Dual Number on Elixir

dual_number.ex

import Kernel, except: [+: 1, +: 2, -: 1, -: 2, *: 2, /: 2]
defprotocol Arithmetic do
  def left + right
  def left - right
  def left * right
  def left / right
  def +value
  def -value
end

defimpl Arithmetic, for: Integer do
  def left + right, do: Kernel.+(left , right)
  def left - right, do: Kernel.-(left , right)
  def left * right, do: Kernel.*(left , right)
  def left / right, do: Kernel./(left , right)
  def +value, do: Kernel.+(value)
  def -value, do: Kernel.-(value)
end

defimpl Arithmetic, for: Integer do
  def left + right, do: Kernel.+(left , right)
  def left - right, do: Kernel.-(left , right)
  def left * right, do: Kernel.*(left , right)
  def left / right, do: Kernel./(left , right)
  def +value, do: Kernel.+(value)
  def -value, do: Kernel.-(value)
end

defmodule DualNumber do
  import Arithmetic
  defstruct a: 0, b: 0

  def conj(d) do
    %DualNumber{ a: d.a, b: -d.b}
  end

  defimpl Inspect do
    def inspect(%DualNumber{a: a, b: b}, _opts) do
      "#{a}+#{b}ε"
    end
  end
end

defimpl Arithmetic, for: DualNumber do
  def left + right do
             %DualNumber{ a: Kernel.+(left.a, right.a),
                          b: Kernel.+(left.b, right.b) }
  end

  def left - right do
             %DualNumber{ a: Kernel.-(left.a, right.a),
                          b: Kernel.-(left.b, right.b) }
  end

  def left * right do
    %DualNumber{ a: Kernel.*(left.a, right.a),
                 b: Kernel.+(Kernel.*(left.a, right.b), Kernel.*(left.b, right.a)) }

  end

  def left / right do
    (left * %DualNumber{a: right.a, b: Kernel.-(right.b)}) / %DualNumber{a: Kernel.*(right.a, right.a), b: 0}
  end

  def +value, do: value
  def -value, do: %DualNumber{a: Kernel.-(value.a), b: Kernel.-(value.b)}
end

defmodule Test do
  import Arithmetic
  import DualNumber
  def f(x) do
    (%DualNumber{a: 4, b: 0} * x + %DualNumber{a: 3, b: 0}) * x + %DualNumber{a: 2, b: 0} 
  end

  def df(x) do
    %DualNumber{a: 8, b: 0} * x + %DualNumber{a: 3, b: 0}
  end

end