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Derivation

1. Split input into 2 regimes

2. if x < 181.43767545048757

   1. Initial program 39.4

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

   2. Taylor expanded around 0 1.3

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

   4. Applied unpow31.3

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

   5. Applied associate-*r*1.3

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

   if 181.43767545048757 < x

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube0.2

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 181.43767545048757:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3}\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018214 
(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))