Error

Derivation

1. Split input into 2 regimes

2. if x < 415.1344189717718

   1. Initial program 38.7

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

   2. Taylor expanded around 0 1.4

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

   4. Applied add-sqr-sqrt1.4

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

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

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

   6. Applied prod-diff1.4

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

   7. Simplified1.4

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

   8. Simplified1.4

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

   if 415.1344189717718 < 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-cube-cbrt0.1

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

4. Final simplification1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 415.1344189717718:\\ \;\;\;\;\frac{(\left(x \cdot x\right) \cdot \left((\frac{2}{3} \cdot x + -1)_*\right) + 2)_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}\right) \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018248 +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))