Split composition

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

      f{⎕SI}
      g{⎕SI}
      h{⎕SI}
      'x' g⍨∘f⍨∘h 'y'
hfg
      'x' gh⍨∘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
APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming (Train, Hook, Split composition)
Array Numeric literalStringStrand notationObject literalArray notation (design considerations)
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionDefined function (traditional)DfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBranchQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloadingControl structure