Roofline-Based Performance Analysis

Updated 28 October 2025

Roofline-based theoretical analysis is a framework that relates the attainable performance of algorithms to hardware compute and memory limits.
It employs both empirical benchmarking and analytical techniques to measure operational intensity and identify performance bottlenecks.
The method guides hardware/software co-design by providing actionable insights for optimization in HPC, deep learning, FPGA, and emerging architectures.

Roofline-based theoretical analysis is a quantitative framework for examining and predicting the performance limitations of compute systems by relating algorithmic operational intensity to the throughput ceilings imposed by hardware compute resources and memory bandwidth. It is widely used for performance modeling and bottleneck analysis in domains spanning classical HPC kernels, deep learning inference/training, FPGA acceleration, and emerging machine learning hardware. The method’s hallmark is its capacity to represent both hardware limits and software characteristics within a single visual and analytic model, often yielding actionable insights for hardware/software co-design and optimization.

1. The Roofline Model: Core Principles and Mathematical Foundations

The roofline model relates the attainable performance of a kernel or algorithm to two architectural limits: peak computational throughput and peak memory bandwidth. The canonical formulation is

$P = \min(P_\text{peak},\; I \times b_s)$

where:

$P$ : attainable performance (FLOP/s or application-relevant throughput)
$P_\text{peak}$ : hardware peak compute rate (from CPU/GPU/FPGA specs or empirical measurement)
$I$ : operational (arithmetic) intensity (FLOP per byte communicated to/from memory, may be layer- or kernel-specific)
$b_s$ : peak attainable memory bandwidth for the relevant hierarchy level (e.g., DRAM or on-chip SRAM)

The model is piecewise: at low $I$ , performance is memory-bound (sloped region); at high $I$ , compute-bound (flat region). The transition (ridge point) is at $I_\text{balance} = P_\text{peak}/b_s$ .

Advanced and domain-specific variants generalize this formulation:

Multi-level rooflines: Separate $b_{L_i}$ and $AI_{L_i}$ for different memory hierarchy levels (Verhelst et al., 22 May 2025)
Communication Rooflines: FLOP/s bounded by network bandwidth × communication intensity in distributed AI (Jiang et al., 2020)
Energy Rooflines: Throughput per unit energy, factoring both compute and memory energy with static (base) power (K. et al., 24 Sep 2025)

Classes of bottlenecks (compute-bound, memory-bound, or communication-bound) are exposed according to where an algorithm or measured kernel's (OI, performance) point falls relative to the roofline.

2. Methodologies for Roofline Construction and Characterization

Construction of a roofline-based model typically requires both hardware and application/software characterization:

Hardware characterization: Empirical benchmarking for peak compute and bandwidth (e.g., microbenchmarks for GEMM, STREAM) (Spear et al., 2015, Verhelst et al., 22 May 2025)
Software characterization: Calculation or measurement of operational intensity per kernel or layer; can be done analytically (counting flops and bytes from algorithm structure (Louboutin et al., 2016)), via instrumentation/profiling ((Batashev, 30 Jul 2025), using LLVM or PMUs), or by empirical runtime monitoring.
Visualization: Interactive tools generate roofline plots, supporting comparison across architectures, overlay of empirical and theoretical limits, and scenario analysis (Spear et al., 2015). Modern frameworks may integrate these directly into development environments (e.g., Eclipse IDE (Spear et al., 2015)) or provide open-source toolchains (e.g., miniperf for RISC-V (Batashev, 30 Jul 2025)).

Several domains leverage specialized methodologies:

Hierarchical/multi-level rooflines: Empirical Roofline Toolkit and Nsight Compute enable analysis at L1/L2/HBM/Tensor Core for GPUs (Yang et al., 2020).
Compiler-based analysis: LLVM-IR–level counting enables hardware-agnostic, PMU-free roofline construction for emerging ISAs (Batashev, 30 Jul 2025).
Time-based and energy-based rooflines: Models explicitly relate performance, runtime, and energy efficiency to operational intensity, incorporating runtime overheads and power modeling (Wang et al., 2020, K. et al., 24 Sep 2025).

3. Domain-specific Roofline Model Variations and Applications

The core roofline methodology has been extended to accommodate the unique architectural and algorithmic patterns in multiple research domains:

ML Accelerators and FPGAs

Multi-level and energy-aware rooflines: Explicit computation of AI and bandwidth/energy efficiency at each level (register, SRAM, DRAM), with utilization factors ( $U_{MAC}$ , $U_{mem}$ ) accounting for underutilization (Verhelst et al., 22 May 2025).
LUT-based computational roof: In FPGAs, LUTMUL redefines the performance roof by mapping multiplications to LUTs rather than DSPs, shifting the computational ceiling from "DSP-bound" to "LUT-bound" (Xie et al., 1 Nov 2024).
Rootline-guided design: Used to select optimal quantization, dataflow, and specialization strategies for DNN accelerators.

Deep Learning Inference and Training

Hierarchical rooflines: Visualize performance/layer bottlenecks at all memory levels and for all precisions (FP16, FP32, Tensor Cores), enabling framework and kernel optimization (Yang et al., 2020).
Time-based extensions: For DL workloads, models plot compute/bandwidth time as axes, explicitly incorporating effect of kernel complexity and launch overheads (Wang et al., 2020). This is crucial for LSTMs and transformers, where frequent low-AI (memory-bound) operations dominate.
LLM inference bottleneck analysis: Roofline models pinpoint why large LMs are often memory-bound during token-by-token decoding, and systematize the evaluation of optimization methods (quantization, batching, operator fusion) (Yuan et al., 26 Feb 2024).
Sparsity roofline: Jointly models network accuracy, sparsity, and theoretical speedup, shifting from per-kernel performance to global model-level tradeoffs (accuracy-speedup curves) (Shinn et al., 2023).

Real-world Systems and Emerging Architectures

Energy and time rooflines for edge devices: Explicit formulae account for both dynamic (compute/memory) and static (idle) power, and enable power mode tuning for latency and energy optimization on devices like NVIDIA Jetson (K. et al., 24 Sep 2025).
RISC-V and PMU-limited systems: LLVM-based roofline profiling enables cross-platform, PMU-independent bottleneck analysis even on unproven hardware (Batashev, 30 Jul 2025).
UAVs and cyber-physical systems: The F-1 model adapts roofline methodology to system-level dynamics, combining sensor, compute, and physical constraints for maximizing UAV operational safety and velocity (Krishnan et al., 2022).

4. Insights and Optimization Guidance from Roofline Analysis

Roofline-based theoretical analysis provides a range of actionable insights:

Algorithm–hardware mapping: Algorithmic choices (e.g., stencil order for PDEs (Louboutin et al., 2016), choice of dataflow/quantization in ML (Verhelst et al., 22 May 2025)) are systematically matched to hardware photos via operational intensity.
Bottleneck diagnosis: Visualization clarifies whether code is compute- or memory-bound (or communication-bound), and distinguishes lack of utilization from true limit attainment.
Implementation refinement: When kernels appear memory-bound, redundant data movement, register pressure, and low arithmetic intensity can be precisely diagnosed and remedied—often requiring code restructuring or algorithm specialization (Owen et al., 22 Jan 2024).
Power/energy optimization: Time/energy rooflines enable principled frequency/voltage tuning and design-space exploration to balance latency and energy efficiency (K. et al., 24 Sep 2025).
Framework/hardware selection: Automatic tools (e.g., LLM-Viewer (Yuan et al., 26 Feb 2024)) leverage model and hardware profiles to guide practitioners in selecting hardware, serving parameters, and architectural optimizations optimally.

Typical recommendations—driven by roofline insights—include increasing arithmetic intensity (via data reuse, quantization, operator fusion), targeting the most limiting bandwidth or compute roof, reducing kernel launch overheads, or shifting workloads to preferred (compute-optimized) regimes.

5. Limitations, Extensions, and Future Research Directions

The roofline model is intentionally optimistic, focusing on best-case bounds. It does not, by default, account for:

Non-overlapping bottlenecks across multiple hierarchy levels simultaneously (ECM model addresses this (Hammer et al., 2017)).
Fine-grained utilization losses, idiosyncratic instruction behavior, or kernel launch overheads unless extended with empirical/per-case analysis (Wang et al., 2020).
Effects of algorithmic structure that yield inhomogeneous operational intensity across a workload (necessitating per-layer or per-kernel analysis).

Open research opportunities identified in the literature include:

Generalized multi-domain/multi-chip rooflines: Extension to chiplet architectures, complex system hierarchies (Verhelst et al., 22 May 2025).
Co-design of sparsity/quantization and hardware: Sparsity roofline modeling as a method for ex-ante hardware–algorithm narrowing (Shinn et al., 2023).
Automated tool integration: Embedding roofline workflows into compiler toolchains for emerging ISA/hardware (Batashev, 30 Jul 2025).
Energy-centric frameworks: Use of time/energy rooflines to guide ML "race-to-idle" strategies and power-aware design (K. et al., 24 Sep 2025).
Performance portability and auto-tuning: Systematic tuning of workload mapping and scheduling guided by roofline-derived bottleneck locations (Anderson et al., 2023).

6. Representative Table of Roofline Model Variants

Model Variant	Key Formulae / Axis	Specialization	References
Classic Roofline	$P = \min(P_\text{peak}, I b_s)$	Single memory, single compute limit	(Spear et al., 2015)
Multi-level Roofline	$AI_{L_i}$ , $b_{L_i}$ at each $L_i$	Register/SRAM/DRAM-level analysis	(Verhelst et al., 22 May 2025)
Communication Roof	COI-based ( $FLOPS/$ Comm. bytes)	Distributed, network-limited AI training	(Jiang et al., 2020)
Energy Roofline	$P_E = 1 / (E_{op} + \sum_i E_{L_i}/AI_{L_i})$	Joint throughput/efficiency roofline	(Verhelst et al., 22 May 2025, K. et al., 24 Sep 2025)
Sparsity Roofline	(Accuracy, speedup) axes, SoL latency	DNN sparsity/accuracy/speedup joint analysis	(Shinn et al., 2023)
Time Roofline	(Compute time, bandwidth time) axes	DL kernel launch overhead, batch effects	(Wang et al., 2020, K. et al., 24 Sep 2025)

7. Impact and Significance in System and Algorithm Design

The adoption of roofline-based theoretical analysis has driven advances in algorithm–hardware co-design and provided a lingua franca for communication between architecture, systems, and applications researchers. Its generalization to energy, communication, and domain-specific bottlenecks makes it foundational for optimizing emerging workloads (deep learning, sparse inference, edge AI), guiding both incremental optimization and fundamental hardware/software architectural choices. Robust, automated, and portable toolchains have further democratized its use outside highly specialized performance engineering communities.

Key references demonstrating these principles, methodological innovations, and broad applicability include (Xie et al., 1 Nov 2024, Verhelst et al., 22 May 2025, K. et al., 24 Sep 2025, Wang et al., 2020, Yang et al., 2020, Shinn et al., 2023, Hammer et al., 2017), and (Batashev, 30 Jul 2025).