StochasticRounding

This package exports Float32sr,Float16sr and BFloat16sr. Three number formats that behave like their deterministic counterparts but with stochastic rounding that is proportional to the distance of the next representable numbers and therefore exact in expectation (see also example below in "Usage"). Although there is currently no known hardware implementation available, Graphcore is working on IPUs with stochastic rounding. Stochastic rounding makes the current Float16/BFloat16 software implementations considerably slower, but less than <10x currently. Xoroshio128Plus, a random number generator from the Xorshift family, is used through the RandomNumbers.jl package.

Stochastic rounding is only applied on arithmetic operations, and not on type conversions or for subnormal numbers (standard round to nearest instead).

Usage

julia> a = BFloat16sr(1.0)
BFloat16sr(1.0)
julia> a/3
BFloat16sr(0.33398438)
julia> a/3
BFloat16sr(0.33203125)

As 1/3 is not exactly representable the rounding will be at 66.6% chance towards 0.33398438 and at 33.3% towards 0.33203125 such that in expectation the result is 0.33333... and therefore exact. You can use BFloat16_chance_roundup(x::Float32) to get the chance that x will be round up.

From v0.3 onwards the random number generator is randomly seeded on every import or using such that running the same calculations twice, will, in general, not yield bit-reproducible results. However, you can seed the random number generator at any time with any integer larger than zero as follows

julia> StochasticRounding.seed(2156712)

Theory

Round nearest (tie to even) is the standard rounding mode for IEEE floats. Stochastic rounding is explained in the following schematic

The exact result x of an arithmetic operation (located at one fifth between x₂ and x₃ in this example) is round down to x₂ for round to nearest rounding mode. For stochastic rounding only at 80% chance x is round down, in 20% chance it is round up to x₃, proportional to the distance of x between x₂ and x₃.

Installation

StochasticRounding.jl is registered in the Julia registry. Hence, simply do

julia>] add StochasticRounding

where ] opens the package manager.

Performance

Define a few random 1000x1000 matrices

julia> using StochasticRounding, BenchmarkTools, BFloat16s
julia> A1 = rand(Float32,1000,1000);
julia> A2 = rand(Float32,1000,1000);   # A1, A2 shouldn't be identical as a+a=2a is not round
julia> B1,B2 = Float32sr.(A1),Float32sr.(A2);

And similarly for the other number types. Then on an Intel(R) Core(R) i5 (Ice Lake) @ 1.1GHz timings via @btime +($A1,$A2) etc. are

Stochastic rounding adds about 3-5ms and imposes a x8 performance decrease for Float32, x9 performance decrease for BFloat16 and x1.3 for Float16.