Let $x \mapsto e^x$ be the D1932: Standard natural real exponential function.
The standard natural hyperbolic sine function is the D4364: Real function \begin{equation} \sinh : \mathbb{R} \to \mathbb{R}, \quad \sinh(x) = \frac{1}{2} (e^x - e^{-x}) \end{equation}

Let $\exp$ be the D1932: Standard natural real exponential function.
The standard natural hyperbolic sine function is the D4364: Real function \begin{equation} \sinh : \mathbb{R} \to \mathbb{R}, \quad \sinh(x) = \frac{1}{2} (\exp(x) - \exp(-x)) \end{equation}