(x+y)^2;
x/(x+y)+y/(x-y);
a:=(x+y)^2;
(x-y)^2;
b:=ws;   % ws - Work Space
% By the way, this is a comment
a-b;

% Free and bound variables
% ------------------------
x:=z+1$ a;
x:=z-1$ a;
clear x; x; a;
x:=2*z$ a:=a;
clear x; a;
clear a,b; a;

x:=x+1; % this would lead to an infinite loop

% Switches
% --------
% mcd - Making Common Denominator
off mcd; x/(x+y)+y/(x-y);
on mcd; ws;

% exp - EXPansion
off exp; (x+y)^2;
on exp; ws;

% gcd - Greatest Common Divisor
(x^2-y^2)/(x-y); (x^3-y^3)/(x-y); (x^3-y^3)/(x^2-y^2);
on gcd; ws;
off gcd;

% Small variables
% ---------------
a:=(x+y+z)^6;
weight x=1,y=2$
% x has first order of smallness, y - second order
wtlevel 4$ % keep terms up to fourth order of smallness
a;
clear x,y,a;

% Elementary functions
% --------------------
sin(-x); cos(pi/4); sin(5*pi/6);
log(e^x); log(1); log(e);
sqrt(0); sqrt(12*x^2*y);
sqrt(x^2-2*x+1);

% Differentiation
% ---------------
f:=e^x*log(y)/sin(x);
df(f,x);           % derivative in x
df(f,x,2);         % second derivative in x
df(f,x,2,y);       % second derivative in x and first one in y

f:=y/x;
depend y,x;        % tell REDUCE that y depends on x
df(f,x);
nodepend y,x;      % remove this dependence
df(f,x);
clear f;

% Integration
% -----------
load_package algint;
% The package algint extends the integrator capabilities
int(1/x+1+x+x^2,x);
int(1/(x^4-1),x); df(ws,x);
int(x*log(x),x); int(x*sin(x),x); int(x^2*e^x,x);
u:=sqrt(x^2-a^2)$ int(u,x); int(1/u,x); clear u;
% This integral cannot be calculated in elementary functions
int(e^(x^2)/x,x);

% Definite integration
% --------------------
%load_package defint;
%int(sin(x)/x,x,0,infinity);
%int(sqrt(x)*e^(-x),x,0,infinity);

% Numerical calculations
% ----------------------
a:=123456789/987654321; % common divisor is cancelled
a^4;          % REDUCE works with arbitrarily long integers
on rounded;   % rounded or real numbers
precision 30; % 30 digits precision
a:=123456789/987654321; a^4;
pi; e;
a:=sin(pi/4); a^2;
precision 12$

% Numerical integration
% ---------------------
load_package numeric;

% This command shows how much CPU time have been used
% since its previous call (or since REDUCE started)
showtime;

% the space between -10 and .. is essential
a:=num_int(exp(-x^2),x=(0 .. 10)); 4*a^2/pi;
showtime;

a:=num_int(exp(-x^2-y^2),x=(0 .. 10),y=(0 .. 10)); 4*a/pi;
showtime;

clear a; off rounded;

% Plots
% -----

% single function
plot(sin(x),x=(-pi..pi));

% several functions
plot(sin(x),x-x^3/6,x-x^3/6+x^5/120,x=(-pi..pi),
    xlabel="x",ylabel="f(x)",title="sin and its Taylor series");

% points
plot({{0,0},{0.1,0.1},{0.2,0},{0.3,0.2},{0.4,0}});

% parametric
plot(point(e^(0.1*t)*cos(t),e^(0.1*t)*sin(t)),t=(-10 .. 10),
    points=500);

% implicit
plot(x^2+y^2-1=0,x=(-1 .. 1),y=(-1 .. 1));

% contour
plot(x^2+y^6={0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},
    x=(-1 .. 1),y=(-1 .. 1));

% surface
plot(sin(sqrt(x^2+y^2))/sqrt(x^2+y^2),
    x=(-10 .. 10),y=(-10 .. 10),hidden3d,points=40);

% space curve
a:=0.1$
plot(point(cos(t),sin(t),a*t),t=(-10 .. 10),points=100);
clear a;

% parametric surface
plot(point(sin(u)*cos(v),sin(u)*sin(v),cos(u)),
    u=(0 .. 2*pi),v=(0 .. 2*pi),points=25);

% implicit surface
plot(x^2+y^2+z^2-1=0,x=(-1 .. 1),y=(-1 .. 1),z=(-1 .. 1),
    points=40);

% Controlling output form
% -----------------------
w:=(x^2*(y^2+2*y)+x*(y^2+z)/(2*a))/(y+2);

% revpri - REVerse PRInt - in increasing degrees order
on revpri; w; off revpri;

order y,x;
% order of variables - y, x, and then all the other ones
w;
order nil; % cancel the order command

% allfac - ALL FACtors - put all simple factors out of brackets
off allfac; w; on allfac;

% div - DIVide all terms of the numerator
% by simple factors of the denominator
on div; w; off div;

% factor - output expressions as powers of x with coefficients
factor x; w;
% rat - RATional - divide all such parts of the numerator by the
% denominator, producing powers of x with rational coefficients
on rat; w;
on div; w;
on revpri; w;
off rat,div,revpri;
remfac x; % cancel the factor command

% nat - NATural mathematical form output:
% powers are raised, fractions: numerator over denominator
off nat; w;   % REDUCE will be able to read such an expression
write "Abcd"; % any output is terminated by $
on nat; clear w;

% Case sensitivity
% ----------------
x+X; % REDUCE does not distinguish small and capital letters
input_case nil$ % makes REDUCE case sensitive
x+X;

end;