clusterphobia | Unassisted clustering algorithms and data | Machine Learning library

I have implemented an approximate natural log function based on a Padé Approximation of a truncated Taylor Series. The accuracy is acceptable (±0.000025) but despite several rounds of optimizations, its exec time is still about 2.5x that of the standard library ln function! If it isn't faster and it isn't as accurate, it is worthless! Nevertheless, I am using this as a way to learn how to optimize my Rust code. (My timings come from using the criterion crate. I used blackbox, summed the values in the loop and created a string from the result to defeat the optimizer.)

On Rust Playground, my code is:

https://play.rust-lang.org/?version=stable&mode=debug&edition=2018&gist=94246553cd7cc0c7a540dcbeff3667b9

An overview of my algorithm, which works on a ratio of unsigned integers:

1. Range reduction to the interval [1, 2] by dividing by the largest power of two not exceeding the value:
   • Change representation of numerator → 2ⁿ·N where 1 ≤ N ≤ 2
   • Change representation of denominator → 2ᵈ·D where 1 ≤ D ≤ 2
2. This makes the result log(numerator/denominator) = log(2ⁿ·N / 2ᵈ·D) = (n-d)·log(2) + log(N) - log(D)
3. To perform log(N), Taylor series does not converge in the neighborhood of zero, but it does near one...
4. ... since N is near one, substitute x = N - 1 so that we now need to evaluate log(1 + x)
5. Perform a substitution of y = x/(2+x)
6. Consider the related function f(y) = Log((1+y)/(1-y))
   • = Log((1 + x/(2+x)) / (1 - x/(2+x)))
   • = Log( (2+2x) / 2)
   • = Log(1 + x)
7. f(y) has a Taylor Expansion which converges must faster than the expansion for Log(1+x) ...
   • For Log(1+x) → x - x²/2 + x³/3 - y⁴/4 + ...
   • For Log((1+y)/(1-y)) → y + y³/3 + y⁵/5 + ...
8. Use the Padé Approximation for the truncated series y + y³/3 + y⁵/5 ...
9. ... Which is 2y·(15 - 4y²)/(15 - 9y²)
10. Repeat for the denominator and combine the results.

Here is the Padé Approximation part of the code: