F# 如何编写通用数字的函数？

我对F#很陌生，发现类型推断真的很酷。但目前看来，它也可能导致代码重复，这不是一件很酷的事情。我想对数字的位数求和，如下所示：

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

crossfoot 123

这样可以正确打印

6

。但现在我的输入数字不适合int 32位，所以我必须将其转换为

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

crossfoot 123L

然后，一个

biginger

来了，猜猜看

当然，我只能使用

bigint

版本，根据需要向上转换输入参数，向下转换输出参数。但是首先我假设在

int

上使用

biginger

会带来一些性能上的损失。第二，让cf=int（crossfoot（bigint 123））读起来不好

---

除了kvb使用数字文字（Brian的链接）的技术之外，还有一种通用的方法来编写这个吗，我已经成功地使用了一种不同的技术，这种技术可以产生更好的推断结构类型签名，还可以用于创建预计算类型特定的函数，以获得更好的性能以及对支持的数字类型的控制（因为您通常希望支持所有整数类型，但不支持有理类型，例如）：

继Daniel和我就不同技术产生的推断类型签名进行的讨论之后，下面是一个概述：

数字文字技术 Crossfoot不添加任何类型注释：

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)

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)

Crossfoot添加一些类型注释：

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)

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)

记录型技术 Crossfoot，无注释，接受G的预计算实例：

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

---

基于Brian和Stephen的答案，这里有一些完整的代码：

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

更新：简单答案 这是一个独立的实现，没有

NumericLiteralG

模块，并且有一个限制性稍小的推断类型：

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

解释

F#中实际上有两种类型的泛型：1）通过.NET接口/继承的运行类型多态性，以及2）编译时泛型。编译时泛型需要适应泛型数字操作和duck类型（）之类的操作。这些特性是F#的一部分，但在.NET中不受支持，因此必须在编译时由F#处理

插入符号（

^

）用于区分普通插入符号（使用撇号）。简而言之，

'a

是在运行时处理的，而

^a

是在编译时处理的——这就是为什么函数必须标记为

内联的原因

我以前从未尝试过写这样的东西。结果比我想象的要笨拙。我看到用F#编写通用数字代码的最大障碍是：创建一个非零或非一的通用数字实例。请参阅中Int32的
实现，了解我的意思
GenericZero
和
GenericOne
是内置的，它们是使用用户代码中不可用的技术实现的。在这个函数中，因为我们只需要一个小数字（10），所以我创建了一个10
GenericOne
s的序列，并将它们相加

我也不能解释为什么需要所有的类型注释，只是说每次编译器遇到泛型类型上的操作时，它似乎认为它在处理一个新类型。因此，它最终推断出一些具有重复大小限制的奇怪类型（例如，它可能需要多次
（+）
）。添加类型注释可以让它知道我们始终在处理相同的类型。没有它们，代码可以正常工作，但是添加它们简化了推断的签名。
既然出现了如何在使用广义数字文本时使类型签名不那么复杂的问题，我想我应该投入2美分。主要的问题是F#的运算符可能是不对称的，因此您可以执行类似于
System.DateTime.Now+System.TimeSpan.FromHours（1.0）
的操作，这意味着F#的类型推断在执行算术运算时添加中间类型变量

在数值算法的情况下，这种潜在的不对称性通常是没有用处的，并且导致类型签名的爆炸是相当丑陋的（尽管它通常不会影响F#在给定具体参数时正确应用函数的能力）。这个问题的一个潜在解决方案是将算术运算符的类型限制在您关心的范围内。例如，如果定义此模块：

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
  ...

然后只要打开
SymmetricOps
模块，就可以让操作符只应用于同一类型的两个参数。现在我们可以定义：

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

及

推断出的类型是相对干净的

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

虽然我们仍然可以从一个好的、可读的定义中得到好处，但是crossfoot是否正是您想要做的，或者它只是对一个长数字的数字求和

因为如果您只想对数字求和，那么：

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

如果您需要输入大整数或int64之类的东西，crossfoot的工作方式是，您可以将大数字拆分为bitesize块（字符串），然后将它们输入此函数，并将它们相加。
我在寻找解决方案时偶然发现了这个主题，我正在发布我的答案，因为我找到了一种表达通用数字的方法，而不需要手工构建数字的非最佳实现

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

这个实现在cast过程中没有复杂的迭代。它用于实例模块

我担心没有，至少从.NET方面来说没有，因为.NET没有一个通用的子类型来表示所有的数字类型（实际上，它根本不能，因为所有的值类型都必须直接从System.ValueType派生）。也就是说，我不知道F#，所以他们可能以某种方式解决了这个问题——但不要抱太高的希望。注意这里使用数字文字t的
crossfoot
的疯狂推断类型签名
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

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

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_')

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