P-Cast Precision in FP8 Attention: Sink-Induced Collapse and the Optimality of S=2^8

Abstract

FP8 (E4M3) acceleration for attention computation offers significant throughput gains, but the 3-bit mantissa introduces precision challenges when the softmax probability matrix~$P$ is cast to FP8 before the $P \cdot V$ matrix multiplication. We analyze two implementation choices that affect output precision under the \emph{Attention Sink} phenomenon: (1)~the KV block iteration order, and (2) the static scaling factor applied to $P$ before casting. We show that forward KV iteration causes \emph{P-collapse} -- to leading order a fraction $Φ(Δ+ δ_k - 6.93 - \ln S)$ of non-sink $P$ values underflow to zero, where the small shift $δ_k \approx 1$ (for $k_{\text{sink}}{=}4$) is the expected within-sink-block score maximum -- and that reverse iteration removes it, with a zero-underflow guarantee when reverse is combined with $S{=}256$. We further give a constructive characterization of $S = 256 = 2^8$ as the static scale that simultaneously satisfies (i)~bit-exact IEEE 754 scaling, (ii) the lower envelope of a sawtooth function $dp(S)$ over the E4M3 number line ($dp = 2^{-4}$, the minimum worst-case quantization step), and (iii)~the maximum normal-range coverage \emph{among bit-exact ($2^k$) scales} (a non-bit-exact scale such as $448$ attains slightly higher coverage; sec.5}). Both optimizations are already deployed in FlashAttention-3/4 on engineering grounds; our contribution is a quantitative account of \emph{why} these choices are good and a closed-form threshold $Δ_c = 6.93 + \ln S - δ_k$ for predicting kernel-level precision loss. Kernel-faithful experiments ($Q, K, V$ in FP32 to isolate the P-cast effect) show $3$-$10\times$ MSE improvement at moderate sink strengths, and paired tests confirm both fixes saturate to the same precision floor when combined -- which motivated updating the hpc-ops kernel from $S{=}1$ to $S{=}256$.