Average Error: 29.3 → 1.0
Time: 3.4m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 259.3686770711884:\\ \;\;\;\;\frac{\left(2 + \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right)\right) - {x}^{2}}{2}\\ \mathbf{if}\;x \le +\infty:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 259.3686770711884

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    6. Applied add-cube-cbrt1.3

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

    8. Applied add-cube-cbrt1.3

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

    if 259.3686770711884 < 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

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

    4. Applied simplify0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))