MXFP8-E4M3: Efficient Block-based FP8 Encoding

• MXFP8-E4M3 is a block-based FP8 encoding format that employs per-block scale factors and E4M3 data representation for efficient tensor quantization.
• It uses a three-step conversion process—blockwise maximum finding, scale factor derivation, and per-element quantization—to maintain robust quantization and model accuracy.
• MXFP8-E4M3 delivers substantial throughput and energy efficiency gains with minimal accuracy loss, outperforming BF16 and INT8 in large-scale neural network applications.

The MXFP8-E4M3 format is a block-based, microscaling floating-point data type designed for efficient representation and computation in large-scale neural network training and inference. It combines FP8 narrow precision (1 sign bit, 4 exponent bits, 3 mantissa bits: E4M3) with per-block scale factors to enable high throughput and memory efficiency, particularly for deep learning workloads on modern AI hardware.

1. Definition and Structure

MXFP8-E4M3 encodes tensors as contiguous blocks of fixed length (typically 32 elements). Each block stores both:

• An 8-bit scale factor $X$, usually in power-of-two format.
• 32 FP8 private elements $\{Q_i\}$ encoded as E4M3: one sign bit, 4 exponent bits, 3 mantissa bits per element.

For a single tensor block, the decoded value of the $i$-th element is $V_i \approx Q_i \times X$, with $Q_i$ the FP8-E4M3 encoded value and $X$ the block's shared scale. This design separates range adaptation (scale) from coefficient representation (private value), enabling robust quantization across a diverse range of activation and weight magnitude distributions (Mishra et al., 30 May 2025).

2. Conversion Algorithm and Hardware Implementation

Conversion Process

As described in FPGA hardware implementations (Gorodecky et al., 5 Nov 2024), the conversion algorithm for FP32 input to MXFP8-E4M3 follows three steps:

A. Blockwise Maximum Finder:

• Among 32 FP32 inputs $\{V_i\}$, find $a_{max} = \max |V_i|$ via pairwise exponent comparison.

B. Scale Factor Derivation:

• Compute floating-point scale candidate $X_{float} = a_{max} / destmax$, where $destmax$ is the largest representable magnitude in E4M3.
• Calculate exponent for scale: $e = \lceil \log_2(X_{float}) \rceil$.
• Convert to shared scale: $X = 2^{\text{clamp}(e, -127, 127) + bias}$ (bias = 127, for UE8M0 intermediate exponent encoding).
• This "round up" approach to scale ensures post-scaling values do not exceed representable FP8-E4M3 range (Mishra et al., 30 May 2025).

C. Per-Element Quantization:

• For each input, store the sign, the quantized exponent, and the quantized mantissa (rounded from the highest FP32 mantissa bits).
• Private value $Q_i$ is the nearest FP8-E4M3 value to $V_i / X$, with round-to-nearest (ties to even).
• Mantissa bits are rounded according to a lookup table as implemented in FPGA conversion logic (Gorodecky et al., 5 Nov 2024).

Hardware Resource Consumption

FPGA measurements report that for a 32-element block E4M3 converter:

• Comparator tree (max finder): ~55% LUT resource
• Scale conversion: ~1%
• Quantization logic: ~44% Typical critical path delays are 80.2 ns on Xilinx Virtex UltraScale; total LUT consumption is 2776 for E4M3 type (Gorodecky et al., 5 Nov 2024).

3. Mathematical Formulation and MXFP8 Arithmetic

FP8-E4M3 Encoding Equation:

$$x = (-1)^S \cdot \left(1 + \frac{M}{8}\right) \cdot 2^{E-7}$$

where $S$ is the sign, $E$ the 4-bit exponent, $M$ the 3-bit mantissa (Micikevicius et al., 2022).

MX Block Dot Product:

For two blocks $A$, $B$ (each $k$ FP8 elements and scales $X^A$, $X^B$):

$C = X^A X^B \sum_{i=1}^k (P^A_i \cdot P^B_i)$

This factorized representation allows efficient accumulation of scaled dot products during matrix multiplication, supporting high-throughput streaming and fusion in dedicated hardware units such as RISC-V MXDOTP (İslamoğlu et al., 19 May 2025).

4. Accuracy and Efficiency in Pre-training and Inference

Quantization with MXFP8-E4M3 enables memory and bandwidth savings without compromising model accuracy:

• Fine-grained blockwise scaling enables matching, within 0.1%, the validation perplexity and downstream scores vs. reference BF16 pre-training across models up to 8B parameters and 15T tokens (Mishra et al., 30 May 2025).
• Comparing to BF16 on NVIDIA Blackwell GPUs, MXFP8 delivers $\sim2\times$ throughput improvement with no measurable degradation in LLM accuracy or stability.
• Empirical results from MXFP8 matrix multiplies reach 356 GFLOPS/W at 0.8V, 1GHz on custom MXDOTP hardware, with a 25$\times$ speedup and 12.5$\times$ energy efficiency gain over software emulated scaling approaches (İslamoğlu et al., 19 May 2025).

5. MXFP8-E4M3 in Mixed-Precision and Adaptive Quantization

MicroMix and related algorithms adapt assignment of MXFP8, MXFP6, MXFP4 formats per layer/channel based on measured quantization error and activation magnitude (Liu et al., 4 Aug 2025):

• Blockwise quantization is governed by maximum within-block value, shared scale $s = 2^{\lfloor \log_2(\max |X_i|)\rfloor - b}$.
• Quantization error for MXFP8 is maintained below INT8 bounds via selection of precision per channel (threshold $T(n)$).
• Mixed-precision kernel design—fused dequantization and MMA on FP4, FP8 Tensor Cores—enables 8–46% GEMM speedups and up to 9% end-to-end throughput gains over TensorRT-FP8, with memory savings ~20%.

Adaptive assignment of MXFP8 is shown to preserve $\geq$95% of FP16 baseline accuracy on Llama, Qwen, and other LLMs in various deployment scenarios.

6. Comparative Evaluation with INT8, BF16, and Other FP8 Formats

Empirical studies demonstrate that MXFP8-E4M3 offers:

• Superior workload coverage and accuracy for NLP and multi-modal tasks: 96.32% pass rate vs. INT8’s 65.87%; marginally higher accuracy for quantized BERT, Llama, and other transformers (Shen et al., 2023).
• Higher dynamic range (18 binades due to E4M3 extension) compared to alternative FP8 formats (E5M2, E3M4) (Micikevicius et al., 2022).
• Robustness against activation outliers due to microscaling; asymmetric scale variants (e.g., AMXFP4) suggest possible future refinements for strictly low-bitwidth formats, but for 8-bit configurations MXFP8-E4M3 is currently optimal (Lee et al., 15 Nov 2024).

7. Practical Deployment and Standardization

MXFP8-E4M3 is now natively supported in accelerator architectures (NVIDIA Blackwell, Intel Gaudi 2). ML software stacks integrate conversion routines and kernel fusions enabling per-block scaling, stochastic rounding, and efficient quantized arithmetic matched to hardware execution pipelines. MXDOTP demonstrates the value of ISA extensions that directly support MX operations. FPGA, ASIC, and custom RISC-V implementations validate resource efficiency and practical deployability (Gorodecky et al., 5 Nov 2024, İslamoğlu et al., 19 May 2025).

High-throughput deployment workflows utilize recipes for scale selection, block partitioning (32 elements), and conversion thresholding—ensuring robust and efficient quantization for multi-billion parameter models and trillion-token datasets.

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In summary, the MXFP8-E4M3 format—characterized by blockwise microscaling, FP8 (E4M3) encoding, and rigorously defined conversion logic—provides a foundational tool for scaling deep learning systems, maintaining accuracy under aggressive quantization, and maximizing hardware and memory efficiency for LLMs and other neural architectures. Empirical and hardware validation confirm its centrality in both research and production settings.