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

↓

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

Derivation

Initial program 58.1
\[\frac{e^{x} - e^{-x}}{2}\]
Taylor expanded around 0 0.6
\[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]
Simplified0.6
\[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}}{2}\]
Final simplification0.6
\[\leadsto \frac{\frac{1}{60} \cdot {x}^{5} + \left(x \cdot \left(x \cdot \frac{1}{3}\right) + 2\right) \cdot x}{2}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))