Average Error: 29.6 → 0.9
Time: 5.1s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 364.77253699098526:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 364.77253699098526

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.1

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    if 364.77253699098526 < 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-cbrt-cube0.1

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

    4. Applied add-cbrt-cube42.9

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

    5. Applied cbrt-unprod42.9

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

    6. Simplified0.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 364.77253699098526:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \sqrt[3]{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (neg (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (neg (* (+ 1.0 eps) x))))) 2.0))