LogarithmicNumbers.jl

A logarithmic number system for Julia.

Provides the signed Logarithmic and unsigned ULogarithmic types for representing real numbers on a logarithmic scale.

This is useful when numbers are too big or small to fit accurately into a Float64 and you only really care about magnitude.

For example, it can be useful to represent probabilities in this form, and you don't need to worry about getting zero when multiplying many of them together.

pkg> add LogarithmicNumbers

julia> using LogarithmicNumbers

julia> ULogarithmic(2.7)
exp(0.9932517730102834)

julia> float(ans)
2.7

julia> x = exp(ULogarithmic, 1000) - exp(ULogarithmic, 998)
exp(999.8545865421312)

julia> float(x) # overflows
Inf

julia> log(x)
999.8545865421312

• ULogarithmic{T} represents a non-negative real number by its logarithm of type T.
• Logarithmic{T} represents a real number by its absolute value as a ULogarithmic{T} and a sign bit.
• LogFloat64 is an alias for Logarithmic{Float64}. There are also ULogFloat16, ULogFloat32, ULogFloat64, LogFloat16, and LogFloat32.

• ULogarithmic(x) and Logarithmic(x) represent the number x.
• exp(ULogarithmic, logx) represents exp(logx), and logx can be huge. Use this when you already know the logarithm logx of your number x.

• Arithmetic: +, -, *, /, ^, inv, prod, sum, sqrt, cbrt, fourthroot.
• Ordering: ==, <, ≤, cmp, isless, isequal, sign, signbit, abs.
• Logarithm: log, log2, log10, log1p. These are returned as the base (non-logarithmic) type.
• Conversion: float, unsigned, signed, widen, big. These also operate on types.
• Special values: zero, one, typemin, typemax.
• Predicates: iszero, isone, isinf, isfinite, isnan.
• IO: show, write, read.
• Random: rand(ULogarithmic) is a random number in the unit interval.
• Misc: nextfloat, prevfloat, hash.
• Note: Any functions not mentioned here might be inaccurate.

It is natural to use this package in conjunction with other packages which return logarithms. The general pattern is that you can use exp(ULogarithmic, logfunc(args...)) instead of func(args...) to get the answer as a logarithmic number. Here are some possibilities for func:

• StatsFuns.jl: normpdf, normcdf, normccdf, plus equivalents for other distributions.
• Distributions.jl: pdf, cdf, ccdf.
• SpecialFunctions.jl: gamma, factorial, beta, erfc, erfcx.

On Julia 1.9+, if you load ForwardDiff.jl, you should be allowed to compute

• derivatives of functions involving exp(Logarithmic, x)
• derivatives of functions evaluated at Logarithmic(x)