# Pi Day 2022 at RelationalAI

To celebrate Pi day, let’s have some fun with π in our modeling language Rel.

We are excited to join the growing community of math lovers to celebrate International Pi Day (opens in a new tab). Pi Day is held on March 14 as its date, 3/14, resembles 3.14 — the first three digits of π.

To celebrate Pi Day, we’re going to demonstrate how π can be approximated in various ways in our modeling language Rel (opens in a new tab), ranging from infinite series expressions to Monte-Carlo simulations. If you make it to the end, a little challenge awaits where you can test your Rel skills.

## Using the Greek Letter π

Rel ships with a system-provided relation `pi_float64`

with the value of π as a Float64 number.

`def output = pi_float64`

Relation: `3.141592653589793`

This is great, but the first thing we want to do is to refer to π using the Greek letter `π`

, which is possible because Rel supports Unicode characters, which are a superset of ASCII.

`def π = pi_float64`

Now we have a relation named `π`

that refers to our Float64 number of π. To test that it works, let’s query for this new relation.

Query: `def`

`output`

`=`

`π`

Output: `3.141592653589793`

It works! Now let’s get to work.

## Best Approximation of π

First, let’s look which of the following approximations (`3.14`

, `3.14159`

, `22/7`

, or `355/113`

) comes closest to the actual irrational value of π.

To do this, we define a relation `pi`

that maps the name of the approximation to its actual value. We can think of `pi`

as a dictionary in other languages that maps a String name to its Float64 value.

Then we are asking the system to give us the name of the best approximation. In mathematical terms, we perform the following task:

In Rel it looks like this:

```
def pi["3.14"] = 3.14
def pi["22/7"] = 22/7
def pi["3.14159"] = 3.14159
def pi["355/113"] = 355/113
def best_approximation = argmin[abs[pi[i]-π] for i]
def output = best_approximation
```

Output: `"355/113"`

The most accurate approximation of the four options is `355/113`

. It’s not surprising that `3.14`

and `22/7`

are not the most accurate options, but it’s interesting to see that `3.14159`

is only the second best. It means that if you want to remember only six digits for π\piπ, the best option is `355/113`

.

## Computing π Using the Madhava-Leibniz Series

The Madhava-Leibniz series (opens in a new tab) is one of the most famous infinite series that involves π. This series has a long history that goes back as far as the 14th century. It reads:

In Rel, we can express this series very naturally,

```
@inline
def leibniz_term[k] = (-1)^k/(2*k+1)
@inline
def leibniz_formula[n] =
4*sum[leibniz_term[x]
for x in range[0, n, 1]]
```

and plot its convergence behavior.

```
// collect convergence data
module leibniz_data
def n_terms = 50
def terms[i] = i, range(0, n_terms-1, 1, i)
def pi_approx[i] = leibniz_formula[terms[i]]
def error[i] = pi_approx[i] - π
end
// plot data
def output = vegalite:plot[
vegalite:line[:terms, :error, {:data, leibniz_data}]
]
```

From the plot, we can see that the Madhava-Leibniz series oscillates around π\piπ and converges slowly towards the exact value.

## Calculate the Gaussian Integral

Another famous mathematical formula involving π\piπ is the Gaussian integral (opens in a new tab),

which is ubiquitous in many STEM areas including probability theory, statistics, and physics.

We can express this integral in Rel by discretizing the x axis and turning the integral into a sum x∫dx→∑Δx. We then use Rel’s aggregation capabilities to perform this sum.

```
// discretization parameters
def n = 100
def x_max = 10.0
def dx = 2*x_max / n
// x points
def x = -x_max + dx * range[0, n-1, 1]
// Gaussian
@inline
def gaussian[x] = natural_exp[-x^2]
// Gaussian Integral
def integral = sum[gaussian[i] for i in x] * dx
def pi = integral^2
def output = pi
```

Output: `3.1415926535897913`

By approximating the integral with 100 points, we can approximate π with a surprisingly high accuracy of around 10^{−15}.

## Sampling a Unit Circle with Monte Carlo

Our last example is another famous one. To estimate π, we can use a Monte Carlo simulation. The key idea is to approximate the area of a circle with a radius of 0.5 — which is π/4 — by placing random points in an enclosing unit square and counting the points that fall within the circle.

The ratio between the number of points within the circle, N_{circle}, and the total number of points placed, N_{total}, approximates the circle area of π/4.

In Rel, we can build this simulation with just a few lines of code:

```
@inline
def rand = random_threefry_float64
module sample
def size = 10^3
def x[i] = rand[i, 1] - 0.5, range(1, size, 1, i)
def y[i] = rand[i, 2] - 0.5, range(1, size, 1, i)
end
def points_in_circle(i) = sample:x[i]^2 + sample:y[i]^2 < 0.5^2
```

We’re using the Threefry pseudorandom number generator to place our random points. The module `sample`

contains all the information about our samples from the sample size `size`

to the `x`

and `y`

positions of the points. The relation `points_in_circle`

holds all the sample IDs where the point falls within the circle.

The only thing left is the calculation of the ratio N_{circle}/N_{total} which we multiply by four to get `π`

.

Now it’s your turn! Complete the following query, and find the definition for `ratio`

such that the relation `pi`

holds the approximate value of π.

```
// def ratio = ???
def pi = 4*ratio
def output = pi
```

Output: `3.164`

`def ratio = count[points_in_circle] / sample:size`

We hope you had fun playing around with π in Rel. Happy Pi Day from all of us at RelationalAI!

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