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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 57.21717512275444:\\ \;\;\;\;\frac{\left(2 + x \cdot {\left(e^{\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)} \cdot \sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}\right)}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \varepsilon - x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 57.21717512275444
1. Initial program 38.7
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
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
  \[\leadsto \frac{\left(\color{blue}{e^{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}} \cdot x + 2\right) - {x}^{2}}{2}\]
8. Using strategy rm
9. Applied add-cube-cbrt1.3
  \[\leadsto \frac{\left(e^{\color{blue}{\left(\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)} \cdot \sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}\right) \cdot \sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}}} \cdot x + 2\right) - {x}^{2}}{2}\]
10. Applied exp-prod1.3
  \[\leadsto \frac{\left(\color{blue}{{\left(e^{\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)} \cdot \sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}\right)}} \cdot x + 2\right) - {x}^{2}}{2}\]
if 57.21717512275444 < x
1. Initial program 0.2
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around -inf 0.2
  \[\leadsto \frac{\color{blue}{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + e^{x \cdot \varepsilon - x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Recombined 2 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 57.21717512275444:\\ \;\;\;\;\frac{\left(2 + x \cdot {\left(e^{\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)} \cdot \sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)}\right)}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \varepsilon - x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}\]

Error

Try it out

Derivation

`if x < 57.21717512275444`

`if 57.21717512275444 < x`

Runtime

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