README.md

Chapter 2: Arithmetic

In this chapter we extend the language grammar to include arithmetic operations such as addition, subtraction, multiplication, division, and unary minus. This allows us to write statements such as:

return 1 + 2 * 3 + -5;

Here is the complete language grammar for this chapter.

You can also read this chapter in a linear Git revision history on the linear branch and compare it to the previous chapter.

Extensions to Intermediate Representation

In Chapter 1 we introduced following nodes.

Node Name	Type	Description	Inputs	Value
Start	Control	Start of function	None	None for now as we do not have function arguments yet
Return	Control	End of function	Predecessor control node, Data node value	Return value of the function
Constant	Data	Constants such as integer literals	None, however Start node is set as input to enable graph walking	Value of the constant

We extend the set of nodes by adding following additional node types.

Node Name	Type	Description	Inputs	Value
Add	Data	Add two values	Two data nodes, values are added, order not important	Result of the add operation
Sub	Data	Subtract a value from another	Two data nodes, values are subtracted, order matters	Result of the subtract
Mul	Data	Multiply two values	Two data nodes, values are multiplied, order not important	Result of the multiply
Div	Data	Divide a value by another	Two data nodes, values are divided, order matters	Result of the division
Minus	Data	Negate a value	One data node, value is negated	Result of the unary minus

Peephole Optimizations

Nodes in the graph can be peephole-optimized. The graph is viewed through a "peephole", a small chunk of graph, and if a certain pattern is detected we locally rewrite the graph.

During parsing, these peephole optimizations are particularly easy to check and apply: there are no uses (yet) of a just-created Node from a just-parsed piece of syntax, so there's no effort to the "rewrite" part of the problem. We just replace in-place before installing Nodes into the graph.

This replacement might allow us to kill the unused inputs from the replaced Node, basically doing a modest Dead Code Elimination during parsing.

E.g. Suppose we already parsed out a constant 1, and a constant 2; then when we parse an Add(1,2), the peephole rule for constant math replaces the Add with a constant 3. At this point, we also kill the unused Add, which recursively may kill the unused constants 1 and 2.

Constant Folding and Constant Propagation

In this chapter and next we focus on a particular peephole optimization: constant folding and constant propagation. Since we do not have non-constant values until Chapter 4, the main feature we demonstrate now is constant folding. However, we introduce some additional ideas into the compiler at this stage, to set the scene for Chapter 4.

It is useful for the compiler to know at various points of the program whether a node's value is a constant. The compiler can use this knowledge to perform various optimizations such as:

• Evaluate expressions at compile time and replace an expression with a constant. This idea can be extended in a number of ways and is called Partial Evaluation.
• The compiler may be able to identify regions of code that are dead and no longer needed, such as when a conditional branch always take one of the branches (Chapter 6).
• Pointers may be known to be not-null, and a null check can be skipped (Chapter 10).
• Array indices may be known to be in-range, and a range check can be skipped.
• Many additional optimizations are possible when the compiler learns more about the possible set of Node values.

In order to achieve above, we need a way to talk about what kinds of values a Node can take on at runtime - we call these Types, and we annotate Nodes with Types.

The Type annotation serves two purposes:

• it defines the set of operations allowed on the Node
• it defines the set of values the Node takes on

The Type itself is represented as a Java class; the Java classes are used as convenient to represent the Type's structure; all types are subtypes of the class Type. For now, we only have the following hierarchy of types:

Type
+-- TypeInteger

It turns out that the set of values associated with a Type at a specific Node can be conveniently represented as a "lattice" https://en.wikipedia.org/wiki/Lattice_(order) . Our lattice has the following structure:

Our lattice elements can be one of three types:

• The highest element is "top", denoted by T; assigning T means that the Node's value may or may not be a compile time constant.
• All elements in the middle are constants.
• The lowest is "bottom", denoted by ⊥; assigning ⊥ means that we know that the Node's value is not a compile time constant.

An invariant of peephole optimizations is that the type of a Node always moves up the lattice (towards "top"); peepholes are pessmistic assuming the worst until they can prove better. A later optimistic optimization will start all Nodes at top and move Types down the lattice as eager assumptions are proven wrong.

In later chapters we will explore extending this lattice, as it frequently forms the heart of core optimizations we want our compiler to do.

We add a _type field to every Node, to store its current computed best Type. We need a field to keep the optimizer runtime linear, and later when doing an optimistic version of constant propagation (called Sparse Conditional Constant Propagation).

We add a _type field even to Start and Return, and later to all control Nodes - because the Sea of Nodes does not distinguish between control and data. Both nodes are equally peepholed and optimized, and this will be covered starting in Chapter 4 and Chapter 5.

There are other important properties of the Lattice that we discuss in Chapter 4 and Chapter 10, such as the "meet" and "join" operators and their rules.

Nodes Pre Peephole Optimization

The following visual shows how the graph looks like pre-peephole optimization:

• Control nodes appear as square boxes with yellow background
• Control edges are in bold red
• The edges from Constants to Start are shown in dotted lines as these are not true control edges
• We label each edge with its position in the node's list of inputs.

Post-peephole

Demonstration

To demonstrate how the optimisation works, let's consider the following code:

return 1+2;

This is parsed as:

        var lhs = parseMultiplication();
        if (match("+")) return new AddNode(lhs, parseAddition()).peephole();

When the peephole is called on the AddNode node it first calls the compute function:

    public final Node peephole( ) {
        // Compute initial or improved Type
        Type type = _type = compute();

    public Type compute() {
        if( in(1)._type instanceof TypeInteger i0 &&
            in(2)._type instanceof TypeInteger i1 ) {
            if (i0.isConstant() && i1.isConstant())
                return TypeInteger.constant(i0.value()+i1.value());
        }
        return Type.BOTTOM;
    }

In our case both of the operands(1, 2) are constants, and instance of TypeInteger. Since compute returns a type, and we know that both of our operands are constants, we can do constant propagation/folding where we compute the result of the expression so that instead of having 1+2 we can have 3.

The 'TypeInteger' interface inherits from type but also has a field that stores the value observed for that type.

    private final long _con

AddNode has the layout:

Since, we already know the value of both of the operands, we can just add them together and replace the old node with a constant node that just holds the result of the expression.

To do this - we have to recognize this happens when the type computed for the node is already a constant, but the node is still not an instance of constant node.

This can be captured by:

// Replace constant computations from non-constants with a constant node
if (!(this instanceof ConstantNode) && type.isConstant()) {
}

Note how we return a constant type when doing constant folding

return TypeInteger.constant(i0.value()+i1.value());

We can create a constant node in the following way:

return new ConstantNode(type).peephole();

This recursively calls the peephole algorithm. Now, we end up with this:

Notice, how we added the constant node holding a constant value of 3. The remaining add node is unused and can be killed.

We can call the kill function before this way:

// Replace constant computations from non-constants with a constant node
if (!(this instanceof ConstantNode) && type.isConstant()) {
     kill();             // Kill `this` because replacing with a Constant
     return new ConstantNode(type).peephole();

This is responsible for deleting the dangling add node.

public void kill( ) {
  assert isUnused();      // Has no uses, so it is dead
  for( int i=0; i<nIns(); i++ )
        setDef(i,null);  // Set all inputs to null, recursively killing unused Nodes

  _inputs.clear();
  _type=null;             // Flag as dead
  assert isDead();        // Really dead now
}

isUnused - This makes sure the current node has no outputs. Ideally, in our case there would be one output as the return statement is using the add expression. However, the optimisation happens before even creating the return node, which means that at this point there are no users of the add node.

We can now freely go ahead and delete the node.

for(int i=0; i<nIns(); i++ )
    setDef(i,null);  // Set all inputs to null, recursively killing unused Nodes

_inputs.clear();
_type=null;        

We finally end up with this: