I am currently porting some code from Java to F# that deals with multidimensional functions. It supports variable dimension, so in the original implementation each point is represented as an array of doubles. The critical function of the code is an optimisation routine, that basically generates a sequence of points based on some criteria, evaluates a given function at these points and looks for a maximum. This works for any dimension. The operations I need are:
check the dimension of a point
create a new point with the same dimension of a given point
set (in procedural or functional sense) a given coordinate of a point
In F# I could obviously also use arrays in the same way. I was wandering though if there is a better way. If the dimension was fixed in advance, the obvious choice would be to use tuples. Is it possible to use tuples in this dynamic setting though?
No, tuples will be fixed by dimension. Also note that .NET tuples are boxed. If you are operating on large collections of points with small dimension (such as arrays of 2d points), using structs may help.
If you really want to push the F#/.NET advantage over Java, have a look at generics. Writing code with generics allows to write code that works for any dimension, and use different representations for different dimensions (say structs for 1-3 dimensions, and vectors for larger dimensions):
let op<'T where 'T :> IVector> (x: 'T) =
...
This is only relevant though if you are willing to go a long way to get the absolutely best performance and generality. Most projects do not need that, stick with the simplest thing that works.
For the fun of it, here is an extended example of how to utilize generics and F# inlining:
open System.Numerics
type IVector<'T,'V> =
abstract member Item : int -> 'T with get
abstract member Length : int
abstract member Update : int * 'T -> 'V
let lift<'T,'V when 'V :> IVector<'T,'V>> f (v: 'V) : 'V =
if v.Length = 0 then v else
let mutable r = v.Update(0, f v.[0])
for i in 1 .. v.Length - 1 do
r <- r.Update(i, f v.[i])
r
let inline norm (v: IVector<_,_>) =
let sq i =
let x = v.[i]
x * x
Seq.sum (Seq.init v.Length sq)
let inline normalize (v: 'V) : 'V =
let n = norm v
lift (fun x -> x / n) v
[<Struct>]
type Vector2D<'T>(x: 'T, y: 'T) =
member this.X = x
member this.Y = y
interface IVector<'T,Vector2D<'T>> with
member this.Item
with get (i: int) =
match i with
| 0 -> x
| _ -> y
member this.Length = 2
member this.Update(i: int, v: 'T) =
match i with
| 0 -> Vector2D(v, y)
| _ -> Vector2D(x, v)
override this.ToString() =
System.String.Format("{0}, {1}", x, y)
[<Sealed>]
type Vector<'T>(x: 'T []) =
interface IVector<'T,Vector<'T>> with
member this.Item with get (i: int) = x.[i]
member this.Length = x.Length
member this.Update(i: int, v: 'T) =
let a = Array.copy x
a.[i] <- v
Vector(a)
override this.ToString() =
x
|> Seq.map (fun e -> e.ToString())
|> String.concat ", "
[<Struct>]
type C(c: Complex) =
member this.Complex = c
static member Zero = C(Complex(0., 0.))
static member ( + ) (a: C, b: C) = C(a.Complex + b.Complex)
static member ( * ) (a: C, b: C) = C(a.Complex * b.Complex)
static member ( / ) (a: C, b: C) = C(a.Complex / b.Complex)
override this.ToString() = string c
let v1 = Vector2D(10., 30.)
normalize v1
|> printfn "%O"
let v2 = Vector2D(C(Complex(1.25, 0.8)), C(Complex(0.5, -1.)))
normalize v2
|> printfn "%O"
let v3 = Vector([| 10.; 30.; 50.|])
normalize v3
|> printfn "%O"
Note that norm and normalize are fairly general, they cope with specialized 2D vectors and generalized N-dimensional vectors, and with different component types such as complex numbers (you can define your own). The use of generics and F# inlining ensure that while general, these algorithms perform well for the special cases, using compact representations. This is where F# and .NET generics shine compared to Java, where you are obliged to create specialized copies of your code to get decent performance.
Related
RE: What is the best way to pass generic function that resolves to multiple types
Please read the referenced link before going further below
I am trying to extend the concept and pass a generic function that takes 2 parameters and does something with them.
The static approach works, however the interface based one causes a compile error (see the code lines marked with //error):
The declared type parameter '?' cannot be used here since the type parameter cannot be resolved at compile time.
Does anyone know how to fix it?
module MyModule
type T = Content of int
with
static member (+) ((Content i1), (Content i2)) = Content (i1 + i2)
static member (*) ((Content i1), (Content i2)) = Content (i1 * i2)
type W = { Content: int }
with
static member (+) ({Content = i1}, {Content = i2}) = { Content = i1 + i2 }
static member (*) ({Content = i1}, {Content = i2}) = { Content = i1 * i2 }
type Sum = Sum with static member inline ($) (Sum, (x, y)) = x + y
type Mul = Mul with static member inline ($) (Mul, (x, y)) = x * y
let inline f1 (la: 'a list) (lb: 'b list) reducer =
let a = la |> List.reduce (fun x y -> reducer $ (x, y))
let b = lb |> List.reduce (fun x y -> reducer $ (x, y))
(a, b)
type I = abstract member Reduce<'a> : 'a -> 'a -> 'a
let f2 (la: 'a list) (lb: 'b list) (reducer: I) =
let a = la |> List.reduce reducer.Reduce
let b = lb |> List.reduce reducer.Reduce
(a, b)
let main ()=
let lt = [Content 2; Content 4]
let lw = [{ Content = 2 }; { Content = 4 }]
let _ = f1 lt lw Sum
let _ = f1 lt lw Mul
let _ = f2 lt lw { new I with member __.Reduce x y = x + y} //error
let _ = f2 lt lw { new I with member __.Reduce x y = x * y} //error
0
The problem with your attempt is that you can't use operators + or * on parameters x and y, because it's not known that their type 'a has those operators defined.
To answer your further question in comments about how to achieve it anyway - if you want to use multiplication and addition on any type 'a that the caller chooses, you have to specify that. For an interface method, the only way to do this is by constraining the type parameter 'a, and the only two kinds of constraints that .NET runtime supports are "has a parameterless constructor" and "implements a given interface or inherits from a given class".
The latter one would be useful in your case: make both types implement the interface and then constrain type parameter 'a to implement that interface:
type IArithmetic<'a> =
abstract member add : 'a -> 'a
abstract member mult : 'a -> 'a
type T = Content of int
with
interface IArithmetic<T> with
member this.add (Content y) = let (Content x) = this in Content (x + y)
member this.mult (Content y) = let (Content x) = this in Content (x * y)
type W = { Content: int }
with
interface IArithmetic<W> with
member this.add y = { Content = this.Content + y.Content }
member this.mult y = { Content = this.Content * y.Content }
type I = abstract member Reduce<'a when 'a :> IArithmetic<'a>> : 'a -> 'a -> 'a
// ^^^^^^^^^^^^^^^^^^^^^^^^^^
// the constraint right here
...
let _ = f2 lt lw { new I with member __.Reduce x y = x.add y }
let _ = f2 lt lw { new I with member __.Reduce x y = x.mult y }
Is this a bit awkward? I guess so, but you're kind of doing the same thing for the SRTP version, so why not?
The core idea is: if you want your Reduce method to work not with just any type, but only with types that can do certain things, you have to specify what those things are. In the SRTP case you're doing that by defining the (+) and (*) operators. In the interface case you're doing that by implementing the interface.
Q: But can I make the interface somehow pick up the (+) and (*) operators?
A: In general, no. The .NET runtime just doesn't support the kind of constraints like "any type that has a method with certain signature". This means that such constraints can't be compiled down to IL, which means they can't be used in an interface implementation.
And this is the price you pay for using SRTPs: all those inline functions - they don't get compiled to IL, they always get expanded (inserted, substituted) at use sites. For small, simple functions, this is no big deal. But if your whole program is like that, you might see some unexpected compiled code bloat, potentially translating to slower startup time etc.
Having said all that, I must note that the code you're showing is toy POC kind of code, not intended to solve any real, practical problem. And as such, most musings on it are in danger of being completely useless.
If you have an actual problem in mind, perhaps try sharing it, and somebody would suggest the best solution for that specific case.
In particular, I have a nagging feeling that you might not actually need higher-rank functions (that's what it's called when a function doesn't lose genericity when passed as parameter).
The MSDN doc on Type Extensions states that "Before F# 3.1, the F# compiler didn't support the use of C#-style extension methods with a generic type variable, array type, tuple type, or an F# function type as the “this” parameter." (http://msdn.microsoft.com/en-us/library/dd233211.aspx)
How can be a Type Extension used on F# function type? In what situations would such a feature be useful?
Here is how you can do it:
[<Extension>]
type FunctionExtension() =
[<Extension>]
static member inline Twice(f: 'a -> 'a, x: 'a) = f (f x)
// Example use
let increment x = x + 1
let y = increment.Twice 5 // val y : int = 7
Now for "In what situations would such a feature be useful?", I honestly don't know and I think it's probably a bad idea to ever do this. Calling methods on a function feels way too JavaScript-ey, not idiomatic at all in F#.
You may simulate the . notation for extension methods with F#'s |> operator. It's a little clumsier, given the need for brackets:
let extension f x =
let a = f x
a * 2
let f x = x*x
> f 2;;
val it : int = 4
> (f |> extension) 2;;
val it : int = 8
> let c = extension f 2;; // Same as above
val c : int = 8
I have a list of functions in F# which are all of type (float -> float -> float -> float). I want to do some kind of fold on the sequence to get a single function which returns the sum of all of the functions.
For instance, I could pass the values 1.0, 2.0, and 3.0 to every function in the list, and get a return value from each one. Then I could compute the sum of all of these values. However, I want to generalize this.
I know how to do this recursively, but I feel like it should be doable in one line. Is there a concise way to accomplish this task?
The solution by #Lee is a one liner you're looking for. If you wanted to save a few characters, you can use List.sumBy which first applies a given function to an element of the list (similar to List.map) and then sums the result (just like List.sum):
let sumAll (fs:(_ -> _ -> _ -> float) list) a b c =
List.sumBy (fun f -> f a b c) fs
Both this and Lee's version uses type annotations to specify that the functions in the list return float. This is needed, because otherwise the compiler does not know what kind of numbers you want to sum using List.sum (floats, integers, etc.). This ambiguity needs to be resolved to compile the function.
Alternatively, you could mark the function as inline and then it would be inlined when you call it (and it would work for multiple different numeric types). You can also pass the fs parameter as the last one and use partial function application:
let inline sumAll a b c = List.sumBy (fun f -> f a b c)
Now you can call it using pipelining as follows: fs |> sumAll 1 2 3.
let sumAll (fs: (float -> float -> float -> float) list) a b c = List.map (fun f -> f a b c) fs |> Seq.sum
The answers by #Lee and #Tomas are great, but there is a shorter way.
If you can afford passing (a, b, c) as a triple upon invocation:
let inline sumAll() = (|||>) >> List.sumBy
// usage
let predicates =
[
fun a b c -> a
fun a b c -> b * 42.0 - c
]
let ret1 = predicates |> sumAll()(1.0, 2.0, 3.0)
It will be also generic:
let predicates2 =
[
fun a b c -> c - 10
fun a b c -> a + c * 42
]
let ret2 = predicates2 |> sumAll()(1, 2, 3)
A more readable way which supports curried arguments:
let sumAllCurried a b c = (a,b,c) |> (|||>) |> List.sumBy<_, float>
// usage
let ret3 = predicates |> sumAllCurried 1.0 2.0 3.0
Note, I'm using a type parameter on List.sumBy since it looks shorter than typing an entire type specification for f.
this simple function:
let sum a b = a + b
will work only for int types
how to make it so that it would also work for float and long ?
Use inline:
let inline sum a b = a + b
UPDATE:
If you're interested in writing your own polymorphic numerical functions, you should use both inline and LanguagePrimitives module.
Here is a polymorphic cosine function from the thread Converting Haskell Polymorphic Cosine function to F#:
let inline cosine n (x: ^a) =
let one: ^a = LanguagePrimitives.GenericOne
Seq.initInfinite(fun i -> LanguagePrimitives.DivideByInt (- x*x) ((2*i+1)*(2*i+2)))
|> Seq.scan (*) one
|> Seq.take n
|> Seq.sum
The example function you give only works for int types because of type inference; the type inference mechanism will automatically infer int because it sees the addition. If you want to make the same function for float and long, you'd either do inline as Pad has said or you could do this:
let sumFloat (a:float) b = a + b
let sumLong (a:int64) b = a + b
But inline is the right mechanism to get the generic "any type that supports addition" behavior that you're looking for.
let f g x y = g x y
f (+) 0.0 1.0;;
f (=) 0 1;;
I like this solution as well.
In my quest to learn more F#, I tried to implement an "accumulator generator" as described by Paul Graham here. My best solution so far is completely dynamically typed:
open System
let acc (init:obj) : obj->obj=
let state = ref init
fun (x:obj) ->
if (!state).GetType() = typeof<Int32>
&& x.GetType() = typeof<Int32> then
state := (Convert.ToInt32(!state) + Convert.ToInt32(x)) :> obj
else
state := (Convert.ToDouble(!state) + Convert.ToDouble(x)) :> obj
!state
do
let x : obj -> obj = acc 1 // the type annotation is necessary here
(x 5) |> ignore
printfn "%A" (x 2) // prints "8"
printfn "%A" (x 2.3) // prints "10.3"
I have three questions:
If I remove the type annotation for x, the code fails to compile because the compiler infers type int -> obj for x - although acc is annotated to return an obj->obj. Why is that and can I avoid it?
Any ideas to improve this dynamically typed version?
Is it possible to implement this with proper static types? Maybe with member constraints? (It is possible in Haskell, but not in OCaml, AFAIK)
In my quest to learn more F#, I tried to implement an "accumulator generator" as described by Paul Graham here.
This problem requires the existence of an unspecified numeric tower. Lisp happens to have one and it happens to be adequate for Paul Graham's examples because this problem was specifically designed to make Lisp look artificially good.
You can implement a numeric tower in F# either using a union type (like type number = Int of int | Float of float) or by boxing everything. The following solution uses the latter approach:
let add (x: obj) (y: obj) =
match x, y with
| (:? int as m), (:? int as n) -> box(m+n)
| (:? int as n), (:? float as x)
| (:? float as x), (:? int as n) -> box(x + float n)
| (:? float as x), (:? float as y) -> box(x + y)
| _ -> failwith "Run-time type error"
let acc x =
let x = ref x
fun (y: obj) ->
x := add !x y
!x
let x : obj -> _ = acc(box 1)
do x(box 5)
do acc(box 3)
do printfn "%A" (x(box 2.3))
However, numeric towers are virtually useless in the real world. Unless you are very careful, trying to learn from these kinds of borked challenges will do you more harm than good. You should leave asking yourself why we do not want a numeric tower, do not want to box and do not want run-time type promotion?
Why didn't we just write:
let x = 1
let x = x + 5
ignore(3)
let x = float x + 2.3
We know the type of x at every step. Every number is stored unboxed. We know that this code will never produce a run-time type error...
I agree with Jon that this is quite artificial example and it is not a good starting point for learning F#. However, you can use static member constraints to get reasonably close without dynamic casts and reflection. If you mark it as inline and add convert both of the parameters using float:
let inline acc x =
let x = ref (float x)
fun y ->
x := (float y) + !x
!x
You'll get a function with the following type:
val inline acc :
^a -> ( ^b -> float)
when ^a : (static member op_Explicit : ^a -> float) and
^b : (static member op_Explicit : ^b -> float)
The function takes any two arguments that can be explicitly converted to float. The only limitation compared to the LISP version (I guess) is that it always returns float (as the most universal numeric type available). You can write something like:
> acc 1 2;; // For two integers, it returns float
val it : float = 3.0
> acc 1 2.1;; // integer + float
val it : float = 3.1
> acc 1 "31";; // It even works with strings!
val it : float = 32.0
It's definitely not possible to implement this with proper static types. You say you can in Haskell, but I don't believe you.
The problem with trying to do this with static typing is in adding two different numbers of possibly different types while preserving the type of the left-hand side. As Jon Harrop says this is possible with a union type. Once you've defined the union type and a corresponding addition operation which works as mentioned, the actual accumulator is very simple. My implementation:
module MyTest
type Numeric =
| NInt of int
| NFloat of float
member this.Add(other : Numeric) : Numeric =
match this with
| NInt x ->
match other with
| NInt y -> NInt (x + y)
| NFloat y -> NInt (x + (int y))
| NFloat x ->
match other with
| NInt y -> NFloat (x + (float y))
| NFloat y -> NFloat (x + y)
override this.ToString() =
match this with
| NInt x -> x.ToString()
| NFloat x -> x.ToString()
let foo (n : Numeric) =
let acc = ref n
fun i ->
acc := (!acc).Add(i)
!acc
let f = foo (NFloat 1.1)
(2 |> NInt |> f).ToString() |> printfn "%s"