# Function definitions

Numbat comes with a large number of predefined functions, but it is also possible to add new functions. A function definition is introduced with the fn keyword:

fn max_distance(v: Velocity, θ: Angle) -> Length = v² · sin(2 θ) / g0


This exemplary function computes the maximum distance of a projectile under the influence of Earths gravity. It takes two parameters (The initial velocity v and the launch angle θ), which are both annotated with their corresponding physical dimension (their type). The function returns a distance, and so the return type is specified as Length.

## Type inference

The return type annotation may be omitted, but it is often desirable to add it for better readability of the code and in order to catch potential errors.

The parameter types can also (sometimes) be omitted, in which case Numbat tries to infer their type. However, this often leads to overly generic function signatures. For example, consider the following function to compute the kinetic energy of a massive object in motion:

fn kinetic_energy(mass, speed) = 1/2 * mass * speed^2


Without any type annotations, this function has an overly generic type where mass and speed can have arbitrary dimensions (and the return type is type(mass) * type(speed)^2). So for this case, it is probably better to add parameter and return types.

## Generic functions

Sometimes however, it is useful to write generic functions. For example, consider max(a, b) — a function that returns the larger of the two arguments. We might want to use that function with dimensionful arguments such as max(1 m, 1 yd). To define such a generic function, you can introduce type parameters in angle brackets:

fn max<T>(a: T, b: T) -> T = if a > b then a else b


This function signature tells us that max takes two arguments of arbitrary type T (but they need to match!), and returns a quantity of the same type T.

Note that you can perform the usual operations with type parameters, such as multiplying/dividing them with other types, or raising to rational powers. For example, consider this cube-root function

fn cube_root<T>(x: T^3) -> T = x^(1/3)


that can be called with a scalar (cube_root(8) == 2) or a dimensionful argument (cube_root(1 liter) == 10 cm).

Note: cube_root can also be defined as fn cube_root<T>(x: T) -> T^(1/3), which is equivalent to the definition above.

## Recursive functions

It is also possible to define recursive functions. In order to do so, you currently need to specify the return type — as the type signature can not (yet) be inferred otherwise.

For example, a naive recursive implementation to compute Fibonacci numbers in Numbat looks like this:

fn fib(n: Scalar) -> Scalar =
if n ≤ 2
then 1
else fib(n - 2) + fib(n - 1)