

PROJECT 10 Completed (24 days) Completed (57 days) Completed (26 days) Completed (47 days) Completed (19 days) Completed (14 days)  Nonfloatingpoint fractional exponentiation approach Completed on 16Nov2016 (24 days) The intended fractional exponentiation approach is expected to account for most of the significant digits of the decimal type. Thus, the corresponding algorithm has to be able to deal with many scenarios involving fractions formed by big numbers (e.g., 0.23456789885 defined by the fraction 23456789885/10^11), what implies that the nroot calculating approach (i.e., NewtonRaphson) has to be able to deal with a huge range of n values (i.e., many integers within the 110^28 range).The complexity of the aforementioned fraction can be reduced by analysing its constituent elements separately: the numerators are used in integer exponentiation (quick and reliable implementation easily dealing with any number) and the denominators in NewtonRaphson (reliability conditioned by the initial guess). Additionally, the ideas in the previous section seem to indicate that somehow reducing the number of n values/denominators (or interrelating them) could be helpful to find good initial guesses. That's why it soon became clear that the best option was creating multipleof10based fractions. Even under the aforementioned conditions, the number of different input scenarios might still be too big (i.e., at least, 28 different trends); but the multipleof10 reliance proved to also be helpful on this front, via noticeable similarities across different n values (e.g., 10, 1000 or 10^10). I was able to come up with relatively simple approaches delivering acceptably good guesses for all the possible n values. Math2_Private_New_PowSqrt_RootIni.cs includes all the code calculating the initial guesses for GetNRoot . These algorithms describe the patterns which I saw after analysing a reasonably big number of different input conditions; for example, after writing to a file the roots for 10, 10^2, 10^3, etc. with n = 10, 100, 1000, etc., a simple visual inspection was enough to extract worthy conclusions. All this information is referred by one of the following two main methods:
Number range and any exponent within the decimal range, by bearing in mind the aforementioned 25firstdecimaldigits limitation), what implies that a valid result for the target accuracy (i.e., 1e28m or, under very specific conditions, 5e28m ) will always be found. 