# Pipelining

Suppose we wanted to compute the sum of squares of the numbers from 0 up to $n$. How might we go about it? Of course (math being the best form of optimization), the most efficient way would be a closed-form formula:

$\frac{n (n+1) (2n+1)}{6}$

But let's imagine you've forgotten that formula. In an imperative language you might use a for loop:

# Python
def sum_sq(n):
sum = 0
for i in range(0,n):
sum += i*i
return sum


The equivalent recursive code in OCaml would be:

let sum_sq n =
let rec loop i sum =
if i>n then sum
else loop (i+1) (sum + i*i)
in loop 0 0


Another, clearer way of producing the same result in OCaml uses higher-order functions and the pipeline operator:

let square x = x*x
let sum = List.fold_left (+) 0

let sum_sq n =
0--n                (* [0;1;2;...;n]   *)
|> List.map square  (* [0;1;4;...;n*n] *)
|> sum              (*  0+1+4+...+n*n  *)


The function sum_sq first constructs a list containing all the numbers 0..n. Then it uses the pipeline operator |> to pass that list through List.map square, which squares every element. Then the resulting list is pipelined through sum, which adds all the elements together.

Pipelining with lists and other data structures is quite idiomatic. The other alternatives that you might consider are somewhat uglier:

(* worse: a lot of extra let..in syntax *)
let sum_sq n =
let l = 0--n in
let sq_l = List.map square l in
sum sq_l

(* maybe worse:  have to read the function applications from right to left
* rather than top to bottom *)
let sum_sq n =
sum (List.map square (0--n))


We could improve our code a little further by using List.rev_map instead of List.map. List.rev_map is a tail-recursive version of map that reverses the order of the list. Since (+) is associative and commutative, we don't mind the list being reversed.

let sum_sq n =
0--n                    (* [0;1;2;...;n]   *)
|> List.rev_map square  (* [n*n;...;4;1;0] *)
|> sum                  (*  n*n+...+4+1+0  *)