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Derivation

1. Split input into 2 regimes

2. if x < 70.21344082412496

   1. Initial program 39.7

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

   2. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]

   if 70.21344082412496 < x

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   4. Taylor expanded around inf 0.2

      \[\leadsto \frac{\color{blue}{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x \cdot \varepsilon + x\right)}\right)\right) - \frac{e^{-\left(x \cdot \varepsilon + x\right)}}{\varepsilon}}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 70.21344082412496:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{-\left(\varepsilon \cdot x + x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{-\left(\varepsilon \cdot x + x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))

Details

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