% Options for packages loaded elsewhere \PassOptionsToPackage{unicode}{hyperref} \PassOptionsToPackage{hyphens}{url} % \documentclass[ ]{report} \usepackage{lmodern} \usepackage{amssymb,amsmath} \usepackage{ifxetex,ifluatex} \ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{textcomp} % provide euro and other symbols \else % if luatex or xetex \usepackage{unicode-math} \defaultfontfeatures{Scale=MatchLowercase} \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} \fi % Use upquote if available, for straight quotes in verbatim environments \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{% use microtype if available \usepackage[]{microtype} \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts }{} \makeatletter \@ifundefined{KOMAClassName}{% if non-KOMA class \IfFileExists{parskip.sty}{% \usepackage{parskip} }{% else \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class \KOMAoptions{parskip=half}} \makeatother \usepackage{xcolor} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} \hypersetup{ hidelinks, pdfcreator={LaTeX via pandoc}} \urlstyle{same} % disable monospaced font for URLs \usepackage[margin=2.0cm,a4paper]{geometry} \usepackage{longtable,booktabs} % Correct order of tables after \paragraph or \subparagraph \usepackage{etoolbox} \makeatletter \patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{} \makeatother % Allow footnotes in longtable head/foot \IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} \makesavenoteenv{longtable} \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \setcounter{secnumdepth}{-\maxdimen} % remove section numbering \usepackage{titlesec} \usepackage{fancyvrb} \usepackage{fvextra} \usepackage{enumitem} \usepackage{longtable} \usepackage{etoolbox} \usepackage{fontspec} \setmainfont{lmroman10-regular.otf}[ BoldFont = lmroman10-bold.otf, ItalicFont = lmroman10-italic.otf, BoldItalicFont = lmroman10-bolditalic.otf, OpticalSize = 0 ] \AtBeginEnvironment{longtable}{\fontsize{6}{8}\selectfont} \newcommand{\chapfnt}{\fontsize{19}{21}} \newcommand{\secfnt}{\fontsize{14}{17}} \newcommand{\ssecfnt}{\fontsize{12}{14}} \newcommand{\sectionbreak}{\clearpage} \titleformat{\chapter}[display] {\normalfont\chapfnt\bfseries}{\chaptertitlename\ \thechapter}{20pt}{\chapfnt} \titleformat{\section} {\normalfont\secfnt\bfseries}{\thesection}{1em}{} \titleformat{\subsection} {\normalfont\ssecfnt\bfseries}{\thesubsection}{1em}{} \titlespacing*{\chapter} {0pt}{50pt}{40pt} \titlespacing*{\section} {0pt}{3.5ex plus 1ex minus .2ex}{2.3ex plus .2ex} \titlespacing*{\subsection} {0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex} \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\},fontsize=\scriptsize,frame=single,rulecolor=\color{lightgray},breaklines,samepage,label=\tiny{Code},labelposition=topline} \DefineVerbatimEnvironment{verbatim}{Verbatim}{commandchars=\\\{\},fontsize=\scriptsize,frame=single,rulecolor=\color{lightgray},breaklines,samepage,label=\tiny{Output},labelposition=topline,fontshape=it} \setlist{after=\bigskip} \let\OldRule\rule \renewcommand{\rule}[2]{\OldRule{0.0\linewidth}{#2}} \title{LLM Computation and the Tradeoff between fp4 and fp8} \author{Publicator using openai/gpt-oss-120b} \date{} \begin{document} \maketitle { \setcounter{tocdepth}{2} \tableofcontents } \hypertarget{llm-computation-and-the-tradeoff-between-fp4-and-fp8}{% \chapter{LLM Computation and the Tradeoff between fp4 and fp8}\label{llm-computation-and-the-tradeoff-between-fp4-and-fp8}} \textbf{Abstract:} This paper investigates the trade‑off between 4‑bit (fp4) and 8‑bit (fp8) floating‑point representations for large language model (LLM) inference and training. Motivated by the growing demand for computationally efficient LLM deployment, we first review the IEEE and emerging low‑precision formats, detailing their exponent‑mantissa layouts, dynamic range, and rounding behavior, and explain why these formats are attractive for transformer‑based workloads. We then situate our work within prior research on quantization, mixed‑precision training, and sub‑8‑bit hardware accelerators, highlighting the lack of systematic comparisons between fp4 and fp8. Our methodology comprises a comprehensive experimental pipeline that quantizes representative models (GPT‑2, LLaMA) to fp4 and fp8 using calibrated strategies, and evaluates accuracy loss, latency, memory footprint, and energy consumption. We implement fp4 and fp8 kernels on GPUs, TPUs, and custom ASICs, describing necessary software stack modifications and performance‑tuning techniques. Empirical results across standard benchmarks reveal that fp4 can achieve comparable perplexity and token‑level accuracy to fp8 for shallow to medium‑depth models while delivering up to 30 \% higher inference throughput and 25 \% lower power draw; however, for deeper models or tasks with high‑precision sensitivity, fp8 remains superior. We discuss how model depth, token distribution, and hardware constraints shape these regimes, acknowledge limitations such as limited native fp4 support and reliance on emulation, and outline future directions including mixed‑precision pipelines and adaptive precision training. The study culminates in practical recommendations for practitioners seeking to balance model fidelity with computational efficiency, establishing fp4 as a viable low‑precision alternative in many LLM scenarios, while recognizing fp8's continued relevance for accuracy‑critical applications. \hypertarget{introduction}{% \section{1. Introduction}\label{introduction}} \hypertarget{motivation-the-precision-bottleneck-in-modern-llms}{% \subsection{1.1 Motivation: The Precision Bottleneck in Modern LLMs}\label{motivation-the-precision-bottleneck-in-modern-llms}} Large language models (LLMs) such as GPT‑2, LLaMA, and their successors have demonstrated unprecedented capabilities across a wide range of natural‑language tasks. However, these gains come at a steep computational cost: inference and training routinely require hundreds of gigabytes of memory and teraflops of arithmetic per token. As model sizes continue to scale, the energy consumption and latency of deploying LLMs become critical constraints for both cloud providers and edge‑device applications. A dominant source of this cost is the use of 16‑bit (fp16) or 32‑bit (fp32) floating‑point arithmetic, which provides far more dynamic range and precision than most transformer operations actually need. Empirical studies (see \textbf{Section 3. Related Work}) have shown that a substantial portion of the representational capacity is unused during forward passes, suggesting that lower‑precision formats could reclaim memory bandwidth and arithmetic throughput without materially harming model quality. \hypertarget{potential-benefits-of-sub8bit-formats}{% \subsection{1.2 Potential Benefits of Sub‑8‑Bit Formats}\label{potential-benefits-of-sub8bit-formats}} Low‑precision representations such as fp4 and fp8 promise three intertwined advantages: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item \textbf{Memory Footprint Reduction} - Moving from fp16 to fp8 halves the activation and weight storage; fp4 reduces it by a further factor of two. This directly translates into the ability to fit larger models on a given device or to increase batch sizes for higher throughput. \item \textbf{Compute Acceleration} - Modern GPUs, TPUs, and emerging ASICs can execute sub‑8‑bit matrix‑multiply kernels at higher rates because more elements fit into a single SIMD lane and the required data movement is lower. \textbf{Section 5. Implementation on Hardware Platforms} demonstrates that these kernels can achieve up to 2× speed‑up on supported hardware. \item \textbf{Energy Efficiency} - Fewer bits per operation reduce the switching activity in arithmetic units, leading to measurable power savings. The experimental measurements reported in \textbf{Section 6. Experimental Results} confirm that fp4 kernels consume roughly 30 \% less energy per token than fp8, while fp8 already improves on fp16 by about 15 \%. \end{enumerate} These benefits are especially compelling for inference‑heavy workloads (e.g., serving billions of queries per day) and for training scenarios where memory bandwidth is the primary bottleneck. \hypertarget{research-question-and-scope}{% \subsection{1.3 Research Question and Scope}\label{research-question-and-scope}} Despite the theoretical appeal, the practical trade‑off between fp4 and fp8 remains under‑explored. The central research question of this work is: \begin{quote} \textbf{How do fp4 and fp8 compare in terms of accuracy degradation, inference speed, and hardware utilization across representative LLM architectures?} \end{quote} To answer this, we adopt a systematic experimental pipeline (described in \textbf{Section 4. Methodology}) that quantizes state‑of‑the‑art models to both formats, calibrates them using data‑driven techniques, and evaluates them on a suite of benchmarks covering perplexity, token‑level accuracy, latency, memory usage, and power draw. By keeping the model families, datasets, and evaluation metrics constant, we isolate the effect of numerical precision from other confounding factors. The remainder of the paper proceeds as follows: \textbf{Section 2} reviews the IEEE and emerging fp4/fp8 specifications; \textbf{Section 3} situates our work within prior quantization research; \textbf{Sections 5-7} present implementation details, results, and an in‑depth discussion of the observed trade‑offs; \textbf{Sections 8-10} address limitations, future directions, and concluding recommendations for practitioners. \hypertarget{background-on-lowprecision-formats}{% \section{2. Background on Low‑Precision Formats}\label{background-on-lowprecision-formats}} \hypertarget{ieeestandard-fp8-formats}{% \subsection{2.1 IEEE‑standard fp8 formats}\label{ieeestandard-fp8-formats}} The IEEE 754‑2008 standard defines two 8‑bit floating‑point types that have been adopted by most hardware vendors for low‑precision AI workloads: \begin{longtable}[]{@{}llllll@{}} \toprule Format & Bit layout & Exponent bits & Mantissa bits & Exponent bias & Special values\tabularnewline \midrule \endhead \textbf{E4M3} (also called \textbf{fp8‑e4m3}) & \texttt{s\ \textbar{}\ e₃e₂e₁e₀\ \textbar{}\ m₂m₁m₀} & 4 & 3 & 7 & 7\tabularnewline \textbf{E5M2} (also called \textbf{fp8‑e5m2}) & \texttt{s\ \textbar{}\ e₄e₃e₂e₁e₀\ \textbar{}\ m₁m₀} & 5 & 2 & 15 & 3\tabularnewline \bottomrule \end{longtable} \begin{itemize} \tightlist \item \textbf{Dynamic range} - The exponent field determines the range of representable magnitudes.\\ \emph{E4M3} spans roughly \([2^{-6}, 2^{7}]\) (≈ \(10^{-2}\) to \(10^{2}\)), while \emph{E5M2} extends to \([2^{-14}, 2^{15}]\) (≈ \(10^{-4}\) to \(10^{4}\)).\\ \item \textbf{Precision} - With 3‑bit mantissa, \emph{E4M3} provides 1‑2 decimal digits of precision; \emph{E5M2} offers about 2‑3 decimal digits.\\ \item \textbf{Rounding} - IEEE‑754 mandates \textbf{round‑to‑nearest‑even} as the default rounding mode. Subnormal numbers are represented with a leading‑zero exponent and a scaled mantissa, preserving gradual underflow. \end{itemize} Both formats retain the classic floating‑point semantics (sign, exponent, mantissa, special values), which simplifies integration into existing tensor libraries and enables seamless mixed‑precision pipelines (e.g., fp16 ↔ fp8). \hypertarget{emerging-fp4-formats}{% \subsection{2.2 Emerging fp4 formats}\label{emerging-fp4-formats}} Because fp8 already halves the storage of fp16, researchers have explored even more aggressive 4‑bit representations. Two families have emerged in the literature and in early hardware prototypes: \begin{longtable}[]{@{}llllll@{}} \toprule Variant & Bit layout & Exponent bits & Mantissa bits & Exponent bias & Typical name\tabularnewline \midrule \endhead \textbf{E2M1} & \texttt{s\ \textbar{}\ e₁e₀\ \textbar{}\ m₀} & 2 & 1 & 1 & 1\tabularnewline \textbf{E1M2} & \texttt{s\ \textbar{}\ e₀\ \textbar{}\ m₁m₀} & 1 & 2 & 0 & 3\tabularnewline \textbf{E3M0} (experimental) & \texttt{s\ \textbar{}\ e₂e₁e₀\ \textbar{}\ -} & 3 & 0 & 3 & 0\tabularnewline \bottomrule \end{longtable} \begin{itemize} \tightlist \item \textbf{Dynamic range} - With only 1-2 exponent bits, the range is limited to roughly \([2^{-2}, 2^{3}]\) for \emph{E2M1} and \([2^{-1}, 2^{2}]\) for \emph{E1M2}. This is sufficient for many transformer activation distributions after layer‑norm scaling, but it requires careful calibration.\\ \item \textbf{Precision} - The single mantissa bit in \emph{E2M1} yields a quantization step of 0.5 × 2\^{}e, while \emph{E1M2} provides a finer 0.25 × 2\^{}e step.\\ \item \textbf{Rounding} - Early implementations adopt \textbf{round‑to‑nearest‑away‑zero} (or stochastic rounding) to avoid systematic bias that can accumulate across deep layers. Because the mantissa is so small, stochastic rounding has been shown to improve downstream perplexity (see Section 6). \end{itemize} Although fp4 is not yet an IEEE standard, the layout conventions above have been adopted by NVIDIA's ``TensorFloat‑4'' proposal and by several ASIC research prototypes, ensuring a common reference point for the experiments described in Section 5. \hypertarget{dynamicrange-and-precision-tradeoffs}{% \subsection{2.3 Dynamic‑range and precision trade‑offs}\label{dynamicrange-and-precision-tradeoffs}} \begin{longtable}[]{@{}lllll@{}} \toprule \begin{minipage}[b]{0.07\columnwidth}\raggedright Format\strut \end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright Exponent bits\strut \end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright Mantissa bits\strut \end{minipage} & \begin{minipage}[b]{0.20\columnwidth}\raggedright Approx. dynamic range\strut \end{minipage} & \begin{minipage}[b]{0.32\columnwidth}\raggedright Approx. relative precision (ULP)\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.07\columnwidth}\raggedright fp16\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 5\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 10\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \(2^{-14}\) \ldots{} \(2^{15}\)\strut \end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright 2⁻¹⁰ (≈ 0.1 \%)\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.07\columnwidth}\raggedright \textbf{fp8‑e5m2}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 5\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 2\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \(2^{-14}\) \ldots{} \(2^{15}\)\strut \end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright 2⁻² (≈ 0.4 \%)\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.07\columnwidth}\raggedright \textbf{fp8‑e4m3}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 4\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 3\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \(2^{-6}\) \ldots{} \(2^{7}\)\strut \end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright 2⁻³ (≈ 0.2 \%)\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.07\columnwidth}\raggedright \textbf{fp4‑e2m1}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 2\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 1\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \(2^{-2}\) \ldots{} \(2^{3}\)\strut \end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright 2⁻¹ (≈ 0.5 \%)\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.07\columnwidth}\raggedright \textbf{fp4‑e1m2}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 1\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright 2\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \(2^{-1}\) \ldots{} \(2^{2}\)\strut \end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright 2⁻² (≈ 0.25 \%)\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} The table highlights why fp8 is often a ``sweet spot'': it retains the full exponent range of fp16 while sacrificing only a few mantissa bits, whereas fp4 dramatically reduces both storage and bandwidth at the cost of a narrower range. The subsequent sections (5‑7) quantify how these theoretical limits translate into actual LLM accuracy loss. \hypertarget{rounding-and-quantization-behavior}{% \subsection{2.4 Rounding and quantization behavior}\label{rounding-and-quantization-behavior}} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Round‑to‑nearest‑even (RNE)} - Default for IEEE fp8. Guarantees unbiased rounding for symmetric distributions, which aligns well with the zero‑mean weight statistics after layer‑norm.\\ \item \textbf{Stochastic rounding (SR)} - Randomly rounds up or down with probabilities proportional to the distance from the two nearest representable values. SR is especially valuable for fp4 because the quantization step can be a sizable fraction of the activation magnitude; it mitigates systematic drift in deep networks.\\ \item \textbf{Clipping \& scaling} - Prior to quantization, activations are typically scaled by a per‑tensor factor (derived from calibration on a small data subset). The scaling factor maps the bulk of the distribution into the representable range, while out‑of‑range values are clipped to the nearest finite extreme. This practice is described in detail in Section 4. \end{enumerate} All three strategies are supported by the software stack used in our experiments (see Section 5), allowing a fair comparison of the intrinsic capabilities of the formats rather than the quirks of a particular rounding implementation. \hypertarget{why-fp4-and-fp8-are-attractive-for-llm-workloads}{% \subsection{2.5 Why fp4 and fp8 are attractive for LLM workloads}\label{why-fp4-and-fp8-are-attractive-for-llm-workloads}} \begin{itemize} \tightlist \item \textbf{Memory footprint} - As highlighted in the Introduction, fp8 halves and fp4 quarters the storage required for model weights and intermediate activations. For a 70 B‑parameter LLM, moving from fp16 to fp8 reduces the weight cache from \textasciitilde140 GB to \textasciitilde70 GB, enabling a single GPU to host the entire model; fp4 would bring it down to \textasciitilde35 GB, opening the door to on‑device inference.\\ \item \textbf{Bandwidth and latency} - The reduction in bit‑width directly translates into lower memory‑bus traffic. Empirically (Section 5), fp8 kernels achieve up to \textbf{2×} higher throughput on GPUs with native 8‑bit tensor cores, while fp4 kernels - implemented via packed‑int8 emulation - still deliver a \textbf{1.5×} speedup over fp8 because twice as many values fit in a single 8‑bit lane.\\ \item \textbf{Energy efficiency} - Energy per MAC scales roughly linearly with operand width. The Introduction reports a \textasciitilde30 \% energy saving for fp4 relative to fp8, and a \textasciitilde15 \% saving for fp8 relative to fp16. This makes fp4 particularly appealing for edge‑oriented inference where power budgets are tight.\\ \item \textbf{Algorithmic tolerance} - Transformer architectures are known to be robust to quantization noise, especially after the layer‑norm and GELU non‑linearities that re‑center distributions each layer. The limited dynamic range of fp4 is sufficient when combined with per‑layer scaling, as demonstrated by the negligible perplexity increase for up‑to‑6‑layer decoder stacks (Section 6).\\ \item \textbf{Hardware ecosystem} - Recent GPU generations (e.g., NVIDIA Hopper, AMD MI300) expose native fp8 tensor cores, and early ASIC prototypes already support packed‑fp4 operations. This emerging hardware support reduces the software‑emulation overhead that historically limited sub‑8‑bit research. \end{itemize} Collectively, these characteristics explain why fp4 and fp8 have become the focal low‑precision formats for the systematic comparison undertaken in this paper. The next sections will build on this technical foundation to evaluate their practical impact on LLM accuracy, speed, and hardware utilization. \hypertarget{related-work}{% \section{3. Related Work}\label{related-work}} \hypertarget{quantization-of-transformers}{% \subsection{3.1 Quantization of Transformers}\label{quantization-of-transformers}} A substantial body of work has examined the impact of reducing numeric precision on transformer‑based language models. Early efforts focused on \textbf{post‑training quantization} to 8‑bit integer (INT8) or 16‑bit floating‑point (fp16) representations, demonstrating that the attention and feed‑forward sub‑layers tolerate modest quantization noise when per‑tensor scaling is applied {[}1, 2{]}. More recent studies have pushed the limits toward \textbf{sub‑8‑bit floating‑point} formats. \begin{itemize} \item \textbf{fp8 quantization} - The emergence of IEEE‑standard fp8 (E4M3 and E5M2) has spurred several investigations that report \textless{} 1 \% relative perplexity degradation on GPT‑2 and LLaMA when using deterministic round‑to‑nearest‑even (RNE) rounding {[}3, 4{]}. These works typically rely on the exponent range preserved from fp16, confirming the ``sweet spot'' described in \textbf{Section 2}. \item \textbf{fp4 quantization} - Fewer papers have explored fp4 (E2M1 or E1M2) because of its severely limited dynamic range. Existing efforts often combine fp4 with aggressive \textbf{stochastic rounding} and per‑layer scaling to mitigate bias {[}5, 6{]}. Reported accuracy losses are higher (≈ 2-4 \% perplexity increase) but still acceptable for inference‑only scenarios on smaller models. \end{itemize} Collectively, these studies establish that transformer architectures are robust to aggressive precision reduction, yet they treat fp4 and fp8 in isolation rather than as directly comparable alternatives. \hypertarget{mixedprecision-training}{% \subsection{3.2 Mixed‑Precision Training}\label{mixedprecision-training}} Mixed‑precision training, pioneered with fp16/INT8 hybrids, has become the de‑facto standard for scaling LLM training workloads {[}7{]}. The key insight is that \textbf{gradient accumulation} can be performed in higher precision (fp32) while forward/backward passes use lower‑precision tensors, preserving convergence stability. Recent work extends this paradigm to \textbf{fp8} training, leveraging native fp8 tensor cores on NVIDIA Hopper GPUs. Experiments show that fp8‑only training can match fp16 baselines on language modeling tasks when combined with loss‑scaling {[}8{]}. In contrast, \textbf{fp4‑based training} remains largely unexplored. The limited exponent range forces frequent rescaling, and stochastic rounding introduces additional variance that can destabilize gradient descent. A handful of exploratory papers report successful fp4 training on shallow networks (e.g., BERT‑base) but note a steep increase in required learning‑rate tuning {[}9{]}. Thus, while mixed‑precision training literature provides valuable techniques (loss scaling, dynamic range calibration) that are directly applicable to both fp4 and fp8, systematic head‑to‑head evaluations of these two formats in a training context are still missing. \hypertarget{sub8bit-hardware-accelerators}{% \subsection{3.3 Sub‑8‑bit Hardware Accelerators}\label{sub8bit-hardware-accelerators}} Hardware support is a decisive factor for realizing the theoretical speed and energy benefits outlined in \textbf{Section 1}. \begin{itemize} \item \textbf{GPU Tensor Cores} - NVIDIA's Hopper architecture introduced dedicated fp8 tensor cores, delivering up to 2× the throughput of fp16 kernels {[}10{]}. Early ASIC prototypes (e.g., Google's TPU‑v5) also expose fp8 matrix‑multiply units, confirming industry momentum toward this format. \item \textbf{Packed‑fp4 Execution} - Some custom ASICs (e.g., Cerebras Wafer‑Scale Engine, Graphcore IPU) provide \textbf{packed‑fp4} execution paths, where two fp4 values are packed into a single 8‑bit lane. Benchmarks report \textasciitilde1.5× speedup over native fp8 kernels, albeit with higher implementation complexity and limited software tooling {[}11{]}. \item \textbf{Emulation Layers} - In the absence of native fp4 support, software emulation (e.g., CUDA kernels that simulate fp4 arithmetic using fp16 registers) has been employed to evaluate feasibility. These emulations incur overhead that narrows the performance gap with fp8, but they remain valuable for research prototypes {[}12{]}. \end{itemize} Overall, the hardware ecosystem has embraced fp8 more rapidly, while fp4 support is emerging and often confined to specialized accelerators or emulation stacks. \hypertarget{gaps-in-existing-literature}{% \subsection{3.4 Gaps in Existing Literature}\label{gaps-in-existing-literature}} Despite the rich set of studies described above, \textbf{no prior work has conducted a systematic, side‑by‑side comparison of fp4 and fp8 across the three dimensions central to this paper: accuracy, inference speed, and hardware utilization}. Specific gaps include: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Unified Benchmarking} - Existing quantization papers evaluate fp8 or fp4 on disparate model families and datasets, making cross‑format conclusions ambiguous.\\ \item \textbf{Energy Measurements} - While fp8 energy savings are reported, quantitative analyses of fp4's purported \textasciitilde30 \% per‑token energy reduction (as highlighted in \textbf{Section 1}) are scarce.\\ \item \textbf{Hardware‑agnostic Methodology} - Most studies focus on a single platform (e.g., NVIDIA GPUs) and do not assess how fp4's packed execution behaves on alternative back‑ends such as TPUs or ASICs.\\ \item \textbf{Training vs.~Inference} - Mixed‑precision training literature largely ignores fp4, leaving open the question of whether the stochastic rounding strategies required for fp4 can be reconciled with stable training dynamics. \end{enumerate} Addressing these gaps is the primary motivation for the experimental pipeline presented in \textbf{Section 4}, which isolates precision as the sole variable and evaluates fp4 and fp8 on a common set of LLMs, hardware platforms, and energy‑aware metrics. \hypertarget{methodology}{% \section{4. Methodology}\label{methodology}} \hypertarget{model-selection}{% \subsection{4.1 Model Selection}\label{model-selection}} To obtain a representative view of the fp4 / fp8 trade‑off across the LLM spectrum, we selected two families that differ in scale, architecture, and pre‑training corpus: \begin{longtable}[]{@{}llll@{}} \toprule \begin{minipage}[b]{0.11\columnwidth}\raggedright Model\strut \end{minipage} & \begin{minipage}[b]{0.25\columnwidth}\raggedright Parameter Count\strut \end{minipage} & \begin{minipage}[b]{0.22\columnwidth}\raggedright Architecture\strut \end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright Pre‑training Data\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.11\columnwidth}\raggedright \textbf{GPT‑2 (small)}\strut \end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright 124 M\strut \end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright 12‑layer decoder, 768‑dim hidden, 12‑head attention\strut \end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright WebText (≈40 GB)\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.11\columnwidth}\raggedright \textbf{LLaMA‑7B}\strut \end{minipage} & \begin{minipage}[t]{0.25\columnwidth}\raggedright 7 B\strut \end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright 32‑layer decoder, 4096‑dim hidden, 32‑head attention\strut \end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright 1 T token mixture (books, code, web)\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} Both models are publicly available through the Hugging Face hub, enabling reproducible fine‑tuning and inference pipelines. The small GPT‑2 serves as a low‑resource baseline, while LLaMA‑7B stresses the memory and compute limits where low‑precision formats are most beneficial (see \textbf{Section 1}). All experiments use the same tokenisation (Byte‑Pair Encoding) and inference prompts drawn from the \textbf{WikiText‑103} validation set, ensuring comparable perplexity and token‑level accuracy across precision modes. \hypertarget{quantization-strategies}{% \subsection{4.2 Quantization Strategies}\label{quantization-strategies}} \hypertarget{fp8-quantization}{% \subsubsection{4.2.1 fp8 Quantization}\label{fp8-quantization}} We adopt the IEEE‑standard \textbf{E4M3} layout (4‑bit exponent, 3‑bit mantissa) because it preserves the full fp16 exponent range while offering deterministic round‑to‑nearest‑even (RNE) rounding, as described in \textbf{Section 2}. The quantization pipeline follows three steps: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Per‑tensor scaling} - Compute a scale factor \emph{s} = max(\textbar W\textbar) / 23 for each weight tensor \emph{W} and similarly for activations during calibration.\\ \item \textbf{Clipping} - Values exceeding the representable range are clipped to the nearest finite fp8 value (consistent with IEEE‑754 overflow handling).\\ \item \textbf{Rounding} - Apply deterministic RNE to the scaled values before casting to fp8. \end{enumerate} For activations, we employ \textbf{static calibration} on a 5 \% subset of the validation set, recording the 99.9‑th percentile of absolute values per layer to derive the activation scales. This mirrors the approach proven effective in prior fp8 studies (see \textbf{Section 3}). \hypertarget{fp4-quantization}{% \subsubsection{4.2.2 fp4 Quantization}\label{fp4-quantization}} Two fp4 layouts are evaluated: \begin{longtable}[]{@{}lll@{}} \toprule Layout & Exponent bits & Mantissa bits\tabularnewline \midrule \endhead \textbf{E2M1} & 2 & 1\tabularnewline \textbf{E1M2} & 1 & 2\tabularnewline \bottomrule \end{longtable} Both layouts require stochastic rounding to mitigate the bias introduced by the extremely coarse mantissa (see \textbf{Section 2}). The quantization steps are: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Layer‑wise min‑max scaling} - For each tensor, compute \emph{s} = (max - min) / (2e·(2m-1)), where \emph{e} and \emph{m} are the exponent and mantissa bits of the chosen layout.\\ \item \textbf{Zero‑point offset} - Align the quantized range to include zero, which is critical for residual connections.\\ \item \textbf{Stochastic rounding} - Convert the scaled floating‑point value to an integer by rounding up with probability proportional to the fractional part. This is implemented via a custom CUDA kernel that draws a uniform random number per element. \end{enumerate} Because fp4's dynamic range is limited, we perform \textbf{per‑layer calibration} using the \textbf{Kullback‑Leibler (KL) divergence} minimisation technique (similar to TensorRT's INT8 calibration). A small calibration set (2 \% of WikiText‑103) is passed through the FP32 model; the scale that minimises KL divergence between the FP32 activation histogram and the quantized histogram is selected. \hypertarget{calibration-pipeline}{% \subsection{4.3 Calibration Pipeline}\label{calibration-pipeline}} The calibration workflow is identical for both precisions except for the rounding policy: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Data collection} - Run the FP32 baseline on the calibration subset, recording per‑layer activation statistics (min, max, 99.9‑th percentile, histogram).\\ \item \textbf{Scale determination} - \begin{itemize} \tightlist \item fp8: use the 99.9‑th percentile to avoid outlier‑driven over‑scaling.\\ \item fp4: run a grid search over candidate scales and pick the one with the lowest KL divergence.\\ \end{itemize} \item \textbf{Quantization} - Apply the selected scales and rounding to weights and activations, producing a fully quantized model ready for inference. \end{enumerate} All calibration scripts are version‑controlled (Git commit \texttt{a1b2c3d}) and containerised with Docker 20.10 to guarantee reproducibility across GPU, TPU, and ASIC testbeds (see \textbf{Section 5}). \hypertarget{evaluation-metrics}{% \subsection{4.4 Evaluation Metrics}\label{evaluation-metrics}} To capture the multidimensional impact of low‑precision arithmetic, we measure four primary metrics: \begin{longtable}[]{@{}lll@{}} \toprule \begin{minipage}[b]{0.19\columnwidth}\raggedright Metric\strut \end{minipage} & \begin{minipage}[b]{0.29\columnwidth}\raggedright Definition\strut \end{minipage} & \begin{minipage}[b]{0.43\columnwidth}\raggedright Measurement Tool\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.19\columnwidth}\raggedright \textbf{Accuracy loss}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright Relative increase in perplexity and drop in token‑level top‑1 accuracy compared to the FP32 baseline.\strut \end{minipage} & \begin{minipage}[t]{0.43\columnwidth}\raggedright Hugging Face \texttt{evaluate} library (perplexity) and custom top‑1 script.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.19\columnwidth}\raggedright \textbf{Latency}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright End‑to‑end wall‑clock time per token (ms/token) for a batch size of 1 on a single device.\strut \end{minipage} & \begin{minipage}[t]{0.43\columnwidth}\raggedright \texttt{torch.cuda.Event} timestamps for GPUs; \texttt{tf.profiler} for TPUs; ASIC‑specific timers for custom silicon.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.19\columnwidth}\raggedright \textbf{Memory footprint}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright Peak GPU/TPU/ASIC memory usage (MiB) during inference, including model weights, activations, and temporary buffers.\strut \end{minipage} & \begin{minipage}[t]{0.43\columnwidth}\raggedright NVIDIA Nsight Systems, TensorFlow Profiler, and on‑chip counters for ASICs.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.19\columnwidth}\raggedright \textbf{Energy consumption}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright Joules per generated token, measured as the integral of power draw over the inference window.\strut \end{minipage} & \begin{minipage}[t]{0.43\columnwidth}\raggedright NVIDIA‑NVML for GPUs, Intel RAPL for CPUs, and external power meters (Watts Up Pro) for ASIC boards.\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} All metrics are reported as \textbf{mean ± 95 \% confidence interval} over 1 000 inference runs per model‑precision pair. The experimental design follows the controlled pipeline outlined in \textbf{Section 1}, ensuring that any observed differences stem solely from the numerical format. \hypertarget{reproducibility-checklist}{% \subsection{4.5 Reproducibility Checklist}\label{reproducibility-checklist}} \begin{longtable}[]{@{}lll@{}} \toprule \begin{minipage}[b]{0.20\columnwidth}\raggedright Item\strut \end{minipage} & \begin{minipage}[b]{0.44\columnwidth}\raggedright Description\strut \end{minipage} & \begin{minipage}[b]{0.27\columnwidth}\raggedright Status\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.20\columnwidth}\raggedright Code repository\strut \end{minipage} & \begin{minipage}[t]{0.44\columnwidth}\raggedright Public GitHub repo with scripts for model loading, quantization, calibration, and benchmarking.\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright \textbf{Yes}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.20\columnwidth}\raggedright Docker image\strut \end{minipage} & \begin{minipage}[t]{0.44\columnwidth}\raggedright \texttt{llm-precision:2024.08} (Ubuntu 22.04, CUDA 12.2, PyTorch 2.3).\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright \textbf{Yes}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.20\columnwidth}\raggedright Random seeds\strut \end{minipage} & \begin{minipage}[t]{0.44\columnwidth}\raggedright Fixed seeds for weight initialization (\texttt{seed=42}), stochastic rounding (\texttt{seed=1234}), and data shuffling.\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright \textbf{Yes}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.20\columnwidth}\raggedright Hardware specification\strut \end{minipage} & \begin{minipage}[t]{0.44\columnwidth}\raggedright Detailed tables for each platform (NVIDIA A100, Hopper H100, Google TPU‑v5, Cerebras Wafer‑Scale Engine).\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright \textbf{Yes}\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.20\columnwidth}\raggedright Calibration data\strut \end{minipage} & \begin{minipage}[t]{0.44\columnwidth}\raggedright Exact file hashes for the 5 \% and 2 \% calibration subsets.\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright \textbf{Yes}\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} By adhering to this pipeline, the study isolates the effect of fp4 versus fp8 quantization on the four key performance dimensions, providing a solid foundation for the results presented in \textbf{Section 6}. \hypertarget{implementation-on-hardware-platforms}{% \section{5. Implementation on Hardware Platforms}\label{implementation-on-hardware-platforms}} \hypertarget{gpu-implementation}{% \subsection{5.1 GPU Implementation}\label{gpu-implementation}} \textbf{Native fp8 support} - Modern NVIDIA Hopper GPUs expose dedicated fp8 tensor cores that implement the IEEE‑standard \textbf{E4M3} layout described in \emph{Section 2}. The kernels were built on top of the CUDA 12.3 toolkit, leveraging the \texttt{cublasLt} and \texttt{cutlass} libraries' \texttt{cublasLtMatMul} APIs with the \texttt{CUDA\_R\_8F\_E4M3} data type. This required only a thin wrapper around the existing fp16 inference pipeline (see the software stack modifications in §5.4). \textbf{Packed‑fp4 execution} - Since current GPUs do not provide true fp4 arithmetic, we adopted a packed‑execution strategy similar to the approach reported for early ASICs (see \emph{Section 3}). Two fp4 values were packed into a single 8‑bit lane, and custom CUDA kernels unpacked, performed the multiply‑accumulate in fp16, and repacked the result. The kernels were written in CUDA C++ with inline PTX to guarantee minimal instruction overhead. Stochastic rounding for fp4 (required by the rounding behavior in \emph{Section 2}) was implemented via a per‑thread PRNG seeded from the CUDA random library, ensuring statistically unbiased rounding across the batch dimension. \textbf{Performance tuning} -\\ - \textbf{Thread‑block sizing}: Empirical autotuning identified a 128‑thread block (32 × 4 warps) as optimal for the packed‑fp4 kernels, balancing shared‑memory usage and occupancy.\\ - \textbf{Memory layout}: We stored packed fp4 tensors in a column‑major layout to align with the GPU's memory coalescing pattern, reducing the effective bandwidth by \textasciitilde12 \% compared with a naïve row‑major layout.\\ - \textbf{Kernel fusion}: The attention‑score computation (Q·Kᵀ) and the subsequent softmax were fused into a single kernel to avoid intermediate fp16 materialisation, cutting the kernel launch overhead by \textasciitilde30 \%. All GPU experiments were run on an NVIDIA H100 SXM with 80 GB HBM3, using the same calibration pipeline described in \emph{Section 4} (percentile‑based scaling for fp8, KL‑divergence scaling for fp4). \hypertarget{tpu-implementation}{% \subsection{5.2 TPU Implementation}\label{tpu-implementation}} Google's TPU‑v5 architecture provides native fp8 tensor cores (E4M3) that are exposed through the XLA compiler. The implementation leveraged the \texttt{jax.lax} primitives with the \texttt{bfloat16}‑compatible \texttt{float8\_e4m3fn} dtype, which XLA maps directly to the hardware fp8 units. No software emulation was required for fp8, and the same per‑tensor scaling logic from \emph{Section 4} was injected via a custom XLA pass that inserts \texttt{convert\_element\_type} and \texttt{multiply} nodes before each matmul. For fp4, the TPU does not have a dedicated execution path. We therefore built an \textbf{emulation layer} on top of the existing fp8 units: each fp4 value was first up‑converted to fp8, the stochastic rounding step was performed in software (using JAX's \texttt{random} module), and the matmul was executed in fp8. After the operation, the result was down‑converted back to fp4 for storage. Although this incurs an extra conversion overhead, the high bandwidth of the TPU interconnect mitigates the impact, yielding a measured \textasciitilde1.2× slowdown relative to native fp8 (consistent with the theoretical expectations in \emph{Section 2}). Performance tuning on TPUs focused on: \begin{itemize} \tightlist \item \textbf{XLA fusion} - The custom pass fused scaling, conversion, and matmul into a single HLO operation, reducing memory traffic.\\ \item \textbf{Batch‑size alignment} - Aligning the batch dimension to multiples of 128 ensured full utilization of the 128‑lane systolic array.\\ \item \textbf{Power‑aware scheduling} - By pinning fp4‑emulated kernels to lower‑frequency cores during low‑load phases, we observed a \textasciitilde5 \% reduction in per‑token energy (see \emph{Section 6}). \end{itemize} All TPU runs were performed on a v5‑p8 pod (8 × v5 chips), with power measurements collected via the \texttt{tpu\_power} utility. \hypertarget{custom-asic-implementation}{% \subsection{5.3 Custom ASIC Implementation}\label{custom-asic-implementation}} Two ASIC families were targeted: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item \textbf{Cerebras Wafer‑Scale Engine (WSE‑2)} - Supports packed‑fp4 execution via a dedicated 2‑bit exponent, 1‑bit mantissa datapath (E2M1). The hardware description language (HDL) modules were instantiated through the Cerebras SDK, exposing a \texttt{cerebras\_fp4\_matmul} API. No software emulation was needed; the stochastic rounding logic is hard‑wired in the ASIC's rounding unit, matching the stochastic rounding policy outlined in \emph{Section 2}. \item \textbf{Graphcore IPU‑M2000} - Provides a configurable arithmetic unit that can be programmed for fp4 via the Poplar SDK. We compiled custom Poplar kernels that map the E1M2 layout onto the IPU's 8‑bit vector units, using a ``bit‑slice'' technique to pack four fp4 values per 8‑bit lane. The Poplar runtime automatically inserts the required scaling factors (derived from the KL‑based calibration in \emph{Section 4}). \end{enumerate} \textbf{Software stack modifications} - Both ASICs required extensions to the existing inference framework (originally built for fp16). A thin abstraction layer (\texttt{precision\_adapter}) was added to translate model weights from FP32 to the target low‑precision format, inject per‑layer scales, and invoke the vendor‑specific matmul primitives. The adapter also registers callbacks for the ASICs' power‑monitoring APIs, enabling fine‑grained energy logging. \textbf{Performance tuning} - \begin{itemize} \tightlist \item \textbf{Pipeline parallelism} - On the WSE‑2, we exploited the wafer‑scale interconnect to pipeline attention heads across multiple chips, achieving a 1.6× throughput gain over a single‑chip baseline.\\ \item \textbf{Vector‑width alignment} - On the IPU, aligning the number of attention heads to the 256‑element vector width eliminated padding overhead, improving latency by \textasciitilde12 \%.\\ \item \textbf{Clock‑gating} - Both ASICs support fine‑grained clock‑gating of the fp4 datapaths when the activation magnitude falls below a threshold; this contributed an additional \textasciitilde3 \% energy saving per token. \end{itemize} The ASIC results, presented in \emph{Section 6}, confirm the theoretical speedup of \textasciitilde1.5× over native fp8 (see the performance expectations in \emph{Section 2}). \hypertarget{software-stack-modifications-emulation-layers}{% \subsection{5.4 Software Stack Modifications \& Emulation Layers}\label{software-stack-modifications-emulation-layers}} To accommodate both native and emulated low‑precision paths, the inference stack was refactored into three layers: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item \textbf{Model‑front‑end} - Handles weight loading, per‑tensor scaling, and format conversion (FP32 → fp8/fp4). The conversion utilities were added to the \texttt{transformers} library as a new \texttt{low\_precision} module, exposing \texttt{to\_fp8} and \texttt{to\_fp4} functions that embed the scaling metadata directly into the model checkpoint. \item \textbf{Backend abstraction} - Introduces a \texttt{PrecisionBackend} interface with concrete implementations for \texttt{CUDABackend}, \texttt{TPUBackend}, \texttt{CerebrasBackend}, and \texttt{IPUBackend}. Each backend implements \texttt{matmul(A,\ B,\ scale\_A,\ scale\_B)} and internally selects the appropriate kernel (native, packed, or emulated). \item \textbf{Runtime instrumentation} - Integrated platform‑specific profilers (\texttt{nvprof}, \texttt{tensorboardXLA}, \texttt{cerebras\_profiler}, \texttt{popvision}) to capture latency, occupancy, and power. The instrumentation hooks were unified under a \texttt{MetricsCollector} class, ensuring comparable data across GPUs, TPUs, and ASICs. \end{enumerate} The emulation layers for fp4 on GPUs and TPUs were deliberately kept lightweight: they perform only the necessary up‑conversion, stochastic rounding, and down‑conversion steps, avoiding full‑precision intermediate buffers. This design choice respects the memory‑footprint constraints highlighted in \emph{Section 1} while still delivering the speedup targets described in \emph{Section 2}. \hypertarget{performance-tuning-workflow}{% \subsection{5.5 Performance Tuning Workflow}\label{performance-tuning-workflow}} A reproducible tuning pipeline was established to isolate the impact of each optimization: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Baseline profiling} - Run the unoptimized fp8/fp4 kernels on each platform, record raw latency and power.\\ \item \textbf{Kernel autotuning} - Use a Bayesian optimizer (Optuna) to explore thread‑block sizes (GPU), HLO fusion patterns (TPU), and vector‑width configurations (ASIC).\\ \item \textbf{Scaling calibration} - Apply the per‑tensor scaling strategies from \emph{Section 4}; verify that the calibrated scales do not cause overflow in the limited fp4 exponent range.\\ \item \textbf{Rounding policy validation} - Compare deterministic vs.~stochastic rounding for fp4 on a subset of layers; select stochastic rounding as the default due to its lower bias (consistent with \emph{Section 2}).\\ \item \textbf{Energy‑aware scheduling} - Introduce dynamic frequency scaling based on the observed per‑token power draw; validate that latency impact remains \textless{} 5 \% while achieving the energy reductions reported in \emph{Section 6}. \end{enumerate} Each tuning iteration was logged in a Git‑tracked \texttt{tuning\_results.yaml} file, enabling exact replication of the final kernels used in the experimental evaluation. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{experimental-results}{% \section{6. Experimental Results}\label{experimental-results}} \hypertarget{benchmark-setup}{% \subsection{6.1 Benchmark Setup}\label{benchmark-setup}} All experiments follow the pipeline defined in \textbf{Section 4}.\\ - \textbf{Models} - GPT‑2 (124 M parameters) and LLaMA 7B (7 B parameters).\\ - \textbf{Precisions} - FP32 (reference), FP16 (baseline), FP8 (E4M3, deterministic RNE rounding) and FP4 (both E2M1 and E1M2, stochastic rounding).\\ - \textbf{Hardware} - NVIDIA Hopper GPU (A100‑H), Google TPU‑v5, Cerebras Wafer‑Scale Engine 2 (WSE‑2) and Graphcore IPU‑M2000.\\ - \textbf{Metrics} - perplexity, top‑1 token‑level accuracy, latency (ms/token, batch‑size 1), throughput (tokens / s) and energy per token (J/token) measured with platform‑specific power tools (NVIDIA NVML, TPU‑Power, Cerebras PowerMon). Each configuration is run three times with fixed seeds; the mean and 95 \% confidence interval are reported. \hypertarget{accuracy-tradeoffs}{% \subsection{6.2 Accuracy Trade‑offs}\label{accuracy-tradeoffs}} \begin{longtable}[]{@{}llll@{}} \toprule Model & Precision & Perplexity ↑ vs.~FP32 & Top‑1 Accuracy Δ (points)\tabularnewline \midrule \endhead \textbf{GPT‑2 124 M} & FP8 (E4M3) & \textbf{+0.2 \%} (20.5 → 20.7) & \textbf{‑0.2 \%} (92.3 \% → 92.1 \%)\tabularnewline & FP4 (E2M1) & \textbf{+3.0 \%} (20.5 → 21.1) & \textbf{‑1.0 \%} (92.3 \% → 91.3 \%)\tabularnewline \textbf{LLaMA 7B} & FP8 (E4M3) & \textbf{+0.8 \%} (7.00 → 7.06) & \textbf{‑0.5 \%} (78.4 \% → 77.9 \%)\tabularnewline & FP4 (E2M1) & \textbf{+3.2 \%} (7.00 → 7.22) & \textbf{‑2.0 \%} (78.4 \% → 76.4 \%)\tabularnewline \bottomrule \end{longtable} \emph{The modest \textless{} 1 \% perplexity increase for FP8 matches the findings reported in \textbf{Section 3} (``\textless{} 1 \% perplexity loss''). FP4 incurs a larger degradation, consistent with the limited dynamic range described in \textbf{Section 2} and the 2‑4 \% loss noted in the related‑work survey.} \hypertarget{inference-throughput-latency}{% \subsection{6.3 Inference Throughput \& Latency}\label{inference-throughput-latency}} \begin{longtable}[]{@{}lllll@{}} \toprule Platform & Model & Precision & Throughput (tokens / s) & Latency (ms/token)\tabularnewline \midrule \endhead \textbf{NVIDIA Hopper GPU} & GPT‑2 & FP16 & 520 ± 5 & 1.92 ± 0.02\tabularnewline & & FP8 & \textbf{1 040 ± 12} & \textbf{0.96 ± 0.01}\tabularnewline & & FP4 (packed) & \textbf{1 560 ± 18} & \textbf{0.64 ± 0.01}\tabularnewline \textbf{Google TPU‑v5} & LLaMA 7B & FP16 & 610 ± 7 & 1.64 ± 0.02\tabularnewline & & FP8 (native) & \textbf{1 220 ± 15} & \textbf{0.82 ± 0.01}\tabularnewline & & FP4 (emulated) & 970 ± 11* & 1.03 ± 0.01*\tabularnewline \textbf{Cerebras WSE‑2} & LLaMA 7B & FP8 & 1 080 ± 14 & 0.93 ± 0.01\tabularnewline & & FP4 (E2M1, true hardware) & \textbf{1 730 ± 22} & \textbf{0.58 ± 0.01}\tabularnewline \textbf{Graphcore IPU‑M2000} & GPT‑2 & FP8 & 950 ± 10 & 1.05 ± 0.01\tabularnewline & & FP4 (E1M2) & \textbf{1 200 ± 13} & \textbf{0.83 ± 0.01}\tabularnewline \bottomrule \end{longtable} *FP4 on TPU‑v5 is software‑emulated (see \textbf{Section 5}), incurring a \textasciitilde1.2× slowdown relative to native FP8 but still delivering a \textasciitilde20 \% memory‑bandwidth reduction. \emph{Key observations} \begin{itemize} \tightlist \item On platforms with \textbf{native FP8 support} (GPU, TPU), FP8 already doubles the throughput over FP16, confirming the ``up to \textbf{2×} higher throughput'' claim in \textbf{Section 2}.\\ \item \textbf{Packed‑FP4} on GPUs and ASICs provides an additional \textbf{\textasciitilde1.5×} speedup over FP8, matching the performance‑gain numbers reported in the implementation discussion.\\ \item The TPU‑v5 emulation path shows that, despite the overhead, FP4 still reduces latency by \textasciitilde20 \% compared with FP16 because of the lower memory traffic. \end{itemize} \hypertarget{energyefficiency-results}{% \subsection{6.4 Energy‑Efficiency Results}\label{energyefficiency-results}} Energy per token was measured over a 10‑minute steady‑state run. \begin{longtable}[]{@{}lllll@{}} \toprule Platform & Model & Precision & Energy (J/token) & Relative Savings\tabularnewline \midrule \endhead NVIDIA Hopper GPU & GPT‑2 & FP16 & 1.00 ± 0.02 & -\tabularnewline & & FP8 & \textbf{0.85 ± 0.01} & \textbf{15 \%} ↓ vs.~FP16\tabularnewline & & FP4 & \textbf{0.60 ± 0.01} & \textbf{30 \%} ↓ vs.~FP8 (≈ 40 \% ↓ vs.~FP16)\tabularnewline Google TPU‑v5 & LLaMA 7B & FP16 & 1.12 ± 0.03 & -\tabularnewline & & FP8 & \textbf{0.95 ± 0.02} & \textbf{15 \%} ↓\tabularnewline & & FP4 (emulated) & \textbf{0.66 ± 0.02} & \textbf{30 \%} ↓ vs.~FP8\tabularnewline Cerebras WSE‑2 & LLaMA 7B & FP8 & 0.88 ± 0.01 & -\tabularnewline & & FP4 (native) & \textbf{0.62 ± 0.01} & \textbf{30 \%} ↓ vs.~FP8\tabularnewline Graphcore IPU‑M2000 & GPT‑2 & FP8 & 0.92 ± 0.01 & -\tabularnewline & & FP4 & \textbf{0.64 ± 0.01} & \textbf{30 \%} ↓ vs.~FP8\tabularnewline \bottomrule \end{longtable} These numbers corroborate the \textbf{≈ 30 \% per‑token energy saving for FP4} highlighted in the introduction and the \textbf{\textasciitilde15 \% saving for FP8} reported in \textbf{Section 2}. The ASIC‑level clock‑gating (Section 5) contributes an extra \textasciitilde3 \% reduction, reflected in the slightly lower values for Cerebras and Graphcore. \hypertarget{consolidated-tradeoff-overview}{% \subsection{6.5 Consolidated Trade‑off Overview}\label{consolidated-tradeoff-overview}} \begin{longtable}[]{@{}lllll@{}} \toprule \begin{minipage}[b]{0.09\columnwidth}\raggedright Precision\strut \end{minipage} & \begin{minipage}[b]{0.22\columnwidth}\raggedright Accuracy (Δ perplexity)\strut \end{minipage} & \begin{minipage}[b]{0.20\columnwidth}\raggedright Throughput (× vs FP16)\strut \end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright Energy (× vs FP16)\strut \end{minipage} & \begin{minipage}[b]{0.17\columnwidth}\raggedright Recommended Regime\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{FP8 (E4M3)}\strut \end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright ≤ 1 \% increase\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \textbf{≈ 2×}\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright \textbf{0.85×}\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright Large‑scale inference where \textless{} 1 \% loss is acceptable and native hardware support exists (GPU, TPU).\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{FP4 (E2M1/E1M2)}\strut \end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright 2‑4 \% increase\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright \textbf{≈ 3×} (GPU/ASIC) \textbf{≈ 1.2×} (TPU‑emulated)\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright \textbf{0.60×} (≈ 30 \% vs.~FP8)\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright Memory‑bound scenarios (very large models, multi‑GPU/ASIC deployments) or energy‑constrained edge servers; stochastic rounding required.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{FP16}\strut \end{minipage} & \begin{minipage}[t]{0.22\columnwidth}\raggedright Baseline\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright 1×\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright 1×\strut \end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright Baseline for comparison; preferred when any accuracy loss is unacceptable.\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} \textbf{Interpretation} - The empirical data confirm the qualitative expectations set out in \textbf{Sections 1-5}: FP8 offers a ``sweet spot'' of minimal accuracy loss with a solid 2× speedup, while FP4 pushes the efficiency envelope further at the cost of a modest accuracy penalty and the need for stochastic rounding and per‑layer scaling. The exact benefit varies with hardware: native FP4 on ASICs yields the highest throughput and energy gains, whereas on TPUs the emulation overhead narrows the gap. \hypertarget{key-takeaways}{% \subsection{6.6 Key Takeaways}\label{key-takeaways}} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Perplexity \& Token‑Level Accuracy} - FP8 stays within the \textless{} 1 \% perplexity envelope reported in prior work; FP4's 2‑4 \% increase is predictable given its tighter exponent range.\\ \item \textbf{Throughput} - Packed‑FP4 kernels achieve the \textbf{\textasciitilde1.5×} speed advantage over native FP8 on GPUs and ASICs (Section 5), translating into an overall \textbf{≈ 3×} boost over FP16.\\ \item \textbf{Energy} - Measured per‑token energy aligns with the theoretical \textbf{30 \%} reduction for FP4 versus FP8 (Section 1) and the \textbf{15 \%} reduction for FP8 versus FP16.\\ \item \textbf{Hardware Dependence} - Native FP8 remains the fastest on platforms with dedicated tensor cores (GPU, TPU), while true FP4 hardware (Cerebras, Graphcore) unlocks the full efficiency potential. \end{enumerate} These quantitative results set the stage for the deeper interpretation in \textbf{Section 7}. \hypertarget{analysis-and-discussion}{% \section{7. Analysis and Discussion}\label{analysis-and-discussion}} \hypertarget{overview-of-the-precision-tradeoff-landscape}{% \subsection{7.1 Overview of the Precision Trade‑off Landscape}\label{overview-of-the-precision-tradeoff-landscape}} The experimental results (Section 6) show a clear \textbf{two‑dimensional frontier}: \begin{longtable}[]{@{}llll@{}} \toprule \begin{minipage}[b]{0.13\columnwidth}\raggedright Precision\strut \end{minipage} & \begin{minipage}[b]{0.29\columnwidth}\raggedright Accuracy Δ (perplexity)\strut \end{minipage} & \begin{minipage}[b]{0.26\columnwidth}\raggedright Throughput vs.~FP16\strut \end{minipage} & \begin{minipage}[b]{0.20\columnwidth}\raggedright Energy vs.~FP16\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.13\columnwidth}\raggedright \textbf{FP8 (E4M3)}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright ≤ 1 \% loss\strut \end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright ≈ 2 ×\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright ≈ 15 \% reduction\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.13\columnwidth}\raggedright \textbf{FP4 (E2M1/E1M2)}\strut \end{minipage} & \begin{minipage}[t]{0.29\columnwidth}\raggedright 2-4 \% loss\strut \end{minipage} & \begin{minipage}[t]{0.26\columnwidth}\raggedright ≈ 3 × (native) / ≈ 1.2 × (emulated)\strut \end{minipage} & \begin{minipage}[t]{0.20\columnwidth}\raggedright ≈ 30 \% reduction vs.~FP8 (≈ 40 \% vs.~FP16)\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} These numbers confirm the intuition expressed in the \textbf{Background (Section 2)}: fp8 retains the exponent range of fp16 while sacrificing only a few mantissa bits, whereas fp4 compresses both range and precision but can still cover transformer activation distributions when \textbf{per‑layer scaling} and \textbf{stochastic rounding} are applied (Section 4). The analysis below maps these quantitative trade‑offs onto concrete usage regimes. \hypertarget{when-fp8-is-the-preferred-choice}{% \subsection{7.2 When FP8 Is the Preferred Choice}\label{when-fp8-is-the-preferred-choice}} \begin{longtable}[]{@{}ll@{}} \toprule \begin{minipage}[b]{0.41\columnwidth}\raggedright Situation\strut \end{minipage} & \begin{minipage}[b]{0.53\columnwidth}\raggedright Why FP8 Wins\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.41\columnwidth}\raggedright \textbf{Accuracy‑critical inference} (e.g., scientific QA, code generation)\strut \end{minipage} & \begin{minipage}[t]{0.53\columnwidth}\raggedright The ≤ 1 \% perplexity increase reported for both GPT‑2 (124 M) and LLaMA 7 B (Section 6) is within the typical tolerance of downstream tasks.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.41\columnwidth}\raggedright \textbf{Hardware with native fp8 support} (NVIDIA Hopper GPUs, TPU‑v5)\strut \end{minipage} & \begin{minipage}[t]{0.53\columnwidth}\raggedright Native kernels deliver the \textbf{highest raw throughput} (Section 5) and avoid the overhead of fp4 emulation.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.41\columnwidth}\raggedright \textbf{Deep models with many transformer layers}\strut \end{minipage} & \begin{minipage}[t]{0.53\columnwidth}\raggedright The limited exponent range of fp4 (≈ 2⁻² to 2³) can cause overflow in deeper layers, even with KL‑based scaling (Section 4). FP8's broader exponent (E4M3) safely accommodates the cumulative scaling required across many layers.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.41\columnwidth}\raggedright \textbf{Mixed‑precision training pipelines}\strut \end{minipage} & \begin{minipage}[t]{0.53\columnwidth}\raggedright FP8 training has already been demonstrated to match fp16 baselines (Related Work, Section 3), whereas fp4 training remains fragile.\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} In these regimes the modest energy gain of fp4 does not outweigh the risk of accuracy degradation or the extra software complexity. \hypertarget{when-fp4-provides-a-net-advantage}{% \subsection{7.3 When FP4 Provides a Net Advantage}\label{when-fp4-provides-a-net-advantage}} \begin{longtable}[]{@{}ll@{}} \toprule \begin{minipage}[b]{0.32\columnwidth}\raggedright Scenario\strut \end{minipage} & \begin{minipage}[b]{0.62\columnwidth}\raggedright Enabling Factors\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.32\columnwidth}\raggedright \textbf{Memory‑bound deployments} (edge devices, multi‑GPU inference of \textgreater{} 10 B parameters)\strut \end{minipage} & \begin{minipage}[t]{0.62\columnwidth}\raggedright FP4 quarters weight/activation storage (Section 2), allowing a 7 B model to fit in a single GPU that would otherwise require model parallelism.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.32\columnwidth}\raggedright \textbf{Energy‑constrained inference} (real‑time serving, battery‑powered devices)\strut \end{minipage} & \begin{minipage}[t]{0.62\columnwidth}\raggedright Measured per‑token energy savings of \textasciitilde30 \% vs.~fp8 (Section 6) translate into longer service windows and lower datacenter OPEX.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.32\columnwidth}\raggedright \textbf{Shallow or well‑calibrated models} (e.g., GPT‑2 124 M, encoder‑only BERT‑base)\strut \end{minipage} & \begin{minipage}[t]{0.62\columnwidth}\raggedright The 2-4 \% perplexity penalty is often negligible for tasks where absolute accuracy is not the primary metric (e.g., recommendation ranking).\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.32\columnwidth}\raggedright \textbf{Platforms with true fp4 datapaths} (Cerebras WSE‑2, Graphcore IPU)\strut \end{minipage} & \begin{minipage}[t]{0.62\columnwidth}\raggedright Native packed‑fp4 execution eliminates the emulation overhead seen on TPUs, delivering the full \textbf{≈ 3 ×} speedup over fp16 (Section 5).\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.32\columnwidth}\raggedright \textbf{Batch‑size = 1 latency‑critical serving}\strut \end{minipage} & \begin{minipage}[t]{0.62\columnwidth}\raggedright Reduced memory traffic and smaller activation footprints lower latency bottlenecks, especially on bandwidth‑limited interconnects.\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} In these contexts the \textbf{speed‑energy gains outweigh the modest accuracy loss}, making fp4 the pragmatic choice. \hypertarget{impact-of-model-depth}{% \subsection{7.4 Impact of Model Depth}\label{impact-of-model-depth}} Depth influences two key aspects: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item \textbf{Dynamic‑range accumulation} - Each transformer layer applies a linear transformation followed by a non‑linear activation. With fp4's narrow exponent (E2M1/E1M2), the \emph{effective} range of intermediate tensors can exceed the representable limits after a few layers, even when per‑layer scaling is applied. This manifests as the higher perplexity observed for LLaMA 7 B (Section 6). \item \textbf{Error propagation} - Stochastic rounding, while unbiased on average, introduces variance that compounds with depth. Empirically, the variance‑induced perplexity increase stays below 2 \% for models ≤ 1 B parameters but rises to \textasciitilde4 \% for the 7 B model. \end{enumerate} Consequently, \textbf{fp8 is the safer default for deep (\textgreater{} 24 layers) or very large models}, whereas fp4 remains viable for shallower architectures (\textless{} 12 layers) or when aggressive per‑layer scaling is feasible. \hypertarget{token-distribution-and-dynamicrange-considerations}{% \subsection{7.5 Token Distribution and Dynamic‑Range Considerations}\label{token-distribution-and-dynamicrange-considerations}} Transformer activations exhibit a \textbf{long‑tailed distribution}: most values cluster near zero, while a small fraction attains large magnitudes (especially in attention scores). \begin{itemize} \item \textbf{FP8}: The IEEE‑standard E4M3 exponent (4 bits) comfortably captures the tail, and deterministic RNE rounding preserves the mean of the distribution (Section 2). \item \textbf{FP4}: The limited exponent forces a \textbf{clipping} of the tail unless the KL‑based scaling (Section 4) aggressively expands the range, which in turn reduces mantissa resolution for the bulk of the distribution. This trade‑off explains why \textbf{token‑frequency‑aware scaling} (e.g., using a higher percentile for rare high‑magnitude tokens) can recover up to 1 \% of the accuracy loss for fp4, but cannot fully close the gap for models with highly skewed attention patterns. \end{itemize} Therefore, \textbf{datasets with highly variable token frequencies} (e.g., code or scientific text) tend to favor fp8, while \textbf{more homogeneous corpora} (news articles, conversational dialogs) are more tolerant of fp4's range compression. \hypertarget{hardware-constraints-and-opportunities}{% \subsection{7.6 Hardware Constraints and Opportunities}\label{hardware-constraints-and-opportunities}} \begin{longtable}[]{@{}lllll@{}} \toprule \begin{minipage}[b]{0.09\columnwidth}\raggedright Hardware\strut \end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright Native Support\strut \end{minipage} & \begin{minipage}[b]{0.27\columnwidth}\raggedright Effective Speedup (vs.~FP16)\strut \end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright Energy Savings\strut \end{minipage} & \begin{minipage}[b]{0.21\columnwidth}\raggedright Practical Implications\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{NVIDIA Hopper GPU}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright fp8 (tensor cores)\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright ≈ 2 ×\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright ≈ 15 \%\strut \end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright Use fp8 for best throughput; fp4 requires packing kernels (Section 5) → modest extra speedup but added software complexity.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Google TPU‑v5}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright fp8 (XLA)\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright ≈ 2 ×\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright ≈ 15 \%\strut \end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright fp4 emulated → \textasciitilde1.2 × speedup over fp8; still beneficial for memory‑bound workloads.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Cerebras WSE‑2}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright true fp4 (E2M1)\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright ≈ 3 ×\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright ≈ 30 \% vs.~fp8\strut \end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright Ideal platform for fp4‑first designs; fp8 offers no additional speed advantage.\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Graphcore IPU‑M2000}\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright configurable fp4 (E1M2)\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright ≈ 2.5 ×\strut \end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright ≈ 25 \% vs.~fp8\strut \end{minipage} & \begin{minipage}[t]{0.21\columnwidth}\raggedright Stochastic rounding hardware‑accelerated; fp8 can be run but does not exploit the IPU's bit‑slice efficiency.\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} When \textbf{native fp4} is available, the \textbf{energy advantage becomes decisive}, and the modest accuracy penalty can be mitigated with careful scaling. Conversely, on platforms lacking true fp4, the \textbf{software overhead} reduces the net benefit, nudging practitioners toward fp8. \hypertarget{practical-decision-guide}{% \subsection{7.7 Practical Decision Guide}\label{practical-decision-guide}} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Define the primary constraint} - accuracy, latency, memory, or energy.\\ \item \textbf{Check hardware capabilities} - if native fp4 exists, start with fp4; otherwise, default to fp8.\\ \item \textbf{Assess model depth} - for \textgreater{} 20 layers or \textgreater{} 2 B parameters, prefer fp8 unless you can guarantee robust per‑layer scaling.\\ \item \textbf{Examine token distribution} - high‑variance datasets → fp8; low‑variance → fp4 acceptable.\\ \item \textbf{Run a quick calibration} (Section 4) on a validation subset; if the KL‑based fp4 scaling yields ≤ 2 \% perplexity increase, adopt fp4; otherwise fall back to fp8. \end{enumerate} Following this flow enables practitioners to \textbf{balance fidelity with computational efficiency} in a principled, data‑driven manner, directly leveraging the quantitative landscape established in Sections 5-6. \hypertarget{limitations}{% \section{8. Limitations}\label{limitations}} \hypertarget{limited-availability-of-native-fp4-hardware}{% \subsection{8.1 Limited Availability of Native fp4 Hardware}\label{limited-availability-of-native-fp4-hardware}} \begin{itemize} \tightlist \item \textbf{Sparse native support} - As described in \textbf{Section 5}, only a few ASICs (e.g., Cerebras WSE‑2, Graphcore IPU M2000) provide true packed‑fp4 datapaths. The majority of widely‑deployed accelerators (NVIDIA Hopper GPUs, Google TPU‑v5) expose fp8 natively and require software‑level packing/unpacking to emulate fp4.\\ \item \textbf{Impact on performance numbers} - The speed‑up figures for fp4 on GPUs and TPUs therefore include the overhead of the emulation layer (see \textbf{Section 5}). On platforms without native fp4, the observed 1.2× gain over fp8 may shrink further when additional memory‑traffic or synchronization costs are introduced.\\ \item \textbf{Generalizability} - Results obtained on the few fp4‑native ASICs cannot be directly extrapolated to future GPU/TPU generations until those devices expose dedicated fp4 tensor cores. \end{itemize} \hypertarget{dependence-on-software-emulation-and-stochastic-rounding}{% \subsection{8.2 Dependence on Software Emulation and Stochastic Rounding}\label{dependence-on-software-emulation-and-stochastic-rounding}} \begin{itemize} \tightlist \item \textbf{Emulation fidelity} - Our fp4 implementation on GPUs and TPUs relies on up‑conversion to fp8, stochastic rounding in software, and down‑conversion back to fp4. While we validated numerical equivalence against the ASIC kernels, subtle differences in PRNG seeding or rounding tie‑break rules can lead to small variance in perplexity (≈0.1 \% on average).\\ \item \textbf{Reproducibility constraints} - Stochastic rounding introduces nondeterminism that must be controlled via fixed random seeds; any deviation in the seed or PRNG library version can affect the reported accuracy loss for fp4.\\ \item \textbf{Tool‑chain maturity} - The custom XLA pass and CUDA kernels used for fp4 are prototype‑level; they are not yet part of the standard vendor SDKs, which may limit adoption in production pipelines. \end{itemize} \hypertarget{scope-of-model-architectures-evaluated}{% \subsection{8.3 Scope of Model Architectures Evaluated}\label{scope-of-model-architectures-evaluated}} \begin{itemize} \tightlist \item \textbf{Model selection} - The experimental suite (see \textbf{Section 4}) focuses on two representative LLMs: GPT‑2 (124 M) and LLaMA‑7B. These models span a lightweight and a medium‑scale regime but do not cover the full spectrum of modern LLMs (e.g., 30 B-175 B parameter models, encoder‑only architectures, or multimodal transformers).\\ \item \textbf{Depth‑related effects} - As highlighted in \textbf{Section 7}, deeper models exacerbate fp4's limited exponent range. Because we did not evaluate models deeper than \textasciitilde32 layers, the reported fp4 accuracy degradation may be optimistic for the largest LLMs.\\ \item \textbf{Training vs.~inference} - Our study is limited to inference‑time quantization; the behavior of fp4 during mixed‑precision training remains an open question (see \textbf{Section 9}). \end{itemize} \hypertarget{calibration-and-scaling-assumptions}{% \subsection{8.4 Calibration and Scaling Assumptions}\label{calibration-and-scaling-assumptions}} \begin{itemize} \tightlist \item \textbf{Per‑tensor scaling} - Both fp8 and fp4 pipelines rely on a calibration step that determines optimal scaling factors (percentile‑based for fp8, KL‑divergence‑based for fp4). The calibration set is a small, fixed subset of the validation data. Different data distributions or larger calibration corpora could shift the optimal scales, potentially altering the observed accuracy‑energy trade‑off.\\ \item \textbf{Static scaling} - Our experiments use static, per‑tensor scales throughout inference. Dynamic, per‑token scaling - while potentially improving fp4's range utilization - was not explored due to the added runtime overhead. \end{itemize} \hypertarget{energy-measurement-granularity}{% \subsection{8.5 Energy Measurement Granularity}\label{energy-measurement-granularity}} \begin{itemize} \tightlist \item \textbf{Platform‑specific tools} - Energy consumption was measured with vendor‑provided power APIs (NVIDIA NVML, TPU power meters, ASIC on‑chip counters). These tools report power at coarse granularity (e.g., per‑kernel or per‑second), introducing measurement noise that may affect the exact percentage savings reported in \textbf{Section 6}.\\ \item \textbf{System‑level factors} - Our energy figures exclude host‑CPU power and cooling overhead, focusing solely on the accelerator die. Real‑world deployments will see additional energy components that could diminish the relative advantage of fp4. \end{itemize} \hypertarget{summary}{% \subsection{8.6 Summary}\label{summary}} In sum, the limitations of this work stem from (1) the scarcity of native fp4 hardware, (2) reliance on software emulation and stochastic rounding, (3) a narrowed set of LLM architectures and depths, (4) calibration choices that may not generalize across all workloads, and (5) the granularity of energy measurements. These constraints should be kept in mind when extrapolating the presented trade‑offs to broader deployment scenarios. \hypertarget{future-work}{% \section{9. Future Work}\label{future-work}} \hypertarget{mixedprecision-pipelines}{% \subsection{9.1 Mixed‑Precision Pipelines}\label{mixedprecision-pipelines}} Building on the \textbf{implementation} described in \emph{Section 5} and the \textbf{accuracy-throughput trade‑offs} highlighted in \emph{Section 7}, future work should explore pipelines that combine fp4, fp8, and higher‑precision formats (fp16/fp32) within a single inference pass. A plausible strategy is to retain fp8 for layers that are most sensitive to exponent range (e.g., early embedding and deep transformer blocks) while delegating fp4 to memory‑bound components such as attention‑score matrices or feed‑forward projections that dominate bandwidth consumption. This hybrid approach can be guided by per‑layer sensitivity analyses (e.g., layer‑wise perplexity impact) and automated by a compiler pass that inserts the appropriate conversion kernels. Expected benefits include: \begin{itemize} \tightlist \item \textbf{Latency reduction} beyond the \textasciitilde1.5× fp4‑only speed‑up reported in \emph{Section 6} by exploiting fp8's native throughput on GPUs/TPUs for the critical path.\\ \item \textbf{Memory savings} comparable to pure fp4 for the bulk of the model, preserving the \textasciitilde30 \% per‑token energy reduction observed for fp4.\\ \item \textbf{Graceful accuracy degradation}, as the most numerically fragile layers remain in fp8, keeping overall perplexity increase within the ≤ 1 \% envelope demonstrated for fp8‑only runs. \end{itemize} A systematic evaluation would require extending the calibration workflow of \emph{Section 4} to jointly optimise scaling factors for both precisions, possibly using multi‑objective optimisation (accuracy vs.~latency). \hypertarget{adaptive-precision-during-training}{% \subsection{9.2 Adaptive Precision During Training}\label{adaptive-precision-during-training}} The current study focuses on inference; however, the \textbf{training limitations} identified in \emph{Section 8} (absence of fp4‑native support, stochastic‑rounding nondeterminism) open a rich research avenue. Adaptive‑precision training would dynamically select the numeric format for each tensor (weights, activations, gradients) based on runtime statistics such as gradient variance, loss‑scale magnitude, or layer depth. Concrete steps include: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Dynamic loss‑scaling for fp4} - extending the static per‑tensor scaling of \emph{Section 4} to a per‑step scheme that reacts to overflow events, thereby mitigating the convergence fragility noted for fp4 training.\\ \item \textbf{Hybrid optimizer state storage} - keeping optimizer moments (e.g., Adam's first/second moments) in fp8 or fp16 while casting model weights to fp4, reducing memory pressure without sacrificing optimizer fidelity.\\ \item \textbf{Curriculum‑style precision scheduling} - starting training with fp8 (or fp16) for stability, then progressively annealing to fp4 once the model reaches a plateau, akin to learning‑rate warm‑up. \end{enumerate} Experimental validation would involve reproducing the \textbf{energy‑efficiency gains} of fp4 (≈ 30 \% vs.~fp8) while measuring any impact on final validation perplexity. Integration with the \textbf{autotuning framework} introduced in \emph{Section 5} could automate the precision‑selection policy. \hypertarget{integration-with-emerging-sub8bit-hardware}{% \subsection{9.3 Integration with Emerging Sub‑8‑Bit Hardware}\label{integration-with-emerging-sub8bit-hardware}} The \textbf{hardware landscape} is rapidly evolving: NVIDIA's Hopper line already provides native fp8, and early ASICs (Cerebras WSE‑2, Graphcore IPU‑M2000) support true fp4 datapaths, as shown in \emph{Section 5}. Future work should therefore target next‑generation accelerators that expose \textbf{native sub‑8‑bit arithmetic} (e.g., fp2, custom logarithmic encodings). Key research directions are: \begin{itemize} \tightlist \item \textbf{Co‑design of kernels and ISA} - collaborating with hardware vendors to expose fused fp4‑fp8 operations (e.g., a single MAC that can accept mixed‑precision operands), reducing the conversion overhead that currently limits fp4 on GPUs/TPUs.\\ \item \textbf{Benchmark suite expansion} - extending the open‑source repository (\texttt{llm-precision:2024.08}) with micro‑benchmarks for emerging formats, enabling fair cross‑platform comparisons of latency, bandwidth, and energy as done in \emph{Section 6}.\\ \item \textbf{Power‑aware scheduling} - leveraging the \textbf{clock‑gating} mechanisms that yielded an extra \textasciitilde3 \% token‑wise energy reduction on ASICs (see \emph{Section 5}), and generalising them to future sub‑8‑bit units. \end{itemize} By aligning software pipelines with hardware that natively handles sub‑8‑bit formats, the community can close the gap between the \textbf{theoretical energy savings} of fp4 and the \textbf{practical performance} observed when fp4 is emulated. \hypertarget{opensource-tooling-and-community-benchmarks}{% \subsection{9.4 Open‑Source Tooling and Community Benchmarks}\label{opensource-tooling-and-community-benchmarks}} To accelerate adoption, the authors plan to release a \textbf{precision‑agnostic profiling library} that automatically instruments kernels for latency, memory traffic, and power (building on the instrumentation layer of \emph{Section 5}). Coupled with a \textbf{public leaderboard} for mixed‑precision LLM inference, this will encourage reproducibility and foster collaborative exploration of the design space outlined above. \hypertarget{conclusion}{% \section{10. Conclusion}\label{conclusion}} \hypertarget{summary-of-findings}{% \subsection{10.1 Summary of Findings}\label{summary-of-findings}} \begin{itemize} \item \textbf{Accuracy vs.~Precision} - As shown in \textbf{Section 6. Experimental Results}, fp8 (E4M3) incurs ≤ 1 \% perplexity increase and ≤ 0.5 \% top‑1 accuracy loss, while fp4 (E2M1/E1M2) leads to a 2-4 \% perplexity rise and 1-2 \% top‑1 drop. The larger degradation of fp4 is directly linked to its limited exponent range and reliance on stochastic rounding (see \textbf{Section 2. Background on Low‑Precision Formats}). \item \textbf{Throughput and Latency} - Native fp8 kernels on GPUs and TPUs deliver roughly a 2× speed‑up over fp16 (see \textbf{Section 5. Implementation on Hardware Platforms}). Packed‑fp4 kernels add an additional ≈ 1.5× boost on platforms with true fp4 datapaths, yielding an overall ≈ 3× improvement versus fp16. When fp4 is emulated (e.g., on TPUs), the gain drops to ≈ 1.2× but still surpasses fp16. \item \textbf{Energy Efficiency} - Per‑token energy is reduced by \textasciitilde15 \% with fp8 and by an additional \textasciitilde30 \% with fp4 relative to fp8 (≈ 40 \% vs.~fp16), confirming the theoretical savings discussed in \textbf{Section 1. Introduction} and measured in \textbf{Section 6}. \item \textbf{Hardware Dependence} - The highest raw throughput is achieved with native fp8 support (NVIDIA Hopper GPUs, TPU‑v5). True fp4 hardware (Cerebras WSE‑2, Graphcore IPU) unlocks the full memory‑bandwidth and energy benefits, while emulation on other platforms incurs modest overhead (see \textbf{Section 5}). \item \textbf{Regime Classification} - \textbf{Section 7. Analysis and Discussion} identifies three practical regimes: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Accuracy‑critical} (deep models, skewed token distributions) → fp8.\\ \item \textbf{Memory/energy‑constrained} (edge inference, large batch serving) → fp4 with proper scaling.\\ \item \textbf{Hybrid} (mixed‑precision pipelines) → combine fp8 for sensitive layers and fp4 for bandwidth‑bound parts (as suggested in \textbf{Section 9. Future Work}). \end{enumerate} \end{itemize} \hypertarget{practical-recommendations}{% \subsection{10.2 Practical Recommendations}\label{practical-recommendations}} \begin{longtable}[]{@{}llll@{}} \toprule \begin{minipage}[b]{0.09\columnwidth}\raggedright Goal\strut \end{minipage} & \begin{minipage}[b]{0.34\columnwidth}\raggedright Recommended Precision\strut \end{minipage} & \begin{minipage}[b]{0.27\columnwidth}\raggedright Key Configuration\strut \end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright When to Use\strut \end{minipage}\tabularnewline \midrule \endhead \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Minimal accuracy loss}\strut \end{minipage} & \begin{minipage}[t]{0.34\columnwidth}\raggedright \textbf{fp8 (E4M3)}\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright Deterministic round‑to‑nearest‑even, per‑tensor scaling based on 99.9‑th percentile activations (Section 4)\strut \end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright Deep LLMs (\textgreater{} 20 layers), code or scientific text, platforms with native fp8 tensor cores\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Maximum throughput \& memory savings}\strut \end{minipage} & \begin{minipage}[t]{0.34\columnwidth}\raggedright \textbf{fp4 (E2M1 or E1M2)}\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright Stochastic rounding, KL‑divergence‑based per‑layer scaling, packed execution (Section 5)\strut \end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright Shallow or well‑calibrated models, inference on edge devices, ASICs with native fp4 support\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Balanced trade‑off}\strut \end{minipage} & \begin{minipage}[t]{0.34\columnwidth}\raggedright \textbf{Hybrid fp8 + fp4}\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright Apply fp8 to attention‑heavy or early‑layer blocks; fp4 to feed‑forward or later layers; use the precision‑agnostic profiler from \textbf{Section 9} to locate low‑sensitivity regions\strut \end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright Large‑scale serving where latency and memory dominate but a ≤ 1 \% accuracy budget is required\strut \end{minipage}\tabularnewline \begin{minipage}[t]{0.09\columnwidth}\raggedright \textbf{Energy‑constrained deployment}\strut \end{minipage} & \begin{minipage}[t]{0.34\columnwidth}\raggedright \textbf{fp4 (native)}\strut \end{minipage} & \begin{minipage}[t]{0.27\columnwidth}\raggedright Enable clock‑gating and per‑tensor scaling; prefer E2M1 on Cerebras WSE‑2 or E1M2 on Graphcore IPU for the extra 3 \% token‑wise savings reported in \textbf{Section 5}\strut \end{minipage} & \begin{minipage}[t]{0.19\columnwidth}\raggedright Battery‑powered or thermally limited environments\strut \end{minipage}\tabularnewline \bottomrule \end{longtable} Additional implementation tips derived from the study: \begin{itemize} \tightlist \item \textbf{Calibration} - Use the unified calibration workflow (Section 4) to collect activation statistics on a representative validation slice; for fp4, prefer KL‑based scale selection to mitigate overflow.\\ \item \textbf{Kernel Fusion} - Fuse Q·K·V, softmax, and subsequent mat‑muls as demonstrated in \textbf{Section 5} to recover up to 30 \% launch‑overhead reduction.\\ \item \textbf{Autotuning} - Leverage Optuna‑driven autotuning of thread‑block sizes and memory layouts (Section 5) to achieve consistent latency improvements across GPUs, TPUs, and ASICs.\\ \item \textbf{Profiling} - Employ the precision‑agnostic profiler introduced in \textbf{Section 9} to monitor per‑layer precision impact and dynamically adjust scaling during serving. \end{itemize} \hypertarget{outlook}{% \subsection{10.3 Outlook}\label{outlook}} The present work establishes a clear decision framework for choosing between fp4 and fp8 in LLM inference. Future extensions - mixed‑precision pipelines, adaptive precision during training, and emerging sub‑8‑bit hardware - will further narrow the accuracy gap for fp4 while preserving its energy and memory advantages (see \textbf{Section 9. Future Work}). Practitioners are encouraged to adopt the open‑source tooling (Docker image \texttt{llm-precision:2024.08}, public GitHub repo) to reproduce the reported numbers and to contribute additional benchmarks, especially for larger models and dynamic scaling strategies. \end{document}