The Missing Fourth Term for the Emulation Tensor Memory Equilibrium (TME) Model: The Residue Deconstruction Cost

Abstract

The Tensor-Memory Equilibrium (TME) model of "FP8 is All You Need (Part 1)" calculates the execution time of Ozaki Scheme II emulation of fp64 as the maximum of a tensor-core term and a High-Bandwidth Memory (HBM) traffic term, plus a per-output reconstruction term. However, it omits the per-input deconstruction cost: every streamed fp64 operand must be scaled, rounded, and reduced modulo each of the $r$ moduli on SIMT pipes before any matrix multiply can issue. In this note we add this fourth term, calibrate its constant from the cuBLAS emulation path, and derive a closed-form operational-intensity threshold $\mathrm{OI}^{*} = c_q r P_{\mathrm{fp64}}/(8P_{\mathrm{int}})$ below which emulation cannot match native fp64 regardless of tensor-core throughput. On the NVIDIA B300 GPU the threshold is $\mathrm{OI}^{*}\approx 0.56$ FLOP/B. As a result, GEMV, SpMV, and low-batch GEMV, which are the memory-bound kernels the original paper claims to accelerate, are limited to 0.3-0.9x of native performance, and the 7-point stencil to 1.8x rather than the claimed 3.1x. Dense GEMM is unaffected as expected. We also show that precomputing and storing the residues moves the same cost into the bandwidth term, and we state the instruction count that an implementation would have to achieve to invalidate the bound.