Lambda lifting

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)

Suppose that you find that the worker function, isqrt', might be useful somewhere else (say, in a context where a better initial estimate is known). You would like to use the Lifting pattern to raise it to the top level. However, isqrt' contains a Free variable, n, which is bound in the outer function. What to do?

The solution is to promote n to be an argument of isqrt'. The refactored code might look like this:

 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

If you want to lift the definition of g, you might be tempted to write:

 f x y = g y x + g y (2*x)
   where
     g y x = sqrt y + x

However, this would mean that sqrt y is evaluated twice, whereas in the first program, it would be evaluated only once. A more efficient approach is not to lift out the variable y, but rather the expression sqrt y:

 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 sqrt y is actually not technically maximal. Lifting out the MFE would give you:

 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.

This converse of lambda lifting is lambda dropping, also known as avoiding parameter passing.