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Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ and De~Moivre's formula

describe rotations generated by an imaginary unit $i$ satisfying $i^2=-1$.

In associative and alternative hypercomplex algebras, however, arbitrary elements

$A$ need not square to $-1$, the exponential map may lose geometric meaning, and

fractional powers $A^x$ 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{$A^x$ has a closed form if and

only if $A^2$ is a nonzero real scalar}. When $A^2<0$ we obtain the elliptic

(rotation) formula with $\alpha^x= (\sqrt{\lvert A^{2} \rvert})^{x} \in \mathbb{R}_{>0}$

\[

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 $A^2>0$, 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

$\log(A)=A\pi/2$ on the generated $1$--$A$ 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 $i^i=e^{-\pi/2}$and its hyperbolic analogues as special</p> <p>cases. When$A^2\notin\mathbb{R}$or$A$is a zero divisor (as in the sedenions),</p> <p>the two-term formulas fail exactly, giving a precise obstruction.</p> <p>This yields an algorithmic framework for fractional exponentiation in all</p> <p>Cayley--Dickson and split algebras: test$A^2$, normalize, and apply the elliptic</p> <p>or hyperbolic closed form when allowed. Applications include hypercomplex</p> <p>cyclotomy, Galois actions along arbitrary imaginary directions, and rotation/boost</p> <p>operators in higher dimensions. All claims are supported by independent numerical</p> <p>validation, with reproducible code provided.</p> <p>Critically, we demonstrate that this algebraic approach eliminates the need for the computationally expensive \texttt{arctan2} function required by standard polar decomposition methods. Benchmarking against standard library implementations reveals a mean computational speedup of approximately$2x+$times faster for quaternion roots than SciPy Rotation as standard library, attributed to the replacement of transcendental inverse trigonometric functions with hardware-accelerated algebraic operations. Numerical validation confirms that this performance gain incurs no penalty in accuracy, maintaining machine precision ($10^{-16}$) 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.