if x < 540.2976f0

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   22.0
2. Using strategy rm
   22.0
3. Applied neg-sub0 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{red}{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.0
4. Applied exp-diff to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{0 - \left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   22.0
5. Applied flip-- to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.2
6. Applied frac-times to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   22.6
7. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.6
8. Applied un-div-inv to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.6
9. Applied frac-sub to get
   \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
   22.6
10. Applied taylor to get
    \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
    0.1
11. 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
12. 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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13. Applied final simplification
14. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}}\]
    0.1

if 540.2976f0 < x

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   0.1
2. Using strategy rm
   0.1
3. Applied neg-mul-1 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{red}{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
   0.1
4. Applied exp-prod to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
   0.1
5. Using strategy rm
   0.1
6. Applied add-sqr-sqrt to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^2}}{2}\]
   0.1
7. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - {\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^2}{2} \leadsto \frac{\color{blue}{{\left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^2} - {\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^2}{2}\]
   0.1
8. Applied difference-of-squares to get
   \[\frac{\color{red}{{\left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}\right)}^2 - {\left(\sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^2}}{2} \leadsto \frac{\color{blue}{\left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} + \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right) \cdot \left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}}{2}\]
   0.1
9. Applied simplify to get
   \[\frac{\color{red}{\left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} + \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)} \cdot \left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}{2} \leadsto \frac{\color{blue}{\left(\sqrt{{\left(e^{-1}\right)}^{\left((\varepsilon * x + x)_*\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} + \sqrt{\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}\right)} \cdot \left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}{2}\]
   0.1
10. Applied simplify to get
    \[\frac{\left(\sqrt{{\left(e^{-1}\right)}^{\left((\varepsilon * x + x)_*\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} + \sqrt{\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}\right) \cdot \color{red}{\left(\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \sqrt{\left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}}{2} \leadsto \frac{\left(\sqrt{{\left(e^{-1}\right)}^{\left((\varepsilon * x + x)_*\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} + \sqrt{\frac{\frac{1}{\varepsilon} + 1}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}\right) \cdot \color{blue}{\left(\sqrt{\frac{1 + \frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}} - \sqrt{{\left(e^{-1}\right)}^{\left((\varepsilon * x + x)_*\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right)}}{2}\]
    0.1