# Phillip Trelford's Array

## POKE 36879,255

Welcome to day 2 of the F# Advent Calendar in English, and don’t miss Scott Wlaschin’s introduction to property-based testing from yesterday.

In A Christmas Carol Charles Dickens wrote of cold winters with snow as a matter of course. White Christmases were common during the Little Ice Age that lasted from the 1550s to the 1850s. Nowadays the chances of a snowfall on Christmas day are much lower, but the imagery of a white Christmas persists.

In this post we’ll generate our snowflakes instead.

Koch Snowflake

The Koch snowflake is a mathematical curve constructed from an equilateral triangle where each line segment is recursively altered:

The picture above was generated in the F# REPL, using WinForms to display a bitmap:

```let snowflake (graphics:Graphics) length =
use pen = new Pen(Color.White)
let angle = ref 0.0
let x = ref ((float width/2.0) - length/2.0)
let y = ref ((float height/2.0) - length/3.0)
let rec segment n depth =
if depth = 0 then
line n
else
segment (n/3.0) (depth-1)
rotate -60.0
segment (n/3.0) (depth-1)
rotate 120.0
segment (n/3.0) (depth-1)
rotate -60.0
segment (n/3.0) (depth-1)
and line n =
let r = !angle * Math.PI / 180.0
let x2 = !x + cos(r) * n
let y2 = !y + sin(r) * n
graphics.DrawLine(pen, float32 !x,float32 !y, float32 x2, float32 y2)
x := x2
y := y2
and rotate a =
angle := !angle + a
let depth = 5
segment length depth
rotate 120.0
segment length depth
rotate 120.0
segment length depth```

The full snippet is available on F# Snippets: http://fssnip.net/oA

Paper and Scissors

Snowflakes can be created by folding paper and cutting holes with scissors. We can get a similar effect using transparent polygons and rotational symmetry:

Here the polygons are selected randomly and like snowflakes each one is different:

```let paperSnowflake () =
let image = new Bitmap(int width, int height)
use graphics = Graphics.FromImage(image)
use brush = new SolidBrush(Color.Black)
graphics.FillRectangle(brush, 0, 0, int width, int height)
graphics.TranslateTransform(float32 (width/2.0),float32 (height/2.0))
let color = Color.FromArgb(128,0,128,255)
use brush = new SolidBrush(color)
let rand = Random()
let polys =
[for i in 1..12 ->
let w = rand.Next(20)+1 // width
let h = rand.Next(20)+1 // height
let m = rand.Next(h)    // midpoint
let s = rand.Next(30)   // start
[|0,s; -w,s+m; 0,s+h; w,s+m|]
]
for i in 0.0..60.0..300.0 do
graphics.RotateTransform(float32 60.0)
let poly points =
let points = [|for (x,y) in points -> Point(x*5,y*5)|]
graphics.FillPolygon(brush,points)
polys |> List.iter poly
image```

The full snippet is on F# Snippets: http://fssnip.net/oB

Another interesting method of generating snowflakes is cellular automata, but I’ll leave that as an exercise for the reader.

Happy holidays!

This weekend Evelina, Yan an I had the pleasure of speaking at f(by) the first dedicated functional conference in Belarus. It was a short hop by train from Vilnius to Minsk, where we had been attending Build Stuff. Sergey Tihon, of F# Weekly fame, was waiting for us at the train station to guide us to the hotel with a short tour of the city.

The venue was a large converted loft space, by the river and not far from the central station, with great views over the city. The event attracted over 100 developers from across the region, and we were treated to tea and tasty local cakes during the breaks.

Evelina was the first of us to speak and got a great response to her talk on Understanding Social Networks with F#.

The slides and samples are available on Evelina’s github repository.

Next up Yan presented Learn you to tame complex APIs with F# powered DSLs:

My talk was another instalment of F# Eye for the C# Guy.

The talk introduces F# from the perspective of a C# developer using live samples covering syntax, F#/C# interop, unit testing, data access via F# Type Providers and F# to JS with FunScript.

In one example we looked at CO2 emissions using World Bank data (using FSharp.Data) in a line chart (using FSharp.Charting):

```[gb;uk;by] => (fun i -> i.``CO2 emissions (kg per 2005 PPP \$ of GDP)``)
```

Many thanks to Alina for inviting us, and the Minsk F# community for making us feel very welcome.

Last week Gian Ntzik gave a great talk at the F#unctional Londoners meetup on the Nessos Streams library. It’s a lightweight F#/C# library for efficient functional-style pipelines on streams of data.

The main difference between LINQ/Seq and Streams is that LINQ is about composing external iterators (Enumerable/Enumerator) and Streams is based on the continuation-passing-style composition of internal iterators, which makes optimisations such as loop fusion easier.

The slides (using FsReveal) and samples are available on Gian’s github repository.

Simple Streams

Gian started the session by live coding a simple implementation of streams in about 20 minutes:

```type Stream<'T> = ('T -> unit) -> unit

let inline ofArray (source: 'T[]) : Stream<'T> =
fun k ->
let mutable i = 0
while i < source.Length do
k source.[i]
i <- i + 1

let inline filter (predicate: 'T -> bool) (stream: Stream<'T>) : Stream<'T> =
fun k -> stream (fun value -> if predicate value then k value)

let inline map (mapF: 'T -> 'U) (stream: Stream<'T>) : Stream<'U> =
fun k -> stream (fun v -> k (mapF v))

let inline iter (iterF: 'T -> unit) (stream: Stream<'T>) : unit =
stream (fun v -> iterF v)

let inline toArray (stream: Stream<'T>) : 'T [] =
let acc = new List<'T>()
stream |> iter (fun v -> acc.Add(v))
acc.ToArray()

let inline fold (foldF:'State->'T->'State) (state:'State) (stream:Stream<'T>) =
let acc = ref state
stream (fun v -> acc := foldF !acc v)
!acc

let inline reduce (reducer: ^T -> ^T -> ^T) (stream: Stream< ^T >) : ^T
when ^T : (static member Zero : ^T) =
fold (fun s v -> reducer s v) LanguagePrimitives.GenericZero stream

let inline sum (stream : Stream< ^T>) : ^T
when ^T : (static member Zero : ^T)
and ^T : (static member (+) : ^T * ^T -> ^T) =
fold (+) LanguagePrimitives.GenericZero stream```

and as you can see only about 40 lines of code.

Sequential Performance

Just with this simple implementation, Gian was able to demonstrate a significant performance improvement over F#’s built-in Seq module for a simple pipeline:

```#time // Turns on timing in F# Interactive

let data = [|1L..1000000L|]

let seqValue =
data
|> Seq.filter (fun x -> x%2L = 0L)
|> Seq.map (fun x -> x * x)
|> Seq.sum
// Real: 00:00:00.252, CPU: 00:00:00.234, GC gen0: 0, gen1: 0, gen2: 0

let streamValue =
data
|> Stream.ofArray
|> Stream.filter (fun x -> x%2L = 0L)
|> Stream.map (fun x -> x * x)
|> Stream.sum
// Real: 00:00:00.119, CPU: 00:00:00.125, GC gen0: 0, gen1: 0, gen2: 0```

Note for operations over arrays, the F# Array module would be more appropriate choice and is slightly faster:

```let arrayValue =
data
|> Array.filter (fun x -> x%2L = 0L)
|> Array.map (fun x -> x * x)
|> Array.sum
// Real: 00:00:00.094, CPU: 00:00:00.093, GC gen0: 0, gen1: 0, gen2: 0```

Also LINQ does quite well here as it has a specialized overloads including one for summing over int64 values:

```open System.Linq

let linqValue =
data
.Where(fun x -> x%2L = 0L)
.Select(fun x -> x * x)
.Sum()
// Real: 00:00:00.058, CPU: 00:00:00.062, GC gen0: 0, gen1: 0, gen2: 0```

However with F# Interactive running in 64-bit mode Streams take back the advantage (thanks to Nick Palladinos for the tip):

```let streamValue =
data
|> Stream.ofArray
|> Stream.filter (fun x -> x%2L = 0L)
|> Stream.map (fun x -> x * x)
|> Stream.sum
// Real: 00:00:00.033, CPU: 00:00:00.031, GC gen0: 0, gen1: 0, gen2: 0```

Looks like the 64-bit JIT is doing some black magic there.

Parallel Performance

Switching to the full Nessos Streams library, there’s support for parallel streams via the ParStream module:

```let parsStreamValue =
data
|> ParStream.ofArray
|> ParStream.filter (fun x -> x%2L = 0L)
|> ParStream.map (fun x -> x + 1L)
|> ParStream.sum
// Real: 00:00:00.069, CPU: 00:00:00.187, GC gen0: 0, gen1: 0, gen2: 0```

which demonstrates a good performance increase with little effort.

For larger computes Nessos Streams supports cloud based parallel operations against Azure.

Overall Nessos Streams looks like a good alternative to the Seq module for functional pipelines.

Nessos LinqOptimzer

For further optimization Gian recommended the Nessos LinqOptimizer:

An automatic query optimizer-compiler for Sequential and Parallel LINQ. LinqOptimizer compiles declarative LINQ queries into fast loop-based imperative code. The compiled code has fewer virtual calls and heap allocations, better data locality and speedups of up to 15x

The benchmarks are impressive:

Reactive Extensions (Rx)

One of the questions in the talk and on twitter later was, given Rx is also a push model, how does the performance compare:

Clearly the Nessos Streams library and Rx have different goals (data processing vs event processing), but I thought it would be interesting to compare them all the same:

```open System.Reactive.Linq

let rxValue =
data
.ToObservable()
.Where(fun x -> x%2L = 0L)
.Select(fun x -> x * x)
.Sum()
.ToEnumerable()
// Real: 00:00:02.895, CPU: 00:00:02.843, GC gen0: 120, gen1: 0, gen2: 0

let streamValue =
data
|> Stream.ofArray
|> Stream.filter (fun x -> x%2L = 0L)
|> Stream.map (fun x -> x * x)
|> Stream.sum
// Real: 00:00:00.130, CPU: 00:00:00.109, GC gen0: 0, gen1: 0, gen2: 0```

In this naive comparison you can see Nessos Streams is roughly 20 times faster than Rx.

Observable module

F# also has a built-in Observable module for operations over IObservable<T> (support for operations over events was added to F# back in 2006). Based on the claims on Rx performance made by Matt Podwysocki I was curious to see how it stacked up:

```let obsValue =
data
|> Observable.ofSeq
|> Observable.filter (fun x -> x%2L = 0L)
|> Observable.map (fun x -> x * x)
|> Observable.sum
|> Observable.first
// Real: 00:00:00.479, CPU: 00:00:00.468, GC gen0: 18, gen1: 0, gen2: 0```

As you can see Observable module comes off roughly 5 times faster.

Note: I had to add some simple combinators to make this work, you can see the full snippet here: http://fssnip.net/ow

Summary

Nessos Streams look like a promising direction for performance of functional pipelines, and for gaining raw imperative performance the Nessos LINQOptimizer is impressive.