I'm learning F# (again) and I'm trying to sum some rows in excel. This is my attempt.
let sumRows (source: seq<double[]>) =
source
|> Seq.reduce (fun a b -> (a, b) ||> Seq.map2 (fun x y -> x + y) |> Seq.toArray)
Can it be done better? I already discovered double forward pipe operator, but now, whole part fun a b -> (a, b) ||> seems quite redundant...
You are right it is redundant, because the double pipe operator is convenient only when you need to convert a tuple into two separate parameters. In this case you already have them as 2 separate parameters so you could just pass them like this:
let sumRows (source: seq<double[]>) =
source
|> Seq.reduce (fun a b -> Seq.map2 (fun x y -> x + y) a b |> Seq.toArray)
We can get rid of the |> Seq.toArray by replacing Seq.map2 with Array.map2:
let sumRows (source: seq<double[]>) =
source
|> Seq.reduce (fun a b -> Array.map2 (fun x y -> x + y) a b)
now we can simplify further by removing the fun a b ->:
let sumRows (source: seq<double[]>) =
source
|> Seq.reduce (Array.map2 (fun x y -> x + y) )
Finally, did you know that an operator like + can be treated as a 2 parameters function
by putting in parenthesis (+)?
let sumRows2 (source: seq<double[]>) =
source
|> Seq.reduce (Array.map2 (+) )
All of this versions have the same signature and they are all correct. You choose whichever suits more your style.
BTW, you maybe tempted to go one further and do this:
let sumRows2 = Seq.reduce (Array.map2 (+) )
but it causes issues with the famous Value restriction error. There are workarounds like adding the type annotation or actually using it somewhere in the code, but the best workaround is to add the parameter, like we had before.
I'm trying to explore the dynamic capabilities of F# for situations where I can't express some function with the static type system. As such, I'm trying to create a mapN function for (say) Option types, but I'm having trouble creating a function with a dynamic number of arguments. I've tried:
let mapN<'output> (f : obj) args =
let rec mapN' (state:obj) (args' : (obj option) list) =
match args' with
| Some x :: xs -> mapN' ((state :?> obj -> obj) x) xs
| None _ :: _ -> None
| [] -> state :?> 'output option
mapN' f args
let toObjOption (x : #obj option) =
Option.map (fun x -> x :> obj) x
let a = Some 5
let b = Some "hi"
let c = Some true
let ans = mapN<string> (fun x y z -> sprintf "%i %s %A" x y z) [a |> toObjOption; b |> toObjOption; c |> toObjOption]
(which takes the function passed in and applies one argument at a time) which compiles, but then at runtime I get the following:
System.InvalidCastException: Unable to cast object of type 'ans#47' to type
'Microsoft.FSharp.Core.FSharpFunc`2[System.Object,System.Object]'.
I realize that it would be more idiomatic to either create a computation expression for options, or to define map2 through map5 or so, but I specifically want to explore the dynamic capabilities of F# to see whether something like this would be possible.
Is this just a concept that can't be done in F#, or is there an approach that I'm missing?
I think you would only be able to take that approach with reflection.
However, there are other ways to solve the overall problem without having to go dynamic or use the other static options you mentioned. You can get a lot of the same convenience using Option.apply, which you need to define yourself (or take from a library). This code is stolen and adapted from F# for fun and profit:
module Option =
let apply fOpt xOpt =
match fOpt,xOpt with
| Some f, Some x -> Some (f x)
| _ -> None
let resultOption =
let (<*>) = Option.apply
Some (fun x y z -> sprintf "%i %s %A" x y z)
<*> Some 5
<*> Some "hi"
<*> Some true
To explain why your approach does not work, the problem is that you cannot cast a function of type int -> int (represented as FSharpFunc<int, int>) to a value of type obj -> obj (represented as FSharpFunc<obj, obj>). The types are the same generic types, but the cast fails because the generic parameters are different.
If you insert a lot of boxing and unboxing, then your function actually works, but this is probably not something you want to write:
let ans = mapN<string> (fun (x:obj) -> box (fun (y:obj) -> box (fun (z:obj) ->
box (Some(sprintf "%i %s %A" (unbox x) (unbox y) (unbox z))))))
[a |> toObjOption; b |> toObjOption; c |> toObjOption]
If you wanted to explore more options possible thanks to dynamic hacks - then you can probably do more using F# reflection. I would not typically use this in production (simple is better - I'd just define multiple map functions by hand or something like that), but the following runs:
let rec mapN<'R> f args =
match args with
| [] -> unbox<'R> f
| x::xs ->
let m = f.GetType().GetMethods() |> Seq.find (fun m ->
m.Name = "Invoke" && m.GetParameters().Length = 1)
mapN<'R> (m.Invoke(f, [| x |])) xs
mapN<obj> (fun a b c -> sprintf "%d %s %A" a b c) [box 1; box "hi"; box true]
I am interested to implement fold3, fold4 etc., similar to List.fold and List.fold2. e.g.
// TESTCASE
let polynomial (x:double) a b c = a*x + b*x*x + c*x*x*x
let A = [2.0; 3.0; 4.0; 5.0]
let B = [1.5; 1.0; 0.5; 0.2]
let C = [0.8; 0.01; 0.001; 0.0001]
let result = fold3 polynomial 0.7 A B C
// 2.0 * (0.7 ) + 1.5 * (0.7 )^2 + 0.8 * (0.7 )^3 -> 2.4094
// 3.0 * (2.4094) + 1.0 * (2.4094)^2 + 0.01 * (2.4094)^3 -> 13.173
// 4.0 * (13.173) + 0.5 * (13.173)^2 + 0.001 * (13.173)^3 -> 141.75
// 5.0 * (141.75) + 0.2 * (141.75)^2 + 0.0001 * (141.75)^3 -> 5011.964
//
// Output: result = 5011.964
My first method is grouping the 3 lists A, B, C, into a list of tuples, and then apply list.fold
let fold3 f x A B C =
List.map3 (fun a b c -> (a,b,c)) A B C
|> List.fold (fun acc (a,b,c) -> f acc a b c) x
// e.g. creates [(2.0,1.5,0.8); (3.0,1.0,0.01); ......]
My second method is to declare a mutable data, and use List.map3
let mutable result = 0.7
List.map3 (fun a b c ->
result <- polynomial result a b c // Change mutable data
// Output intermediate data
result) A B C
// Output from List.map3: [2.4094; 13.17327905; 141.7467853; 5011.963942]
// result mutable: 5011.963942
I would like to know if there are other ways to solve this problem. Thank you.
For fold3, you could just do zip3 and then fold:
let polynomial (x:double) (a, b, c) = a*x + b*x*x + c*x*x*x
List.zip3 A B C |> List.fold polynomial 0.7
But if you want this for the general case, then you need what we call "applicative functors".
First, imagine you have a list of functions and a list of values. Let's assume for now they're of the same size:
let fs = [ (fun x -> x+1); (fun x -> x+2); (fun x -> x+3) ]
let xs = [3;5;7]
And what you'd like to do (only natural) is to apply each function to each value. This is easily done with List.map2:
let apply fs xs = List.map2 (fun f x -> f x) fs xs
apply fs xs // Result = [4;7;10]
This operation "apply" is why these are called "applicative functors". Not just any ol' functors, but applicative ones. (the reason for why they're "functors" is a tad more complicated)
So far so good. But wait! What if each function in my list of functions returned another function?
let f1s = [ (fun x -> fun y -> x+y); (fun x -> fun y -> x-y); (fun x -> fun y -> x*y) ]
Or, if I remember that fun x -> fun y -> ... can be written in the short form of fun x y -> ...
let f1s = [ (fun x y -> x+y); (fun x y -> x-y); (fun x y -> x*y) ]
What if I apply such list of functions to my values? Well, naturally, I'll get another list of functions:
let f2s = apply f1s xs
// f2s = [ (fun y -> 3+y); (fun y -> 5+y); (fun y -> 7+y) ]
Hey, here's an idea! Since f2s is also a list of functions, can I apply it again? Well of course I can!
let ys = [1;2;3]
apply f2s ys // Result: [4;7;10]
Wait, what? What just happened?
I first applied the first list of functions to xs, and got another list of functions as a result. And then I applied that result to ys, and got a list of numbers.
We could rewrite that without intermediate variable f2s:
let f1s = [ (fun x y -> x+y); (fun x y -> x-y); (fun x y -> x*y) ]
let xs = [3;5;7]
let ys = [1;2;3]
apply (apply f1s xs) ys // Result: [4;7;10]
For extra convenience, this operation apply is usually expressed as an operator:
let (<*>) = apply
f1s <*> xs <*> ys
See what I did there? With this operator, it now looks very similar to just calling the function with two arguments. Neat.
But wait. What about our original task? In the original requirements we don't have a list of functions, we only have one single function.
Well, that can be easily fixed with another operation, let's call it "apply first". This operation will take a single function (not a list) plus a list of values, and apply this function to each value in the list:
let applyFirst f xs = List.map f xs
Oh, wait. That's just map. Silly me :-)
For extra convenience, this operation is usually also given an operator name:
let (<|>) = List.map
And now, I can do things like this:
let f x y = x + y
let xs = [3;5;7]
let ys = [1;2;3]
f <|> xs <*> ys // Result: [4;7;10]
Or this:
let f x y z = (x + y)*z
let xs = [3;5;7]
let ys = [1;2;3]
let zs = [1;-1;100]
f <|> xs <*> ys <*> zs // Result: [4;-7;1000]
Neat! I made it so I can apply arbitrary functions to lists of arguments at once!
Now, finally, you can apply this to your original problem:
let polynomial a b c (x:double) = a*x + b*x*x + c*x*x*x
let A = [2.0; 3.0; 4.0; 5.0]
let B = [1.5; 1.0; 0.5; 0.2]
let C = [0.8; 0.01; 0.001; 0.0001]
let ps = polynomial <|> A <*> B <*> C
let result = ps |> List.fold (fun x f -> f x) 0.7
The list ps consists of polynomial instances that are partially applied to corresponding elements of A, B, and C, and still expecting the final argument x. And on the next line, I simply fold over this list of functions, applying each of them to the result of the previous.
You could check the implementation for ideas:
https://github.com/fsharp/fsharp/blob/master/src/fsharp/FSharp.Core/array.fs
let fold<'T,'State> (f : 'State -> 'T -> 'State) (acc: 'State) (array:'T[]) =
checkNonNull "array" array
let f = OptimizedClosures.FSharpFunc<_,_,_>.Adapt(f)
let mutable state = acc
for i = 0 to array.Length-1 do
state <- f.Invoke(state,array.[i])
state
here's a few implementations for you:
let fold2<'a,'b,'State> (f : 'State -> 'a -> 'b -> 'State) (acc: 'State) (a:'a array) (b:'b array) =
let mutable state = acc
Array.iter2 (fun x y->state<-f state x y) a b
state
let iter3 f (a: 'a[]) (b: 'b[]) (c: 'c[]) =
let f = OptimizedClosures.FSharpFunc<_,_,_,_>.Adapt(f)
if a.Length <> b.Length || a.Length <> c.Length then failwithf "length"
for i = 0 to a.Length-1 do
f.Invoke(a.[i], b.[i], c.[i])
let altIter3 f (a: 'a[]) (b: 'b[]) (c: 'c[]) =
if a.Length <> b.Length || a.Length <> c.Length then failwithf "length"
for i = 0 to a.Length-1 do
f (a.[i]) (b.[i]) (c.[i])
let fold3<'a,'b,'State> (f : 'State -> 'a -> 'b -> 'c -> 'State) (acc: 'State) (a:'a array) (b:'b array) (c:'c array) =
let mutable state = acc
iter3 (fun x y z->state<-f state x y z) a b c
state
NB. we don't have an iter3, so, implement that. OptimizedClosures.FSharpFunc only allow up to 5 (or is it 7?) params. There are a finite number of type slots available. It makes sense. You can go higher than this, of course, without using the OptimizedClosures stuff.
... anyway, generally, you don't want to be iterating too many lists / arrays / sequences at once. So I'd caution against going too high.
... the better way forward in such cases may be to construct a record or tuple from said lists / arrays, first. Then, you can just use map and iter, which are already baked in. This is what zip / zip3 are all about (see: "(array1.[i],array2.[i],array3.[i])")
let zip3 (array1: _[]) (array2: _[]) (array3: _[]) =
checkNonNull "array1" array1
checkNonNull "array2" array2
checkNonNull "array3" array3
let len1 = array1.Length
if len1 <> array2.Length || len1 <> array3.Length then invalidArg3ArraysDifferent "array1" "array2" "array3" len1 array2.Length array3.Length
let res = Microsoft.FSharp.Primitives.Basics.Array.zeroCreateUnchecked len1
for i = 0 to res.Length-1 do
res.[i] <- (array1.[i],array2.[i],array3.[i])
res
I'm working with arrays at the moment, so my solution pertained to those. Sorry about that. Here's a recursive version for lists.
let fold3 f acc a b c =
let mutable state = acc
let rec fold3 f a b c =
match a,b,c with
| [],[],[] -> ()
| [],_,_
| _,[],_
| _,_,[] -> failwith "length"
| ahead::atail, bhead::btail, chead::ctail ->
state <- f state ahead bhead chead
fold3 f atail btail ctail
fold3 f a b c
i.e. we define a recursive function within a function which acts upon/mutates/changes the outer scoped mutable acc variable (a closure in functional speak). Finally, this gets returned.
It's pretty cool how much type information gets inferred about these functions. In the array examples above, mostly I was explicit with 'a 'b 'c. This time, we let type inference kick in. It knows we're dealing with lists from the :: operator. That's kind of neat.
NB. the compiler will probably unwind this tail-recursive approach so that it is just a loop behind-the-scenes. Generally, get a correct answer before optimising. Just mentioning this, though, as food for later thought.
I think the existing answers provide great options if you want to generalize folding, which was your original question. However, if I simply wanted to call the polynomial function on inputs specified in A, B and C, then I would probably do not want to introduce fairly complex constructs like applicative functors with fancy operators to my code base.
The problem becomes a lot easier if you transpose the input data, so that rather than having a list [A; B; C] with lists for individual variables, you have a transposed list with inputs for calculating each polynomial. To do this, we'll need the transpose function:
let rec transpose = function
| (_::_)::_ as M -> List.map List.head M :: transpose (List.map List.tail M)
| _ -> []
Now you can create a list with inputs, transpose it and calculate all polynomials simply using List.map:
transpose [A; B; C]
|> List.map (function
| [a; b; c] -> polynomial 0.7 a b c
| _ -> failwith "wrong number of arguments")
There are many ways to solve this problem. Few are mentioned like first zip3 all three list, then run over it. Using Applicate Functors like Fyodor Soikin describes means you can turn any function with any amount of arguments into a function that expects list instead of single arguments. This is a good general solution that works with any numbers of lists.
While this is a general good idea, i'm sometimes shocked that so few use more low-level tools. In this case it is a good idea to use recursion and learn more about recursion.
Recursion here is the right-tool because we have immutable data-types. But you could consider how you would implement it with mutable lists and looping first, if that helps. The steps would be:
You loop over an index from 0 to the amount of elements in the lists.
You check if every list has an element for the index
If every list has an element then you pass this to your "folder" function
If at least one list don't have an element, then you abort the loop
The recursive version works exactly the same. Only that you don't use an index to access the elements. You would chop of the first element from every list and then recurse on the remaining list.
Otherwise List.isEmpty is the function to check if a List is empty. You can chop off the first element with List.head and you get the remaining list with the first element removed by List.tail. This way you can just write:
let rec fold3 f acc l1 l2 l3 =
let h = List.head
let t = List.tail
let empty = List.isEmpty
if (empty l1) || (empty l2) && (empty l3)
then acc
else fold3 f (f acc (h l1) (h l2) (h l3)) (t l1) (t l2) (t l3)
The if line checks if every list has at least one element. If that is true
it executes: f acc (h l1) (h l2) (h l3). So it executes f and passes it the first element of every list as an argument. The result is the new accumulator of
the next fold3 call.
Now that you worked on the first element of every list, you must chop off the first element of every list, and continue with the remaining lists. You achieve that with List.tail or in the above example (t l1) (t l2) (t l3). Those are the next remaining lists for the next fold3 call.
Creating a fold4, fold5, fold6 and so on isn't really hard, and I think it is self-explanatory. My general advice is to learn a little bit more about recursion and try to write recursive List functions without Pattern Matching. Pattern Matching is not always easier.
Some code examples:
fold3 (fun acc x y z -> x + y + z :: acc) [] [1;2;3] [10;20;30] [100;200;300] // [333;222;111]
fold3 (fun acc x y z -> x :: y :: z :: acc) [] [1;2;3] [10;20;30] [100;200;300] // [3; 30; 300; 2; 20; 200; 1; 10; 100]
I'm trying to do some mathematical operations with decimal options on a custom type:
type LineItem = {Cost: decimal option; Price: decimal option; Qty: decimal option}
let discount = 0.25M
let createItem (c, p, q) =
{Cost = c; Price = p; Qty = q}
let items =
[
(Some 1M , None , Some 1M)
(Some 3M , Some 2.0M , None)
(Some 5M , Some 3.0M , Some 5M)
(None , Some 1.0M , Some 2M)
(Some 11M , Some 2.0M , None)
]
|> List.map createItem
I can do some very simple arithmetic with
items
|> Seq.map (fun line -> line.Price
|> Option.map (fun x -> discount * x))
which gives me
val it : seq<decimal option> =
seq [null; Some 0.500M; Some 0.750M; Some 0.250M; ...]
If I try to actually calculate the thing I need
items
|> Seq.map (fun line -> line.Price
|> Option.map (fun x -> discount * x)
|> Option.map (fun x -> x - (line.Cost
|> Option.map (fun x -> x)))
|> Option.map (fun x -> x * (line.Qty
|> Option.map (fun x -> x))))
I get the error
error FS0001: Type constraint mismatch. The type
'a option
is not compatible with type
decimal
The type ''a option' is not compatible with the type 'decimal'
where I would have expected a seq<decimal option>.
I must be missing something but I can't seem to spot whatever it is I'm missing.
You are mixing decimal with decimal option.
If you're trying to solve everything with Option.map you may want to try with Option.bind instead, so your code will be 'linearly nested':
items
|> Seq.map (
fun line ->
Option.bind(fun price ->
Option.bind(fun cost ->
Option.bind(fun qty ->
Some ((discount * price - cost ) * qty)) line.Qty) line.Cost) line.Price)
which can be an interesting exercise, especially if you want to understand monads, then you will be able to use a computation expression, you can create your own or use one from a library like F#x or F#+:
open FSharpPlus.Builders
items |> Seq.map (fun line ->
monad {
let! price = line.Price
let! cost = line.Cost
let! qty = line.Qty
return ((discount * price - cost ) * qty)
}
)
but if you link F#+ you'll have Applicative Math Operators available:
open FSharpPlus.Operators.ApplicativeMath
items |> Seq.map (fun line -> ((discount *| line.Price) |-| line.Cost ) |*| line.Qty)
That's nice stuff to learn but otherwise I would suggest to use F# built-in features instead, like pattern-match, it would be easier:
items
|> Seq.map (fun line -> match line.Price, line.Qty, line.Cost with
| Some price, Some qty, Some cost ->
Some ((discount * price - cost ) * qty)
| _ -> None)
Then since you can also pattern-match over records it can be further reduced to:
items
|> Seq.map (function
| {Cost = Some cost; Price = Some price; Qty = Some qty} ->
Some ((discount * price - cost ) * qty)
| _ -> None)
Note that Option.map (fun x -> x) doesn't transform anything.
One problem you have is the following code:
(line.Cost |> Option.map (fun x -> x))
The lambda function (fun x -> x) already exists. This is the id function. It just returns whatever you have unchanged. You also could write the code you have like this:
(line.Cost |> Option.map id)
And the next thing. Mapping over the id function makes no sense. You unwrap whatever is inside the option, apply the id function to it. What didn't change the decimal at all. Then you wrap the decimal again in an option. You also can just write:
line.Cost
and completely remove the Option.map as it does nothing.
So the code you have here:
|> Option.map (fun x -> x - (line.Cost |> Option.map (fun x -> x)))
is identical to:
|> Option.map (fun x -> x - line.Cost)
This obviously does not work because here you try to subtract x a decimal with line.Cost a option decimal. So you get a type-error.
I guess what you really want to do is to subtract line.Cost from line.Price if line.Cost is present, otherwise you want to preserve line.Price unchanged.
One way would be just to provide a default value for line.Costs that can be used and have no effect on the subtraction. For example you could use the value 0 for subtraction if line.Costs is None.
So you also could write something like this instead:
|> Option.map (fun x -> x - (defaultArg line.Cost 0m))
A default value for multiplication would be 1m. So you overall end with.
items
|> Seq.map (fun line ->
line.Price
|> Option.map (fun x -> discount * x)
|> Option.map (fun x -> x - (defaultArg line.Cost 0m))
|> Option.map (fun x -> x * (defaultArg line.Qty 1m)))
The above code for example returns:
[None; Some -2.500M; Some -21.250M; Some 0.500M; Some -10.500M]
If your goal is that a whole computation turns into None as soon a single
value is None. I would just add map2 as a helper function.
module Option =
let map2 f x y =
match x,y with
| Some x, Some y -> Some (f x y)
| _ -> None
then you just can write:
items
|> List.map (fun line ->
line.Price
|> Option.map (fun price -> price * discount)
|> Option.map2 (fun cost price -> price - cost) line.Cost
|> Option.map2 (fun qty price -> price * qty) line.Qty)
and it will return:
[None; None; Some -21.250M; None; None]
Within the Option.map the x is actually a decimal, but the signature for Option.map is 'T option -> 'U option. So here:
Option.map (fun x -> x - (line.Cost |> Option.map (fun x -> x)))
you have the following:
Option.map (fun x -> /*decimal*/ x - /*decimal option*/(line.Cost |> Option.map (fun x -> x)))
So the decimal option has to be converted to decimal to be compatible with what's in the first Option.map. But now you have to deal with None result.
Below a quick (and nasty) fix that just is using an if statement to extract Value (which will be a decimal) or if None then return 0.
items
|> Seq.map (fun line -> line.Price
|> Option.map (fun x -> discount * x)
|> Option.map (fun x -> x - if line.Cost.IsSome then line.Cost.Value else 0m)
|> Option.map (fun x -> x * if line.Qty.IsSome then line.Qty.Value else 0m))
For a more sophisticated solution I recommend this answer.
For completeness sake, you may also leverage the monadic properties of the option type by "lifting" the values outside of the option. This is a somewhat simpler variant of the applicative approach linked by #PiotrWolkowski and those shown by #Gustavo. Applicatives not only wrap values in the monad, but also the functions applied to them.
We start by making the option type amenable to monadic operations in terms of bind and return. Thankfully, those functions are already defined, there's only a little tweak in the argument order.
let (>>=) ma f = Option.bind f ma
let ``return`` = Some
On top of this there's the lift function and a couple of operators for convenience. If needed, they could be generalized by marking them inline.
let liftOpt op x y =
x >>= fun a ->
y >>= fun b ->
``return`` (op a b)
let (.*) = liftOpt (*)
let (.-) = liftOpt (-)
Now your calculation becomes
items
|> Seq.map (fun line ->
(line.Price .* Some discount .- line.Cost) .* line.Qty )
|> Seq.iter (printfn "%A")
which will print
<null>
<null>
Some -21.250M
<null>
<null>
I have asked a related question here. I want to do a similar thing but this time thread an accumulator though the array of functions. I immediately thought of Array.Reduce or Array.Fold but they are not working for me:
let AddTen x =
x + 10
let MultiplyFive x =
x * 5
let SubtractTwo x =
x - 2
let functionArray = [| AddTen; MultiplyFive; SubtractTwo |]
let calculateAnswer functionArray x = functionArray |>Array.reduce(fun acc f -> f acc)
The last line throws this exception:
Type mismatch. Expecting a
'a -> 'b but given a
'b The resulting type would be infinite when unifying ''a' and ''b -> 'a'
Am I thinking about the problem incorrectly?
Take a look at these two:
let calculateReduce = functionArray |> Array.reduce (fun f g -> f >> g)
let calculateFold x = functionArray |> Array.fold (fun acc f -> f acc) x
In the reduce version, you take an array of functions and compose them into a single function which you can later call on x.
In the fold version you fold over the array of functions, threading the accumulator through and applying each function to it in sequence. x is the initial value of the accumulator here.
Your original code didn't work, because a reduce expects a 'a -> 'a -> 'a function, which in case of an array of functions would imply composition, while you were trying to apply one function of type int -> int to another.