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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 10.836919516792397:\\ \;\;\;\;\frac{\frac{{\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}}{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\left((x \cdot \frac{2}{3} + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_*\right) + \left((e^{\log_* (1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right))} - 1)^*\right))_*}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 10.836919516792397
1. Initial program 39.4
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
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 log1p-expm1-u1.3
  \[\leadsto \frac{\left(\color{blue}{\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*)} + 2\right) - {x}^{2}}{2}\]
5. Using strategy rm
6. Applied flip3--1.3
  \[\leadsto \frac{\color{blue}{\frac{{\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) \cdot \left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) + \left({x}^{2} \cdot {x}^{2} + \left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) \cdot {x}^{2}\right)}}}{2}\]
7. Applied simplify1.3
  \[\leadsto \frac{\frac{{\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\color{blue}{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\left((x \cdot \frac{2}{3} + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right))_*}}}{2}\]
8. Using strategy rm
9. Applied expm1-log1p-u1.3
  \[\leadsto \frac{\frac{{\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}}{(\left((\frac{2}{3} \cdot \left({x}^{3}\right) + 2)_*\right) \cdot \left((\left((x \cdot \frac{2}{3} + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_*\right) + \color{blue}{\left((e^{\log_* (1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right))} - 1)^*\right)})_*}}{2}\]
if 10.836919516792397 < 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. Using strategy rm
3. Applied *-un-lft-identity0.3
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{1 \cdot \left(-\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
4. Applied exp-prod0.3
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
5. Applied simplify0.3
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\color{blue}{e}}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\]
Recombined 2 regimes into one program.

Error

Derivation

`if x < 10.836919516792397`

`if 10.836919516792397 < x`

Runtime