if x < 112873629.87382902

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   48.5
2. Using strategy rm
   48.5
3. Applied add-cube-cbrt to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{{\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   48.6
4. Applied add-cube-cbrt to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot {\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}}\right)}^3} \cdot {\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   48.9
5. Applied cube-unprod to get
   \[\frac{\color{red}{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}}\right)}^3 \cdot {\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   48.9
6. Using strategy rm
   48.9
7. Applied flip-- to get
   \[\frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   49.0
8. Applied associate-*l/ to get
   \[\frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^3 - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2}\]
   49.0
9. Applied neg-sub0 to get
   \[\frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\color{red}{-\left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\color{blue}{0 - \left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
   49.0
10. Applied exp-diff to get
    \[\frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{\color{red}{e^{0 - \left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{\color{blue}{\frac{e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    49.0
11. Applied cbrt-div to get
    \[\frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \color{red}{\sqrt[3]{\frac{e^{0}}{e^{\left(1 - \varepsilon\right) \cdot x}}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \color{blue}{\frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    49.0
12. Applied flip-+ to get
    \[\frac{{\left(\sqrt[3]{\color{red}{1 + \frac{1}{\varepsilon}}} \cdot \frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\left(\sqrt[3]{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}}} \cdot \frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    48.8
13. Applied cbrt-div to get
    \[\frac{{\left(\color{red}{\sqrt[3]{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}}} \cdot \frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\left(\color{blue}{\frac{\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}}{\sqrt[3]{1 - \frac{1}{\varepsilon}}}} \cdot \frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    48.8
14. Applied frac-times to get
    \[\frac{{\color{red}{\left(\frac{\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}}{\sqrt[3]{1 - \frac{1}{\varepsilon}}} \cdot \frac{\sqrt[3]{e^{0}}}{\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}}{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}}^3 - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    48.8
15. Applied cube-div to get
    \[\frac{\color{red}{{\left(\frac{\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}}{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}}\right)}^3} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}\right)}^3}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
    48.9
16. Applied frac-sub to get
    \[\frac{\color{red}{\frac{{\left(\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}\right)}^3}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2} \leadsto \frac{\color{blue}{\frac{{\left(\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right) - {\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right)}}}{2}\]
    48.9
17. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left(\sqrt[3]{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2} \cdot \sqrt[3]{e^{0}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right) - {\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon \cdot \varepsilon}\right) - \frac{1 - \frac{1}{\varepsilon}}{\frac{e^{x}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}} \cdot \frac{\frac{1}{\varepsilon \cdot \varepsilon} - 1}{e^{\varepsilon \cdot x}}}}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
    49.2
18. Applied simplify to get
    \[\frac{\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon \cdot \varepsilon}\right) - \frac{1 - \frac{1}{\varepsilon}}{\frac{e^{x}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}} \cdot \frac{\frac{1}{\varepsilon \cdot \varepsilon} - 1}{e^{\varepsilon \cdot x}}}{\color{red}{{\left(\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}^3 \cdot \left(\frac{1}{\varepsilon} + 1\right)}}}{2} \leadsto \frac{\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon \cdot \varepsilon}\right) - \frac{1 - \frac{1}{\varepsilon}}{\frac{e^{x}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}} \cdot \frac{\frac{1}{\varepsilon \cdot \varepsilon} - 1}{e^{\varepsilon \cdot x}}}{\color{blue}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}}}{2}\]
    48.3
19. Applied taylor to get
    \[\frac{\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon \cdot \varepsilon}\right) - \frac{1 - \frac{1}{\varepsilon}}{\frac{e^{x}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}} \cdot \frac{\frac{1}{\varepsilon \cdot \varepsilon} - 1}{e^{\varepsilon \cdot x}}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
    0.1
20. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
    0.1
21. Applied simplify to get
    \[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
    0.1

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22. Applied final simplification
23. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
    0.1

if 112873629.87382902 < x

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   0