nn2) assert unchanged(Pow, x, 1 - nn2) assert unchanged(Pow, x, -1 + nn2)