(onnx-detail-float4)=

Float stored in 4 bits

Papers

4 bit floating point formats have emerged as a solution to the rising cost and deployment challenges of large language models. The S1E2M1 format has been part of the Open Compute Project (OCP) standard.

As a result, a new data type was introduced in onnx==1.18.0 to support a limited set of operators to enable computation with float4.

• FLOAT4E2M1: 1 bit for the sign, 2 bits for the exponents, and 1 bit for the mantissa. No nan or infinities.

E2M1

$S$ stands for the sign. $10_2$ describe a number base 2.

.. list-table:: Float4 type
   :widths: 10 10
   :header-rows: 1

   * -
     - E2M1
   * - Exponent bias
     - 1
   * - Infinities
     -
   * - NaN
     -
   * - Zeros
     - :math:`S.00.0_2`
   * - Max
     - :math:`S.11.1_2`
   * - Min
     - :math:`S.00.1_2 = 2^{-1}`

Let's denote the bit representation as $S.b_2 b_1 b_0$. The float value is defined by the following expressions:

.. list-table:: Float4 type values
   :widths: 10 10
   :header-rows: 1

   * -
     - E2M1
   * - exponent :math:`\neq` 0
     - :math:`(-1)^S 2^{\sum_{i=1}^2 b_i 2^{i-1} - 1} \left( 1 + b_0 2^{-1} \right)`
   * - exponent :math:`=` 0
     - :math:`(-1)^S  b_0 2^{-1}`

The following table lists all the representable values by float4 E2M1, ignoring the sign bit:

.. list-table:: Float4 type values
   :widths: 10 10
   :header-rows: 1

   * - bits (ignoring sign bit)
     - E2M1
   * - 000
     - 0
   * - 001
     - 0.5
   * - 010
     - 1
   * - 011
     - 1.5
   * - 100
     - 2
   * - 101
     - 3
   * - 110
     - 4
   * - 111
     - 6

Cast

Upcasting from float4 to float32, float16, bfloat16, and float8 is exact. The behavior for downcasting to float 4 is summarized below

Packing and Unpacking

Float4 is stored as 2x4bit in a single byte. The first element is stored in the 4 LSB and the second element is stored in the 4 MSB, i.e. for elements x and y that are consecutive elements in the array:

pack(x,y): y << 4 | x & 0x0F
unpack(z): x = z & 0x0F, y = z >> 4

In case the total number of elements is odd, padding of 4 bits will be appended. The storage size of a 4 bit tensor of size N is ceil(N/2).