Leibniz Approximation of Pi

Mix.install([
  {:vega_lite, "~> 0.1.6"},
  {:kino_vega_lite, "~> 0.1.10"}
])

Introduction

Implementation of the Leibniz series for approximation of π.

In short, it states that:

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \ldots = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{2k + 1}$

Implementation

defmodule Pi do
  defp generate_minus(acc, current, last_step) when current > last_step do
    acc
  end

  defp generate_minus(acc, current, last_step) do
    generate_plus(acc - 1 / current, current + 2, last_step)
  end

  defp generate_plus(acc, current, last_step) when current > last_step do
    acc
  end

  defp generate_plus(acc, current, last_step) do
    generate_minus(acc + 1 / current, current + 2, last_step)
  end

  def generate(last_step) do
    4 * generate_minus(1, 3, last_step)
  end
end

Pi.generate(1_000_000_000)

Visualization

alias VegaLite, as: Vl

pi = :math.pi()

data =
  0..100
  |> Enum.map(fn value -> value * 2 + 1 end)
  |> Enum.map(fn value ->
    [
      %{"legend" => "Pi", "x" => value, "y" => pi},
      %{"legend" => "Approximation", "x" => value, "y" => Pi.generate(value)}
    ]
  end)
  |> List.flatten()

Vl.new(width: 660, height: 300)
|> Vl.data_from_values(data)
|> Vl.mark(:line)
|> Vl.encode_field(:x, "x", type: :temporal)
|> Vl.encode_field(:y, "y", type: :quantitative)
|> Vl.encode_field(:color, "legend", type: :nominal)