Error

Derivation

1. Split input into 2 regimes

2. if x < 10.844456534739852

   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. Simplified1.3

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

   5. Applied log1p-expm1-u1.3

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

   7. Applied add-cube-cbrt1.3

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

   if 10.844456534739852 < x

   1. Initial program 0.3

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

   3. Applied add-cbrt-cube0.3

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

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 10.844456534739852:\\ \;\;\;\;\frac{\log_* (1 + \sqrt[3]{(e^{(\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_* - x \cdot x} - 1)^*} \cdot \left(\sqrt[3]{(e^{(\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_* - x \cdot x} - 1)^*} \cdot \sqrt[3]{(e^{(\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_* - x \cdot x} - 1)^*}\right))}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}\right) \cdot \left(\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}\right) \cdot \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}\right)\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018214 +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))