Not knowing exactly what FibersFailedNew and FibersFailedNewT2 are, I'm going to hazard a guess that they are MachinePrecision numbers. I'm also going to guess that your use of SetPrecision does not achieve what you want it to do. The code
SetPrecision[FibersFailedNew, 40]
does not change the precision of FibersFailedNew; it displays the value of converting FibersFailedNew to 40-digit precision. And in the code,
SetPrecision[FibersRemainingNewT2 = 1 - FibersFailedNewT2, 55]
the calculation inside SetPrecision is done first at MachinePrecision and the result is converted to 55-digit precision. In all the OP's code currently(currently posted), the underlying arithmetic iswould be done at machine precision. The explanation of what is going on lies in the granularity of machine precision numbers.
As you may know, [$MachineEpsilon](http://reference.wolfram.com/mathematica/ref/$MachineEpsilon.html) is the smallest number that may be added to ``1.` `` to yield a distinct number. Basically, on a binary machine, it changes the lowest-order bit. For numbers between `0.5` and `1.` the smallest change would be `$MachineEpsilon/2`http://reference.wolfram.com/mathematica/ref/$MachineEpsilon.html) is the smallest number that may be added to ``1.` `` to yield a distinct number. Basically, on a binary machine, it changes the lowest-order bit. For numbers between `0.5` and `1.` the smallest change would be `$MachineEpsilon/2`. The difference 1 - FibersFailedNew will differ from 1. by a multiple of
$MachineEpsilon/2
(* 1.11022*10^-16 *)
which will be determined by rounding. In the OP's numbers, both FibersFailedNew and FibersFailedNewT2 round to the same multiple of $MachineEpsilon/2, namely 3:
FibersFailedNew = N[3.612171938234261799295174910010284760488*10^-16];
FibersFailedNewT2 = N[3.612171938234886971562562561866073555376*10^-16];
1. - N[FibersFailedNew];
1. - %
(* 3.33067*10^-16 *)
1. - N[FibersFailedNewT2];
1. - %
(* 3.33067*10^-16 *)
You may want to think about how these numbers are calculated and whether it might be better to use arbitrary precision numbers from the start.
How to use SetPrecision
Assuming that it is desired to carry out the calculations with arbitrary precision numbers, then here is a way to proceed. Assign the result of SetPrecision to a new or the same variable and use it for your calculations. Below I used the precision [$MachinePrecision](http://reference.wolfram.com/mathematica/ref/$MachinePrecision.html), because it seems advisable to me to keep track of how precise my numbers actually are. If they came from `MachinePrecision`, then certainly no more than `$MachinePrecision`http://reference.wolfram.com/mathematica/ref/$MachinePrecision.html), because it seems advisable to me to keep track of how precise my numbers actually are. If they came from `MachinePrecision` numbers, then certainly no more than `$MachinePrecision` digits are accurate. However, higher precision may be used.
FibersFailedNew = SetPrecision[FibersFailedNew, $MachinePrecision];
FibersFailedNewT2 = SetPrecision[FibersFailedNewT2, $MachinePrecision];
1 - FibersFailedNew
1 - FibersFailedNewT2
(*
0.9999999999999996387828061765738
0.9999999999999996387828061765113
- or in FullForm -
0.9999999999999996387828061765738200704825089989715239512322`31.39682135577845
0.9999999999999996387828061765113028437437438133926444624335`31.39682135577837
*)
Display of numbers
Concerning the implicit question about what is shown on the screen:
The number of digits displayed by the Front End for MachinePrecision numbers is controlled by the system option
SystemOptions[MachineRealPrintPrecision]
(* {"MachineRealPrintPrecision" -> 6} *)
The Front End displays an arbitrary precision number according to the number of digits of precision of the number.