Improve Precision and Performance for All Functions

[mark-schultz-wu]

Hey, I saw that atan and ln were using very large Taylor series and that some tests were running at a reduced precision. I wanted to see if I could get everything to pass the brute-force test suite at a high precision.

Here are the main changes:

• atan: I implemented a range reduction using the identity atan(x) = 2 * atan(x / (1 + sqrt(1+x^2))). This repeatedly shrinks the input x to a small interval (e.g., [0, 1/8]). By reducing the interval, the Taylor series converges much faster and now passes the high-precision tests.

• ln: I used the identity ln(x) = 2 * atanh((x-1)/(x+1)). I then added a similar range reduction for atanh and approximated it with a small Taylor series.

• Other Fixes: I rewrote the exp implementation using the identity exp(x) = 2^k * exp(r) to improve its precision. I also rewrote sin(x) and cos(x) with more numerically stable argument reduction to resolve the test failures in csc and cot.

• Testing & Performance: I added criterion as a dev dependency to benchmark the changes. The atan function saw a ~1500x speedup, and all functions now pass the brute-force tests with an absolute tolerance of 1e-10. There was a minor, acceptable performance regression in cos and sin due to the more complex, correct logic.

[michaelciraci]

Thanks so much for taking the time to contribute. Sorry I'm just seeing this now (I need to check my GitHub notifications).

A few days I found that there are several functions from Rust's libm port that require minimal work to make it const: https://github.com/rust-lang/compiler-builtins/tree/master/libm

Here's an example of a function (atan) from libm that required minimal work to put into this crate: https://github.com/michaelciraci/Trig-Const/blob/cd046bc4256b7662b5e36e4528bea38f78b660b5/src/atan.rs

I'll go through this MR and compare precision to std trig functions. There were some functions from libm that couldn't be made const, so it may be cherry picking different functions from different sources. My ultimate goal is highest precision as possible.

[michaelciraci]

You made significant improvements over the algorithms in this crate from a week ago. Although using a port from libm still shows a pretty substantial improvement over using a series. This is what I get for benchmarking:

Function       | Total Tests| Diff Count|       Max Diff
Current atan   |      502655|      32840|    2.22045e-16
Mark    atan   |      502655|     129988|    5.55112e-16
Current atanh  |       19999|       6706|    1.95399e-13
Mark    atanh  |       19999|      13062|    1.74083e-13
Current ln     |       99999|       3502|    8.88178e-16
Mark    ln     |       99999|      79094|    2.13163e-14
Current exp    |       20001|       2028|    3.63798e-12
Mark    exp    |       20001|      15434|    1.81899e-11
Current sin    |      502655|      22582|    1.11022e-16
Mark    sin    |      502655|     449397|    1.11022e-15
Current cos    |      502655|      21844|    1.11022e-16
Mark    cos    |      502655|     440723|    1.11022e-15