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aljabar/node_modules/algebrite/tests/simplify.coffee

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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)",
]
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