\(\frac{\left(e^{-(\varepsilon * x + x)_*} + e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right) + \left(\frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{e^{(\varepsilon * x + x)_*}}\right)}{2}\)
- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
12.6
- Using strategy
rm 12.6
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
12.6
- Applied un-div-inv to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
12.6
- Using strategy
rm 12.6
- Applied expm1-log1p-u to get
\[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{(e^{\log_* (1 + \left(\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right))} - 1)^*}}{2}\]
12.7
- Applied simplify to get
\[\frac{(e^{\color{red}{\log_* (1 + \left(\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right))}} - 1)^*}{2} \leadsto \frac{(e^{\color{blue}{\log_* (1 + \left(\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}\right))}} - 1)^*}{2}\]
12.7
- Applied taylor to get
\[\frac{(e^{\log_* (1 + \left(\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}\right))} - 1)^*}{2} \leadsto \frac{(e^{\log_* (1 + \left(\left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + \left(\frac{1}{e^{(\varepsilon * x + x)_*}} + \frac{1}{\varepsilon \cdot e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) - \frac{1}{\varepsilon \cdot e^{(\varepsilon * x + x)_*}}\right))} - 1)^*}{2}\]
9.5
- Taylor expanded around 0 to get
\[\frac{(e^{\color{red}{\log_* (1 + \left(\left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + \left(\frac{1}{e^{(\varepsilon * x + x)_*}} + \frac{1}{\varepsilon \cdot e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) - \frac{1}{\varepsilon \cdot e^{(\varepsilon * x + x)_*}}\right))}} - 1)^*}{2} \leadsto \frac{(e^{\color{blue}{\log_* (1 + \left(\left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + \left(\frac{1}{e^{(\varepsilon * x + x)_*}} + \frac{1}{\varepsilon \cdot e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) - \frac{1}{\varepsilon \cdot e^{(\varepsilon * x + x)_*}}\right))}} - 1)^*}{2}\]
9.5
- Applied simplify to get
\[\frac{(e^{\log_* (1 + \left(\left(\frac{1}{e^{x \cdot \left(1 - \varepsilon\right)}} + \left(\frac{1}{e^{(\varepsilon * x + x)_*}} + \frac{1}{\varepsilon \cdot e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) - \frac{1}{\varepsilon \cdot e^{(\varepsilon * x + x)_*}}\right))} - 1)^*}{2} \leadsto \frac{\left(e^{-(\varepsilon * x + x)_*} + e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right) + \left(\frac{\frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon}}{e^{(\varepsilon * x + x)_*}}\right)}{2}\]
1.8
- Applied final simplification
