BFloat16s

This package defines the BFloat16 data type.

Hardware implementation of this datatype is available in Google's Cloud TPUs as well as in a growing number of CPUs, GPUs, and more specialized processors. See the wikipedia entry for more information.

This package is suitable to evaluate whether using BFloat16 would cause precision problems for any particular algorithm, even without access to supporting hardware. Note that this package is designed for functionality, not performance, so this package should be used for precision experiments only, not performance experiments.

Usage

This package exports the BFloat16 data type. This datatype should behave just like any builtin floating point type (e.g. you can construct it from other floating point types - e.g. BFloat16(1.0)). In addition, this package provides the LowPrecArray type. This array is supposed to emulate the kind of matmul operation that TPUs do well (BFloat16 multiply with Float32 accumulate). Broadcasts and scalar operations are peformed in Float32 (as they would be on a TPU) while matrix multiplies are performed in BFloat16 with Float32 accumulates, e.g.

julia> A = LowPrecArray(rand(Float32, 5, 5))
5×5 LowPrecArray{2,Array{Float32,2}}:
 0.252818  0.619702   0.553199  0.75225   0.30819
 0.166347  0.976339   0.399945  0.589101  0.526253
 0.350232  0.0447034  0.490874  0.525144  0.841436
 0.903734  0.879541   0.706704  0.304369  0.951702
 0.308417  0.645731   0.65906   0.636451  0.765263

julia> A^2
5×5 LowPrecArray{2,Array{Float32,2}}:
 1.13603   1.64932  1.39712  1.27283  1.82597
 1.03891   1.93298  1.44455  1.42625  1.86842
 0.998384  1.28403  1.37666  1.24076  1.68507
 1.18951   2.33245  2.04367  2.26849  2.35588
 1.22636   1.90367  1.70848  1.63986  2.1826

julia> A.storage^2
5×5 Array{Float32,2}:
 1.13564  1.64708  1.39399  1.27087  1.82128
 1.03924  1.93216  1.44198  1.42456  1.86497
 1.00201  1.28786  1.37826  1.24295  1.6882
 1.19089  2.33262  2.04094  2.26745  2.354
 1.22742  1.90498  1.70653  1.63928  2.18076

julia> Float64.(A.storage)^2
5×5 Array{Float64,2}:
 1.13564  1.64708  1.39399  1.27087  1.82128
 1.03924  1.93216  1.44198  1.42456  1.86497
 1.00201  1.28786  1.37826  1.24295  1.6882
 1.19089  2.33262  2.04094  2.26745  2.354
 1.22742  1.90498  1.70653  1.63928  2.18076

Note that the low precision result differs from (is less precise than) the result computed in Float32 arithmetic (which matches the result in Float64 precision).