Simple Applicative Functor Example

Question

I'm reading the Learn You a Haskell book. I'm struggling to understand this applicative functor code:

(*) <$> (+3) <*> (*2) $ 2

This boils down to: (3+2) * (2*2) = 20

I don't follow how. I can expand the above into the less elegant but more explicit for newbie comprehension version:

((fmap (*) (+3)) <*> (*2)) 2

I understand the basics of the <*> operator. This makes perfect sense:

class (Functor f) => Applicative f where
    pure :: a -> f a
    (<*>) :: f (a -> b) -> f a -> f b

But I don't see how the command works? Any tips?

Answer 1

One method to approach this types of questions is to use substitution. Take the operator, in this case (<*>) or function, get its implementation and insert it to the code in question.

In the case of (*) <$> (+3) <*> (*2) $ 2 you are using the ((->) a) instance of Applicative found in the Applicative module in base, you can find the instance by clicking the source link on the right and searching for "(->":

instance Applicative ((->) a) where
    pure = const
    (<*>) f g x = f x (g x)

Using the definition for (<*>) we can continue substituting:

((fmap (*) (+3)) <*> (*2)) 2 == (fmap (*) (+3)) 2 ((*2) 2)
== (fmap (*) (+3)) 2 4

Ok now we need the Functor instance for ((->) a). You can find this by going to the haddock info for Functor, here clicking on the source link on the right and searching for "(->" to find:

instance Functor ((->) r) where
    fmap = (.)

Now continuing substituting:

(fmap (*) (+3)) 2 4 == ((*) . (+3)) 2 4
== (*) ((+3) 2) 4
== (*) 5 4
== 20

A more symbolic appraoch

Many people report better long term sucess with these types of problems when thinking about them symbolically. Instead of feeding the 2 value through the problem lets focus instead on (*) <$> (+3) <*> (*2) and only apply the 2 at the end.

(*) <$> (+3) <*> (*2)
== ((*) . (+3)) <*> (*2)
== (\x -> ((*) . (+3)) x ((*2) x))
== (\x -> ((*) . (+3)) x (x * 2))
== (\x -> (*) (x + 3) (x * 2))
== (\x -> (x + 3) * (x * 2))
== (\x -> 2 * x * x + 6 * x)

Ok now plug in 2 for x

2 * 2 * 2 + 6 * 2
8 + 12
20