\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 24.86374916342476:\\ \;\;\;\;\frac{\frac{\sqrt{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*}}{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \varepsilon - x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 24.86374916342476
1. Initial program 39.0
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.3
  \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied flip--1.3
  \[\leadsto \frac{\color{blue}{\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}}{2}\]
5. Using strategy rm
6. Applied add-sqr-sqrt1.3
  \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}{2}\]
7. Applied associate-/l*1.3
  \[\leadsto \frac{\color{blue}{\frac{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}{\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}}}}{2}\]
8. Simplified1.3
  \[\leadsto \frac{\frac{\color{blue}{\sqrt{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*}}}{\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}}}{2}\]
if 24.86374916342476 < x
1. Initial program 0.3
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around -inf 0.3
  \[\leadsto \frac{\color{blue}{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + e^{x \cdot \varepsilon - x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Recombined 2 regimes into one program.
Final simplification1.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 24.86374916342476:\\ \;\;\;\;\frac{\frac{\sqrt{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) + \left(-{x}^{4}\right))_*}}{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \varepsilon - x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]

Error

Derivation

`if x < 24.86374916342476`

`if 24.86374916342476 < x`

Runtime