ml_dtypes

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
8-bit floating point representations, parameterized by number of exponent and mantissa bits, as well as the bias (if any) and representability of infinity, NaN, and signed zero.
- float8_e3m4
- float8_e4m3
- float8_e4m3b11fnuz
- float8_e4m3fn
- float8_e4m3fnuz
- float8_e5m2
- float8_e5m2fnuz
- float8_e8m0fnu
Microscaling (MX) sub-byte floating point representations:
- float4_e2m1fn
- float6_e2m3fn
- float6_e3m2fn
Narrow integer encodings:
- int1
- int2
- int4
- uint1
- uint2
- uint4

See below for specifications of these number formats.

Installation

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 .

Example Usage

>>> 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)

Specifications of implemented floating point formats

`bfloat16`

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.

`float4_e2m1fn`

Exponent: 2, Mantissa: 1, bias: 1.

Extended range: no inf, no NaN.

Microscaling format, 4 bits (encoding: 0bSEEM) using byte storage (higher 4 bits are unused). NaN representation is undefined.

Possible absolute values: [0, 0.5, 1, 1.5, 2, 3, 4, 6]

`float6_e2m3fn`

Exponent: 2, Mantissa: 3, bias: 1.

Extended range: no inf, no NaN.

Microscaling format, 6 bits (encoding: 0bSEEMMM) using byte storage (higher 2 bits are unused). NaN representation is undefined.

Possible values range: [-7.5; 7.5]

`float6_e3m2fn`

Exponent: 3, Mantissa: 2, bias: 3.

Extended range: no inf, no NaN.

Microscaling format, 4 bits (encoding: 0bSEEEMM) using byte storage (higher 2 bits are unused). NaN representation is undefined.

Possible values range: [-28; 28]

`float8_e3m4`

Exponent: 3, Mantissa: 4, bias: 3. IEEE 754, with NaN and inf.

`float8_e4m3`

Exponent: 4, Mantissa: 3, bias: 7. IEEE 754, with NaN and inf.

`float8_e4m3b11fnuz`

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

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

`float8_e4m3fn`

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.

`float8_e4m3fnuz`

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

`float8_e5m2`

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

`float8_e5m2fnuz`

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

`float8_e8m0fnu`

OpenCompute MX scale format E8M0, which has the following properties:

Unsigned format
8 exponent bits
Exponent range from -127 to 127
No zero and infinity
Single NaN value (0xFF).

`int1`, `uint1`, `int2`, `int4`, `uint2` and `uint4`

1, 2 and 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 as an integer number of bytes. Relaxing this restriction would be a considerable engineering project. These types therefore use an unpacked representation, where each element of the array is padded up to a byte in memory. The lower two or four bits of each byte contain the representation of the number, whereas the remaining upper bits are ignored.

Quirks of low-precision Arithmetic

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)

License

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).

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ml-dtypes 0.6.0

ml_dtypes

Installation

Example Usage

Specifications of implemented floating point formats

bfloat16

float4_e2m1fn

float6_e2m3fn

float6_e3m2fn

float8_e3m4

float8_e4m3

float8_e4m3b11fnuz

float8_e4m3fn

float8_e4m3fnuz

float8_e5m2

float8_e5m2fnuz

float8_e8m0fnu

int1, uint1, int2, int4, uint2 and uint4

Quirks of low-precision Arithmetic

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`bfloat16`

`float4_e2m1fn`

`float6_e2m3fn`

`float6_e3m2fn`

`float8_e3m4`

`float8_e4m3`

`float8_e4m3b11fnuz`

`float8_e4m3fn`

`float8_e4m3fnuz`

`float8_e5m2`

`float8_e5m2fnuz`

`float8_e8m0fnu`

`int1`, `uint1`, `int2`, `int4`, `uint2` and `uint4`

Algorithm	Hash digest
SHA256	`57ed0d6b4ac5e7868361303a9c57fbcf63b768236ee14456f585dfcf260d0292`
MD5	`9ada4e02940e9f10a4453aaa7a46fd71`
BLAKE2b-256	`b15d6a01538e507ef0ed5e879985b13a92467bf8960696fb1131f8b8cadc60ff`

Algorithm	Hash digest
SHA256	`fb87f46b4f7ad7b5d3ad8f4b452b024bd4229d44c8ff934798c1fe656210387a`
MD5	`bb959c26da5bcfa054a77a78e92e9c04`
BLAKE2b-256	`e2554561acefa00fa4bcbfb82ca6a48578b41f372cd7dd7cdd6eb4720abc2e5f`

Algorithm	Hash digest
SHA256	`26b1f1fa4f0435a2946859823f6e2bf06796f1e9f10f5a05b08a5e3c8f46ff69`
MD5	`3327a4044f9df0cf94cb0975f03d0b7e`
BLAKE2b-256	`89a5da8ae6c6f1babe4b68e3e55d43d39b529e29774f10e0910671a6b8c86eb8`

Algorithm	Hash digest
SHA256	`28d676428b104bb9717b0928bc5c5129f2d6b51b6727587cc4289e7bf8713cb5`
MD5	`29ca1db0c3750489978d2ed6aba95356`
BLAKE2b-256	`d22220fd70ca6ed12446cb92d5b2a7745bd185f9d8b8cdeeadad976574398e6b`

Algorithm	Hash digest
SHA256	`084dfe51a7ad58b171f05115f8226ed4233a454a1611371947e806e76f0c638d`
MD5	`168e1f53605b9c8d633f9086b0972dc1`
BLAKE2b-256	`5051fd1582b8f5ed8a9e7be0e161a6ea0dff70cb280479a12178df0b3a72700e`

Algorithm	Hash digest
SHA256	`6eaed129a4afe90694b8685e2f9b6294849f5eda4af9a15be83a4326eeebd775`
MD5	`e79d163a2d2103591b074bfb7f84b717`
BLAKE2b-256	`6a57780ca3e5ab135b9fbdd8e5441abf5f801b30398371b691291e05ab9834c0`

Algorithm	Hash digest
SHA256	`2a3e9d53925597fbffafd2a37048dadeddd0bdaba58058f6ae0869ed709a184d`
MD5	`f2fbce81e4be04f03427ee85fe3fe9a3`
BLAKE2b-256	`badb9c61ec2760b5cbfb1c6558d5c991a6d8fd3271053c32db20506a9a90272b`

Algorithm	Hash digest
SHA256	`3b4a480aa8fd54a1805b8ac10f3f91763926a74f73c0c364c10f9231854f4170`
MD5	`da9c8c5b30a530cdff9227bb9ed64f77`
BLAKE2b-256	`c7f97d76c1eae866f5d4636401b31b6d6dd90e4b4ced1fa7cfdfcca9c60e4bd3`

Algorithm	Hash digest
SHA256	`37da32aa97749251025666d62372775019594577b9c9e9cfda83bed48d778fdb`
MD5	`28fcd71c23f7fa06b621edb14f51f00b`
BLAKE2b-256	`edcf87e8a6c57eed63a91782a0d229856ddf73e138ce004dd71e2799a9dcdb33`

Algorithm	Hash digest
SHA256	`5359c588cc62de6f78d7430f06b65853d884955494d86d6ad90b6dd64a3f3a08`
MD5	`ab7b1d6c104150b6f866b243f54d3582`
BLAKE2b-256	`846a441eb053b078954f7fea284dfb288701884d0a1404d39babb858e1649023`