Correctly rounded elementary and special functions

Ziv's strategy cannot decide the rounding when the exact result lies extremely close to a representable number. This happens systematically in three situations, which the decimal functions treat separately:

• Exact values: \Gamma(n) = (n-1)!, 1/\Gamma(n), \zeta(-n), \zeta(0) = -1/2, \log_2(2^n), \log_{10}(10^n), \sin(\pi k/4), \tan(\pi k/4), \cot(\pi k/4), \csc(\pi k/4), \sec(\pi k/4), \mathrm{asin}(1/2)/\pi = 1/6 and the other rational values of the inverse trigonometric functions divided by \pi, \mathrm{acos}(1) = \mathrm{acosh}(1) = 0, \log \Gamma(1) = \log \Gamma(2) = 0, G(n), \log G(n) for n = 1, 2, 3, f(0) for functions whose arb implementation may not return an exact zero or one, powers x^n, x^{p/q} and 2^x, 10^x with exact results, and \mathrm{atan2} on the axes. (arb does not detect most of these.) These are detected from the sign, the scientific exponent, the limb exponent and individual digits of the argument, in time independent of its length and exponent (for instance \sin(\pi x) depends on three digits, and all trivial zeros \zeta(-2k) and poles of \Gamma are found for any k); the argument is only converted to an integer or rational number when it is small or a genuine candidate (\log_2(x) first checks the digit count, the last digit and a residue modulo a word-size prime).
• Tiny arguments: when the leading Taylor term S (x, 1, \pm 1/x, x - 1 for \log, 1/(x-1) for \zeta near 1, -1/2 for \zeta near 0, 1/2 or \operatorname{sgn}(x)/2 for \mathrm{acos}(x)/\pi, \mathrm{acot}(x)/\pi near 0 and \mathrm{atan}(x)/\pi, \mathrm{asec}(x)/\pi near infinity, x for the Barnes G-function) is an exact decimal and the remaining tail, of known sign, lies more than prec + 4e digits below it, the result is S rounded with the tail acting as a sticky bit. For example \sin(10^{-10^9}), \Gamma(10^{-10^9}) = 10^{10^9} - \gamma + \ldots, \zeta(1 + 10^{-10^9}) and \cot(10^{-10^9}) are all correctly rounded at negligible cost.
• Large arguments: \tanh, \coth, \mathrm{erf}, \mathrm{erfc}, \mathrm{expm1} and \zeta converge exponentially fast to \pm 1, 2, -1 or 1, and \mathrm{acot}, \mathrm{acoth}, \mathrm{acsc}, \mathrm{acsch} behave like 1/x; these are handled in the same way.

The remaining special functions (Bessel functions, hypergeometric functions, ...) rely on Ziv's strategy alone and can in principle return GR_UNABLE for arguments at which the exact value is a representable number not detected by arb.

Some arguments are rejected with GR_UNABLE before any evaluation (for balls as well), since the arb/acb algorithms would need time or memory far beyond what the result is worth: for the non-elementary functions, an argument with a part of magnitude at least 10^{10^6} (and for the Barnes G-function also at most 10^{-10^6}, and at least 10^{50}); for the polylogarithm, the Hurwitz zeta function and the polygamma function, an order s with |\operatorname{Re}(s)| \ge 10^{10} or |\operatorname{Im}(s)| \ge 10^{7} (there is no Riemann-Siegel type formula); and for the zeta-type functions (\zeta, Hurwitz \zeta, \eta, \beta, \operatorname{Li}_s, \xi) in the complex contexts, |\operatorname{Re}(s)| \ge 10^{30}, |\operatorname{Im}(s)| \ge 10^{15} (the number of terms grows with |\operatorname{Im}(s)|) and, for \xi, an imaginary part whose cancellation (\xi(s) decays like e^{-\pi |\operatorname{Im}(s)|/4}) exceeds the maximal working precision.