@Carl Love  Yes you probably want to emphasize it has a computer science related causality before someone starts a new field of math called catasrtrophe theory.

Ok i will get around at some point to looking at all of the case examples for which i have archived that the erroraneous output is recieved or remains unevaluated. 

It happens equally as often for the floor function as it does for frac in my experience, and is mostly trancendental functions. Sometimes it occurs just for particular expressions that involve integer powers of pi and rational coefficients, a while ago i did post a way that i was able to deterministically express the the errors in evalf using certain interpolations of the curve fitting function taking the number of digits as a variable, it will be some where in the history of questions on this forum no doubt.

Most are able to be fixed it really isn't major bug at all, just alot more frequently occuring may have previously been anticipated by those responsible for the development of the floating point approximation inbuilt code, I guess a combination of me being quite naive with respect to the inner workings of maple and also quite pedantic might explain why i keep coming across them.

Once I have spent a reasonable amount of time looking at the cumulative content i will post them here.

@Carl Love  I think I will put things like this on hold until my book arrives in the mail.

@tomleslie  Yes you are quite correct mate I should have said that my intentions were to endow a set with a specific ordering, hence what I am really referring to was a Group. 

But in the handleing of such a group, there is always an associated set, so really what I am saying is that i want to create a system where by the multiset that is created when applying a given ordering "rule" or in the simple case, operator, and have some kind of generic method of doing so that allows me to clearly display output indicating that this particular ordering was applied to produce the "multiset" or group.

@vv  ok well nobody expects frac  of anything to be greater than 1 unless they are riding to school on the short bus for starters.

And a "psychological" bug would be  something like the state of confusion that you are placed in if i know tell you, with all sincerity, that you were in a horrible accident, and are in a coma, and the maple forum was used in the brain signalling technology as the mental image projection we are using in our attempts to wake you, so please sir, wake up.

@Carl Love well ok I concede i pulled that 10^4 out of one's posterier. But, I also require an explicit definition of a "Catastrophic Cancellation" from you Good Sir. 

@vv 

Maple 16:

evalf[100](frac(23*Pi));

0.25663103256524448464079781542856633653489618562743388242372562307977734458280696844480098286869256

evalf[100](frac(20*Pi));
 

0.83185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136

@vv did you try folding the higher digit demand inside the lower one as i did above? of course nobody wants 10000 digits, but this seems to work for the example type above.

And i have archived a number of them that are solved just as readily that way, if not, using both evalf and convert(x,'rational') to differing levels of precision according to a case by case basis, obviously no interface can be expected to achieve complete precision for every single function in an infinitude of functions possible with an infinitude of differing parameterizations, so like I repeatedly tell collegues that ask me to change to another one like MATLAB or mathematica, If I had of chosen YOUR interface I am certain I would find just as many due to the nature of these bugs 

@Carl Love 

So I do concede it's a much grander effort than my lazy afternoon bit bash with the floor function, but they are around the same growth rate tbh. 

What can be said in terms of Asymtotics in terms of the direction for which the limit of the ratio of the two functions approaches 1?  I just don't understand Carl you tell me to respect the big-O but then your code just omits it? Like is being ignored how to show respect? Finally it makes sense now why nobody likes my facebook posts, the are so good, people hold them in to high esteem to notice them

@Carl Love

@Carl Love

(evalf[10](frac((8*k+1)^(8*k+1)*frac(((8*k+1)*Pi-frac((8*k+1)*Pi))/(8*k+1+Pi)))));

Digits := 10000;
evalf[10](evalf[10000](frac((8*k+1)^(8*k+1)*frac(((8*k+1)*Pi-frac((8*k+1)*Pi))/(8*k+1+Pi)))));
 

@Carl Love  the consistent between the two functions is the arithmetic progression 22,23,24..,28 for which both have zeros. 

@Carl Love

DATA0 := [seq([k, frac(k^k*frac((k*Pi-frac(k*Pi))/(k+1)))], k = 1 .. 200)];

DATA1 := [seq([k, frac(Pi*k^k*frac((k*Pi-frac(k*Pi))/(k+1)))], k = 1 .. 200)];

@Carl Love 

The latter appears to have distinct regions of significant change in the degree of pathological behavior, which i have separated into seperate graphs to illustrate this.

x=[0,2]

x=[2,3.5]

x=[3.5,3.8]

x=[3.8,4]

x=[4,infinity]

@Carl Love  i see exactly what you mean, i have archived quite a large number of these that occur and are very easily fixed, I just dont see why they haven't been, but i am on maple 16, so they very well might have been already. But for the time being I am happy that they are easily fixed. But it did make me predict quite an embarassing asymptotic relation that i submitted to the facebook page "mathematical theorems you've never heard of because they are false"

@Carl Love 

In my study of one of Pierre Dusart's papers "Estimates Of Some Functions without R.H" he quotes an asymptotic expansion found by Cesaro, and I was unclear as to what he meant in the use of the curly brackets. 

I just need you to confirm that what I have written here is equivalent and correct. Thanks Carl.