# Program transformations in Hakaru

## Coalesce

Coalesce is an internal transformation that works on the untyped Hakaru AST. It takes recursive NAryOp terms that have the same type and combines them into a single term. For instance:

3.0 + 1.5 + 0.3


is parser as:

NaryOp Sum [3.0, NaryOp Sum [1.5, NaryOp Sum [0.3]]]


which when coalesced becomes:

NaryOp Sum [3.0,1.5,0.3]


## Optimizations

The Hakaru AST has a suite of standard compiler optimizations which have a substantial effect on the runtime of the resulting program. The current pipeline is described by the optimizations variable in Language.Hakaru.Syntax.Transforms. In order, the optimizations performed are:

1. A-normalization
2. Uniquification of variables (needed for let-floating)
3. Let-floating
4. Common subexpression elimination
6. Uniquification of variables (for the C backend)
7. Constant Propagation

Each pass is described in more detail below.

### A-normalization

Found in Language.Hakaru.Syntax.ANF

See The Essence of Compiling with Continuations by Flannigan, Sabry, Duba, and Felleisen

A-normalization converts expressions into administrative normal form (ANF). This ensures that all intermediate values are named and all arguments to functions or primitive operations are either literals or variables. ANF is a common program representation for functional language compilers which can simplify some compiler passes and make others more effective. As an example, consider

(add1 (let ([x (f y)]) 5))


This expression in ANF looks like the following

(let ([x (f y)]) (add1 5))


which opens up the opportunity for constant folding to eliminate the (add1 5) expression. This pass exists mostly to simplify the implementation of CSE, but is useful for other passes as well.

### Uniquification

Found in Language.Hakaru.Syntax.Uniquify

Ensures all variables in the program have unique variable identifiers. This is not strictly necessary, but simplifies the implementation of other passes, several of which rely on this property.

### Let-floating

Found in Language.Hakaru.Syntax.Hoist

See Let-Floating: Moving Bindings to Give Faster Programs (1996) by Simon Peyton Jones , Will Partain , André Santos

Let-floating alters the bindings structure of the program in order to improve performance. Typically, this entails moving definitions into or out of lambda expressions. When a lambda expression encodes a loop, this effectively accomplishes loop invariant code motion. This pass only moves definitions upward in the AST. For the most part, we are only interested in looping constructs like summate and product, and moving summate expressions out of other summate or product expressions when they do not depend on the index. This can radically alter the asymptotics of the resulting program, as nested loops are converted into sequentially executed loops.

The only assumption this pass makes about the input AST is that all variable identifiers are unique. This is to handle the case where two branches of a match statement introduce the same variable. If both binders are hoisted out of the match statement, they one binding will shadow the other.

This pass, as implemented, unconditionally floats expression to where their data dependencies are fulfilled. This is not safe in a general purpose language, and we may need to layer some heuristics on top of this pass to make it less aggressive if we end up introducing performance regressions.

### Common Subexpression Elimination

Found in Language.Hakaru.Syntax.CSE

Common subexpression elimination eliminates redundant computation by reusing results for equivalent expressions. The current implementation of this pass relies on the program being in ANF.

ANF simplifies the implementation of CSE greatly by ensuring all expressions are named and that if two expressions may be shared, one of them is let-bound so that it dominates the other. In short, ANF simplifies the program to a simple top-down traversal of the AST. Consider the example

(+ (add1 z) (add1 z))


Eliminating the common expression (add1 z) requires us to traverse the expression in evaluation order, track expression which have already been evaluated, recognize when an expression is duplicated, and introduce it with a new name that dominates all use sites of that expression. However, an expression in ANF allows us to perform CSE simply by keeping track of let-bound expressions and propagating those expressions downward into the AST. Consider the example in ANF

(let ([t1 (add1 z)])
(+ t1 t2)))


To remove the common subexpression, we simply have to note that the (add1 z) bound to t2 is equivalent to the expression bound to t1 and replace it with the variable t1.

(let ([t1 (add1 z)])
(let ([t2 t1])
(+ t1 t2)))


Trivial bindings can then be eliminated, if desired, giving

(let ([t1 (add1 z)])
(+ t1 t1)))


A major goal of CSE is to cleanup any work which is duplicated by the let-floating pass.

### Pruning

Found in Language.Hakaru.Syntax.Prune

This is essentially a limited form of dead code elimination. If an expression is bound to a variable which is never referenced, then that expression need never be executed, as the code language has no side effects. This pass serves to clean up some of the junk introduced by other passes.

Cases which are handled

1. (let ([x e1]) e2) => e2 if x not in fv(e2)
2. (let ([x e1]) x) => e1

### Constant Propagation

Found in Language.Hakaru.Evalutation.ConstantPropagation

Performs simple constant propagation and constant folding. The current implementation does not do that much work, mostly just evaluating primitive operations when their arguments are constant.

## Unused Passes

### Loop Peeling

Found in Language.Hakaru.Syntax.Unroll

Loop peeling was an initial attempt at performing loop invariant code motion by leveraging CSE to do most of the heavy lifting. Peeling is a common strategy to make other optimization passes “loop-aware”. The idea is to peel off one iteration of a loop and then apply the existing suite of optimizations. Consider the following summate whose body e is some loop-invariant computation.

(summate lo hi (λ x -> e))


After peeling we obtain

(if (= lo hi)
0
(let ([x lo])
(let ([t1 e])
(let ([t2 (summate (+ lo 1) hi (λ x -> e))])
(+ t1 t2)))))


After applying CSE, the loop invariant body is simply reused on each iteration

(if (= lo hi)
0
(let ([x lo])
(let ([t1 e])
(let ([t2 (summate (+ lo 1) hi (λ x -> t1))])
(+ t1 t2)))))


ANF ensures that all subexpression in the e bound to t1 are shareable with the copy of e used in the body of the summate, allowing us to hoist out subexpressions of e and not just the entire summate body.

This pass is currently disabled in favor of the let-floating pass, which does a better job without causing an exponential blow up in code size. Some of Hakaru’s looping constructs, such as array, cannot be peeled, so we cannot move loop invariant operations out of array statements.