Applicative functor evaluation is not clear to me

The other two answers have given the detail of how this is calculated – but I thought I might chime in with a more “intuitive” answer to explain how, without going through a detailed calculation, one can “see” that the result must be 508.

As you implied, every Applicative (in fact, even every Functor) can be viewed as a particular kind of “context” which holds values of a given type. As simple examples:

• Maybe a is a context in which a value of type a might exist, but might not (usually the result of a computation which may fail for some reason)
• [a] is a context which can hold zero or more values of type a, with no upper limit on the number – representing all possible outcomes of a particular computation
• IO a is a context in which a value of type a is available as a result of interacting with “the outside world” in some way. (OK that one isn’t so simple…)

And, relevant to this example:

• r -> a is a context in which a value of type a is available, but its particular value is not yet known, because it depends on some (as yet unknown) value of type r.

The Applicative methods can be very well understood on the basis of values in such contexts. pure embeds an “ordinary value” in a “default context” in which it behaves as closely as possible in that context to a “context-free” one. I won’t go through this for each of the 4 examples above (most of them are very obvious), but I will note that for functions, pure = const – that is, a “pure value” a is represented by the function which always produces a no matter what the source value.

Rather than dwell on how <*> can best be described using the “context” metaphor though, I want to dwell on the particular expression:

f <$> a <*> b

where f is a function between 2 “pure values” and a and b are “values in a context”. This expression in fact has a synonym as a function: liftA2. Although using the liftA2 function is generally considered less idiomatic than the “applicative style” using <$> and <*>, the name emphasies that the idea is to “lift” a function on “ordinary values” to one on “values in a context”. And when thought of like this, I think it is usually very intuitive what this does, given a particular “context” (ie. a particular Applicative instance).

So the expression:

(+) <$> a <*> b

for values a and b of type say f Int for an Applicative f, behaves as follows for different instances f:

• if f = Maybe, then the result, if a and b are both Just values, is to add up the underlying values and wrap them in a Just. If either a or b is Nothing, then the whole expression is Nothing.
• if f = [] (the list instance) then the above expression is a list containing all sums of the form a' + b' where a' is in a and b' is in b.
• if f = IO, then the above expression is an IO action that performs all the I/O effects of a followed by those of b, and results in the sum of the Ints produced by those two actions.

So what, finally, does it do if f is the function instance? Since a and b are both functions describing how to get a given Int given an arbitrary (Int) input, it is natural that lifting the (+) function over them should be the function that, given an input, gets the result of both the a and b functions, and then adds the results.

And that is, of course, what it does – and the explicit route by which it does that has been very ably mapped out by the other answers. But the reason why it works out like that – indeed, the very reason we have the instance that f <*> g = \x -> f x (g x), which might otherwise seem rather arbitrary (although in actual fact it’s one of the very few things, if not the only thing, that will type-check), is so that the instance matches the semantics of “values which depend on some as-yet-unknown other value, according to the given function”. And in general, I would say it’s often better to think “at a high level” like this than to be forced to go down to the low-level details of exactly how computations are performed. (Although I certainly don’t want to downplay the importance of also being able to do the latter.)

[Actually, from a philosophical point of view, it might be more accurate to say that the definition is as it is just because it’s the “natural” definition that type-checks, and that it’s just happy coincidence that the instance then takes on such a nice “meaning”. Mathematics is of course full of just such happy “coincidences” which turn out to have very deep reasons behind them.]