README.md

MATLAB Tensor & Matrix Core models

Overview

This repository provides accurate tensor (NVIDIA) and matrix (AMD) core models written in MATLAB. It also includes parts of the model validation data which is used to refine the models as shown in [1].

The models directory contains the MATLAB models of tensor cores of several NVIDIA GPUs and matrix cores of three arch of AMD GPUs, all of which are build on the parameterised model in Generic_BFMA_TC.m. For example the B200TC.m models the General Matrix Multiply (GEMM) based on the accurate model of a tensor core in the NVIDIA Blackwell B200 GPUs. In the current version of the toolbox, the models take matrices and input and output floating-point formats as inputs and multiply the matrices by using a recursive summation algorithm to accummulate the results of several tensor core invocations.

The initial analysis of the behaviour of GPU tensor cores is performed with the code available at IEEE_HPEC2025_block_FMA_tests. It is based on the generalised testing methodology [2] which determines the following features of hardware computing mixed-precision inner products:

• Support for subnormal numbers in input and output
• Presence of extra bits for significand alignment in multi-term addition
• Availability of extra carry bits
• Normalization patterns in multi-term floating-point addition
• Supported rounding modes
• Effective FMA size (i.e., number of terms accumulated before a single normalization)

The model_validation contains part of the model validation data that was used in [1] to refine the models and verify the bit-accurate behaviour against the corresponding GPUs. Full-sized experiments and data is not stored in this repository due to size limitations.

The experiments directory contains various experiments with some of the models that were performed to plot the results in [1]. These can serve as examples on how to utilise the models.

Dependencies and installation

1. Set up the custom precision floating-point format simulator CPFloat.
2. Add models/ to the MATLAB search path.
3. Add models/tools to the MATLAB search path.

Compatibility & Validation in Octave GNU and Python

1. Fully compatible with GNU Octave (no code modifications required) and validated on one million samples for every model.
2. Package can easily be used in Python either via (a). MATLAB Engine API or (b) Oct2Py
3. Consistent and produce same results across all three.

Example: Using in-built models

The following example rounds two matrices to fp16 and multiplies them using the model of the B200 tensor core. Note that B200TC computes the GEMM, with alpha and beta scale factors set to 1.

>> inopts.format = 'binary16';
>> outopts.format = 'binary32';
>> A = cpfloat(rand(4,4), inopts);
>> B = cpfloat(rand(4,4), inopts);
>> B200TC(1, A, B, 1, 0, inopts.format, outopts.format)

ans =

   0.995566666126251   1.208170533180237   1.368334889411926   1.017799258232117
   0.991239666938782   1.084852933883667   1.350871562957764   1.328557014465332
   1.190854787826538   1.693876862525940   1.763551592826843   1.278026223182678
   0.901759386062622   1.838499188423157   1.608222723007202   1.265371918678284

The following example uses an 8-bit floating-point format as the input format in the B200 tensor core model.

>> inopts.format = 'fp8-e4m3';
>> A = cpfloat(rand(4,4), inopts);
>> B = cpfloat(rand(4,4), inopts);
>> B200TC(1, A, B, 1, 0, inopts.format, outopts.format)

ans =

   0.390136718750000   0.589843750000000   0.625976562500000   0.748046875000000
   1.180175781250000   1.117187500000000   1.220703125000000   1.935546875000000
   1.267822265625000   0.752929687500000   0.867187500000000   1.813476562500000
   1.007812500000000   1.242187500000000   1.395996093750000   1.740234375000000

Example: Setting up the NVIDIA B200 model

While the B200 tensor core model comes with this toolbox, below is a minimal example for setting it up. The input matrices are assumed to be rounded to the appropriate formats with CPFloat. The model in B200TC.m provides a more detailed set up that changes the parameters of a generalised model based on all possible input/output format combinations. Few parameters are shown below:

def_params.fma  = 16;        % Number of products in one FMA group
def_params.neab = 2;        % Number of extra alignment bits (guard precision)
%---------------- Rounding configuration ----------------%
def_params.frmode = 'rz';  % Final rounding mode:
%---------------- Accumulation architecture ----------------%
def_params.global_alignment  = 1;   % Align all products (and optionally c) to a common exponent
def_params.late_partial_sum  = 0;   % Add accumulation term 'c' after product summation; (products sum kept in denormalised form)
def_params.odd_even_grouping = 0;   % Enable separate accumulation of odd/even उत्पाद
def_params.pair_wise_sum     = 0;   % Enable pair-wise summation (not implemented)
%---------------- Exponent handling ----------------%
def_params.min_exp_limit   = -133; % Minimum exponent allowed for product alignment
def_params.c_min_exp_limit = 0;     % Control minimum exponent for c: 1 → clamp to -126 (FP32 subnormal boundary); 0 → allow special handling when c = 0
%---------------- Accuracy / reference model ----------------%
def_params.correct_rounding = 0;    % Enable exact (Kulisch-style) accumulation                                
%---------------- Subnormal handling ----------------%
def_params.in_subnormals  = 1;   % Input subnormal support: 1 → preserve and process subnormals; 0 → flush subnormals to zero (FTZ)
def_params.out_subnormals = 1;   % Output subnormal support: 1 → generate subnormal outputs; 0 → flush subnormal results to zero

D = GEMM(alpha, A, B, beta, C, informat, outformat, def_params);

Supported Architectures

Tensor core models of following NVIDIA GPU archs are supported:

1. Volta (V100)
2. Ampere (A100, A2, A30, A40)
3. Hopper (H100, GH200)
4. Ada Lovelace (RTX-1000, L40S)
5. Blackwell (B200)

Matrix core models of

1. CDNA 1 (M100)
2. CDNA 2 (MI210, MI250)
3. CDNA 3 (MI300, MI300A, MI300X)

References

[1] F. A. Khattak and M. Mikaitis, Accurate Models of NVIDIA Tensor Cores. arXiv:2512.07004 [cs.MS]. Dec. 2025 (Updated Apr. 2026).
[2] F. A. Khattak and M. Mikaitis, Generalized Methodology for Determining Numerical Features of Hardware Floating-Point Matrix Multipliers: Part I. 2025 IEEE High Performance Extreme Computing Conference (HPEC). Sep. 2025.