In quad:
maxpos3 = 2^16384*(1-2^-113)
sin(maxpos3) = 0.951914854078820481136324892937573
sin(maxpos3) contained in
[4942624506426098432520117030049803/5192296858534827628530496329220096,
9885249012852196865040234060099607/10384593717069655257060992658440192]
These things are just not that difficult to compute any more.
And haven't been for decades. To continue to assume that
they are& deprecate the specifications of 1788 accordingly
does a disservice to all.
(Indeed, I discovered a fast& efficient method of what was
called 'range reduction' around 1980 or 1981. It turns out
I had rediscovered a result of Mary Payne's. Which, in turn,
was a variation of an earlier result by Gosper, I believe.
And this result has been rediscovered& improved upon many
times since. BTW, I won $3 from an MIT professor who bet me
I couldn't do it at the time. Best bet I ever made. :-)
But, while 754 requires these functions to round correctly
throughout their entire domain, the situation is somewhat
easier for 1788. And while it may be necessary, from time
to time, to use the specifications of 754 to correctly
contain a result within a narrow interval for an unreasonably
huge argument, it is only needed in the case of zero width
intervals.
For, indeed, if zero width intervals for unreasonably large
arguments are themselves unreasonable, then one need only
consider efficient domain reduction for the domain of a
circular function for which an ULP is less than 2*pi. And
that only happens for much MUCH smaller arguments.
For single: ulp1 ~ 2^27 ~ 10^8
For double: ulp2 ~ 2^56 ~ 10^17
For quad: ulp3 ~ 2^116 ~ 10^35
As for numbers as large as 2^2^64, except for unreasonable
zero width arguments, one need only consider them when the
PRECISION of the result is ALSO around 2^2^64. Something
that's not going to happen until memory gets much cheaper.
For if the precision required is a small as 10,000 bits, we
have that ulp10kb ~ 2^10003 ~ 10^3011. Which is STILL well
within the range of quad precision numbers.
To conclude: There may be problems in which an interval
computation is intractable when the corresponding
floating-point problem is not.
But computing narrow interval results reliably for 754
transcendental functions is NOT one of them.
Please feel free to consider reproducible versions of
these functions as both feasible& efficient.