Register Allocation in Cretonne

Cretonne uses a decoupled, SSA-based register allocator. Decoupled means that register allocation is split into two primary phases: spilling and coloring. SSA-based means that the code stays in SSA form throughout the register allocator, and in fact is still in SSA form after register allocation.

Before the register allocator is run, all instructions in the function must be legalized, which means that every instruction has an entry in the encodings table. The encoding entries also provide register class constraints on the instruction’s operands that the register allocator must satisfy.

After the register allocator has run, the locations table provides a register or stack slot location for all SSA values used by the function. The register allocator may have inserted spill, fill, and copy instructions to make that possible.

SSA-based register allocation

The phases of the SSA-based register allocator are:

Liveness analysis
For each SSA value, determine exactly where it is live.
Spilling

The process of deciding which SSA values go in a stack slot and which values go in a register. The spilling phase can also split live ranges by inserting copy instructions, or transform the code in other ways to reduce the number of values kept in registers.

After spilling, the number of live register values never exceeds the number of available registers.

Coloring
The process of assigning specific registers to the live values. It’s a property of SSA form that this can be done in a linear scan of the dominator tree without causing any additional spills.
EBB argument fixup

The coloring phase does not guarantee that EBB arguments are placed in the correct registers and/or stack slots before jumping to the EBB. It will try its best, but not making this guarantee is essential to the speed of the coloring phase. (EBB arguments correspond to PHI nodes in traditional SSA form).

The argument fixup phase inserts ‘shuffle code’ before jumps and branches to place the argument values in their expected locations.

The contract between the spilling and coloring phases is that the number of values in registers never exceeds the number of available registers. This sounds simple enough in theory, but in pratice there are some complications.

Real-world complications to SSA coloring

In practice, instruction set architectures don’t have “K interchangable registers”, and register pressure can’t be measured with a single number. There are complications:

Different register banks
Most ISAs separate integer registers from floating point registers, and instructions require their operands to come from a specific bank. This is a fairly simple problem to deal with since the register banks are completely disjoint. We simply count the number of integer and floating-point values that are live independently, and make sure that each number does not exceed the size of their respective register banks.
Instructions with fixed operands

Some instructions use a fixed register for an operand. This happens on the Intel ISAs:

  • Dynamic shift and rotate instructions take the shift amount in CL.
  • Division instructions use RAX and RDX for both input and output operands.
  • Wide multiply instructions use fixed RAX and RDX registers for input and output operands.
  • A few SSE variable blend instructions use a hardwired XMM0 input operand.
Operands constrained to register subclasses
Some instructions can only use a subset of the registers for some operands. For example, the ARM NEON vmla (scalar) instruction requires the scalar operand to be located in D0-15 or even D0-7, depending on the data type. The other operands can be from the full D0-31 register set.
ABI boundaries

Before making a function call, arguments must be placed in specific registers and stack locations determined by the ABI, and return values appear in fixed registers.

Some registers can be clobbered by the call and some are saved by the callee. In some cases, only the low bits of a register are saved by the callee. For example, ARM64 callees save only the low 64 bits of v8-15, and Win64 callees only save the low 128 bits of AVX registers.

ABI boundaries also affect the location of arguments to the entry block and return values passed to the return instruction.

Aliasing registers

Different registers sometimes share the same bits in the register bank. This can make it difficult to measure register pressure. For example, the Intel registers RAX, EAX, AX, AL, and AH overlap.

If only one of the aliasing registers can be used at a time, the aliasing doesn’t cause problems since the registers can simply be counted as one unit.

Early clobbers
Sometimes an instruction requires that the register used for an output operand does not alias any of the input operands. This happens for inline assembly and in some other special cases.

Liveness Analysis

Both spilling and coloring need to know exactly where SSA values are live. The liveness analysis computes this information.

The data structure representing the live range of a value uses the linear layout of the function. All instructions and EBB headers are assigned a program position. A starting point for a live range can be one of the following:

  • The instruction where the value is defined.
  • The EBB header where the value is an EBB argument.
  • An EBB header where the value is live-in because it was defined in a dominating block.

The ending point of a live range can be:

  • The last instruction to use the value.
  • A branch or jump to an EBB where the value is live-in.

When all the EBBs in a function are laid out linearly, the live range of a value doesn’t have to be a contiguous interval, although it will be in a majority of cases. There can be holes in the linear live range.

The part of a value’s live range that falls inside a single EBB will always be an interval without any holes. This follows from the dominance requirements of SSA. A live range is represented as:

  • The interval inside the EBB where the value is defined.
  • A set of intervals for EBBs where the value is live-in.

Any value that is only used inside a single EBB will have an empty set of live-in intervals. Some values are live across large parts of the function, and this can often be represented with coalesced live-in intervals covering many EBBs. It is important that the live range data structure doesn’t have to grow linearly with the number of EBBs covered by a live range.

This representation is very similar to LLVM’s LiveInterval data structure with a few important differences:

  • The Cretonne LiveRange only covers a single SSA value, while LLVM’s LiveInterval represents the union of multiple related SSA values in a virtual register. This makes Cretonne’s representation smaller because individual segments don’t have to annotated with a value number.

  • Cretonne stores the def-interval separately from a list of coalesced live-in intervals, while LLVM stores an array of segments. The two representations are equivalent, but Cretonne optimizes for the common case of a value that is only used locally.

  • It is simpler to check if two live ranges are overlapping. The dominance properties of SSA form means that it is only necessary to check the def-interval of each live range against the intervals of the other range. It is not necessary to check for overlap between the two sets of live-in intervals. This makes the overlap check logarithmic in the number of live-in intervals instead of linear.

  • LLVM represents a program point as SlotIndex which holds a pointer to a 32-byte IndexListEntry struct. The entries are organized in a double linked list that mirrors the ordering of instructions in a basic block. This allows ‘tombstone’ program points corresponding to instructions that have been deleted.

    Cretonne uses a 32-bit program point representation that encodes an instruction or EBB number directly. There are no ‘tombstones’ for deleted instructions, and no mirrored linked list of instructions. Live ranges must be updated when instructions are deleted.

A consequence of Cretonne’s more compact representation is that two program points can’t be compared without the context of a function layout.

Spilling algorithm

There is no one way of implementing spilling, and different tradeoffs between compilation time and code quality are possible. Any spilling algorithm will need a way of tracking the register pressure so the colorability condition can be satisfied.

Coloring algorithm

The SSA coloring algorithm is based on a single observation: If two SSA values interfere, one of the values must be live where the other value is defined.

We visit the EBBs in a topological order such that all dominating EBBs are visited before the current EBB. The instructions in an EBB are visited in a top-down order, and each value define by the instruction is assigned an available register. With this iteration order, every value that is live at an instruction has already been assigned to a register.

This coloring algorithm works if the following condition holds:

At every instruction, consider the values live through the instruction. No matter how the live values have been assigned to registers, there must be available registers of the right register classes available for the values defined by the instruction.

We’ll need to modify this condition in order to deal with the real-world complications.

The coloring algorithm needs to keep track of the set of live values at each instruction. At the top of an EBB, this set can be computed as the union of:

  • The set of live values before the immediately dominating branch or jump instruction. The topological iteration order guarantees that this set is available. Values whose live range indicate that they are not live-in to the current EBB should be filtered out.
  • The set of arguments to the EBB. These values should all be live-in, although it is possible that some are dead and never used anywhere.

For each live value, we also track its kill point in the current EBB. This is the last instruction to use the value in the EBB. Values that are live-out through the EBB terminator don’t have a kill point. Note that the kill point can be a branch to another EBB that uses the value, so the kill instruction doesn’t have to be a use of the value.

When advancing past an instruction, the live set is updated:

  • Any values whose kill point is the current instruction are removed.
  • Any values defined by the instruction are added, unless their kill point is the current instruction. This corresponds to a dead def which has no uses.