The table-maker's dilemma

William Kahan coined the appellation "The Table-Maker's Dilemma" for the alien amount of rounding abstruse functions:

"Nobody knows how abundant it would amount to compute y^w accurately angled for every two floating-point arguments at which it does not over/underflow. Instead, acclaimed algebraic libraries compute elementary abstruse functions mostly aural hardly added than bisected an ulp and about consistently able-bodied aural one ulp. Why can't Y^W be angled aural bisected an ulp like SQRT? Because cipher knows how abundant ciphering it would cost... No accepted way exists to adumbrate how abounding added digits will accept to be agitated to compute a abstruse announcement and annular it accurately to some preassigned amount of digits. Even the actuality (if true) that a bound amount of added digits will ultimately answer may be a abysmal theorem."9

The IEEE amphibian point accepted guarantees that add, subtract, multiply, divide, aboveboard root, and amphibian point butt will accord the accurately angled aftereffect of the absolute attention operation. No such agreement was accustomed in the 1985 accepted for added circuitous functions and they are about alone authentic to aural the endure bit at best. About the 2008 accepted guarantees that befitting implementations will accord accurately angled after-effects which account the alive rounding mode, accomplishing of the functions is about optional.

Using the Gelfond–Schneider assumption and Lindemann–Weierstrass assumption abounding of the accepted elementary functions can be accepted to acknowledgment abstruse after-effects if accustomed rational non-zero arguments; accordingly it is consistently accessible to accurately annular such functions. About free a absolute for a accustomed attention on how authentic after-effects needs to be computed afore a accurately angled aftereffect can be affirmed may appeal a lot of ciphering time.10

There are some bales about now that action actual rounding. The GNU MPFR amalgamation gives accurately angled approximate attention results. Some added libraries apparatus elementary functions with actual rounding in bifold precision:

IBM's libultim, in rounding to abutting only.11

Sun Microsystems's libmcr, in the 4 rounding modes.12

CRlibm, accounting in the Arénaire aggregation (LIP, ENS Lyon). It supports the 4 rounding modes and is proved.13

There abide accountable numbers which a angled amount can never be bent no amount how abounding digits are calculated. Specific instances cannot be accustomed but this follows from the undecidability of the awkward problem. For instance if Goldbach's assumption is authentic but unprovable again the aftereffect of rounding the afterward amount up to the next accumulation cannot be determined: 10-n area n is the aboriginal even amount greater than 4 which is not the sum of two primes, or 0 if there is no such number. The aftereffect is 1 if such a amount exists and 0 if no such amount exists. The amount afore rounding can about be approximated to any accustomed attention even if the assumption is unprovable.