Turning free variables into arguments.
As an example, consider the following Worker wrapper function, which computes the truncated square root of an integer:
isqrt :: Integer -> Integer isqrt n | n < 0 = error "isqrt" | otherwise = isqrt' ((n+1) `div` 2) where isqrt' s | s*s <= n && n < (s+1)*(s+1) = s | otherwise = isqrt' ((s + (n `div` s)) `div` 2)
isqrt :: Integer -> Integer isqrt n | n < 0 = error "isqrt" | otherwise = isqrt' n ((n+1) `div` 2) where isqrt' n s | s*s <= n && n < (s+1)*(s+1) = s | otherwise = isqrt' n ((s + (n `div` s)) `div` 2)
The isqrt' function may now be safely lifted to the top-level.
Note that naive lambda lifting can cause a program to be less lazy. Consider, for example:
f x y = g x + g (2*x) where g x = sqrt y + x
f x y = g y x + g y (2*x) where g y x = sqrt y + x
f x y = let sy = sqrt y in g sy x + g sy (2*x) where g sy x = sy + x
An expression of this sort which only mentions Free variables is called a Free expression. If a free expression is as large as it can be, it is called a Maximal free expression, or MFE for short. Note that
f x y = let psy = (+) (sqrt y) in g psy x + g psy (2*x) where g psy x = psy x
However, you save no more work here than the second version, and in addition, the resulting function is harder to read. In general, it only makes sense to abstract out a Free expression if it is also a Reducible expression.