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Derivation

1. Initial program 58.0

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

2. Taylor expanded around 0 0.7

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

3. Simplified0.7

   \[\leadsto \frac{\color{blue}{(\left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}}{2}\]
4. Using strategy rm

5. Applied fma-udef0.7

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

6. Taylor expanded around -inf 0.7

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

7. Final simplification0.7

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

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 3.3s)Debug log