ml_dtypes is a stand-alone implementation of several NumPy dtype extensions used in machine learning libraries, including:

• bfloat16: an alternative to the standard float16 format
• float8_*: several experimental 8-bit floating point representations including:
  • float8_e4m3b11fnuz
  • float8_e4m3fn
  • float8_e4m3fnuz
  • float8_e5m2
  • float8_e5m2fnuz
• int4 and uint4: low precision integer types.

See below for specifications of these number formats.

The ml_dtypes package is tested with Python versions 3.9-3.12, and can be installed with the following command:

pip install ml_dtypes

To test your installation, you can run the following:

pip install absl-py pytest
pytest --pyargs ml_dtypes

To build from source, clone the repository and run:

git submodule init
git submodule update
pip install .

>>> from ml_dtypes import bfloat16
>>> import numpy as np
>>> np.zeros(4, dtype=bfloat16)
array([0, 0, 0, 0], dtype=bfloat16)

Importing ml_dtypes also registers the data types with numpy, so that they may be referred to by their string name:

>>> np.dtype('bfloat16')
dtype(bfloat16)
>>> np.dtype('float8_e5m2')
dtype(float8_e5m2)

A bfloat16 number is a single-precision float truncated at 16 bits.

Exponent: 8, Mantissa: 7, exponent bias: 127. IEEE 754, with NaN and inf.

Exponent: 4, Mantissa: 3, bias: 11.

Extended range: no inf, NaN represented by 0b1000'0000.

Exponent: 4, Mantissa: 3, bias: 7.

Extended range: no inf, NaN represented by 0bS111'1111.

The fn suffix is for consistency with the corresponding LLVM/MLIR type, signaling this type is not consistent with IEEE-754. The f indicates it is finite values only. The n indicates it includes NaNs, but only at the outer range.

8-bit floating point with 3 bit mantissa.

An 8-bit floating point type with 1 sign bit, 4 bits exponent and 3 bits mantissa. The suffix fnuz is consistent with LLVM/MLIR naming and is derived from the differences to IEEE floating point conventions. F is for "finite" (no infinities), N for with special NaN encoding, UZ for unsigned zero.

This type has the following characteristics:

• bit encoding: S1E4M3 - 0bSEEEEMMM
• exponent bias: 8
• infinities: Not supported
• NaNs: Supported with sign bit set to 1, exponent bits and mantissa bits set to all 0s - 0b10000000
• denormals when exponent is 0

Exponent: 5, Mantissa: 2, bias: 15. IEEE 754, with NaN and inf.

8-bit floating point with 2 bit mantissa.

An 8-bit floating point type with 1 sign bit, 5 bits exponent and 2 bits mantissa. The suffix fnuz is consistent with LLVM/MLIR naming and is derived from the differences to IEEE floating point conventions. F is for "finite" (no infinities), N for with special NaN encoding, UZ for unsigned zero.

This type has the following characteristics:

• bit encoding: S1E5M2 - 0bSEEEEEMM
• exponent bias: 16
• infinities: Not supported
• NaNs: Supported with sign bit set to 1, exponent bits and mantissa bits set to all 0s - 0b10000000
• denormals when exponent is 0

4-bit integer types, where each element is represented unpacked (i.e., padded up to a byte in memory).

NumPy does not support types smaller than a single byte. For example, the distance between adjacent elements in an array (.strides) is expressed in bytes. Relaxing this restriction would be a considerable engineering project. The int4 and uint4 types therefore use an unpacked representation, where each element of the array is padded up to a byte in memory. The lower four bits of each byte contain the representation of the number, whereas the upper four bits are ignored.

If you're exploring the use of low-precision dtypes in your code, you should be careful to anticipate when the precision loss might lead to surprising results. One example is the behavior of aggregations like sum; consider this bfloat16 summation in NumPy (run with version 1.24.2):

>>> from ml_dtypes import bfloat16
>>> import numpy as np
>>> rng = np.random.default_rng(seed=0)
>>> vals = rng.uniform(size=10000).astype(bfloat16)
>>> vals.sum()
256

The true sum should be close to 5000, but numpy returns exactly 256: this is because bfloat16 does not have the precision to increment 256 by values less than 1:

>>> bfloat16(256) + bfloat16(1)
256

After 256, the next representable value in bfloat16 is 258:

>>> np.nextafter(bfloat16(256), bfloat16(np.inf))
258

For better results you can specify that the accumulation should happen in a higher-precision type like float32:

>>> vals.sum(dtype='float32').astype(bfloat16)
4992

In contrast to NumPy, projects like JAX which support low-precision arithmetic more natively will often do these kinds of higher-precision accumulations automatically:

>>> import jax.numpy as jnp
>>> jnp.array(vals).sum()
Array(4992, dtype=bfloat16)

This is not an officially supported Google product.

The ml_dtypes source code is licensed under the Apache 2.0 license (see LICENSE). Pre-compiled wheels are built with the EIGEN project, which is released under the MPL 2.0 license (see LICENSE.eigen).