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# 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](docs/02-grammar.md) for this chapter.
## Extensions to Intermediate Representation
In [Chapter 1](../chapter01/README.md) 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](../chapter04/README.md), 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](https://en.wikipedia.org/wiki/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](../chapter06/README.md)).
* Pointers may be known to be not-null, and a null check can be skipped ([Chapter 10](../chapter10/README.md)).
* 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](https://en.wikipedia.org/wiki/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](../chapter04/README.md) and [Chapter
5](../chapter05/README.md).
There are other important properties of the Lattice that we discuss in [Chapter
4](../chapter04/README.md) and [Chapter 10](../chapter10/README.md), 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
|
