Memory-Bound vs Compute-Bound: The Roofline Model, Why Fusion Helps One but Not the Other

The problem, from first principles

01-kernel-fusion showed fusion cuts memory traffic dramatically. That's not automatically the same thing as "makes the computation faster" — a computation's actual speed is limited by whichever resource (compute throughput or memory bandwidth) it exhausts first, and cutting memory traffic only helps when memory bandwidth was the actual bottleneck to begin with.

From theory to code

Implement arithmetic_intensity(flops, bytes_moved), ridge_point(peak_flops_per_sec, peak_bytes_per_sec), is_memory_bound(...), and achievable_flops_per_sec(...) — together, the classic "roofline model" for predicting a computation's actual achievable throughput on given hardware.

Constraints

• arithmetic_intensity returns flops / bytes_moved (FLOPs per byte of memory traffic).
• ridge_point returns peak_flops_per_sec / peak_bytes_per_sec — the intensity at which compute and memory bandwidth saturate simultaneously.
• is_memory_bound returns True iff the computation's own intensity is strictly below the hardware's ridge point.
• achievable_flops_per_sec returns min(peak_flops_per_sec, arithmetic_intensity * peak_bytes_per_sec).

Hints

Open one at a time. Each gives away a little more than the last.