LogFixPoint16s.jl

Provides LogFixPoint16 - a (software-implemented) 16-bit logarithmic fixed-point number format with adjustable numbers of integer and fraction bits.

Example use

julia> using LogFixPoint16s
julia> v = LogFixPoint16.(rand(Float32,5))
5-element Array{LogFixPoint16,1}:
 LogFixPoint16(0.8925083)
 LogFixPoint16(0.4919428)
 LogFixPoint16(0.69759846)
 LogFixPoint16(0.25616693)
 LogFixPoint16(0.57248604)

julia> sum(v)
LogFixPoint16(2.9139352)

Features

Exports LogFixPoint16, iszero, isnan, signbit, zero, nan, floatmin, floatmax, one, -, inv, *, / , +, -, sqrt, nextfloat, prevfloat, ==, <=, >, >=, show, bitstring as well as conversions to and from Float64, Float32, Float16, Int.

Although LogFixPoint16 is always a 16-bit format, the number of fraction bits (in exchange for integer bits) can be adjusted between 7 and 11. For 7 fraction bits, LogFixPoint16 has a similar dynamic range-precision trade-off as BFloat16; 10 fraction bits are similar to Float16.

julia> LogFixPoint16s.set_nfrac(7)
┌ Warning: LogFixPoint16 was changed to 8 integer and 7 fraction bits.
└ @ Main.LogFixPoint16s ~/git/LogFixPoint16s.jl/src/LogFixPoint16s.jl:24

Furthermode the rounding mode can be changed from round-to-nearest in linear space (default) to log space

julia> LogFixPoint16s.set_rounding_mode(:log)
┌ Warning: LogFixPoint16 rounding mode changed to round to nearest in log-space.
└ @ LogFixPoint16s ~/.julia/packages/LogFixPoint16s/TGYbV/src/change_format.jl:48

The two arguments :lin and :log are allowed.

Theory

A real number x is encoded in LogFixPoint16 as

x = (-1)^s * 2^k

with s being the sign bit and k = i+f the fixed-point number in the exponent, consisting of a signed integer i and a fraction f, which is defined as the significant bits for floating-point numbers. E.g. the number 3 is encoded as

julia> bitstring(LogFixPoint16(3),:split)
"0 1000001 10010110"

The sign bit is 0, the sign bit of the signed integer is 1 (meaning + due to the biases excess representation) such that the integer bits equal to 1. The fraction bits are 1/2 + 1/16 + 1/64 + 1/128. Together this is

0 1000001 10010110 = +2^(1 + 1/2 + 1/16 + 1/64 + 1/128) = 2^1.5859375 ≈ 3

The only exceptions are the bitpatterns 0x0000 (zero) and 0x8000 (Not-a-Real, NaR). The smallest/largest representable numbers are (6 integer bits, 9 fraction bits)

julia> floatmin(LogFixPoint16)
LogFixPoint16(2.3314606e-10)

julia> floatmax(LogFixPoint16)
LogFixPoint16(4.2891566e9)

Decimal precision

Logarithmic fixed-point numbers are placed equi-distantly on a log-scale. Consequently, their decimal precision is perfectly flat throughout the dynamic range of representable numbers. In contrast, floating-point numbers are only equi-distant in logarithmic space when the significand is held fixed; the significant bits, however, are linearly spaced.

As a consequence there is no rounding error for logarithmic fixed-point numbers in multiplication, division, power of 2 or square root - similarly as there is no rounding error for fixed-point numbers for addition and subtraction - as long as no over or underflow occurs.

LogFixPoint16 with 10 fraction bits have a similar decimal precision / dynamic range trade-off as Float16, and 7 fraction bits are similar to BFloat16. However, these decimal precision only apply to additions, as multiplications are rounding error-free. LogFixPoint16s.jl also allows additionally for 8,9 or 11 fraction bits, which are not shown.

Benchmarks

Although LogFixPoint16s are software-emulated, they are considerably fast. Define some matrices

julia> using LogFixPoint16s, BenchmarkTools
julia> A = rand(Float32,1000,1000);
julia> B = rand(Float32,1000,1000);
julia> C,D = Float16.(A),Float16.(B);
julia> E,F = LogFixPoint16.(A),LogFixPoint16.(B);

And then benchmark via @btime +($A,$B): and so on. Then relative to Float64 performance for addition:

Operation	Float64	Float32	BFloat16	Float16	LogFixPoint16
Addition (+)	1200μs	500μs	400μs	1800μs	6500μs
Multiplication (.*)	1200μs	500μs	400μs	2500μs	250μs
Power (.^2)	700μs	300μs	2800μs	1000μs	700μs (*)
Square-root (sqrt.)	1800μs	900μs	1800μs	1200μs	170μs

On an Intel i5 (Ice Lake). (*) via power2.

Installation

LogFixPoint16s.jl is registered in the Julia Registry, so simply do

julia> ] add LogFixPoint16s

where ] opens the package manager.