I have a function getFullFitness
let getFullFitness population =
ResizeArray(population |> Seq.map snd) |> Seq.sum
and I pattern match in the function realTick
let realTick population =
match(getFullfitness population) with
|(50) -> population
| _ -> childGeneration population
Question is on the line |(50) -> population. Since getFullFitness returns an integer sum, how do i match values between 0 and 50 in realTick?
One way is to use a guard -
|t when t < 50 -> ...
In F#, a recommended way to pattern match on ranges is to use active patterns. If we carefully manipulate pattern names, it will look quite close to what we want:
let (|``R0..50``|_|) i =
if i >= 0 && i <= 50 then Some() else None
let realTick population =
match(getFullfitness population) with
| ``R0..50`` -> population
| _ -> childGeneration population
Pattern matching on ranges is supported in OCaml, but it's unlikely to be added to F#. See the related User Voice request at http://fslang.uservoice.com/forums/245727-f-language/suggestions/6027309-allow-pattern-matching-on-ranges.
If you are choosing between two ranges of numbers like in your example, I would
just use a if-then-else expression:
let realTick population =
let fitness = getFullFitness population
if 0 <= fitness && fitness <= 50 then
population
else
childGeneration population
or a simple guard:
let realTick population =
match getFullFitness population with
| fitness when 0 <= fitness && fitness <= 50 ->
population
| _ ->
childGeneration population
If your actual choice is much more complicated, then you might want to use
active patterns. Unlike #pad, I would use a parameterized active pattern:
let (|BetweenInclusive|_|) lo hi x =
if lo <= x && x <= hi then Some () else None
let realTick population =
match getFullFitness population with
| BetweenInclusive 0 50 ->
population
| _ ->
childGeneration population
One higher-order active pattern I've found occasionally useful is a general
purpose predicate:
let (|Is|_|) predicate x =
if predicate x then Some () else None
Using Is you could write something like this:
let lessEq lo x = x <= lo
let greaterEq hi x = hi <= x
let realTick population =
match getFullFitness population with
| Is (greaterEq 0) & Is (lessEq 50) ->
population
| _ ->
childGeneration population
Note that, while something like this is overkill in a simple example like this,
it can be convenient in more complicated scenarios. I personally used active
patterns similar to this to implement a simplification pass in an optimizing
compiler that pattern matched over a large number of cases of primitive
operations and properties of the parameters given to those primitives.
Related
I'm trying to create an infinite Stream in F# that contains armstrong numbers. An armstrong number is one whose cubes of its digits add up to the number. For example, 153 is an armstrong number because 1^3 + 5^3 + 3^3 = 153. so far, I have created several functions to help me do so. They are:
type 'a stream = Cons of 'a * (unit -> 'a stream);;
let rec upfrom n = Cons (n, fun() -> upfrom (n+1));;
let rec toIntArray = function
| 0 -> []
| n -> n % 10 :: toIntArray (n / 10);;
let rec makearmstrong = function
| [] -> 0
| y::ys -> (y * y * y) + makearmstrong ys;;
let checkarmstrong n = n = makearmstrong(toIntArray n);;
let rec take n (Cons(x,xsf)) =
match n with
| 0 -> []
| _ -> x :: take (n-1)(xsf());;
let rec filter p (Cons (x, xsf)) =
if p x then Cons (x, fun() -> filter p (xsf()))
else filter p (xsf());;
And finally:
let armstrongs = filter (fun n -> checkarmstrong n)(upfrom 1);;
Now, when I do take 4 armstrongs;;, (or any number less than 4) this works perfectly and gives me [1;153;370;371] but if I do take 5 armstrongs;;nothing happens, it seems like the program freezes.
I believe the issue is that there are no numbers after 407 that are the sums of their cubes (see http://oeis.org/A046197), but when your code evaluates the equivalent of take 1 (Cons(407, filter checkarmstrong (upfrom 408))) it's going to force the evaluation of the tail and filter will recurse forever, never finding a matching next element. Also note that your definition of Armstrong numbers differs from, say, Wikipedia's, which states that the power the digits are raised to should be the number of digits in the number.
I'm trying to build the nth power function in F#. (Yes, there's already Math.Pow in .Net). Here is my attempt:
let rec nthPower x n =
match n with
| 0 -> 1
| _ -> x * (nthPower x (n-1))
This works fine when n >= 0; however, I don't know how to handle the negative case: when n < 0.
Question:
How to handle the negative case? (n<0)
Is this recursive algorithm efficient? or are there any efficient ways in F#?
You can implement it like this:
let rec nthPower x n =
match n with
| 0 -> 1m
| t when t < 0 -> 1m / (nthPower x -n)
| _ -> decimal x * (nthPower x (n - 1));;
The t when t < 0 allows the pattern matching to match a range of values. I would say that the RHS of this line is self-explanatory, but let me know if it's unclear.
Regarding question #2, I don't think there's anything particularly inefficient about this approach and there's probably not a much simpler way to do it. I'm not sure what the most efficient approach is, but hopefully some mathematicians can chime in.
Edit: I have found an approach that is more efficient for exponents > ~10. It uses memoization and divide-and-conquer to compute the result in O(log n) time instead of O(n):
let rec nthPower x n =
match n with
| 0 -> 1.0
| 1 -> double x
| t when t < 0 -> 1.0 / (nthPower x -n)
| _ ->
let p = nthPower x (n / 2)
p * p * nthPower x (n % 2)
In ML-family languages, people tend to prefer pattern matching to if/else construct. In F#, using guards within pattern matching could easily replace if/else in many cases.
For example, a simple delete1 function could be rewritten without using if/else (see delete2):
let rec delete1 (a, xs) =
match xs with
| [] -> []
| x::xs' -> if x = a then xs' else x::delete1(a, xs')
let rec delete2 (a, xs) =
match xs with
| [] -> []
| x::xs' when x = a -> xs'
| x::xs' -> x::delete2(a, xs')
Another example is solving quadratic functions:
type Solution =
| NoRoot
| OneRoot of float
| TwoRoots of float * float
let solve1 (a,b,c) =
let delta = b*b-4.0*a*c
if delta < 0.0 || a = 0.0 then NoRoot
elif delta = 0.0 then OneRoot (-b/(2.0*a))
else
TwoRoots ((-b + sqrt(delta))/(2.0*a), (-b - sqrt(delta))/(2.0*a))
let solve2 (a,b,c) =
match a, b*b-4.0*a*c with
| 0.0, _ -> NoRoot
| _, delta when delta < 0.0 -> NoRoot
| _, 0.0 -> OneRoot (-b/(2.0*a))
| _, delta -> TwoRoots((-b + sqrt(delta))/(2.0*a),(-b - sqrt(delta))/(2.0*a))
Should we use pattern matching with guards to ignore ugly if/else construct?
Is there any performance implication against using pattern matching with guards? My impression is that it seems to be slow because pattern matching has be checked at runtime.
The right answer is probably it depends, but I surmise, in most cases, the compiled representation is the same. As an example
let f b =
match b with
| true -> 1
| false -> 0
and
let f b =
if b then 1
else 0
both translate to
public static int f(bool b)
{
if (!b)
{
return 0;
}
return 1;
}
Given that, it's mostly a matter of style. Personally I prefer pattern matching because the cases are always aligned, making it more readable. Also, they're (arguably) easier to expand later to handle more cases. I consider pattern matching an evolution of if/then/else.
There is also no additional run-time cost for pattern matching, with or without guards.
Both have their own place. People are more used to If/else construct for checking a value where as pattern matching is like a If/else on steroids. Pattern matching allows you to sort of compare against the decomposed structure of the data along with using gaurds for specifying some additional condition on the parts of the decomposed data or some other value (specially in case of recursive data structures or so called discriminated unions in F#).
I personally prefer to use if/else for simple values comparisons (true/false, ints etc), but in case you have a recursive data structure or something which you need to compare against its decomposed value than there is nothing better than pattern matching.
First make it work and make it elegant and simple and then if you seem some performance problem then check for performance issues (which mostly will be due to some other logic and not due to pattern matching)
Agree with #Daniel that pattern matching is usually more flexible.
Check this implementation:
type Solution = | Identity | Roots of float list
let quadraticEquation x =
let rec removeZeros list =
match list with
| 0.0::rest -> removeZeros rest
| _ -> list
let x = removeZeros x
match x with
| [] -> Identity // zero constant
| [_] -> Roots [] // non-zero constant
| [a;b] -> Roots [ -b/a ] // linear equation
| [a;b;c] ->
let delta = b*b - 4.0*a*c
match delta with
| delta when delta < 0.0 ->
Roots [] // no real roots
| _ ->
let d = sqrt delta
let x1 = (-b-d) / (2.0*a)
let x2 = (-b+d) / (2.0*a)
Roots [x1; x2]
| _ -> failwithf "equation is bigger than quadratic: %A" x
Also notice in https://fsharpforfunandprofit.com/learning-fsharp/ that it is discouraged to use if-else. It is considered a bid less functional.
I did some testing on a self writen prime number generator, and as far as i can say there is "if then else" is significantly slower than pattern matching, can't explain why though, but I as far as I have tested the imperativ part of F# have a slower run time than recursive functional style when it come to optimal algorithms.
I'm new to F# and functional and am working on some HTML parsing code. I want to remove from a HTML document elements that match some criteria. Here I have a sequence of objects (HtmlNodes) and want to remove them from the document.
Is this idiomatic way of using pattern matching? Also as HtmlNode.Remove() has a side-effect on the original HtmlDocument object, is there any particular way of structuring the code to make the side-effect obvious or how should this be handled. You can be as pedantic as you like with the code.
open HtmlAgilityPack
let removeNodes (node : HtmlNode) =
let (|HiddenNodeCount|) (n : HtmlNode) =
match n.SelectNodes("*[#style[contains(.,'visibility:hidden')]]") with
| null -> 0
| _ as x -> Seq.length x
match node with
| x when x.Name.ToLower() = "script" -> node.Remove()
| x when x.NodeType = HtmlNodeType.Comment -> node.Remove()
| HiddenNodeCount x when x > 0 -> node.Remove()
| _ -> ()
let html = "some long messy html code would be here"
let dom = new HtmlDocument(OptionAutoCloseOnEnd=true)
dom.LoadHtml(html)
let nodes = dom.DocumentNode.DescendantNodes()
nodes |> Seq.toArray |> Array.iter removeNodes
Personally, I prefer if elif else over pattern matching when you don't have a data structure to decompose (it's just less typing, and may also serve to differentiate between when a structure is being decomposed versus simpler case testing).
There are some odd things in your code. The Active Pattern isn't very helpful here for two reasons: first, its scope is limited to removeNodes so it is only used once. I'll address the second issues later, but first I will show how I would write this by eliminating the Active Pattern and, for me at least, making the side-effects more obvious (by separating the code which tests whether a node should be removed from the code that does the removing):
let shouldRemoveNode (node : HtmlNode) =
if node.Name.ToLower() = "script" then true
elif node.NodeType = HtmlNodeType.Comment then true
else match node.SelectNodes("*[#style[contains(.,'visibility:hidden')]]") with
| null -> false
| x -> Seq.length x > 0
let removeNode (node: HtmlNode) =
if shouldRemoveNode(node) then node.Remove() else ()
Notice I do use a pattern match in the visibility hidden query since I do get to match against null and bind to x otherwise (rather than binding to x, and then testing x with if else).
The second odd thing with your Active Pattern is that you are using it for converting a node to an int, but the length you obtain isn't immediately useful (you still need to perform a test against it). Whereas the more powerful use of an Active Pattern here would be to carve up nodes into different kinds (assuming this isn't ad-hoc, which was may first point). So you could have:
//expand to encompass several other kinds of nodes
let (|Script|Comment|Hidden|Other|) (node : HtmlNode) =
if node.Name.ToLower() = "script" then Script
elif node.NodeType = HtmlNodeType.Comment then Comment
else match node.SelectNodes("*[#style[contains(.,'visibility:hidden')]]") with
| null -> Other
| x -> if Seq.length x > 0 then Hidden
else Other
let removeNode (node: HtmlNode) =
match node with
| Script | Comment | Hidden -> node.Remove()
| Other -> ()
Edit:
#Pascal made the observation in the comments that shouldRemoveNode can be further condensed into one big boolean expression:
let shouldRemoveNode (node : HtmlNode) =
node.Name.ToLower() = "script" ||
node.NodeType = HtmlNodeType.Comment ||
match node.SelectNodes("*[#style[contains(.,'visibility:hidden')]]") with
| null -> false
| x -> Seq.length x > 0
It isn't clear to me that this is any better than using functions and if-then-else, e.g.
let HiddenNodeCount (n : HtmlNode) =
match n.SelectNodes("*[#style[contains(.,'visibility:hidden')]]") with
| null -> 0
| x -> Seq.length x
if node.Name.ToLower() = "script" then
node.Remove()
elif node.NodeType = HtmlNodeType.Comment then
node.Remove()
elif HiddenNodeCount node > 0 then
node.Remove()
I am trying to learn F# so I paid a visit to Project Euler and I am currently working on Problem 3.
The prime factors of 13195 are 5, 7,
13 and 29.
What is the largest prime
factor of the number 600851475143?
Some things to consider:
My first priority is to learn good functional habits.
My second priority is I would like it to be fast and efficient.
Within the following code I have marked the section this question is regarding.
let isPrime(n:int64) =
let rec check(i:int64) =
i > n / 2L or (n % i <> 0L && check(i + 1L))
check(2L)
let greatestPrimeFactor(n:int64) =
let nextPrime(prime:int64):int64 =
seq { for i = prime + 1L to System.Int64.MaxValue do if isPrime(i) then yield i }
|> Seq.skipWhile(fun v -> n % v <> 0L)
|> Seq.hd
let rec findNextPrimeFactor(number:int64, prime:int64):int64 =
if number = 1L then prime else
//************* No variable
(fun p -> findNextPrimeFactor(number / p, p))(nextPrime(prime))
//*************
//************* Variable
let p = nextPrime(prime)
findNextPrimeFactor(number / p, p)
//*************
findNextPrimeFactor(n, 2L)
Update
Based off some of the feedback I have refactored the code to be 10 times faster.
module Problem3
module private Internal =
let execute(number:int64):int64 =
let rec isPrime(value:int64, current:int64) =
current > value / 2L or (value % current <> 0L && isPrime(value, current + 1L))
let rec nextPrime(prime:int64):int64 =
if number % prime = 0L && isPrime(prime, 2L) then prime else nextPrime(prime + 1L)
let rec greatestPrimeFactor(current:int64, prime:int64):int64 =
if current = 1L then prime else nextPrime(prime + 1L) |> fun p -> greatestPrimeFactor(current / p, p)
greatestPrimeFactor(number, 2L)
let execute() = Internal.execute(600851475143L)
Update
I would like to thank everyone for there advice. This latest version is a compilation of all the advice I received.
module Problem3
module private Internal =
let largestPrimeFactor number =
let rec isPrime value current =
current > value / 2L || (value % current <> 0L && isPrime value (current + 1L))
let rec nextPrime value =
if number % value = 0L && isPrime value 2L then value else nextPrime (value + 1L)
let rec find current prime =
match current / prime with
| 1L -> prime
| current -> nextPrime (prime + 1L) |> find current
find number (nextPrime 2L)
let execute() = Internal.largestPrimeFactor 600851475143L
Functional programming becomes easier and more automatic with practice, so don't sweat it if you don't get it absolutely right on the first try.
With that in mind, let's take your sample code:
let rec findNextPrimeFactor(number:int64, prime:int64):int64 =
if number = 1L then prime else
//************* No variable
(fun p -> findNextPrimeFactor(number / p, p))(nextPrime(prime))
//*************
//************* Variable
let p = nextPrime(prime)
findNextPrimeFactor(number / p, p)
//*************
Your no variable version is just weird, don't use it. I like your version with the explicit let binding.
Another way to write it would be:
nextPrime(prime) |> fun p -> findNextPrimeFactor(number / p, p)
Its ok and occasionally useful to write it like this, but still comes across as a little weird. Most of the time, we use |> to curry values without needing to name our variables (in "pointfree" style). Try to anticipate how your function will be used, and if possible, re-write it so you can use it with the pipe operator without explicit declared variables. For example:
let rec findNextPrimeFactor number prime =
match number / prime with
| 1L -> prime
| number' -> nextPrime(prime) |> findNextPrimeFactor number'
No more named args :)
Ok, now that we have that out of the way, let's look at your isPrime function:
let isPrime(n:int64) =
let rec check(i:int64) =
i > n / 2L or (n % i <> 0L && check(i + 1L))
check(2L)
You've probably heard to use recursion instead of loops, and that much is right. But, wherever possible, you should abstract away recursion with folds, maps, or higher order functions. Two reasons for this:
its a little more readable, and
improperly written recursion will result in a stack overflow. For example, your function is not tail recursive, so it'll blow up on large values of n.
I'd rewrite isPrime like this:
let isPrime n = seq { 2L .. n / 2L } |> Seq.exists (fun i -> n % i = 0L) |> not
Most of the time, if you can abstract away your explicit looping, then you're just applying transformations to your input sequence until you get your results:
let maxFactor n =
seq { 2L .. n - 1L } // test inputs
|> Seq.filter isPrime // primes
|> Seq.filter (fun x -> n % x = 0L) // factors
|> Seq.max // result
We don't even have intermediate variables in this version. Coolness!
My second priority is I would like it
to be fast and efficient.
Most of the time, F# is going to be pretty comparable with C# in terms of speed, or it's going to be "fast enough". If you find your code takes a long time to execute, it probably means you're using the wrong data structure or a bad algorithm. For a concrete example, read the comments on this question.
So, the code I've written is "elegant" in the sense that its concise, gives the correct results, and doesn't rely on any trickery. Unfortunately, its not very fast. For start:
it uses trial division to create a sequence of primes, when the Sieve of Eratosthenes would be much faster. [Edit: I wrote a somewhat naive version of this sieve which didn't work for numbers larger than Int32.MaxValue, so I've removed the code.]
read Wikipedia's article on the prime counting function, it'll give you pointers on calculating the first n primes as well as estimating the upper and lower bounds for the nth prime.
[Edit: I included some code with a somewhat naive implementation of a sieve of eratosthenes. It only works for inputs less than int32.MaxValue, so it probably isn't suitable for project euler.]
Concerning "good functional habit" or rather good practice I see three minor things. Using the yield in your sequence is a little harder to read than just filter. Unnecessary type annotations in a type inferred language leads to difficult refactoring and makes the code harder to read. Don't go overboard and try to remove every type annotation though if you're finding it difficult. Lastly making a lambda function which only takes a value to use as a temp variable reduces readability.
As far as personal style goes I prefer more spaces and only using tupled arguments when the data makes sense being grouped together.
I'd write your original code like this.
let isPrime n =
let rec check i =
i > n / 2L || (n % i <> 0L && check (i + 1L))
check 2L
let greatestPrimeFactor n =
let nextPrime prime =
seq {prime + 1L .. System.Int64.MaxValue}
|> Seq.filter isPrime
|> Seq.skipWhile (fun v -> n % v <> 0L)
|> Seq.head
let rec findNextPrimeFactor number prime =
if number = 1L then
prime
else
let p = nextPrime(prime)
findNextPrimeFactor (number / p) p
findNextPrimeFactor n 2L
Your updated code is optimal for your approach. You would have to use a different algorithm like Yin Zhu answer to go faster. I wrote a test to check to see if F# makes the "check" function tail recursive and it does.
the variable p is actually a name binding, not a variable. Using name binding is not a bad style. And it is more readable. The lazy style of nextPrime is good, and it actually prime-test each number only once during the whole program.
My Solution
let problem3 =
let num = 600851475143L
let rec findMax (n:int64) (i:int64) =
if n=i || n<i then
n
elif n%i=0L then
findMax (n/i) i
else
findMax n (i+1L)
findMax num 2L
I basically divides num from 2, 3, 4.. and don't consider any prime numbers. Because if we divides all 2 from num, then we won't be able to divide it by 4,8, etc.
on this number, my solution is quicker:
> greatestPrimeFactor 600851475143L;;
Real: 00:00:01.110, CPU: 00:00:00.702, GC gen0: 1, gen1: 1, gen2: 0
val it : int64 = 6857L
>
Real: 00:00:00.001, CPU: 00:00:00.000, GC gen0: 0, gen1: 0, gen2: 0
val problem3 : int64 = 6857L
I think that the code with the temporary binding is significantly easier to read. It's pretty unusual to create an anonymous function and then immediately apply it to a value as you do in the other case. If you really want to avoid using a temporary value, I think that the most idiomatic way to do that in F# would be to use the (|>) operator to pipe the value into the anonymous function, but I still think that this isn't quite as readable.