I don’t have a disqus account so I’m commenting here on the blog post:

The reason \log(1+x) is inaccurate when x is small is because 1+x will round the result to fit in a floating point representation. For example, 1.0 + 1E-16 = 1.0. Thus, instead of getting the correct answer, ~1E-16, you’ll just get 0. This page goes into a bit more detail about this function and others.

I updated the blog post with benchmarks. TL;DR: Base.Math.JuliaLibm.log1p is about 30-40% faster, except for very small values, where it is 10-20% slower. On average, there should be a performance gain.