
Use("testers.mpi");


DoNext(_string) <--
[
 NextTest("<font color=0000ff>" : string : "</font>");
 NewLine();
];

Echo({"<HTML><BODY BGCOLOR=\"ffffff\"><PRE><font size=4>"});

Echo({"An assorted selection of the tests found in the Wester benchmark"});

StartTests();

/*
*/

DoNext("Compute 50!");
BenchShow(50!);

DoNext("Compute the prime decomposition of 6!.");
BenchShow(ans:=Factors(6!));



Echo({"This list contains lists of two elements. This list should
be interpreted as "});
BenchShow(PrettyForm(FW(ans)));

DoNext("Compute 1/2 + ... + 1/10.");
BenchShow(Sum(i,2,10,1/i));

DoNext("Compute a numerical approximation of e^(Pi*sqrt(163)) to 50 digits.");
BenchCall(BuiltinPrecisionSet(50));
BenchShow(N(Exp(Pi*Sqrt(163))));

DoNext("Compute an infinite decimal representation of 1/7");
BenchShow(Decimal(1/7));

DoNext("Compute the first terms of the continued fraction of Pi.");
BenchShow(PrettyForm(ContFrac(Internal'Pi())));

DoNext("Simplify sqrt(2*sqrt(3)+4).");
BenchShow(RadSimp(Sqrt(2*Sqrt(3)+4)));

DoNext("Simplify sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))).");
BenchShow(RadSimp(Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))));

DoNext("Simplify 2*infinity-3.");
BenchShow(2*Infinity-3);

Echo({"Infinity is also defined for comparisons like a &lt Infinity"});

DoNext("Compute the normal form of (x^2-4)/(x^2+4x+4).");
BenchShow(PrettyForm(GcdReduce((x^2-4)/(x^2+4*x+4),x)));

DoNext("Expand (x+1)^5, then differentiate and factorize.");
BenchCall(ans:=Factors(D(x)Expand((x+1)^5)));
BenchShow(PrettyForm(FW(ans)));

DoNext("Simplify sqrt(997) - (997^3)^(1/6).");
BenchShow(RadSimp(Sqrt(997) - (997^3)^(1/6)));

DoNext("Simplify sqrt(999983) - (999983^3)^(1/6).");
BenchShow(RadSimp(Sqrt(999983) - (999983^3)^(1/6)));

DoNext("Recognize that (2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3)) - 6 is 0.");
BenchShow(RadSimp( (2^(1/3)+4^(1/3))^3-6*(2^(1/3)+4^(1/3)) - 6));


DoNext("Simplify log e^z into z only for -Pi < Im(z) <= Pi.");
BenchShow(Simplify(Ln(Exp(z))));

DoNext("Invert the 2x2 matrix [[a,b],[1,ab]]. ");
BenchCall(A:={{a,b},{1,a*b}});
BenchShow(ans:=Inverse(A));
BenchShow(TableForm(Simplify(ans)));

DoNext("Find the eigenvalues of the matrix [[5, -3, -7],[-2, 1, 2],[ 2, -3, -4]].");
BenchCall(A:={{5,-3,-7},{-2,1,2},{2,-3,-4}});
BenchShow(EigenValues(A));


DoNext("Compute the limit of (1-cos x)/x^2 when x goes to zero. ");
BenchShow(Limit(x,0)(1-Cos(x))/(x^2));

DoNext("Compute the derivative of |x|.");
BenchShow(D(x)Abs(x));

DoNext("Compute an antiderivative of |x|.");
BenchShow(AntiDeriv(x,Abs(x)));

DoNext("Compute the derivative of |x| (piecewise defined).");
BenchShow(D(x)(if (x<0) (-x) else x));

DoNext("Compute the antiderivative of |x| (piecewise defined).");
BenchShow( AntiDeriv(x,if (x<0) (-x) else x));

DoNext("Compute the first terms of the Taylor expansion of 1/sqrt(1-v^2/c^2) at v=0. ");
BenchCall(ans:=Taylor(v,0,4)Sqrt(1/(1-v^2/c^2)));
BenchCall(ans:=Simplify(ans));
BenchShow(PrettyForm(ans));

DoNext("Compute the inverse of the square of the above expansion. ");
BenchCall(ans:=Taylor(v,0,4)(1/ans)^2);
BenchShow(PrettyForm(Simplify(ans)));



DoNext("Compute the Taylor expansion of tan(x) at x=0 by dividing the expansion of sin(x) by that of cos(x). ");
BenchCall(ans1:=Taylor(x,0,5)(Sin(x)/Cos(x)));
BenchCall(PrettyForm(ans1));
BenchCall(ans2:=Taylor(x,0,5)Tan(x));
BenchCall(PrettyForm(ans2));
BenchShow(ans1-ans2);


DoNext("Compute the Legendre polynomials directly.");

BenchCall(10 # Legendre(0,_x) <-- 1);
BenchCall(20 # Legendre(n_IsInteger,_x) <--
[
 Local(result);
 result:=[
   Local(x);
   Expand((1/(2^n*(n!))) * Deriv(x,n)Expand((x^2-1)^n,x));
 ];
 Eval(result);
]);

BenchCall(ForEach(item,Table(Legendre(i,x),i,0,4,1))PrettyForm(item));


DoNext("Compute the Legendre polynomials recursively, using their recurrence of order 2. ");

BenchCall(10 # LegendreRecursive(0,_x) <-- 1);
BenchCall(20 # LegendreRecursive(1,_x) <-- x);
BenchCall(30 # LegendreRecursive(n_IsPositiveInteger,_x) <--
     Expand(((2*n - 1)*x*LegendreRecursive(n-1,x)-(n - 1)*LegendreRecursive(n-2,x))/n));

BenchCall(ForEach(item,Table(LegendreRecursive(i,x),i,0,4,1))PrettyForm(item));

DoNext("Evaluate the fourth Legendre polynomial at 1. ");
BenchShow(Legendre(4,1));

DoNext("Define the polynomial p = sum( i=1..5, ai*x^i ). ");
BenchCall(ans:=Sum(MakeVector(a,5)*(FillList(x,5)^(1 .. 5))));
BenchCall(PrettyForm(ans));

DoNext("Apply Horner\'s rule to the above polynomial.");
BenchCall(ans:=Sum(MakeVector(a,5)*(FillList(x,5)^(1 .. 5))));
BenchCall(PrettyForm(Horner(ans,x)));


DoNext("Compute the first terms of the continued fraction of Pi. ");
BenchCall(pi:=N(Pi,20));
BenchCall(a:=ContFrac(pi,6));
BenchCall(PrettyForm(a));

DoNext("Compute an infinite decimal representation of 1/7. ");
BenchShow(Decimal(1/7));
Echo({"This result means that the decimal expansion of 1/7 is
0.142857142857142...."});

DoNext("Evaluate TRUE and FALSE. ");
BenchShow(True And False);

DoNext("Solve the equation tan(x) = 1. ");
BenchShow(Solve(Tan(x)==1,x));

DoNext("Revert the Taylor expansion of sin(y) + cos(y) at y=0. ");
BenchCall(t:=InverseTaylor(y,0,6)Sin(y)+Cos(y));
BenchCall(PrettyForm(t));
Echo({"And check that it is in fact the inverse up to degree 5:"});
BenchCall(s:=Taylor(y,0,6)Sin(y)+Cos(y));
BenchShow(BigOh(Subst(y,s)t,y,6));

DoNext("Solve the linear (dependent) system x+y+z=6,2x+y+2z=10,x+3y+z=10.");
BenchShow(OldSolve({x+y+z==6,2*x+y+2*z==10,x+3*y+z==10},{x,y,z}));



DoNext("Evaluate True And False");
BenchShow(True And False);
BenchShow(CanProve(True And False));

DoNext("Simplify x or (not x)");
BenchShow(CanProve(x Or Not x));

DoNext("Simplify x or y or (x and y)");
BenchShow(CanProve(x Or y Or (x And y)));

Echo({curline," examples done"});

Echo({"</FONT></PRE></BODY></HTML>"});
