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Derivation

1. Split input into 2 regimes

2. if (/ (+ (/ 1 eps) (+ eps (+ 1 (/ x eps)))) 2) < -2.0831341995178896e+17 or 2.9853194332290654e+93 < (/ (+ (/ 1 eps) (+ eps (+ 1 (/ x eps)))) 2)

   1. Initial program 30.7

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

   2. Taylor expanded around -inf 30.6

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

   4. Applied add-exp-log30.3

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

   6. Applied pow130.3

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

   7. Applied log-pow30.3

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

   8. Applied exp-prod30.5

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

   9. Applied simplify30.5

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

   if -2.0831341995178896e+17 < (/ (+ (/ 1 eps) (+ eps (+ 1 (/ x eps)))) 2) < 2.9853194332290654e+93

   1. Initial program 24.7

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

   3. Applied exp-neg24.7

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

   4. Applied flip3--26.0

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

   5. Applied frac-times26.1

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

   6. Applied simplify26.8

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

   7. Applied simplify26.7

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

4. Applied simplify29.8

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{1}{\varepsilon} + \left(\varepsilon + \left(\frac{x}{\varepsilon} + 1\right)\right)}{2} \le -2.0831341995178896 \cdot 10^{+17} \lor \neg \left(\frac{\frac{1}{\varepsilon} + \left(\varepsilon + \left(\frac{x}{\varepsilon} + 1\right)\right)}{2} \le 2.9853194332290654 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{{e}^{\left(\log \left(\left(e^{x \cdot \varepsilon - x} + e^{-\left(x + x \cdot \varepsilon\right)}\right) - \left(\frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon \cdot \varepsilon}\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 6.1m)Debug logProfile

herbie shell --seed 2018195 
(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))