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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 382.71997061196936:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\left(2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) - {x}^{2}}\right) + \log \left(\sqrt{\left(2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) - {x}^{2}}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\sqrt{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}}\right) - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 382.71997061196936
1. Initial program 38.8
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
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 unpow31.3
  \[\leadsto \frac{\left(\frac{2}{3} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 2\right) - {x}^{2}}{2}\]
5. Applied associate-*r*1.3
  \[\leadsto \frac{\left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x} + 2\right) - {x}^{2}}{2}\]
6. Using strategy rm
7. Applied add-exp-log1.3
  \[\leadsto \frac{\color{blue}{e^{\log \left(\left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2\right) - {x}^{2}\right)}}}{2}\]
8. Using strategy rm
9. Applied add-sqr-sqrt2.3
  \[\leadsto \frac{e^{\log \color{blue}{\left(\sqrt{\left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2\right) - {x}^{2}} \cdot \sqrt{\left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2\right) - {x}^{2}}\right)}}}{2}\]
10. Applied log-prod1.3
  \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{\left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2\right) - {x}^{2}}\right) + \log \left(\sqrt{\left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x + 2\right) - {x}^{2}}\right)}}}{2}\]
if 382.71997061196936 < x
1. Initial program 0.1
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.1
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Recombined 2 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 382.71997061196936:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\left(2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) - {x}^{2}}\right) + \log \left(\sqrt{\left(2 + \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right) - {x}^{2}}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\sqrt{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}}\right) - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Try it out

Derivation

`if x < 382.71997061196936`

`if 382.71997061196936 < x`

Runtime

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