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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 27.535461960897777:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}\right) \cdot \left(\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right)}}{\sqrt[3]{2} + \left(\sqrt[3]{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{6} + x \cdot \frac{1}{9}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 27.535461960897777
1. Initial program 39.1
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.2
  \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-cbrt-cube1.2
  \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)}}}{2}\]
5. Using strategy rm
6. Applied flip--1.2
  \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \color{blue}{\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}}}{2}\]
7. Applied associate-*r/1.2
  \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}}}{2}\]
8. Applied cbrt-div1.2
  \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}}{\sqrt[3]{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}}}{2}\]
9. Taylor expanded around 0 2.2
  \[\leadsto \frac{\frac{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}}{\color{blue}{\frac{1}{6} \cdot \left({x}^{2} \cdot {2}^{\frac{1}{3}}\right) + \left(\frac{1}{9} \cdot \left({x}^{3} \cdot {2}^{\frac{1}{3}}\right) + {2}^{\frac{1}{3}}\right)}}}{2}\]
10. Simplified1.2
  \[\leadsto \frac{\frac{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}}{\color{blue}{\sqrt[3]{2} + \left(\frac{1}{9} \cdot x + \frac{1}{6}\right) \cdot \left(\left(x \cdot x\right) \cdot \sqrt[3]{2}\right)}}}{2}\]
if 27.535461960897777 < 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. Using strategy rm
3. Applied add-cube-cbrt0.2
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Recombined 2 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 27.535461960897777:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}\right) \cdot \left(\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right)}}{\sqrt[3]{2} + \left(\sqrt[3]{2} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{6} + x \cdot \frac{1}{9}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}\]

Error

Try it out

Derivation

`if x < 27.535461960897777`

`if 27.535461960897777 < x`

Runtime

In
Out