Nature of approximations

Many of the added functions utilize approximations where the precision is accurate up to ULP (Unit in the last place) of 1. The default precision is 32 decimal places and uses the default rounding mode when returning the result.

The approximations themselves utilize specific formulas and various types of mathematical series. For values close to one after argument reduction, convergence rates are relatively fast, although convergence rates for some functions will never truly converge for irrational numbers, as they can be infinitely approximated to any point of precision. To limit these operations, a tolerance within the desired precision is used as such:

$$\displaylines{\left| X_{n-1} - X_{n} \right| \lt \epsilon \\ X_{n} \approx f_{n}\left( x \right) \text{; where } 0\le n \le \infty \\ \epsilon = 10^{-32}}$$

$ X_{n}$ represents the approximation value of the loop at step $n$. When the difference between the previous and next values are less than the tolerance, any further iterations will be inconsequential to the desired precision, ending the loop.

The current bigO complexity has yet to be evaluated, but many of the algorithms have been modeled after IEEE 754 and other resources. More optimizations will likely be implemented, and evaluation of performance will follow.