algebrite/tests/simplify.coffee

Computer Algebra System in Coffeescript

test_simplify = ->
  run_test [

    # the system normally tries to
    # arrange polynomials in a normal
    # form, without the need for simplify
    "x+a*x",
    "(1+a)*x",

    "simplify(A)",
    "A",

    "simplify(A+B)",
    "A+B",

    "simplify(A B)",
    "A*B",

    "simplify(A^B)",
    "A^B",

    "simplify(A/(A+B)+B/(A+B))",
    "1",

    "simplify((A-B)/(B-A))",
    "-1",

    "A=[[A11,A12],[A21,A22]]",
    "",

    "simplify(det(A) inv(A) - adj(A))",
    "0",

    "A=quote(A)",
    "",

    "simplify(-3 exp(-1/3 r + i phi) cos(theta) / sin(theta)\
     + 3 exp(-1/3 r + i phi) cos(theta) sin(theta)\
     + 3 exp(-1/3 r + i phi) cos(theta)^3 / sin(theta))",
    "0",

    "simplify((A^2 C^2 + A^2 D^2 + B^2 C^2 + B^2 D^2)/(A^2+B^2)/(C^2+D^2))",
    "1",

    "simplify(d(arctan(y/x),y))",
    "x/(x^2+y^2)",

    "simplify(d(arctan(y/x),x))",
    "-y/(x^2+y^2)",

    "simplify(1-sin(x)^2)",
    "cos(x)^2",

    "simplify(1-cos(x)^2)",
    "sin(x)^2",

    "simplify(sin(x)^2-1)",
    "-cos(x)^2",

    "simplify(cos(x)^2-1)",
    "-sin(x)^2",

    # tries to get rid of sin and cos if there are more
    # compact clockforms or exponential forms
    "simplify(-cos(2/5*pi)*(k/a)^(1/5)-i*(k/a)^(1/5)*sin(2/5*pi))",
    "((k/a)^(2/5))^(1/2)/((-1)^(3/5))",

    #"simfac(n!/n)-(n-1)!",
    #"0",

    #"simfac(n/n!)-1/(n-1)!",
    #"0",

    #"simfac(rationalize((n+k+1)/(n+k+1)!))-1/(n+k)!",
    #"0",

    #"simfac(condense((n+1)*n!))-(n+1)!",
    #"0",

    #"simfac(1/((n+1)*n!))-1/(n+1)!",
    #"0",

    #"simfac((n+1)!/n!)-n-1",
    #"0",

    #"simfac(n!/(n+1)!)-1/(n+1)",
    #"0",

    #"simfac(binomial(n+1,k)/binomial(n,k))",
    #"(1+n)/(1-k+n)",

    #"simfac(binomial(n,k)/binomial(n+1,k))",
    #"(1-k+n)/(1+n)",

    #"F(nn,kk)=kk*binomial(nn,kk)",
    #"",

    #"simplify(simfac((F(n,k)+F(n,k-1))/F(n+1,k))-n/(n+1))",
    #"0",

    #"F=quote(F)",
    #"",

    "simplify(n!/n)-(n-1)!",
    "0",

    "simplify(n/n!)-1/(n-1)!",
    "0",

    "simplify(rationalize((n+k+1)/(n+k+1)!))-1/(n+k)!",
    "0",

    "simplify(condense((n+1)*n!))-(n+1)!",
    "0",

    "simplify(1/((n+1)*n!))-1/(n+1)!",
    "0",

    "simplify((n+1)!/n!)-n-1",
    "0",

    "simplify(n!/(n+1)!)-1/(n+1)",
    "0",

    "simplify(binomial(n+1,k)/binomial(n,k))",
    "(1+n)/(1-k+n)",

    "simplify(binomial(n,k)/binomial(n+1,k))",
    "(1-k+n)/(1+n)",

    "F(nn,kk)=kk*binomial(nn,kk)",
    "",

    "simplify((F(n,k)+F(n,k-1))/F(n+1,k))-n/(n+1)",
    "0",

    "F=quote(F)",
    "",

    "simplify((n+1)/(n+1)!)-1/n!",
    "0",

    "simplify(a*b+a*c)",
    "a*(b+c)",

    # Symbol's binding is evaluated, undoing simplify

    "x=simplify(a*b+a*c)",
    "",

    "x",
    "a*b+a*c",

    "x=quote(x)",
    "",

    "simplify((6 - 4*2^(1/2))^(1/2))",
    "2-2^(1/2)",

    "4-4*(-1)^(1/3)+4*(-1)^(2/3)",
    "0",

    "simplify(4-4*(-1)^(1/3)+4*(-1)^(2/3))",
    "0",

    # this requires some simplification to be
    # further done after the de-nesting
    "simplify(14^(1/2) - (16 - 4*7^(1/2))^(1/2))",
    "2^(1/2)",


    "simplify(-(2^(1/2)*(-1+7^(1/2)))+2^(1/2)*7^(1/2))",
    "2^(1/2)",


    "simplify((9 + 6*2^(1/2))^(1/2))",
    "3^(1/2)*(1+2^(1/2))",


    "simplify((7 + 13^(1/2))^(1/2))",
    "(1+13^(1/2))/(2^(1/2))",

    # two nested radicals at the same time
    "simplify((17 + 12*2^(1/2))^(1/2) + (17 - 12*2^(1/2))^(1/2))",
    "6",

    "simplify((2 + 3^(1/2))^(1/2))",
    "(1+3^(1/2))/(2^(1/2))",

    "simplify((1/2 + (39^(1/2)/16))^(1/2))",
    "(3^(1/2)+13^(1/2))/(4*2^(1/2))",

    # there would be a slightly better presentation for this,
    # where 108 is factored and some parts get out of the
    # radical but there is no way to de-nest this.
    "simplify((-108+108*(-1)^(1/2)*3^(1/2))^(1/3))",
    "6*(-1)^(2/9)",
    # also: "(-108+108*i*3^(1/2))^(1/3)" is a possible result

    # you can take that 4 out of the radical
    # but other than that there is no
    # "sum or radicals" form of this
    "simplify((-4+4*(-1)^(1/2)*3^(1/2))^(1/3))",
    "2*(-1)^(2/9)",
    # also: "(-4+4*i*3^(1/2))^(1/3)" is a possible result


    # scrambling the order of things a little
    # and checking whether the nested radical
    # still gets simplified.
    "simplify((((-3)^(1/2) + 1)/2)^(1/2))",
    #"(-1)^(1/6)",
    "1/2*(i+3^(1/2))",

    "simplify((1/2 + (-3)^(1/2)/2)^(1/2))",
    #"(-1)^(1/6)",
    "1/2*(i+3^(1/2))",

    # no possible de-nesting, should
    # leave unchanged.
    "simplify((2 +2^(1/2))^(1/2))",
    "(2+2^(1/2))^(1/2)",

    "simplify((1 +3^(1/2)/2)^(1/2) + (1 -3^(1/2)/2)^(1/2))",
    "3^(1/2)",

    "simplify((1 +3^(1/2)/2)^(1/2))",
    "1/2*(1+3^(1/2))",

    # not quite perfect as there is a radical at the
    # denominator, but the de-nesting happens.
    "simplify(((1 +39^(1/2)/8)/2)^(1/2))",
    "(3^(1/2)+13^(1/2))/(4*2^(1/2))",

    "simplify((5 +24^(1/2))^(1/2))",
    "2^(1/2)+3^(1/2)",

    "simplify((3 +4*i)^(1/2))",
    "2+i",

    "simplify((3 -4*i)^(1/2))",
    "2-i",

    "simplify((-2 +2*3^(1/2)*i)^(1/2))",
    #"2*(-1)^(1/3)",
    "1+i*3^(1/2)",

    "simplify((9 - 4*5^(1/2))^(1/2))",
    "-2+5^(1/2)",

    "simplify((61 - 24*5^(1/2))^(1/2))",
    "-4+3*5^(1/2)",

    "simplify((-352+936*(-1)^(1/2))^(1/3))",
    "2*(4+3*i)",


    "simplify((3 - 2*2^(1/2))^(1/2))",
    "-1+2^(1/2)",

    "simplify((27/2+27/2*(-1)^(1/2)*3^(1/2))^(1/3))",
    "3*(-1)^(1/9)",
    # also good: (27/2+27/2*i*3^(1/2))^(1/3)

    # this nested radical is also equal to
    # (-1)^(1/9)
    # but there is no "sum of radicals" form
    # for this.
    "simplify((1/2+1/2*(-1)^(1/2)*3^(1/2))^(1/3))",
    "(-1)^(1/9)",
    # also good: (1/2+1/2*i*3^(1/2))^(1/3)

    "simplify((2 + 5^(1/2))^(1/3))",
    "1/2*(1+5^(1/2))",

    "simplify((-3 + 10*3^(1/2)*i/9)^(1/3))",
    "1+2/3*i*3^(1/2)",

    "simplify((1-3*x^2+3*x^4-x^6)^(1/2))",
    "(-x^6+3*x^4-3*x^2+1)^(1/2)",

    "simplify(subst((-1)^(1/2),i,(-3 + 10*3^(1/2)*i/9)^(1/3)))",
    "1+2/3*i*3^(1/2)",

    "simplify(rationalize(-3 + 10*3^(1/2)*i/9)^(1/3))",
    "1+2/3*i*3^(1/2)",

    # note that sympy doesn't give a straight symbolic answer to
    # this one, the result to this is numeric instead, and with
    # a near-zero imaginary part.
    # In Sympy one can get to the answer obliquely with minpoly instead,
    # as minpoly((-1)^(1/6) - (-1)^(5/6)) -> x^2−3
    "simplify((-1)^(1/6) - (-1)^(5/6))",
    "3^(1/2)",

    "simplify((7208+2736*5^(1/2))^(1/3))",
    "17+3*5^(1/2)",

    "simplify((901+342*5^(1/2))^(1/3))",
    "1/2*(17+3*5^(1/2))",

    "-i*(-2*(-1)^(1/6)/(3^(1/2))+2*(-1)^(5/6)/(3^(1/2)))^(1/4)*(2*(-1)^(1/6)/(3^(1/2))-2*(-1)^(5/6)/(3^(1/2)))^(1/4)/(2^(1/2))",
    "1/2^(1/2)-i/(2^(1/2))",

    "simplify(-i*(-2*(-1)^(1/6)/(3^(1/2))+2*(-1)^(5/6)/(3^(1/2)))^(1/4)*(2*(-1)^(1/6)/(3^(1/2))-2*(-1)^(5/6)/(3^(1/2)))^(1/4)/(2^(1/2)))",
    # this one simplifies to any of these two, these are all the same:
    "(1-i)/(2^(1/2))",
    #    -(-1)^(3/4)
    #"-(-1)^(3/4)",

    "(-1)^(-5/a)",
    #"(-1)^(-5/a)",
    "1/(-1)^(5/a)",


    # -----------------------
    "simplify((-1)^(-5))",
    "-1",

    "simplify((-1)^(5))",
    "-1",

    "simplify((1)^(-5))",
    "1",

    "simplify((1)^(5))",
    "1",

    # seems here that the simplification
    # has more nodes than the result but
    # it's not the case: the 1/... inversion
    # is just done at the print level for
    # legibility
    "simplify((-1)^(-5/a))",
    #"(-1)^(-5/a)",
    "1/(-1)^(5/a)",

    "simplify((-1)^(5/a))",
    "(-1)^(5/a)",

    "simplify((1)^(-5/a))",
    "1",

    "simplify((1)^(5/a))",
    "1",

    # -----------------------
    "simplify((-1)^(-6))",
    "1",

    "simplify((-1)^(6))",
    "1",

    "simplify((1)^(-6))",
    "1",

    "simplify((1)^(6))",
    "1",

    # clockform can be much more compact than the
    # rectangular format so we return that one,
    # the user can always do a rect or a circexp on
    # the result if she desires other forms
    "simplify(i*2^(1/4)*sin(1/8*pi)+2^(1/4)*cos(1/8*pi))",
    "(-1)^(1/8)*2^(1/4)",
    # the circexp of the above is
    # 2^(1/4) exp(1/8 i pi), which is less compact


    # seems here that the simplification
    # has more nodes than the result but
    # it's not the case: the 1/... inversion
    # is just done at the print level for
    "simplify((-1)^(-6/a))",
    #"(-1)^(-6/a)",
    "1/(-1)^(6/a)",

    "simplify((-1)^(6/a))",
    "(-1)^(6/a)",

    "simplify((1)^(-6/a))",
    "1",

    "simplify((1)^(6/a))",
    "1",

    "simplify(transpose(A)*transpose(x))",
    "transpose(A*x)",

    "simplify(inner(transpose(A),transpose(x)))",
    "transpose(inner(x,A))",

    # ---------------------------------------------
    # checking that simplify doesn't make incorrect
    # simplifications

    "simplify(sqrt(-1/2 -1/2 * x))",
    "(-1/2*x-1/2)^(1/2)",

    "simplify(sqrt(x*y))",
    "(x*y)^(1/2)",

    "simplify(sqrt(1/x))",
    "(1/x)^(1/2)",

    "simplify(sqrt(x^y))",
    "(x^y)^(1/2)",

    "simplify(sqrt(x)^2)",
    "x",

    "simplify(sqrt(x^2))",
    "abs(x)",

    # simplifying rational expressions

    "simplify((x+1)*(x+1)/(x+1))",
    "x+1",

    "simplify(x*(x+1)/x)",
    "x+1",

    "simplify((x^2+7x+6)/(x^2-5x-6))",
    "(x+6)/(x-6)",

    "simplify(x*(x+1)/(x+1)+1)",
    "x+1",

    "simplify(x*(x+1)/(x+1))",
    "x",

    "simplify((x^2+3x)/(x^2+5x))",
    "(x+3)/(x+5)",

    "simplify((6x+20)/(2x+10))",
    "(3*x+10)/(x+5)",

    "simplify((x^3-3x^2)/(4x^2-5x))",
    "x*(x-3)/(4*x-5)",

    "simplify((x^2-9)/(x^2+5x+6))",
    "(x-3)/(x+2)",

    "simplify((x^2-3x+2)/(x^2-1))",
    "(x-2)/(x+1)",

    "simplify((x^2-2x-15)/(x^2+x-6))",
    "(x-5)/(x-2)",

    "simplify(10x^3/(2x^2-18x))",
    "5*x^2/(x-9)",

    "simplify(6x^2/(12x^4-9x^3))",
    "1/(x*(2*x-3/2))",

    "simplify(((3-x)*(x-1))/((x-3)*(x+1)))",
    "(-x+1)/(x+1)",

    "simplify(((x-2)*(x-5))/((2-x)*(x+5)))",
    "(-x+5)/(x+5)",

    "simplify((15-10x)/(8x^3-12x^2))",
    "-5/(4*x^2)",

    "simplify(3x/(15x^2-6x))",
    "1/(5*x-2)",

    "simplify((3x^3-15x^2+12x)/(3x-3))",
    "x*(x-4)",

    "simplify((6x^2-12x)/(6x-3x^2))",
    "-2",

    "simplify((2x^2+13x+20)/(2x^2+17x+30))",
    "(x+4)/(x+6)",

    "simplify((x^4+8x^2+7)/(3x^5-3x))",
    "(x^2+7)/(3*x*(x^2-1))",

    "simplify((x^2+8x*k+16*k^2)/(x^2-16k^2))",
    "(x+4*k)/(x-4*k)",

    "simplify((x^2-2x-8)/(x^2-9x+20))",
    "(x+2)/(x-5)",

    "simplify((x^2-25)/(5x-x^2))",
    "-1-5/x",

    "simplify((x^7+2x^6+x^5)/(x^3*(x+1)^8))",
    "x^2/((x+1)^6)",

    "simplify((x^2+2*x+1)/((x+1)^8))",
    "1/((x+1)^6)",

    "simplify(((x^2-5x-14)/(x^2-3x+2))*((x^2-4)/(x^2-14x+49)))",
    "(x+2)^2/((x-7)*(x-1))",

    "simplify(((x^2-9)/(x^2+5x+6))/((3-x)/(x+2)))",
    "-1",

    "simplify((x^2+5x+4)/((x^2-1)/(x+5)))",
    "(x^2+9*x+20)/(x-1)",

    "simplify((x^2-6x-7)/(x^2-10x+21))",
    "(x+1)/(x-3)",

    "simplify((x^2+6x+9)/(x^2-9))",
    "(x+3)/(x-3)",

    "simplify((2x^2-x-28)/(20-x-x^2))",
    "(-2*x-7)/(x+5)",

    "simplify(((x^2+5x-24)/(x^2+6x+8))*((x^2+4x+4)/(x^2-3x)))",
    "(10+x+16/x)/(x+4)",

    "simplify(((x^2-49)/(2x^2-3x-5))/((x^2-x-42)/(x^2+7x+6)))",
    "(x+7)/(2*x-5)",

    "simplify(((x^2-2x-8)/(2x^2-8x-24))/((x^2-9x+20)/(x^2-11x+30)))",
    "1/2",

    "simplify((3/(x+1))/((x+4)/(x^2+11x+10)))",
    "3*(x+10)/(x+4)",

    "simplify((x^3+10x^2)/(x^2+6x-40))",
    "x^2/(x-4)",

    "simplify((x^2+18x+72)/(2x^2+11x-6))",
    "(x+12)/(2*x-1)",

    "simplify((x^2-3x-28)/(49-x^2))",
    "(-x-4)/(x+7)",

    "simplify((6x^2+13x+5)/(3x^2+26x+35))",
    "(2*x+1)/(x+7)",

    "simplify((-x^2+10x-9)/(-x^2+6x+27))",
    "(x-1)/(x+3)",

    "simplify((x-6)*(x^3+x^2-20x)/(x^4-12x^3+36x^2))",
    "(1+x-20/x)/(x-6)",

    # somewhat strange simplification but it's correct
    "simplify(((4x^3-x^2-3x)/(x^2-10x+25))*((10+3x-x^2)/(x^4-x^3)))",
    "(-4-6/(x^2)-11/x)/(x-5)",

    "simplify(((x^2+5x-24)/(x^2-5x+4))/((x^2+x-12)/(x-1)))",
    "(x+8)/(x^2-16)",

    "simplify(((6x^2+x^3-x^4)/(x^2-4))/((3x^3-9x^2)/(x^2+6x-16)))",
    "1/3*(-x-8)",

    "simplify(((3x^2+23x+14)/(x^2+4x+3))/((6x^2+13x+6)/(x^2+2x+1)))",
    "(x^2+8*x+7)/((x+3)*(2*x+3))",

    "simplify((5x^2-18x-8)/((x-4)/(x+6)))",
    "5*x^2+32*x+12",

    "simplify((2/(x+4))/((6x^3+17x^2)/(x^2+3x-4)))",
    "(x-1)/(x^2*(3*x+17/2))",

    # sum of rational expressions
    "simplify(((x^2+1)/(2x^2-4x+2))+((x)/((x-1)^2))+((-x-1)/(x^2-2x+1)))",
    "(x+1)/(2*(x-1))",

    "simplify((4/(6x^2))-(1/(3x^5))+(5/(2x^3)))",
    "(2/3*x^3+5/2*x^2-1/3)/(x^5)",

    "simplify((x/(x^2-2x+1))-(2/(x-1))+(3/(x+2)))",
    "(2*x^2-6*x+7)/(x^3-3*x+2)",

    "simplify((2x/(x^2-9))-(1/(x+3))-(2/(x-3)))",
    "-1/(x-3)",

    "simplify((4/(x+2)-(1/x)+1)*(x+2)*x)",
    "x^2+5*x-2",

    "simplify(((3/(x-4))+x/(2x+7))*(x-4)*(2x+7))",
    "x^2+2*x+21",

    "simplify((x/(x^2+12x+36))-((x-8)/(x+6)))",
    "(-x^2+3*x+48)/((x+6)^2)",

    "simplify(((x^2+14x+40)/(x^2+2x-8))*((x^2+5x-14)/(x^2+7x-30)))",
    "(x+7)/(x-3)",

    # could be cleaner but it's correct
    "simplify(1/(x^2-13x+42)+(x+1)/(x-6)-x^2/(x-7))",
    "1/(x-7)+x/(x-6)-x^2/(x-7)",

    "simplify((x+10)/((3x+8)^3)+x/((3x+8)^2))",
    "(3*x^2+9*x+10)/((3*x+8)^3)",

    "simplify(2/(3x^2)-1/(4x^7)+7/(6x^3))",
    "(2/3*x^5+7/6*x^4-1/4)/(x^7)",

    "simplify(2x/(x+9)-(x-1)/(x))",
    "-1+1/x+2*x/(x+9)",

    "simplify((x+1)/(x-1)+6/(x-7))",
    "(x^2-13)/((x-7)*(x-1))",

    "simplify(9/(x^2-4)-7x/(x^2-4x+4))",
    "(-7*x^2-5*x-18)/((x-2)^2*(x+2))",

    "simplify((2x+1)/(4x^2-3x-7)-(x+3)/(x+1)+x/(4x-7))",
    "x/(4*x-7)-x/(x+1)+2*(-5*x+11)/((4*x-7)*(x+1))",

    "simplify(simplify((2x+1)/(4x^2-3x-7)-(x+3)/(x+1)+x/(4x-7))*(x+1)*(4x-7))",
    "-3*x^2-2*x+22",

    # could be cleaner but it's correct
    "simplify(3/(6x-x^2)-x/(x^2-5x-6))",
    "(-3-x-3/x)/((x-6)*(x+1))",

    # could be cleaner but it's correct
    "simplify(3/(x^2)+(x+9)/(x^2+5x)-2/(x^2+10x+25))",
    "(15+x+75/(x^2)+75/x)/((x+5)^2)",

    "simplify(1/(x+1)-2/((x+1)^2)-3/((x+1)^3))",
    "(x^2-4)/((x+1)^3)",

  ]