LLVM Block Frequency Terminology

Introduction

Block Frequency is a metric for estimating the relative frequency of different basic blocks. This document describes the terminology that the BlockFrequencyInfo and MachineBlockFrequencyInfo analysis passes use.

Branch Probability

Blocks with multiple successors have probabilities associated with each outgoing edge. These are called branch probabilities. For a given block, the sum of its outgoing branch probabilities should be 1.0.

Branch Weight

Rather than storing fractions on each edge, we store an integer weight. Weights are relative to the other edges of a given predecessor block. The branch probability associated with a given edge is its own weight divided by the sum of the weights on the predecessor’s outgoing edges.

For example, consider this IR:

define void @foo() {
    ; ...
    A:
        br i1 %cond, label %B, label %C, !prof !0
    ; ...
}
!0 = metadata !{metadata !"branch_weights", i32 7, i32 8}

and this simple graph representation:

A -> B  (edge-weight: 7)
A -> C  (edge-weight: 8)

The probability of branching from block A to block B is 7/15, and the probability of branching from block A to block C is 8/15.

See LLVM Branch Weight Metadata for details about the branch weight IR representation.

Block Frequency

Block frequency is a relative metric that represents the number of times a block executes. The ratio of a block frequency to the entry block frequency is the expected number of times the block will execute per entry to the function.

Block frequency is the main output of the BlockFrequencyInfo and MachineBlockFrequencyInfo analysis passes.

Implementation: a series of DAGs

The implementation of the block frequency calculation analyses each loop, bottom-up, ignoring backedges; i.e., as a DAG. After each loop is processed, it’s packaged up to act as a pseudo-node in its parent loop’s (or the function’s) DAG analysis.

Block Mass

For each DAG, the entry node is assigned a mass of UINT64_MAX and mass is distributed to successors according to branch weights. Block Mass uses a fixed-point representation where UINT64_MAX represents 1.0 and 0 represents a number just above 0.0.

After mass is fully distributed, in any cut of the DAG that separates the exit nodes from the entry node, the sum of the block masses of the nodes succeeded by a cut edge should equal UINT64_MAX. In other words, mass is conserved as it “falls” through the DAG.

If a function’s basic block graph is a DAG, then block masses are valid block frequencies. This works poorly in practice though, since downstream users rely on adding block frequencies together without hitting the maximum.

Loop Scale

Loop scale is a metric that indicates how many times a loop iterates per entry. As mass is distributed through the loop’s DAG, the (otherwise ignored) backedge mass is collected. This backedge mass is used to compute the exit frequency, and thus the loop scale.

Implementation: Getting from mass and scale to frequency

After analysing the complete series of DAGs, each block has a mass (local to its containing loop, if any), and each loop pseudo-node has a loop scale and its own mass (from its parent’s DAG).

We can get an initial frequency assignment (with entry frequency of 1.0) by multiplying these masses and loop scales together. A given block’s frequency is the product of its mass, the mass of containing loops’ pseudo nodes, and the containing loops’ loop scales.

Since downstream users need integers (not floating point), this initial frequency assignment is shifted as necessary into the range of uint64_t.

Block Bias

Block bias is a proposed absolute metric to indicate a bias toward or away from a given block during a function’s execution. The idea is that bias can be used in isolation to indicate whether a block is relatively hot or cold, or to compare two blocks to indicate whether one is hotter or colder than the other.

The proposed calculation involves calculating a reference block frequency, where:

• every branch weight is assumed to be 1 (i.e., every branch probability distribution is even) and

• loop scales are ignored.

This reference frequency represents what the block frequency would be in an unbiased graph.

The bias is the ratio of the block frequency to this reference block frequency.