Julia SSA-form IR

    With the new IR representation, the compiler learned to handle four new IR nodes, Phi nodes, Pi nodes as well as PhiC nodes and Upsilon nodes (the latter two are only used for exception handling).

    Phi nodes are part of generic SSA abstraction (see the link above if you’re not familiar with the concept). In the Julia IR, these nodes are represented as:

    where we ensure that both vectors always have the same length. In the canonical representation (the one handled by codegen and the interpreter), the edge values indicate come-from statement numbers (i.e. if edge has an entry of , there must be a goto, gotoifnot or implicit fall through from statement 15 that targets this phi node). Values are either SSA values or constants. It is also possible for a value to be unassigned if the variable was not defined on this path. However, undefinedness checks get explicitly inserted and represented as booleans after middle end optimizations, so code generators may assume that any use of a Phi node will have an assigned value in the corresponding slot. It is also legal for the mapping to be incomplete, i.e. for a Phi node to have missing incoming edges. In that case, it must be dynamically guaranteed that the corresponding value will not be used.

    PiNodes encode statically proven information that may be implicitly assumed in basic blocks dominated by a given pi node. They are conceptually equivalent to the technique introduced in the paper ABCD: Eliminating Array Bounds Checks on Demand or the predicate info nodes in LLVM. To see how they work, consider, e.g.

    1. %x::Union{Int, Float64} # %x is some Union{Int, Float64} typed ssa value
    2. if isa(x, Int)
    3.     # use x
    4. else
    5.     # use x
    6. end

    Pi nodes are generally ignored in the interpreter, since they don’t have any effect on the values, but they may sometimes lead to code generation in the compiler (e.g. to change from an implicitly union split representation to a plain unboxed representation). The main usefulness of PiNodes stems from the fact that path conditions of the values can be accumulated simply by def-use chain walking that is generally done for most optimizations that care about these conditions anyway.

    PhiC nodes and Upsilon nodes

    Exception handling complicates the SSA story moderately, because exception handling introduces additional control flow edges into the IR across which values must be tracked. One approach to do so, which is followed by LLVM, is to make calls which may throw exceptions into basic block terminators and add an explicit control flow edge to the catch handler:

    1. invoke @function_that_may_throw() to label %regular unwind to %catch
    3. # Control flow continues here
    5. catch:
    6. # Exceptions go here

    However, this is problematic in a language like Julia, where at the start of the optimization pipeline, we do not know which calls throw. We would have to conservatively assume that every call (which in Julia is every statement) throws. This would have several negative effects. On the one hand, it would essentially reduce the scope of every basic block to a single call, defeating the purpose of having operations be performed at the basic block level. On the other hand, every catch basic block would have n*m phi node arguments (n, the number of statements in the critical region, m the number of live values through the catch block).

    To work around this, we use a combination of Upsilon and PhiC nodes (the C standing for , written φᶜ in the IR pretty printer, because unicode subscript c is not available). There are several ways to think of these nodes, but perhaps the easiest is to think of each PhiC as a load from a unique store-many, read-once slot, with Upsilon being the corresponding store operation. The PhiC has an operand list of all the upsilon nodes that store to its implicit slot. The Upsilon nodes however, do not record which PhiC node they store to. This is done for more natural integration with the rest of the SSA IR. E.g. if there are no more uses of a PhiC node, it is safe to delete it, and the same is true of an Upsilon node. In most IR passes, PhiC nodes can be treated like Phi nodes. One can follow use-def chains through them, and they can be lifted to new PhiC nodes and new Upsilon nodes (in the same places as the original Upsilon nodes). The result of this scheme is that the number of Upsilon nodes (and PhiC arguments) is proportional to the number of assigned values to a particular variable (before SSA conversion), rather than the number of statements in the critical region.

    The corresponding IR (with irrelevant types stripped) is:

    1. 1 ─       nothing::Nothing
    2. 2 ─ %2  = $(Expr(:enter, #4))
    3. 3 ─ %3  = ϒ (false)
    4. │   %4  = ϒ (#undef)
    5. │   %5  = ϒ (1)
    6. │   %6  = ϒ (true)
    8. │         invoke Main.opaque()::Any
    9. │   %9  = ϒ (true)
    10. │         invoke Main.error()::Union{}
    11. └──       $(Expr(:unreachable))::Union{}
    12. 4 ┄ %13 = φᶜ (%3, %6, %9)::Bool
    13. │   %14 = φᶜ (%4, %7, %10)::Core.Compiler.MaybeUndef(Int64)
    14. │   %15 = φᶜ (%5)::Core.Const(1)
    15. └──       $(Expr(:leave, 1))
    16. 5 ─       $(Expr(:pop_exception, :(%2)))::Any
    17. │         $(Expr(:throw_undef_if_not, :y, :(%13)))::Any
    18. │   %19 = Core.tuple(%15, %14)
    19. └──       return %19

    Note in particular that every value live into the critical region gets an upsilon node at the top of the critical region. This is because catch blocks are considered to have an invisible control flow edge from outside the function. As a result, no SSA value dominates the catch blocks, and all incoming values have to come through a φᶜ node.

    The main SSAIR data structure is worthy of discussion. It draws inspiration from LLVM and Webkit’s B3 IR. The core of the data structure is a flat vector of statements. Each statement is implicitly assigned an SSA value based on its position in the vector (i.e. the result of the statement at idx 1 can be accessed using SSAValue(1) etc). For each SSA value, we additionally maintain its type. Since, SSA values are definitionally assigned only once, this type is also the result type of the expression at the corresponding index. However, while this representation is rather efficient (since the assignments don’t need to be explicitly encoded), it of course carries the drawback that order is semantically significant, so reorderings and insertions change statement numbers. Additionally, we do not keep use lists (i.e. it is impossible to walk from a def to all its uses without explicitly computing this map–def lists however are trivial since you can look up the corresponding statement from the index), so the LLVM-style RAUW (replace-all-uses-with) operation is unavailable.

    Instead, we do the following:

    • We keep a separate buffer of nodes to insert (including the position to insert them at, the type of the corresponding value and the node itself). These nodes are numbered by their occurrence in the insertion buffer, allowing their values to be immediately used elsewhere in the IR (i.e. if there are 12 statements in the original statement list, the first new statement will be accessible as SSAValue(13)).
    • RAUW style operations are performed by setting the corresponding statement index to the replacement value.
    • Statements are erased by setting the corresponding statement to nothing (this is essentially just a special-case convention of the above.

    The idea behind this arrangement is that, since the optimization passes need to touch the corresponding memory anyway and incur the corresponding memory access penalty, performing the extra housekeeping should have comparatively little overhead (and save the overhead of maintaining these data structures during IR modification).