BFloat16s
This package defines the BFloat16 data type. The only currently available hardware implementation of this datatype are Google's Cloud TPUs. As such, this package is suitable to evaluate whether using TPUs would cause precision problems for any particular algorithm, even without access to TPU 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).