Half-precision floating-point format


In computing, half precision is a binary floating-point computer number format that occupies 16 bits in computer memory.
They can express values in the range ±65,504, with precision up to 0.0000000596046.
In the IEEE 754-2008 standard, the 16-bit base-2 format is referred to as binary16. It is intended for storage of floating-point values in applications where higher precision is not essential for performing arithmetic computations.
Although implementations of the IEEE Half-precision floating point are relatively new, several earlier 16-bit floating point formats have existed including that of Hitachi's HD61810 DSP of 1982, Scott's WIF and the 3dfx Voodoo Graphics processor.
Nvidia and Microsoft defined the half datatype in the Cg language, released in early 2002, and implemented it in silicon in the GeForce FX, released in late 2002. ILM was searching for an image format that could handle a wide dynamic range, but without the hard drive and memory cost of floating-point representations that are commonly used for floating-point computation. The hardware-accelerated programmable shading group led by John Airey at SGI invented the s10e5 data type in 1997 as part of the 'bali' design effort. This is described in a SIGGRAPH 2000 paper and further documented in US patent 7518615.
This format is used in several computer graphics environments including MATLAB, OpenEXR, JPEG XR, GIMP, OpenGL, Cg, Direct3D, and D3DX. The advantage over 8-bit or 16-bit binary integers is that the increased dynamic range allows for more detail to be preserved in highlights and shadows for images. The advantage over 32-bit single-precision binary formats is that it requires half the storage and bandwidth.
The F16C extension allows x86 processors to convert half-precision floats to and from single-precision floats.
Depending on the computer half-precision can be over an order of magnitude faster than double precision, e.g. 37 PFLOPS vs. for half 550 "AI-PFLOPS ".

IEEE 754 half-precision binary floating-point format: binary16

The IEEE 754 standard specifies a binary16 as having the following format:
The format is laid out as follows:
The format is assumed to have an implicit lead bit with value 1 unless the exponent field is stored with all zeros. Thus only 10 bits of the significand appear in the memory format but the total precision is 11 bits. In IEEE 754 parlance, there are 10 bits of significand, but there are 11 bits of significand precision.

Exponent encoding

The half-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 15; also known as exponent bias in the IEEE 754 standard.
Thus, as defined by the offset binary representation, in order to get the true exponent the offset of 15 has to be subtracted from the stored exponent.
The stored exponents 000002 and 111112 are interpreted specially.
The minimum strictly positive value is
2−24 ≈ 5.96 × 10−8.
The minimum positive normal value is 2−14 ≈ 6.10 × 10−5.
The maximum representable value is × 215 = 65504.

Half precision examples

These examples are given in bit representation
of the floating-point value. This includes the sign bit, exponent, and significand.
0 00000 00000000012 = 000116 = ≈ 0.000000059604645

0 00000 11111111112 = 03ff16 = ≈ 0.000060975552

0 00001 00000000002 = 040016 = ≈ 0.000061035156

0 11110 11111111112 = 7bff16 = = 65504

0 01110 11111111112 = 3bff16 = ≈ 0.99951172

0 01111 00000000002 = 3c0016 = = 1

0 01111 00000000012 = 3c0116 = ≈ 1.00097656

0 01101 01010101012 = 355516 = = 0.33325195

1 10000 00000000002 = c00016 = −2
0 00000 00000000002 = 000016 = 0
1 00000 00000000002 = 800016 = −0
0 11111 00000000002 = 7c0016 = infinity
1 11111 00000000002 = fc0016 = −infinity
By default, 1/3 rounds down like for double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are 0101... which is less than 1/2 of a unit in the last place.

Precision limitations on decimal values in 0, 1

ARM processors support an "alternative half-precision" format, which does away with the special case for an exponent value of 31. It is almost identical to the IEEE format, but there is no encoding for infinity or NaNs; instead, an exponent of 31 encodes normalized numbers in the range 65536 to 131008.

Uses

Hardware and software for machine learning or neural networks tend to use half precision: such applications usually do a large amount of calculation, but don't require a high level of precision.
On older computers that access 8 or 16 bits at a time, half precision arithmetic is faster than single precision, and substantially faster than double precision. On systems with instructions that can handle multiple floating point numbers with in one instruction, half-precision often offers a higher average throughput.