I would like to use functions (for example exponential functions), which only works on Floating types. My input can be any type of Num: Integer, Float, Fractional ...
How is it possible to convert all of them to a Floating?
numToFloating :: (Num a, Floating b) => a -> b
...
I have no idea, where to start
You cannot. The following is a valid specialization
numToFloating :: Complex Float -> Float
since ComplexFloat is an instance of Num. What are we supposed to do with the imaginary part?
So what you really want to say is that the input is any Real type, giving the signature
:: (Real a, Floating b) => a -> b
which turns out to be a little stronger than necessary. There is already realToFrac
realToFrac :: (Real a, Fractional b) => a -> b
which will do what you need, since any Floating type is already Fractional.
Related
If i have a value which is restricted to be fractional, is there a way i can check it either for being a zero value or for some neutral value? I'm trying to implement safe division with signature like this:
safe_idv :: (Fractional q) => q -> q -> Maybe q
I've checked at Hoogle if there is some method in minimal definition which can help, but to no avail.
Thanks in advance
Update: Since the question caused some confusion, I want to avoid changing the constraints on q, and pattern matching still requires Eq (i guess implicitly).
For following definition:
safe_div :: (Fractional q) => q -> q -> Result q
safe_div a 0 = Err ["dvision by zero"]
safe_div a b = Ok (a / b)
following error is raised:
* Could not deduce (Eq q) arising from the literal `0'
from the context: Fractional q
bound by the type signature for:
safe_div :: forall q. Fractional q => q -> q -> Result q
at AST2.hs:11:1-48
Possible fix:
add (Eq q) to the context of
the type signature for:
safe_div :: forall q. Fractional q => q -> q -> Result q
* In the pattern: 0
In an equation for `safe_div':
safe_div a 0 = Err ["dvision by zero"]
With the signature safe_div :: (Fractional q) => q -> q -> Result q, you can only use methods from Fractional or its superclasses on your q values.
(Or other predefined functions that impose no more constraints than Fractional, but those will ultimately have to be implemented by the class methods. So they can't do anything you couldn't do directly with the methods yourself.)
From Fractional itself that gives us:
(/) :: Fractional a => a -> a -> a
recip :: Fractional a => a -> a
fromRational :: Fractional a => Rational -> a
Well none of those look terribly helpful. But Num is a superclass of Fractional, so we have those methods too:
(+) :: Num a => a -> a -> a
(-) :: Num a => a -> a -> a
(*) :: Num a => a -> a -> a
negate :: Num a => a -> a
abs :: Num a => a -> a
signum :: Num a => a -> a
fromInteger :: Num a => Integer -> a
These also aren't going to help us. abs and signum at first seem like they might, since their "purpose" is telling us certain properties about the number; signum even says this:
For real numbers, the signum is either -1 (negative), 0 (zero) or 1 (positive).
Which sounds exactly the kind of thing we want! The only trouble is that signum communicates the result of its inspection as a value of the same type. If we couldn't tell if our number x is equal to zero, how are we going to tell whether signum x is equal to zero? We're right back at the problem we started with.
The fact is every single method of both Fractional and Num only ever returns a value of the (unknown) type implementing the class. That basically means if you don't know what they type actually is, it's impossible to get any information out of them; the only thing you can do with a value of an unknown Fractional type is pass it to another Fractional (or Num) method, which will also only give you back a value of the same unknown type. There's no way to compare it to anything (which would require something returning a Bool, Ordering, or at least a Maybe, Either, etc). There's no way to convert it to text so it can be shown to a user (which would require something returning a String, Text, etc). You can do further calculations, but the only thing we can ever learn is that the calculation didn't error out, and only by trying it and hoping (which is exactly what you're trying to avoid!).
The only way you can implement your desired function is to add more constraints. Eq is exactly the class of types which can be compared, and you want to compare values of your type, so it just makes sense that you will have to constrain your function to operate within this class of types.
However, anyone calling this function polymorphically is in the same boat. It's very useful for intermediate functions (like this one) to work with unknown Fractional types, so that they can be called with any fractional type. But at the outermost level where someone first decided to call one of these functions, it's only ever useful to call these with a concrete type they actually know something about. Nobody wants to do calculations on numbers where they can't inspect the result in any way! This means that even though your safe_div function (as it is currently written) cannot assume any details of any particular type (such as whether it can be compared for equality), it in fact will only ever realistically be called with specific types like Double, Float, etc, all of which do support Eq. So in practice adding the Eq constraint is hardly limiting who can call it.
I imagine the reason you don't want to change the constraint is that you've already coded up functions where you use this, and they only have Fractional constraints (meaning they can't call safe_div :: (Eq a, Fractional a) => a -> a -> Result a). Unfortunately they'll have to be updated to add the Eq constraint too. The fact is that the interface of just Fractional gives only the ability to do basic arithmetic. To do comparisons and branching calculations you need more. So all your functions that want to do more than basic arithmetic (and that includes any that call anything that does more than basic arithmetic, not just ones that do the comparison and branching themselves) need more constraints than just Fractional. Fortunately the same reasoning as above applies: it is extremely unlikely that you would ever need to call any of these functions with a type that doesn't support Eq, so there really is very little point in resisting the additional constraint.
I am trying to use fromIntegral to convert from Integer to a real-fractional type. The idea is to have a helper method to later be used for the comparison between two instances of a Fraction (whether they are the same).
I have read some of the documentation related to:
fromIntegral :: (Integral a, Num b) => a -> b
realToFrac :: (Real a, Fractional b) => a -> b
Where I am having trouble is taking the concept and make an implementation of the helper method that takes a Num data type with fractions (numerator and denominator) and returns what I think is a real-fractional type value. Here is what I have been able to do so far:
data Num = Fraction {numerator :: Integer, denominator :: Integer}
helper :: Num -> Fractional
helper (Fraction num denom) = realToFrac(num/denom)
You need to learn about the difference between types and type classes. In OO languages, both are kind of the same concept, but in Haskell they're not.
A type contains concrete values. E.g. the type Bool contains the value True.
A class contains types. E.g. the Ord class doesn't contain any values, but it does contain the types which contain values that can be compared.
In case of numbers in Haskell it's a bit confusing that you can't really tell from the name whether you're dealing with a type or a class. Fractional is a class, whereas Rational is a type (which is an instance of Fractional, but so is e.g. Float).
In your example... first let's give that type a better name
data MyRational = Fraction {numerator :: Integer, denominator :: Integer}
...you have two possibilities what helper could actually do: convert to a concrete Rational value
helper' :: MyRational -> Rational
or a generic Fractional-type one
helper'' :: Fractional r => MyRational -> r
The latter is strictly more general, because Rational is an instance of Fractional (i.e. you can in fact use helper'' as a MyRational -> Rational function, but also as a MyRational -> Double function).
In either case,
helper (Fraction num denom) = realToFrac(num/denom)
does not work because you're trying to carry out the division on integer values and only then converting the result. Instead, you need to convert the integers to something fractional and then carry out the division in that type.
following the signature:
sqrt :: Floating a => a -> a
why is (sqrt 2) legal? Isn't the number 2 a Integer which definitely doesn't satisfy Floating? Same Question about round, round (sqrt 2) is legal, sqrt returns type Floating but round needs ReadFrac.
round :: (RealFrac a, Integral b) => a -> b
The literal 2 is of type Num a => a. That is, it can be any numerical type. Specifically, Haskell will take the integer value 2 and call fromInteger on it (which is defined for any Num). So when you write
sqrt 2
Internally, what's happening is
sqrt (fromInteger 2) :: Floating a => a
And if you force the value, such as in GHCi, you'll get a Double since that's the default for Floating.
Likewise, the type of round (sqrt 2) is going to be Integral b => b and is going to require the sqrt 2 type to be RealFrac a => a. There exist types which are both RealFrac and Floating, so that's not a contradiction. In particular, GHC will happily default to Double here for the same reason as before. If you force the value to be printed, the entire result (Integral b => b) will default to Integer.
It's important to remember that all of these are universally quantified. Floating a => a doesn't mean "this is some floating type and that's all I know". It means "if you have any floating type, I can produce a value of that type". You get to choose which floating type to use, so if a constraint comes along later and says the value is also RealFrac a => a, that's fine because we've simply constrained ourselves to be both Floating and RealFrac. This is contrary to a language like Java, where if I have a value of some interface type, say Comparable, then all I can conclude is that it's some Comparable type, not that it works for all of them.
In Haskell, I just know that
:type ((+)(1))
((+)(1)) :: Num a => a -> a
((+)(1） 2
3
But how about
:type abs(sqrt)
abs(sqrt) :: (Floating a, Num (a -> a)) => a -> a
Actually, I try many times but fail to use the function 'abs(sqrt)'. Then I have a few questions. What is the type(class?) '(Floating a, Num (a -> a))'? Is it possible to use the function 'abs(sqrt)'? How?
A type class is a way to generalize functions so that they can be polymorphic and others can implement those functions for their own types. Take as an example the type class Show, which in a simplified form looks like
class Show a where
show :: a -> String
This says that any type that implements the Show typeclass can be converted to a String (there's some more complication for more realistic constraints, but the point of having Show is to be able to convert values to Strings).
In this case, the function show has the full type Show a => a -> String.
If we examine the function sqrt, its type is
> :type sqrt
sqrt :: Floating a => a -> a
And for abs:
> :type abs
abs :: Num b => b -> b
If you ask GHCi what the types are it will use the type variable a in both cases, but I've used b in the type signature for abs to make it clear that these are different type variables of the same name, and it will help avoid confusion in the next step.
These type signatures mean that sqrt takes a value whose type implements the Floating typeclass (use :info Floating to see all the members) and returns a value of that same type, and that the abs function takes a value whose type implements the Num typeclass and returns a value of that same type.
The expression abs(show) is equivalently parsed as abs sqrt, meaning that sqrt is the first and only argument passed to abs. However, we just said that abs takes a value of a Num type, but sqrt is a function, not a number. Why does Haskell accept this instead of complaining? The reason can be seen a little more clearly when we perform substitution with the type signatures. Since sqrt's type is Floating a => a -> a, this must match the argument b in abs's type signature, so by substituting b with Floating a => a -> a we get that abs sqrt :: (Floating a, Num (a -> a)) => a -> a.
Haskell actually allows the function type to implement the Num typeclass, you could do it yourself although it would likely be nonsensical. However, just because something wouldn't seem to make sense to GHC, so long as the types can be cleanly solved it will allow it.
You can't really use this function, it just doesn't really make sense. There is no built-in instance of Num (a -> a) for any a, so you'd have to define your own. You can, however, compose the functions abs and sqrt using the composition operator .:
> :type abs . sqrt
abs . sqrt :: Floating c => c -> c
And this does make sense. This function is equivalent to
myfunc x = abs (sqrt x)
Note here that x is first applied to sqrt, and then the result of that computation is passed to abs, rather than passing the function sqrt to abs.
When you see Num (a -> a) it generally means you made a mistake somewhere.
Perhaps you really wanted: abs . sqrt which has type Floating c => c -> c - i.e. it's a function of a Floating type (e.g. Float, Double) to the same Floating type.
It is probably not possible to use this function.
What's likely happening here is that the type is saying that abs(sqrt) has the constraints that a must be of type class Floating and (a -> a) must be of type class Num. In other words, the sqrt function needs to be able to be treated as if it was a number.
Unfortunately, sqrt is not of type class Num so there won't be any input that will work here (not that it would make sense anyway). However, some versions of GHCi allow you to get the type of as if it were possible.
Have a look at Haskell type length + 1 for a similar type problem.
As ErikR has said, perhaps you meant to write abs . sqrt instead.
For example, one bad way is to factor through a string:
toReadableNum :: (Num a, Num b, Read b) => a -> b
toReadableNum = read . show
If there are no good ways, are there other bad ways? Implementation specific? Requiring language extension?
You can't go (sanely) from Num to Num, as Num provides no mechanism for extracting information about the value held other than its spurious Eq and Show machinery, but if you are willing to assume a bit more on the behalf of the number you are coming from, then you can have recourse.
In particular
fromIntegral :: (Integral a, Num b) => a -> b
and the composition of
toRational :: Real a => a -> Rational
with
fromRational :: Fractional a => Rational -> a
are both good candidates for doing what you mean, if not exactly what you asked for.
While read . show is well typed and has the signature you propose, the meaning is gobbledigook. There is nothing at all that says the text emitted by one Show instance will be compatible with a completely different Read instance, and there are plenty of counter examples.
The (implied) contract on Read and Show only apply when you use them with the same type!
There are no good ways. Some numbers contain more information that other, so how could you expect to convert between two arbitrary numbers in a good way. Some simple examples: How do you convert a Double to an Int? A Rational to on Int8? A Complex Double to a Float?
All this involve information loss and then there is no obvious right way.
And as #hammar says, the operations in Num simply don't allow you to construct such a function.
You cannot write any useful function of the type (Num a, Num b) => a -> b. Since a and b are type variables, the only useful operations we can use on them are the ones in the Num class. (Eq and Show won't help us much here).
class (Eq a, Show a) => Num a where
(+), (-), (*) :: a -> a -> a
negate :: a -> a
abs :: a -> a
signum :: a -> a
fromInteger :: Integer -> a
The only function here that allows you to make an b if you didn't have one to start with is fromInteger, but you have no way of turning a into an Integer, so the only functions you can write of this type return fromInteger of some constant, or bottom. Not very useful.
As augustss pointed out, there is no obvious way of making this conversion anyway. Remember lots of types can be Num. Not only the various types of real numbers, but also complex numbers, matrices, polynomials, etc. There is no meaningful conversion that would work between all of them.
The good way is to make specific kind of conversion, like round or clamp. Such function does what it say it does.