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Derivation

1. Split input into 2 regimes

2. if x < 743.0021431928681

   1. Initial program 39.1

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

   2. Taylor expanded around 0 1.5

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

   4. Applied add-exp-log1.5

      \[\leadsto \frac{\color{blue}{e^{\log \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)}}}{2}\]
   5. Using strategy rm

   6. Applied flip--1.5

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

   7. Applied log-div1.5

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

   8. Applied exp-diff1.5

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

   9. Taylor expanded around 0 1.5

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

   10. Simplified1.5

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

   if 743.0021431928681 < x

   1. Initial program 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. Taylor expanded around -inf 0

      \[\leadsto \frac{\color{blue}{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + e^{x \cdot \varepsilon - x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 743.0021431928681:\\ \;\;\;\;\frac{\frac{e^{\log \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}\right)}}{e^{\left(x \cdot x\right) \cdot \left(\frac{1}{2} + x \cdot \frac{1}{3}\right) + \log 2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \varepsilon - x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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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