using Symata;

(x+y)^3

Julia()

length(zeros(10)) ==  10   # A Julia expression

isymata()   # we enter Symata mode again

Cos(π * x)

x = 1/3

Out(6)

x = 1/6

Out(6)

Clear(x)

Out(6)

x = 1
a := x
b = x
c = a
d := a

[x,a,b,c,d]

ClearAll(x)

[x,a,b,c,d]

(a = z, [x,a,b,c,d])

ClearAll(x,a,b,c,d)

[a,b] = [x,y]

a

b

[a,b]

[a,b] = [b,a]

[a,b]

Map(Head, [x, x + y, [x,y], Cos(x), f(x), 3, 3.0, BI(3), BF(3)])  # apply the fun

expr = Expand((x+y)^3)

FullForm(expr)   # This shows the internal form. The tree is explicit

expr[2,2,1]   # Return a part of the expression by index into the tree

expr[2,2,1] = z;  # Replace a part of the expression

expr

Part(expr,2,2,1)  # You can do the same thing with Part

Expand((x+y)^3)[4,1]   # You can get parts of expressions directly

expr = Expand((x+y)^20);

expr[14:18:2]  # Parts 14 through 18 with step 2

ClearAll(expr)

headargs(f_(args__)) := [f,args]

headargs(a + b^2 + 3)

Integrate(f(x),x)

headargs(Integrate(f(x),x))

ClearAll(a,b,c,d)

rotheadargs(f_(args__)) := (Last([args])(f,Splat(Most([args]))))

rotheadargs( a + b + c + d)

rotheadargs( a + b + c + d + g(x))

ClearAll(x,a,b,c,d)  # delete definitions from the previous example

a = 1

Definition(a)

a := x

Definition(a)  # This overwrites the previous definition

f(x_) := x^2
f(x_, y_) := x + y
Definition(f)

Definition(f)

ClearAll(f,a)

Timing((Range(10^6), Null ) )   # time a single expression

Time(True)  #  toggle timing all expressions. returns the previous value

Range(10^6);

Time(False);

Trace(True);  # Trace evaluation

(a+b)*(a+b)

Trace(False);

? LeafCount

LeafCount(Expand((a+b)^3))

ByteCount(Expand((a+b)^3))

? Depth

Depth(Expand((a+b)^3))

FullForm(Expand((a+b)^3))  # Examine the tree

Expand((a+b)^3)[2,2,1]    # One of the deepest parts

Integrate( (1+x^2)^(-1), x)

expr = 1/(1+x^2)
Integrate(expr, x)

f(x_) := 1/(1+x^2)
Integrate(expr, x)

g(x_) = expr # Note we do not use ":="

ClearAll(expr)  # We did not use SetDelay, so we can delete expr

Integrate(g(y),y)

ClearAll(f,g,expr)

Integrate(f(y), y)  # The integral can no longer be reduced

MatchQ(z,_)

Map(MatchQ(_), [1,"string", a+b, 1/3])  # MatchQ does Currying with the first argument

FullForm(_Integer)  # underscore is shorthand for Blank

MatchQ(1, _Integer)

myintq = MatchQ(_Integer);

Map(myintq, Range(1/2,5,1/2))

MatchQ(b^2, _^2)  # Match power with exponent equal to 2

MatchQ(b^3, _^_)   # Match any power

MatchQ((b+c)^3, _^_)

MatchQ(b^1, _^_)

Map(MatchQ(f(x_^2)), [f(b^2), f(b^3), g(b^2)])

Map( MatchQ(_gg), [gg(x+y), gg(x), g(x)])

m = MatchQ(_Integer:?(Positive))

Map(m, [1,100, 0, -1, 1.0, x])

m = MatchQ(Condition([x_, y_], x < y))

[ m([2,1]), m([1,2]), m([1,2,3]), m(1)]

m = MatchQ(_Integer | _String)

[m(1), m("zebra"), m(1.0)]

MatchQ([a,a,a,b], [Repeated(a),b])

MatchQ([b], [RepeatedNull(a),b])

ClearAll(m)

f([x_,y_]) => p(x+y)

expr = f([x+y,y]) + f(c) + g([a,b])

ReplaceAll( expr, f([x_,y_]) => p(x+y))

ReplaceAll([a/b, 1/b^2, 2/b^2] , b^n_ => d(n))

ClearAll(expr)

ReplaceAll( [b,a,[a,b]] , [x_,y_,[x_,y_]] => 1 )  # This does not match

ReplaceAll( [a,b,[a,b]] , [x_,y_,[x_,y_]] => 1 )  # This does match

 ReplaceAll( [a, b, c, d, a, b, b, b],  a | b => x)

[1,2,Sequence(a,b)]

f(x_, x_ | y_String) := [x,y]

f(2,2)  # `y` does not match, so it is removed.

f(2,"cat")  # Here the second Alternative matches

f(2,3)      # Here the Pattern fails to match.

( h(a | b) := p, [h(a), h(b), h(c), h(d)] )

 Replace(1 + a + f(a) + g(f(a)), a => b, 2)

 Replace(1 + a + f(a) + g(f(a)), a => b, [2]) == 1 + a + f(b) + g(f(a))

ReplaceAll( [x,x,x,x,x],  x  => RandomReal() )

ReplaceAll( [x,x,x,x,x],  RuleDelayed(x ,RandomReal()))

Replace([1, 7, "Hi", 3, Indeterminate], Except(_:?(NumericQ)) => 0, 1)

 ReplaceRepeated(x^2 + y^6 , List(x => 2 + a, a => 3))

 ReplaceAll( b^c, a::(_^_) => g(a))

f(x_, y_:3) := x + y

[f(a+b,z), f(a+b)]

ClearAll(f)

f(x_) :=  Condition(x^2, x > 3)

[f(4),f(3)]



ReplaceAll( z*y + b , x_ + y_ => x * y )

ReplaceAll( z*y + b +c , x_ + y_ => x * y)

ClearAll(f,h,a,b,x,y)

mylog = :( log )

mylog(2,2)

f = :( (x,y)  ->  x^2 + y^2 )

f(3.0,4.0)

[f(3,4), f(3, 1/2)]

(a = Range(0.0,100.,.01), ccossq = :( x -> cos(x)^2 ), cossq(x_) := cos(x)^2);

Time(True);

Map(cossq, a);

Map(cossq, a);

Map(ccossq, a);

Map(ccossq, a);

(Time(False), ClearAll(f,a,ccossq,cossq,mylog))

expr = Integrate( x^2 * Exp(x)* Cos(x), x)

expr = Collect(expr, Exp(x))

cexpr = Compile(expr)
a = Range(0.0,10.0,.01);

Timing((Map(cexpr, a), Null))

Timing((Map(cexpr, a), Null))

ClearAll(a,expr,cexpr)