Write biSp in point-free form using as few terms as possible
7 terms

We again define a helper combinator BBQ = B B (C B), which has reduction rule

BBQ w x y z = x y (w z)


Then biSp = BBQ BBQ BBQ

  BBQ BBQ BBQ g h f x y
= BBQ g (BBQ h) f x y
= BBQ h f (g x) y
= f (g x) (h y)


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By brute force, this is the only 7-term solution and there is no smaller solution.

8 terms

There are a handful of 8-term solutions. Firstly, the obvious

C BBQ BBQ BBQ


which reduces in one step to BBQ BBQ BBQ.

Then there are three solutions of the form X X X with a five-term X:


X = B (B B) C B which reduces in one step to BBQ
X = C B (C B) B which reduces in one step to BBQ and allows the further extraction, for no net gain or loss, of a helper combinator Q = C B
X = C (B C (B B))


There is one solution of the form X X with a six-term X:


X = C B (B B (C B)) (again, with an equivalent version which extracts Q = C B)


And there is one solution which can only be written in 8 terms with two helper combinators:

Q = C B
X = Q (Q Q B)
biSp = X X