\[\frac{e^{x} - e^{-x}}{2}\]

↓

\[\frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}{2}\]

Derivation

Initial program 58.1
\[\frac{e^{x} - e^{-x}}{2}\]
Taylor expanded around 0 0.7
\[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
Simplified0.7
\[\leadsto \frac{\color{blue}{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}}{2}\]
Final simplification0.7
\[\leadsto \frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right))_*}{2}\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))