# Dex: Prelude

Source file: dex-prelude.futTranslation of various functions from prelude.dx. Not all of them, but only what is needed to support the other Dex examples. We also skip some functions that already exist in Futhark, but under different names - we’ll keep writing `f64.i32`

instead of `FToI`

.

```
let sq (x: f64) = x * x
let mean [n] (xs: [n]f64) : f64 =
f64.sum xs / f64.i64 n
let std [n] (xs: [n]f64) =
f64.sqrt (mean (map sq xs) - sq (mean xs))
let linspace (n: i64) (start: f64) (end: f64) : [n]f64 =
tabulate n (\i -> start + f64.i64 i * ((end-start)/f64.i64 n))
```

Some Dex programs use this sequential scan.

```
let scan' n x0 f =
0) <|
(.loop (arr, acc) = (replicate n x0, x0) for i < n do
let acc' = f i acc
in (arr with [i] = acc', acc')
```

## Random numbers

The random numbers defined in random-numbers.fut are based on the idea of having functions take and return random number states. Dex’s approach to random numbers is based on splitting and never returning the final state. Both work fine in Futhark. The biggest difference is that the Dex implementation uses a high-quality hash algorithm, and we use a hash function found on StackOverflow:

```
type Key = #Key u32
let hash (k : Key) (y: i32): Key =
match k case #Key x ->
let x = x ^ u32.i32 y
let x = ((x >> 16) ^ x) * 0x45d9f3b
let x = ((x >> 16) ^ x) * 0x45d9f3b
let x = ((x >> 16) ^ x)
in #Key x
let newKey = hash (#Key 0)
let splitKey k = (hash k 1, hash k 2)
let splitKey3 k =
let (a, k') = splitKey k
let (b, c) = splitKey k'
in (a,b,c)
let many '^a (f: Key -> a) (k: Key) (i: i64) = f (hash k (i32.i64 i))
let ixkey (k: Key) (i: i64) : Key = hash k (i32.i64 i)
let ixkey2 (k: Key) (i: i64) (j: i64) : Key =
hash (hash k (i32.i64 i)) (i32.i64 j)let rand (k: Key) : f64 =
match k case #Key x ->
f64.u32 x / f64.u32 u32.highestlet randVec 'a (n: i64) (f: Key -> a) (k: Key) : [n]a =
tabulate n (\i -> f (ixkey k i))
let randn (k: Key) : f64 =
let (k1, k2) = splitKey k
let u1 = rand k1
let u2 = rand k2
in f64.sqrt ((-2.0) * f64.log u1) * f64.cos (2.0 * f64.pi * u2)
let bern (p: f64) (k: Key) = rand k < p
let randnVec (n: i64) (k: Key) : [n]f64 =
tabulate n (ixkey k >-> randn)
```

The `randIdx`

function computes a random index into an array. In Dex, where indexes are types, this is done by passing in the size of the array as an implicit parameter, and using type inference to determine the right size for any given application. In Futhark, `randIdx`

is just an elaborate way of generating an integer up to an explicitly given bound.

```
let randIdx (n: i64) (k: Key) =
let unif = rand k
in i64.f64 (f64.floor (unif * f64.i64 n))
```