This package defines the BFloat16 data type, a floating-point format with 1 sign bit, 8 exponent bits and 7 mantissa bits.

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.

This package exports the `BFloat16`

data type. This datatype behaves
just like any built-in floating-point type

```
julia> using BFloat16s
julia> a = BFloat16(2)
BFloat16(2.0)
julia> sqrt(a)
BFloat16(1.4140625)
```

However, in practice you may hit a `MethodError`

indicating that this package does not implement
a method for `BFloat16`

although it should. In this case, please raise an issue so that we can
close the gap in support compared to other low-precision types like `Float16`

. The usage
of `BFloat16`

should be as smooth as the following example, solving a linear equation system

```
julia> A = randn(BFloat16,3,3)
3×3 Matrix{BFloat16}:
1.46875 -1.20312 -1.0
0.257812 -0.671875 -0.929688
-0.410156 -1.75 -0.0162354
julia> b = randn(BFloat16,3)
3-element Vector{BFloat16}:
-0.26367188
-0.14160156
0.77734375
julia> A\b
3-element Vector{BFloat16}:
-0.24902344
-0.38671875
0.36328125
```

In addition, this package provides the `LowPrecArray`

type.
This array is supposed to emulate the kind
of matrix multiplications 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).