Euler's identity and De~Moivre's formula
describe rotations generated by an imaginary unit satisfying .
In associative and alternative hypercomplex algebras, however, arbitrary elements
need not square to , the exponential map may lose geometric meaning, and
fractional powers often lack closed forms.
This paper systematizes existing exponential formulas into a unified algorithmic framework with computational advantages.F
admits a two-term Euler-type fractional power: \emph{ has a closed form if and
only if is a nonzero real scalar}. When we obtain the elliptic
(rotation) formula with
\[
A^x = \alpha^x\!\left[\cos\Bigl(\tfrac{\pi x}{2}\Bigr)
+ E\sin\Bigl(\tfrac{\pi x}{2}\Bigr)\right],
\qquad E=\frac{A}{\sqrt{-A^2}},\; E^2=-1,
\]
extending Euler and De~Moivre to all alternative algebras, including quaternions
and octonions. When , we obtain the symmetric hyperbolic (boost) formula
\[
A^x = \alpha^x\!\left[\cosh\Bigl(\tfrac{\pi x}{2}\Bigr)
+ H\sinh\Bigl(\tfrac{\pi x}{2}\Bigr)\right],
\qquad H=\frac{A}{\sqrt{A^2}},\; H^2=+1,
\]
which covers split-quaternions and other hyperbolic directions.
Both formulas follow from the principal logarithm
on the generated $1A$ plane and remain valid for all real
fractional exponents. A collapse identity is proved:
\[
A^{A x}= \exp(A^2\pi x/2),
\]
giving the classical $i^i=e^{-\pi/2}A^2\notin\mathbb{R}AA^22x+10^{-16}$) and exhibiting superior stability near the identity element. This method offers a unified, high-performance primitive for geometric rotations in computer graphics, robotics, and hypercomplex neural networks.