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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

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

   4. Applied cube-mult1.3

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

   5. Applied associate-*r*1.3

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

   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-sqr-sqrt0.2

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 70.21344082412496:\\ \;\;\;\;\frac{\left(2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(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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