ozaki-jax 0.1.1

FP64-accurate matrix multiplication from BF16 hardware via Ozaki Extract in JAX

ozaki-jax

FP64-accurate matrix multiplication built from BF16 GEMMs using Ozaki Extract in JAX.

For square test matrices, the default implementation uses 36 BF16 GEMMs and reaches about 1e-16 relative error against FP64 reference results.

Usage

import numpy as np
from ozaki_jax import matmul

A = np.random.randn(256, 256)
B = np.random.randn(256, 256)
C = matmul(A, B)

Without JAX runtime dependency:

from ozaki_jax import matmul_numpy
C = matmul_numpy(A, B)

Method summary

Given C = A @ B:

1. Split A and B into n_slices magnitude-controlled slices with Extract.
2. Compute BF16-derived GEMM pairs where i + j <= n_slices - 1.
3. Rescale and accumulate products in FP64.

With n_slices = 8, this keeps 8 * 9 / 2 = 36 slice-pair GEMMs.

Exactness condition

For BF16 slice values bounded by 2^p - 1, exact FP32 accumulation requires:

K * (2^p - 1)^2 < 2^24

In the default BF16 setting (p = 7), this gives K <= 1040.

Notes on rho for BF16

For FP64 source values (m1=53) and BF16 storage (m2=8), storage exactness requires:

rho >= m1 + 1 - m2 = 46

The accumulation constraint is also at least 46 for typical K values in this project, so the implementation uses rho=46 as the lower bound.

Limitations

• K <= 1040 for the default unblocked exactness condition.
• JAX on CPU does not represent TPU BF16 execution behavior.
• Blocking for larger K is not implemented.

Benchmarks

python benchmarks/precision.py
python benchmarks/tpu_validate.py

References

• Mukunoki, D., Ogita, T., & Imamura, T. (2020). "DGEMM using Tensor Cores, and Its Accurate and Reproducible Versions." ISC High Performance 2020.
• Mukunoki, D. (2025). "Ozaki Scheme-Based Accurate Matrix Multiplication on FP8 Tensor Cores." arXiv:2508.00441
• Ozaki, K., Ogita, T., Oishi, S., & Rump, S.M. (2012). "Error-Free Transformations of Matrix Multiplication by Using Fast Routines of Matrix Multiplication and Its Applications." Numerical Algorithms, 59(1).

License

Apache 2.0