Average Error: 29.1 → 1.0
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 335.9456460966649:\\ \;\;\;\;\frac{(\frac{2}{3} \cdot \left({x}^{3}\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 335.9456460966649

    1. Initial program 38.3

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    5. Applied *-un-lft-identity1.3

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

    6. Applied prod-diff1.3

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

    7. Simplified1.3

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

    8. Simplified1.3

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

    if 335.9456460966649 < x

    1. Initial program 0.2

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

    3. Applied add-exp-log0.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 335.9456460966649:\\ \;\;\;\;\frac{(\frac{2}{3} \cdot \left({x}^{3}\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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))