Building Expressions

Supported operators

• :plus
• :minus
• :times
• :divided_by
• :raised_to

Supported functions

• :log10
• :log
• :log2
• :cos
• :sin
• :tan

Supported constants

• :e
• :pi The atom :e represents Euler's number. :pi represents the mathemetical constant π.

Examples

Representing 2x

iex> Expression.new(2, :times, :x) |> IO.puts
(2 * x)

Representing 2*cos(x^2)

iex> Expression.new(2, :times, Expression.new(:cos, Expression.new(:x, :raised_to, 2))) |> IO.puts
(2 * cos((x ^ 2)))

Setting variables

If your expression contains variables, you can give those variables values with the Expression.set_variable function.

Examples

iex> Expression.new(:sin, Expression.new(:x, :raised_to, 2)) |> Expression.set_variable(:x, :pi) |> IO.puts
sin((pi ^ 2))

Evaluating expressions

If an expression does not contain any variables, the expression can be evaluated.

Examples

iex> Expression.new(:sin, Expression.new(:x, :raised_to, 2)) |> Expression.set_variable(:x, :pi) |> Expression.evaluate |> IO.puts
-0.43030121700009166

Simplifying expressions

The Expression.Simplifier module allows for basic simplifications like 0 * x => 0, x^1 => x. As of now, expressions like 2x + x don't get simplified to 3x.

Examples

Simplifying 1^1^1:

iex> Expression.new(1, :raised_to, Expression.new(1, :raised_to, 1)) |> Expression.Simplifier.simplify |> IO.puts
1.0

Computing derivatives

Examples

Computing the derivative of cos(sin(x)) with respect to x and simplifying the result:

iex> Expression.new(:cos, Expression.new(:sin, :x)) |> Expression.Differentiator.differentiate(:x) |> Expression.Simplifier.simplify |> IO.puts
((-1 * sin(sin(x))) * cos(x))
:ok

Computing Taylor/Maclaurin series

The Expression.TaylorSeries module allows you to compute Taylor and McLaurin series of functions.

Examples

First five terms for the Maclaurin series of sin(x):

iex> Expression.new(:sin, :x) |> Expression.TaylorSeries.compute_maclaurin_series(:x, 5) |> IO.puts
((((sin(x) + (cos(x) * x)) + (((-1 * sin(x)) / 2) * (x ^ 2))) + (((-1 * cos(x)) / 6) * (x ^ 3))) + (((-1 * (-1 * sin(x))) / 24) * (x ^ 4)))

Future work

• Add more functions (and the corresponding differentiation rules).
• Add simplifications rules for more complicated rules like a*x + b*x => (a+b)*x or maybe even sin(2pi*x) => 0
• Write unit tests.