# System library

haku comes with a set of built-in functions, called the _system library._
This is a reference for these functions.

## Preamble: how to read this reference

Each function comes with a _signature description_.
These descriptions read like this:

```haku
stroke
  thickness : number
  color : rgba
  position : vec
  -> scribble
```

The first element is always the function's name - in this case `stroke`.
Following this are `argumentName : argumentType` pairs, which describe the arguments that need to be passed to the function.
The last element is always the type of data this function produces.

Function names which are made up of symbols instead of letters are _operators_.
Operators may have one or two arguments, where one argument corresponds to a prefix form `-x`, and two arguments correspond to an infix form `x - y`.
Note that this documentation lists a unary and binary operator of the same spelling as _two separate functions_, not overloads of a single function.

The argument name usually does not matter when calling the function - it is only used for documentation purposes.
The one exception is arguments called `...`, which signify that zero or more arguments can be passed to the function at that position.
(Currently there are no functions that accept any number of arguments, though.)

The argument _type_ however is important.
If you try to use a function with the wrong type of value as its argument, it will fail with an error.
For example, consider a brush where we pass a number as `stroke`'s `color` and `position` arguments.

```haku
stroke 1 1 1
```

This brush will fail to render, since `stroke` expects an `rgba` as its 2nd argument.

With that said, there are several types of values in haku that can be passed into, and returned by functions.

- `_` - special type used to signify that any value may be passed into the function.
- `()` - also known as _nil_, means _no value._
- `boolean` - either `False` or `True`. Indicates truth or falsehood, used in `if` conditions.
- `number` - a real number, with 32 bits of precision.
- `vec` - a 4-dimensional vector, composed of four `number`s.
- `rgba` - an RGBA color, composed of four `number`s.
- `\a -> r` - a function taking in the parameter `a` and returning `r`, as returned by `\x -> x` literals.
- `list t` - a list of values, where each value is of the type `t`.
- `shape` - a mathematical shape.

  - `shapeLike` - anything that can be turned into a `shape` using `toShape`.

- `scribble` - something that can be drawn on the wall.
- `reticle` - an interaction the user can make with the wall.

The syntax `a | b` may be used to signify that one of the listed types is accepted or returned. 

The following syntax:

```haku
{
  A : t
  B : u
}
```

is used for _records_---functions which take in a tag as a parameter, and return a value depending on what tag was passed.
In the case above, if `A` is passed in a value of type `t` is returned, and if `B` is passed in a value of type `u` is returned.
Passing in any other tag is an error.


## Math

```haku
-
  a : number
  -> number

-
  a : vector
  -> vector
```

`-`, when used in its unary form `-x`, returns the number `x` with the opposite sign.
When used on vectors, returns the same vector facing the reverse direction (the individual components are negated.)

This operation is not defined for colors, because it doesn't make sense to have a color with negative RGBA values.

```haku
+
  a : number
  b : number
  -> number

+
  a : vector
  b : vector
  -> vector

+
  a : rgba
  b : rgba
  -> rgba
```

`+` adds two numbers, vectors, or colors together.

```haku
-
  a : number
  b : number
  -> number

-
  a : vector
  b : vector
  -> vector

-
  a : rgba
  b : rgba
  -> rgba
```

`-`, when used in its binary form `x - y`, subtracts two numbers, vectors, or colors from one another.

```haku
*
  a : number
  b : number
  -> number

*
  a : vector
  b : vector
  -> vector

*
  a : rgba
  b : rgba
  -> rgba
```

`*` multiplies two numbers together.
When used on vectors, it scales them component-wise.
Likewise for colors.

```haku
/
  a : number
  b : number
  -> number

/
  a : vector
  b : vector
  -> vector

/
  a : rgba
  b : rgba
  -> rgba
```

`/` divides a number by another number.
When used on vectors, it divides them component-wise.
Likewise for colors.

```haku
floor
  x : number
  -> number

ceil
  x : number
  -> number

round
  x : number
  -> number
```

`floor`, `ceil`, and `round` are rounding functions.
Each of them rounds a little differently:

- `floor` rounds towards -∞.
- `ceil` rounds towards +∞.
- `round` rounds half towards +∞.

```haku
abs
  x : number
  -> number
```

`abs` returns the absolute value of the given number.

If `x` is less than zero, returns `-x`.
Otherwise returns `x`.

```haku
mod
  x : number
  y : number
  -> number
```

`mod` is the modulo operation.

It returns the remainder of dividing `x` by `y`.
haku uses the Euclidean definition, which means the remainder returned by `mod` is _always non-negative_.

```haku
pow
  base : number
  exponent : number
  -> number
```

`pow` raises `base` to the given `exponent`.

```haku
sqrt
  x : number
  -> number

cbrt
  x : nunber
  -> number
```

`sqrt` returns the square root of the given number.

`cbrt` returns the cubic root of the given number.

Other roots may be obtained using `pow x (1 / base)`.

```haku
exp
  x : number
  -> number

ln
  x : number
  -> number
```

In the following functions, `e` is the base of the natural logarithm (approximately `2.7128`.)
The `e` constant (Euler's number) is currently not exposed to haku source code; you have to define it yourself.

`exp` is the exponential function `pow e x`.

`ln` is the natural logarithm (logarithm base `e` of `x`.)

```haku
expMinus1
  x : number
  -> number

ln1Plus
  x : number
  -> number
```

`expMinus1` is `pow e x - 1`, but accurate even when `x` is close to zero.

`ln1Plus` is `ln (1 + x)`, but more accurate than if the operations are performed separately.

```haku
exp2
  x : number
  -> number
```

`exp2` is the exponential function `pow 2 x`.

```haku
log2
  x : number
  -> number

log10
  x : number
  -> number
```

`log2` is the logarithm base `2` of `x`.

`log10` is the logarithm base `10` of `x`.

```haku
hypot
  x : number
  y : number
  -> number
```

`hypot` is the hypotenuse of the Pythagorean triangle with right angle-adjacent sides `x` and `y`.

```haku
sin
  x : number
  -> number

cos
  x : number
  -> number

tan
  x : number
  -> number
```

`sin`, `cos`, and `tan` are the [trigonometric functions][] sine, cosine, and tangent.
Their argument `x` is counted in radians.

[trigonometric functions]: https://wikipedia.org/Trigonometric_functions

```haku
asin
  x : number
  -> number

acos
  x : number
  -> number

atan
  x : number
  -> number
```

`asin`, `acos`, and `atan` are the [inverse trigonometric functions][] arc sine, arc cosine, and arc tangent.
Their argument `x` is counted in radians.

[inverse trigonometric functions]: https://wikipedia.org/Inverse_trigonometric_functions

```haku
atan2
  y : number
  x : number
  -> number
```

`atan2` is the angle between the positive X axis and a line that passes through `(0, 0)` and `(x, y)`.

Note the reverse argument order---`y` comes first, due to `atan2` being a convenience function over `atan (y / x)` that is defined for all arguments.

```haku
sinh
  x : number
  -> number

cosh
  x : number
  -> number

tanh
  x : number
  -> number

asinh
  x : number
  -> number

acosh
  x : number
  -> number

atanh
  x : number
  -> number
```

`sinh`, `cosh`, `tanh`, `asinh`, `acosh`, and `atanh`, are the six [hyperbolic functions][].

[hyperbolic functions]: https://en.wikipedia.org/wiki/Hyperbolic_functions


## Logic

The following functions are used to compare values and work with `boolean`s.

```haku
!
  a : _
  -> boolean
```

If `b` is `()` or `False`, `not` returns `True`.
Otherwise it returns `False`.

```haku
==
  a : _
  b : _
  -> boolean

!=
  a : _
  b : _
  -> boolean
```

`==` returns `True` if `a` and `b` are equal.
Whether two values are considered equal depends on their type:

- If the type of the two values differs, `False` is returned.
- If the two values are `number`s:

  - If any of the values are `NaN`, `False` is returned.
  - Otherwise `True` is returned if the two numbers have the exact same bit representation.

- If the two values are `vec`s, `True` is returned if each of their `number` components is equal to each other using the rules above.
- Likewise with `rgba`s.
- All other types of values use _reference_ equality - `True` is returned only if `a` and `b` are located in the same place in memory.
  This more or less means that the values are considered equal if they are produced by the same call to a system function, in time.

`!=` returns `!(a == b)`.

```haku
<
  a : _
  b : _
  -> boolean

<=
  a : _
  b : _
  -> boolean

>
  a : _
  b : _
  -> boolean

>=
  a : _
  b : _
  -> boolean
```

`<` returns `True` if `a` is less than `b`, and `<=` returns `True` if `a` is less than _or_ equal to `b`.

Order is only well-defined for numbers.
Other types may assume an arbitrary but consistent ordering - `()` may be less than `True`, or it may not be less than `True`, but this will not change between executions of the program.

`a > b` is the same as `b < a`.
`a >= b` is the same as `b <= a`.

---

Note that `and` and `or` are currently missing from this list, but are reserved keywords.
You can implement them using regular functions as a replacement.

```haku
boolAnd = \a, b ->
  if (a)
    if (b) True
    else False
  else False

boolOr = \a, b ->
  if (a)
    True
  else
    if (b) True
    else False
```


## Vectors

```haku
vec
  x : number
  -> vec

vec
  x : number
  y : number
  -> vec

vec
  x : number
  y : number
  z : number
  -> vec

vec
  x : number
  y : number
  z : number
  w : number
  -> vec
```

Creates a new `vec` from one to four number values.

A `vec` always has four dimensions.
If any of the arguments are omitted, its corresponding dimension is initialized to zero.

```haku
vecX
  v : vec
  -> number

vecY
  v : vec
  -> number

vecZ
  v : vec
  -> number

vecW
  v : vec
  -> number
```

`vecX`, `vecY`, `vecZ`, and `vecW` extract the individual components of a `vec`.

---

## Colors

```haku
rgba
  r : number
  g : number
  b : number
  a : number
  -> rgba
```

Creates a new `rgba` with the given color channels.
Note that unlike `vec`, all color channels have to be provided to form an `rgba`.

```haku
rgbaR
  color : rgba
  -> number

rgbaG
  color : rgba
  -> number

rgbaB
  color : rgba
  -> number

rgbaA
  color : rgba
  -> number
```

`rgbaR`, `rgbaG`, `rgbaB`, `rgbaA` extract color channels out of an `rgba`.

---

haku uses RGBA values in a normalized `0` to `1` range rather than `0` to `255`, which may be unfamiliar if you're coming from other image editing software.
This is because it's easier to do math on normalized colors.
For example, consider multiplicatively blending two colors.

```haku
-- This is how you can multiply two colors together.
-- Note that the `*` operator works for colors, so you don't need to define this in your brushes.
mulRgba = \a, b ->
  rgba (rgbaR a * rgbaR b) (rgbaG a * rgbaG b) (rgbaB a * rgbaB b) (rgbaA a * rgbaA b)
```

If haku represented colors using an 8-bit `0` to `255` range instead, to multiply two colors together, you would have to divide them by `255` to get them back into the correct range.

```haku
-- NOTE: This example does NOT work correctly.
mulRgba = \a, b ->
  let red = (rgbaR a * rgbaR b) / 255
  let green = (rgbaG a * rgbaG b) / 255
  let blue = (rgbaB a * rgbaB b) / 255
  let alpha = (rgbaA a * rgbaA b) / 255
  rgba red green blue alpha
```

Note that haku does not clamp colors to the `0` to `1` range.
It is perfectly valid to have a color that is out of range or even `NaN`, but when drawing scribbles:

- any number less than `0` is clamped to `0`.
- any number greater than `1` is clamped to `1`.
- `∞` is clamped back to `1`.
- `-∞` is clamped back to `0`.
- any scribble with a `NaN` color is ignored.

Before scribbles are drawn to the wall, colors are converted to 8-bit integers for more efficient rasterization and storage.
This means some loss of precision will happen, which may cause issues with brushes like this one:

```haku
stroke 128 #00000004 (vec 0 0)
```

If you try to to use this brush to fill up a single spot with black, you will notice that despite all the math suggesting so, the color will end up gray instead.


## Lists

```haku
len
  l : list _
  -> number
```

Returns the number of elements in the given list (its `len`gth.)

```haku
index
  l : list t
  i : number
  -> t
```

Returns an element of the list, located at the given index `i`.
The first element is located at index 0.

Indexing out of bounds is an error.

```haku
range
  min : number
  max : number
  -> list number
```

Generates a list of integers from `min` to `max` (inclusive).

- When `max > min`, the integers go in an increasing order.
- When `min > max`, the integers go in a decreasing order.

Note that this function consumes fuel proportional to the amount of elements in the generated list.

```haku
map
  l : list t
  f : \t -> u
  -> list u
```

Produces a new list containing all the elements of `l`, run through the function `f` one by one.

```haku
filter
  l : list t
  f : \t -> boolean
  -> list t
```

Produces a new list with only those elements of `l` for which `f` returns `True`.

```haku
reduce
  l : list t
  init : a
  f : \a, t -> a
  -> a
```

Reduces the list to a single element, by repeatedly applying a reducing operation `f` on its elements.

This reducing operation receives the current value of an _accumulator_ variable (first argument), and the current element being iterated (second argument).
Its task is to return the _next_ value of the accumulator variable, which then gets passed into the next element, and so on and so forth.

```haku
flatten
  l : list _
  -> list _
```

Returns a new list with a single level of nesting flattened.

---

Some of these operations may be a bit confusing, so here are some examples.

```haku
-- To add two to all elements in a list:
list = range 1 4  -- [1, 2, 3, 4]
twoAdded = map list \x ->
  x + 2
```

```haku
-- To filter out only even numbers in a list:
list = range 1 10
isEven = \x -> mod x 2 == 0
onlyEven = filter list isEven
```

```haku
-- To sum all the numbers in a list:
list = [1, 3, 10, 2, 30, 4, 1]
sum = reduce list 0 \acc, value -> acc + value
```

```haku
-- To flatten a singly-nested list:
list = [[1, 2], [3, 4], [5, 6]]
flatList = flatten list  -- [1, 2, 3, 4, 5, 6]

-- Note that this only applies to a single level of nesting:
deepList = [[[1, 2, 3, 4]]]
lessDeepList = flatten deepList  -- [[1, 2, 3, 4]]
```


## Shapes

```haku
toShape
  value : _
  -> () | shape
```

Converts the given value to a shape.

- For `shape`, clones the shape that was passed in.
- For `vec`, returns a point `shape`.
- For anything else, returns `()`.

```haku
line
  start : vec
  end : vec
  -> shape
```

Creates a line segment shape with the provided `start` and `end` points.

```haku
rect
  position : vec
  size : vec
  -> shape

rect
  x : number
  y : number
  width : number
  height : number
  -> shape
```

Creates a rectangle shape with its top-left corner at `position`, with a given `size` stretching from the top-left corner.

The alternative 4-argument version takes in the rectangle's X/Y coordinates, width, and height as separate arguments instead of aggregating them into a `vec`.

```haku
circle
  center : vec
  radius : number
  -> shape

circle
  x : number
  y : number
  radius : number
  -> shape
```

Creates a circle shape, with its center at `center`, with the provided radius.

The alternative 3-argument version takes in the circle's center X/Y coordinates as separate arguments instead of aggregating them into a `vec`.

## Scribbles

```haku
stroke
  thickness : number
  color : rgba
  shape : shapeLike
  -> scribble
```

Creates a stroke scribble, which outlines the provided shape with a stroke of the given thickness and color.

Point shapes are drawn as circles, and `line` shapes have round caps at the line's endpoints.

```haku
fill
  color : rgba
  shape : shapeLike
  -> scribble
```

Creates a fill scribble, which fills in the entire area of the provided shape with a solid color.

Since this requires the shape to have a surface area, this does not do anything when point and `line` shapes are passed in.


## Reticles

```haku
withDotter
  cont : \{
    To   : vec
    From : vec
    Num  : number
  } -> scribble
  -> reticle
```

The dotter is a reticle that allows the user to draw things on the wall directly under the mouse cursor.
Once the user makes the interaction (presses the left mouse button), `cont` is called repeatedly with every movement of the mouse cursor, until the mouse button is released.

::: aside

It's not called a _plotter_ since the dotter does not plot lines or curves; it only places dots under the mouse cursor.

:::

During the interaction, `cont` is called with a record containing the following fields:

- `To` - the current position of the mouse cursor
- `From` - the previous position of the mouse cursor. May be equal to `To` if the user just pressed the mouse button.
- `Num` - the number of times `cont` has been called since the mouse cursor was pressed. Always greater or equal to `0`.

Since the dotter reticle finishes immediately (there is no extra interaction the user needs to take after pressing the mouse button for the action to be taken,) `cont` may only return a scribble; never another reticle.

The drawing area is a large, square, chunk-aligned perimeter around `To`.
