deccfloat_exp, deccfloat_expm1, deccfloat_log, deccfloat_log1p, deccfloat_sin, deccfloat_cos, deccfloat_tan, deccfloat_sinh, deccfloat_cosh, deccfloat_tanh, deccfloat_asin, deccfloat_acos, deccfloat_atan, deccfloat_asinh, deccfloat_acosh, deccfloat_atanh, deccfloat_sin_cos, deccfloat_sinh_cosh, deccfloat_pi

Each function is evaluated by the first applicable of three methods.

• Real and imaginary arguments. For a real argument $t$ the real decfloat function is called, inheriting its exact cases and special-argument handling; if the value is not real, the standard reductions are applied: $\log(-t) = \log t + \pi i$, $\operatorname{acos} t = \pm i\operatorname{acosh} t$ and $\pi - i\operatorname{acosh}(-t)$ for $t > 1$ and $t < -1$, $\operatorname{acosh} t = i\operatorname{acos} t$ for $-1 \le t < 1$ and $\operatorname{acosh}(-t) + \pi i$ for $t < -1$, $\exp(\pi i t) = \cos(\pi t) + i\sin(\pi t)$, and $\sqrt{-t} = i\sqrt t$. For an imaginary argument $it$ the corresponding real function is used where one exists: $\sin(it) = i\sinh t$, $\cos(it) = \cosh t$, $\tan(it) = i\tanh t$, $\cot(it) = -i\coth t$, $\operatorname{asin}(it) = i\operatorname{asinh} t$, $\operatorname{atan}(it) = i\operatorname{atanh} t$ for $|t| < 1$, $\operatorname{erf}(it) = i\operatorname{erfi} t$ and so on, and $\exp(it) = \cos t + i\sin t$. So $\sqrt{-4} = 2i$, $\log(-1) = \pi i$, $\Gamma(5) = 24$ and $\sin(10^{-10^9} i) = i\sinh(10^{-10^9})$ are correctly rounded at negligible cost.
• Tiny arguments. For $\exp$, $\operatorname{expm1}$, $\log(1 + z)$, $\log$ near 1, the trigonometric and hyperbolic functions and their inverses, $\operatorname{sinc}$, $\Gamma$, $1/\Gamma$, $\psi$, $\zeta$ near 1, $W$, $\operatorname{Li}_2$, $\operatorname{Si}$, $\operatorname{Shi}$ and (through $1/z$) the inverse cotangent, cosecant and their hyperbolic versions at large arguments, the leading part $S$ ($z$, $1$, $\pm 1/z$, $z - 1$ or $1/(z - 1)$) is computed exactly and the sign of each component of $f(z) - S$ from the first Taylor terms; a component whose remainder lies more than prec $+ 4e$ digits below $S$ is $S$ rounded with the remainder as a sticky bit. This works for arbitrarily unbalanced arguments: $\sin(10^{-10^9} + 10^{-2 \cdot 10^9} i)$ rounded toward zero has both parts one ulp below $10^{-10^9}$ and $10^{-2 \cdot 10^9}$.
• Ziv's strategy through acb. The argument is converted to an acb ball and the gr method of the acb ring is evaluated at increasing precision until the rounding of each remaining component is decided; GR_UNABLE is returned if this has not happened at 64 times the working precision.

The last method is the only one for the other special functions. It fails when a component is a representable number or extremely close to one and the case is not detected, for example the real part of $\operatorname{erf}(300 + i)$, which is $1 - O(10^{-39000})$, and for the functions without an acb method at non-real arguments (sec_pi, asin_pi, acos_pi, atan_pi, acot_pi, asec_pi, acsc_pi, erfinv, erfcinv).

@misc{flint-doc-decimal-deccfloat-exp,
  author       = {{The FLINT team}},
  title        = {{deccfloat\_exp, deccfloat\_expm1, deccfloat\_log, deccfloat\_log1p, deccfloat\_sin, deccfloat\_cos, deccfloat\_tan, deccfloat\_sinh, deccfloat\_cosh, deccfloat\_tanh, deccfloat\_asin, deccfloat\_acos, deccfloat\_atan, deccfloat\_asinh, deccfloat\_acosh, deccfloat\_atanh, deccfloat\_sin\_cos, deccfloat\_sinh\_cosh, deccfloat\_pi (decimal.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/decimal/deccfloat_exp.html},
  note         = {Generated 2026-11-02}
}