Error

Derivation

1. Split input into 2 regimes

2. if x < 10.26670485440381

   1. Initial program 38.9

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

   2. Taylor expanded around 0 1.2

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

   4. Applied *-un-lft-identity1.2

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

   5. Applied *-un-lft-identity1.2

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

   6. Applied prod-diff1.2

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

   7. Simplified1.2

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

   8. Simplified1.2

      \[\leadsto \frac{(\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_* + \color{blue}{0}}{2}\]
   9. Using strategy rm

   10. Applied log1p-expm1-u1.2

       \[\leadsto \frac{\color{blue}{\log_* (1 + (e^{(\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_*} - 1)^*)} + 0}{2}\]

   if 10.26670485440381 < x

   1. Initial program 0.4

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

   3. Applied prod-diff0.4

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

   4. Simplified0.4

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

   5. Simplified0.4

      \[\leadsto \frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_* + \color{blue}{0}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 10.26670485440381:\\ \;\;\;\;\frac{\log_* (1 + (e^{(\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_*} - 1)^*)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{\left(-1 + \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) + \left(\frac{\frac{-1}{\varepsilon} + 1}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 +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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