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.

Installation

pkg> add LogarithmicNumbers

Example

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

Documentation

Exported types

• 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.

Constructors

• 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.

Functions in Base

• Arithmetic: +, -, *, /, ^, inv, prod, sum.
• 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.

Interoperability with other packages

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.

Autodiff

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)

This functionality is experimental, please report any bug or unexpected behavior.