How to write a function for generic numbers?

Question

I'm quite new to F# and find type inference really is a cool thing. But currently it seems that it also may lead to code duplication, which is not a cool thing. I want to sum the digits of a number like this:

let rec crossfoot n =
  if n = 0 then 0
  else n % 10 + crossfoot (n / 10)

crossfoot 123

This correctly prints 6. But now my input number does not fit int 32 bits, so I have to transform it to.

let rec crossfoot n =
  if n = 0L then 0L
  else n % 10L + crossfoot (n / 10L)

crossfoot 123L

Then, a BigInteger comes my way and guess what…

Of course, I could only have the bigint version and cast input parameters up and output parameters down as needed. But first I assume using BigInteger over int has some performance penalities. Second let cf = int (crossfoot (bigint 123)) does just not read nice.

Isn't there a generic way to write this?

Answer 1

See

Hidden Features of F#

Answer 2

Building on Brian's and Stephen's answers, here's some complete code:

module NumericLiteralG = 
    let inline FromZero() = LanguagePrimitives.GenericZero
    let inline FromOne() = LanguagePrimitives.GenericOne
    let inline FromInt32 (n:int) =
        let one : ^a = FromOne()
        let zero : ^a = FromZero()
        let n_incr = if n > 0 then 1 else -1
        let g_incr = if n > 0 then one else (zero - one)
        let rec loop i g = 
            if i = n then g
            else loop (i + n_incr) (g + g_incr)
        loop 0 zero 

let inline crossfoot (n:^a) : ^a =
    let (zero:^a) = 0G
    let (ten:^a) = 10G
    let rec compute (n:^a) =
        if n = zero then zero
        else ((n % ten):^a) + compute (n / ten)
    compute n

crossfoot 123
crossfoot 123I
crossfoot 123L

UPDATE: Simple Answer

Here's a standalone implementation, without the NumericLiteralG module, and a slightly less restrictive inferred type:

let inline crossfoot (n:^a) : ^a =
    let zero:^a = LanguagePrimitives.GenericZero
    let ten:^a = (Seq.init 10 (fun _ -> LanguagePrimitives.GenericOne)) |> Seq.sum
    let rec compute (n:^a) =
        if n = zero then zero
        else ((n % ten):^a) + compute (n / ten)
    compute n

Explanation

There are effectively two types of generics in F#: 1) run-type polymorphism, via .NET interfaces/inheritance, and 2) compile time generics. Compile-time generics are needed to accommodate things like generic numerical operations and something like duck-typing (explicit member constraints). These features are integral to F# but unsupported in .NET, so therefore have to be handled by F# at compile time.

The caret (^) is used to differentiate statically resolved (compile-time) type parameters from ordinary ones (which use an apostrophe). In short, 'a is handled at run-time, ^a at compile-time–which is why the function must be marked inline.

I had never tried to write something like this before. It turned out clumsier than I expected. The biggest hurdle I see to writing generic numeric code in F# is: creating an instance of a generic number other than zero or one. See the implementation of FromInt32 in this answer to see what I mean. GenericZero and GenericOne are built-in, and they're implemented using techniques that aren't available in user code. In this function, since we only needed a small number (10), I created a sequence of 10 GenericOnes and summed them.

I can't explain as well why all the type annotations are needed, except to say that it appears each time the compiler encounters an operation on a generic type it seems to think it's dealing with a new type. So it ends up inferring some bizarre type with duplicated resitrictions (e.g. it may require (+) multiple times). Adding the type annotations lets it know we're dealing with the same type throughout. The code works fine without them, but adding them simplifies the inferred signature.

Answer 3

Since the question of how to make the type signatures less hairy when using the generalized numeric literals has come up, I thought I'd put in my two cents. The main issue is that F#'s operators can be asymmetric so that you can do stuff like System.DateTime.Now + System.TimeSpan.FromHours(1.0), which means that F#'s type inference adds intermediary type variables whenever arithmetic operations are being performed.

In the case of numerical algorithms, this potential asymmetry isn't typically useful and the resulting explosion in the type signatures is quite ugly (although it generally doesn't affect F#'s ability to apply the functions correctly when given concrete arguments). One potential solution to this problem is to restrict the types of the arithmetic operators within the scope that you care about. For instance, if you define this module:

module SymmetricOps =
  let inline (+) (x:'a) (y:'a) : 'a = x + y
  let inline (-) (x:'a) (y:'a) : 'a = x - y
  let inline (*) (x:'a) (y:'a) : 'a = x * y
  let inline (/) (x:'a) (y:'a) : 'a = x / y
  let inline (%) (x:'a) (y:'a) : 'a = x % y
  ...

then you can just open the SymmetricOps module whenever you want have the operators apply only to two arguments of the same type. So now we can define:

module NumericLiteralG = 
  open SymmetricOps
  let inline FromZero() = LanguagePrimitives.GenericZero
  let inline FromOne() = LanguagePrimitives.GenericOne
  let inline FromInt32 (n:int) =
      let one = FromOne()
      let zero = FromZero()
      let n_incr = if n > 0 then 1 else -1
      let g_incr = if n > 0 then one else (zero - one)
      let rec loop i g = 
          if i = n then g
          else loop (i + n_incr) (g + g_incr)
      loop 0 zero

and

open SymmetricOps
let inline crossfoot x =
  let rec compute n =
      if n = 0G then 0G
      else n % 10G + compute (n / 10G)
  compute x

and the inferred type is the relatively clean

val inline crossfoot :
   ^a ->  ^a
    when  ^a : (static member ( - ) :  ^a *  ^a ->  ^a) and
          ^a : (static member get_One : ->  ^a) and
          ^a : (static member ( % ) :  ^a *  ^a ->  ^a) and
          ^a : (static member get_Zero : ->  ^a) and
          ^a : (static member ( + ) :  ^a *  ^a ->  ^a) and
          ^a : (static member ( / ) :  ^a *  ^a ->  ^a) and  ^a : equality

while we still get the benefit of a nice, readable definition for crossfoot.

Answer 4

I stumbled upon this topic when I was looking for a solution and I am posting my answer, because I found a way to express generic numeral without the less than optimal implementation of building up the number by hand.

open System.Numerics
// optional
open MathNet.Numerics

module NumericLiteralG = 
    type GenericNumber = GenericNumber with
        static member instance (GenericNumber, x:int32, _:int8) = fun () -> int8 x
        static member instance (GenericNumber, x:int32, _:uint8) = fun () -> uint8 x
        static member instance (GenericNumber, x:int32, _:int16) = fun () -> int16 x
        static member instance (GenericNumber, x:int32, _:uint16) = fun () -> uint16 x
        static member instance (GenericNumber, x:int32, _:int32) = fun () -> x
        static member instance (GenericNumber, x:int32, _:uint32) = fun () -> uint32 x
        static member instance (GenericNumber, x:int32, _:int64) = fun () -> int64 x
        static member instance (GenericNumber, x:int32, _:uint64) = fun () -> uint64 x
        static member instance (GenericNumber, x:int32, _:float32) = fun () -> float32 x
        static member instance (GenericNumber, x:int32, _:float) = fun () -> float x
        static member instance (GenericNumber, x:int32, _:bigint) = fun () -> bigint x
        static member instance (GenericNumber, x:int32, _:decimal) = fun () -> decimal x
        static member instance (GenericNumber, x:int32, _:Complex) = fun () -> Complex.op_Implicit x
        static member instance (GenericNumber, x:int64, _:int64) = fun () -> int64 x
        static member instance (GenericNumber, x:int64, _:uint64) = fun () -> uint64 x
        static member instance (GenericNumber, x:int64, _:float32) = fun () -> float32 x
        static member instance (GenericNumber, x:int64, _:float) = fun () -> float x
        static member instance (GenericNumber, x:int64, _:bigint) = fun () -> bigint x
        static member instance (GenericNumber, x:int64, _:decimal) = fun () -> decimal x
        static member instance (GenericNumber, x:int64, _:Complex) = fun () -> Complex.op_Implicit x
        static member instance (GenericNumber, x:string, _:float32) = fun () -> float32 x
        static member instance (GenericNumber, x:string, _:float) = fun () -> float x
        static member instance (GenericNumber, x:string, _:bigint) = fun () -> bigint.Parse x
        static member instance (GenericNumber, x:string, _:decimal) = fun () -> decimal x
        static member instance (GenericNumber, x:string, _:Complex) = fun () -> Complex(float x, 0.0)
        // MathNet.Numerics
        static member instance (GenericNumber, x:int32, _:Complex32) = fun () -> Complex32.op_Implicit x
        static member instance (GenericNumber, x:int32, _:bignum) = fun () -> bignum.FromInt x
        static member instance (GenericNumber, x:int64, _:Complex32) = fun () -> Complex32.op_Implicit x
        static member instance (GenericNumber, x:int64, _:bignum) = fun () -> bignum.FromBigInt (bigint x)
        static member instance (GenericNumber, x:string, _:Complex32) = fun () -> Complex32(float32 x, 0.0f)
        static member instance (GenericNumber, x:string, _:bignum) = fun () -> bignum.FromBigInt (bigint.Parse x)

    let inline genericNumber num = Inline.instance (GenericNumber, num) ()

    let inline FromZero () = LanguagePrimitives.GenericZero
    let inline FromOne () = LanguagePrimitives.GenericOne
    let inline FromInt32 n = genericNumber n
    let inline FromInt64 n = genericNumber n
    let inline FromString n = genericNumber n

this implementation comes by without complicated iteration during the cast. It uses FsControl for the Instance module.

http://www.fssnip.net/mv

Answer 5

In addition to kvb's technique using Numeric Literals (Brian's link), I've had a lot of success using a different technique which can yield better inferred structural type signatures and may also be used to create precomputed type-specific functions for better performance as well as control over supported numeric types (since you will often want to support all integral types, but not rational types, for example): F# Static Member Type Constraints.

Following up on the discussion Daniel and I have been having about the inferred type signatures yielded by the different techniques, here is an overview:

NumericLiteralG Technique

module NumericLiteralG = 
    let inline FromZero() = LanguagePrimitives.GenericZero
    let inline FromOne() = LanguagePrimitives.GenericOne
    let inline FromInt32 (n:int) =
        let one = FromOne()
        let zero = FromZero()
        let n_incr = if n > 0 then 1 else -1
        let g_incr = if n > 0 then one else (zero - one)
        let rec loop i g = 
            if i = n then g
            else loop (i + n_incr) (g + g_incr)
        loop 0 zero 

Crossfoot without adding any type annotations:

let inline crossfoot1 n =
    let rec compute n =
        if n = 0G then 0G
        else n % 10G + compute (n / 10G)
    compute n

val inline crossfoot1 :
   ^a ->  ^e
    when ( ^a or  ^b) : (static member ( % ) :  ^a *  ^b ->  ^d) and
          ^a : (static member get_Zero : ->  ^a) and
         ( ^a or  ^f) : (static member ( / ) :  ^a *  ^f ->  ^a) and
          ^a : equality and  ^b : (static member get_Zero : ->  ^b) and
         ( ^b or  ^c) : (static member ( - ) :  ^b *  ^c ->  ^c) and
         ( ^b or  ^c) : (static member ( + ) :  ^b *  ^c ->  ^b) and
          ^c : (static member get_One : ->  ^c) and
         ( ^d or  ^e) : (static member ( + ) :  ^d *  ^e ->  ^e) and
          ^e : (static member get_Zero : ->  ^e) and
          ^f : (static member get_Zero : ->  ^f) and
         ( ^f or  ^g) : (static member ( - ) :  ^f *  ^g ->  ^g) and
         ( ^f or  ^g) : (static member ( + ) :  ^f *  ^g ->  ^f) and
          ^g : (static member get_One : ->  ^g)

Crossfoot adding some type annotations:

let inline crossfoot2 (n:^a) : ^a =
    let (zero:^a) = 0G
    let (ten:^a) = 10G
    let rec compute (n:^a) =
        if n = zero then zero
        else ((n % ten):^a) + compute (n / ten)
    compute n

val inline crossfoot2 :
   ^a ->  ^a
    when  ^a : (static member get_Zero : ->  ^a) and
         ( ^a or  ^a0) : (static member ( - ) :  ^a *  ^a0 ->  ^a0) and
         ( ^a or  ^a0) : (static member ( + ) :  ^a *  ^a0 ->  ^a) and
          ^a : equality and  ^a : (static member ( + ) :  ^a *  ^a ->  ^a) and
          ^a : (static member ( % ) :  ^a *  ^a ->  ^a) and
          ^a : (static member ( / ) :  ^a *  ^a ->  ^a) and
          ^a0 : (static member get_One : ->  ^a0)

Record Type Technique

module LP =
    let inline zero_of (target:'a) : 'a = LanguagePrimitives.GenericZero<'a>
    let inline one_of (target:'a) : 'a = LanguagePrimitives.GenericOne<'a>
    let inline two_of (target:'a) : 'a = one_of(target) + one_of(target)
    let inline three_of (target:'a) : 'a = two_of(target) + one_of(target)
    let inline negone_of (target:'a) : 'a = zero_of(target) - one_of(target)

    let inline any_of (target:'a) (x:int) : 'a =
        let one:'a = one_of target
        let zero:'a = zero_of target
        let xu = if x > 0 then 1 else -1
        let gu:'a = if x > 0 then one else zero-one

        let rec get i g = 
            if i = x then g
            else get (i+xu) (g+gu)
        get 0 zero 

    type G<'a> = {
        negone:'a
        zero:'a
        one:'a
        two:'a
        three:'a
        any: int -> 'a
    }    

    let inline G_of (target:'a) : (G<'a>) = {
        zero = zero_of target
        one = one_of target
        two = two_of target
        three = three_of target
        negone = negone_of target
        any = any_of target
    }

open LP

Crossfoot, no annotations required for nice inferred signature:

let inline crossfoot3 n =
    let g = G_of n
    let ten = g.any 10
    let rec compute n =
        if n = g.zero then g.zero
        else n % ten + compute (n / ten)
    compute n

val inline crossfoot3 :
   ^a ->  ^a
    when  ^a : (static member ( % ) :  ^a *  ^a ->  ^b) and
         ( ^b or  ^a) : (static member ( + ) :  ^b *  ^a ->  ^a) and
          ^a : (static member get_Zero : ->  ^a) and
          ^a : (static member get_One : ->  ^a) and
          ^a : (static member ( + ) :  ^a *  ^a ->  ^a) and
          ^a : (static member ( - ) :  ^a *  ^a ->  ^a) and  ^a : equality and
          ^a : (static member ( / ) :  ^a *  ^a ->  ^a)

Crossfoot, no annotations, accepts precomputed instances of G:

let inline crossfootG g ten n =
    let rec compute n =
        if n = g.zero then g.zero
        else n % ten + compute (n / ten)
    compute n

val inline crossfootG :
  G< ^a> ->  ^b ->  ^a ->  ^a
    when ( ^a or  ^b) : (static member ( % ) :  ^a *  ^b ->  ^c) and
         ( ^c or  ^a) : (static member ( + ) :  ^c *  ^a ->  ^a) and
         ( ^a or  ^b) : (static member ( / ) :  ^a *  ^b ->  ^a) and
          ^a : equality

I use the above in practice since then I can make precomputed type specific versions which don't suffer from the performance cost of Generic LanguagePrimitives:

let gn = G_of 1  //int32
let gL = G_of 1L //int64
let gI = G_of 1I //bigint

let gD = G_of 1.0  //double
let gS = G_of 1.0f //single
let gM = G_of 1.0m //decimal

let crossfootn = crossfootG gn (gn.any 10)
let crossfootL = crossfootG gL (gL.any 10)
let crossfootI = crossfootG gI (gI.any 10)
let crossfootD = crossfootG gD (gD.any 10)
let crossfootS = crossfootG gS (gS.any 10)
let crossfootM = crossfootG gM (gM.any 10)

Answer 6

Is crossfoot exactly what you want to do, or is it just summing the digits of a long number?

because if you just want to sum the digits, then:

let crossfoot (x:'a) = x.ToString().ToCharArray()
                       |> (Array.fold(fun acc x' -> if x' <> '.' 
                                                    then acc + (int x')
                                                    else acc) 0)

... And you are done.

Anyways, Can you convert stuff to a string, drop the decimal point, remember where the decimal point is, interpret it as an int, run crossfoot?

Here is my solution. I am not sure exactly how you want "crossfoot" to work when you have a decimal point added.

For instance, do you want: crossfoot(123.1) = 7 or crossfoot(123.1) = 6.1? (I'm assuming you want the latter)

Anyways, the code does allow you to work with numbers as generics.

let crossfoot (n:'a) = // Completely generic input

    let rec crossfoot' (a:int) =       // Standard integer crossfoot
        if a = 0 then 0 
        else a%10 + crossfoot' (a / 10)

    let nstr = n.ToString()

    let nn   = nstr.Split([|'.'|])    // Assuming your main constraint is float/int

    let n',n_ = if nn.Length > 1 then nn.[0],nn.[1] 
                else nn.[0],"0"

    let n'',n_' = crossfoot'(int n'),crossfoot'(int n_)

    match n_' with
    | 0 -> string n''
    | _ -> (string n'')+"."+(string n_')

If you need to input big integers or int64 stuff, the way crossfoot works, you can just split the big number into bitesize chunks (strings) and feed them into this function, and add them together.