Split composition

Split-compose is a tacit construct, used to pre-process its argument(s) with the left and right-most operand before applying the middle operand between the result. Given functions f, g, and h, the split composition on arguments x and y is defined as (f x) g (h y).

The name was introduced by the I language, where it is represented with O, a higher-order function that applies first to the middle function and then the two outer functions (O also represents the Over operator). It doesn't appear as a primitive in any APL, nor can it, because it is a composition of three functions, while a compositional operator can take no more than two operands. This situation is identical to that of the fork. Both split-compose and fork can be constructed using two companion operators, tying together the three involved functions.

In Extended Dyalog APL and dzaima/APL, the construct can be formed using Reverse Compose (⍛) and Compose (∘). In this example, we multiply the interval (integers up until) of the left argument, with the Magnitude of the right:

      5 ⍳⍛×∘| 5 ¯8 ¯2 ¯5 3
5 16 6 20 15

This is evaluated as (⍳5) × (|5 ¯8 ¯2 ¯5 3). A further example concatenates the reciprocal of the left argument with the negation of the right:

      2(,⍨∘÷⍨∘-⍨⍨)4
0.5 ¯4

This is evaluated as (÷2) × (-4).

Alternatives

In dialects that lack Reverse Compose (and even Compose), split-compose can be written either by defining the missing operator(s), or as a single derived function or fork, if this is supported. For example, in Dyalog APL the expression can be formed with Compose and Commute (⍨) as g⍨∘f⍨∘h:

      5 ×⍨∘⍳⍨∘| 5 ¯8 ¯2 ¯5 3
5 16 6 20 15
      2(,⍨∘÷⍨∘-)4
0.5 ¯4

Note that g∘h⍨∘f⍨ applies f before h which can matter for functions with side effects. For example, consider the following where 'x' f⍛g∘h 'y' would print hfg:

      f←{⍞←⊃⎕SI}
      g←{⍞←⊃⎕SI}
      h←{⍞←⊃⎕SI}
      'x' g⍨∘f⍨∘h 'y'
hfg
      'x' g∘h⍨∘f⍨ 'y'
fhg

The equivalent fork is f⍤⊣ g h⍤⊢, for example:

      5 (⍳⍤⊣×|⍤⊢) 5 ¯8 ¯2 ¯5 3
5 16 6 20 15
      2(÷⍤⊣,-⍤⊢)4
0.5 ¯4