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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 0.6224483931737788:\\ \;\;\;\;\frac{\frac{\left(4 + \left(\frac{1}{16} \cdot {x}^{12} + \left(\frac{1}{6} \cdot {x}^{11} + \frac{8}{3} \cdot {x}^{3}\right)\right)\right) - \left({x}^{4} + \frac{1}{4} \cdot {x}^{8}\right)}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\sqrt[3]{{\left(x \cdot \varepsilon + x\right)}^{3}}}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 0.6224483931737788
1. Initial program 39.3
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.2
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied flip--1.2
  \[\leadsto \frac{\color{blue}{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}}}{2}\]
5. Using strategy rm
6. Applied flip--1.2
  \[\leadsto \frac{\frac{\color{blue}{\frac{\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) - \left({x}^{2} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2} \cdot {x}^{2}}}}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}}{2}\]
7. Applied simplify1.2
  \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}^{\left(3 + 1\right)} - {\left(x \cdot x\right)}^{\left(3 + 1\right)}}}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2} \cdot {x}^{2}}}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}}{2}\]
8. Taylor expanded around 0 1.2
  \[\leadsto \frac{\frac{\color{blue}{\left(4 + \left(\frac{1}{16} \cdot {x}^{12} + \left(\frac{1}{6} \cdot {x}^{11} + \frac{8}{3} \cdot {x}^{3}\right)\right)\right) - \left({x}^{4} + \frac{1}{4} \cdot {x}^{8}\right)}}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}}{2}\]
if 0.6224483931737788 < x
1. Initial program 0.7
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Using strategy rm
3. Applied add-cbrt-cube0.7
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\color{blue}{\sqrt[3]{\left(\left(\left(1 + \varepsilon\right) \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}}}{2}\]
4. Applied simplify0.7
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\sqrt[3]{\color{blue}{{\left(x \cdot \varepsilon + x\right)}^{3}}}}}{2}\]
Recombined 2 regimes into one program.

Error

Try it out

Derivation

`if x < 0.6224483931737788`

`if 0.6224483931737788 < x`

Runtime

In
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