LogExpFunctions

Various special functions based on log and exp moved from StatsFuns.jl into a separate package, to minimize dependencies. These functions only use native Julia code, so there is no need to depend on librmath or similar libraries. See the discussion at StatsFuns.jl#46.

The original authors of these functions are the StatsFuns.jl contributors.

LogExpFunctions supports InverseFunctions.inverse and ChangesOfVariables.test_with_logabsdet_jacobian for log1mexp, log1pexp, log1pexp, log2mexp, log2mexp, logexpm1, logistic, logistic, logit, and logcosh (no inverse).

LogExpFunctions.xlogxFunction
xlogx(x)

Return x * log(x) for x ≥ 0, handling $x = 0$ by taking the downward limit.

julia> xlogx(0)
0.0
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LogExpFunctions.xlog1pyFunction
xlog1py(x, y)

Return x * log(1 + y) for y ≥ -1 with correct limit at $x = 0$.

julia> xlog1py(0, -1)
0.0
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LogExpFunctions.logisticFunction
logistic(x)

The logistic sigmoid function mapping a real number to a value in the interval $[0,1]$,

\[\sigma(x) = \frac{1}{e^{-x} + 1} = \frac{e^x}{1+e^x}.\]

Its inverse is the logit function.

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LogExpFunctions.logitFunction
logit(x)

The logit or log-odds transformation, defined as

\[\operatorname{logit}(x) = \log\left(\frac{x}{1-x}\right)\]

for $0 < x < 1$.

Its inverse is the logistic function.

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LogExpFunctions.logcoshFunction
logcosh(x)

Return log(cosh(x)), carefully evaluated without intermediate calculation of cosh(x).

The implementation ensures logcosh(-x) = logcosh(x).

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LogExpFunctions.log1pmxFunction
log1pmx(x)

Return log(1 + x) - x.

Use naive calculation or range reduction outside kernel range. Accurate ~2ulps for all x. This will fall back to the naive calculation for argument types different from Float64.

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LogExpFunctions.logmxp1Function
logmxp1(x)

Return log(x) - x + 1 carefully evaluated. This will fall back to the naive calculation for argument types different from Float64.

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LogExpFunctions.logaddexpFunction
logaddexp(x, y)

Return log(exp(x) + exp(y)), avoiding intermediate overflow/undeflow, and handling non-finite values.

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LogExpFunctions.logsumexpFunction
logsumexp(X)

Compute log(sum(exp, X)).

X should be an iterator of real or complex numbers. The result is computed in a numerically stable way that avoids intermediate over- and underflow, using a single pass over the data.

See also logsumexp!.

References

Sebastian Nowozin: Streaming Log-sum-exp Computation

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logsumexp(X; dims)

Compute log.(sum(exp.(X); dims=dims)).

The result is computed in a numerically stable way that avoids intermediate over- and underflow, using a single pass over the data.

See also logsumexp!.

References

Sebastian Nowozin: Streaming Log-sum-exp Computation

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