Average Error: 29.4 → 1.1
Time: 5.6s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 127.863394544870303:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 127.863394544870303

    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 1.3

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

    4. Applied add-cbrt-cube1.3

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

    5. Simplified1.3

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)}} - 1 \cdot {x}^{2}}{2}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube1.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right) \cdot \left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right)\right) \cdot \left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right)}}}{2}\]

    9. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right)}^{3}}}}{2}\]

    if 127.863394544870303 < 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-cube-cbrt0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 127.863394544870303:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{2.666666666666667 \cdot {x}^{6} + \left(8 \cdot {x}^{3} + 8\right)} - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(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))