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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 400.2976489691012:\\ \;\;\;\;\frac{\frac{8 + \left(8 \cdot {x}^{3} + \frac{5}{3} \cdot {x}^{6}\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x + 2\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 400.2976489691012
1. Initial program 39.6
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.1
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-log-exp1.2
  \[\leadsto \frac{\left(2 + \color{blue}{\log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)}\right) - {x}^{2}}{2}\]
5. Using strategy rm
6. Applied flip3--1.2
  \[\leadsto \frac{\color{blue}{\frac{{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) \cdot \left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) + \left({x}^{2} \cdot {x}^{2} + \left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) \cdot {x}^{2}\right)}}}{2}\]
7. Applied simplify1.2
  \[\leadsto \frac{\frac{{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x + 2\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}}{2}\]
8. Taylor expanded around 0 1.2
  \[\leadsto \frac{\frac{\color{blue}{8 + \left(8 \cdot {x}^{3} + \frac{5}{3} \cdot {x}^{6}\right)}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x + 2\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}{2}\]
if 400.2976489691012 < x
1. Initial program 0.0
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.0
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(\sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
4. Applied associate-*r*0.0
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
Recombined 2 regimes into one program.

Error

Derivation

`if x < 400.2976489691012`

`if 400.2976489691012 < x`

Runtime