Average Error: 29.4 → 1.0
Time: 3.0m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.301278737300282095645798108307644724846:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667406815349750104360282421 + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}}\right)}{2}\\ \end{array}\]
\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 5.301278737300282095645798108307644724846:\\
\;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667406815349750104360282421 + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r441817 = 1.0;
        double r441818 = eps;
        double r441819 = r441817 / r441818;
        double r441820 = r441817 + r441819;
        double r441821 = r441817 - r441818;
        double r441822 = x;
        double r441823 = r441821 * r441822;
        double r441824 = -r441823;
        double r441825 = exp(r441824);
        double r441826 = r441820 * r441825;
        double r441827 = r441819 - r441817;
        double r441828 = r441817 + r441818;
        double r441829 = r441828 * r441822;
        double r441830 = -r441829;
        double r441831 = exp(r441830);
        double r441832 = r441827 * r441831;
        double r441833 = r441826 - r441832;
        double r441834 = 2.0;
        double r441835 = r441833 / r441834;
        return r441835;
}
↓
double f(double x, double eps) {
        double r441836 = x;
        double r441837 = 5.301278737300282;
        bool r441838 = r441836 <= r441837;
        double r441839 = 3.0;
        double r441840 = pow(r441836, r441839);
        double r441841 = 0.6666666666666667;
        double r441842 = r441840 * r441841;
        double r441843 = 2.0;
        double r441844 = r441836 * r441836;
        double r441845 = 1.0;
        double r441846 = r441844 * r441845;
        double r441847 = r441843 - r441846;
        double r441848 = r441842 + r441847;
        double r441849 = r441848 / r441843;
        double r441850 = eps;
        double r441851 = r441845 / r441850;
        double r441852 = r441851 + r441845;
        double r441853 = -r441836;
        double r441854 = r441845 - r441850;
        double r441855 = r441853 * r441854;
        double r441856 = exp(r441855);
        double r441857 = r441852 * r441856;
        double r441858 = r441845 + r441850;
        double r441859 = r441858 * r441853;
        double r441860 = exp(r441859);
        double r441861 = r441851 - r441845;
        double r441862 = r441860 * r441861;
        double r441863 = cbrt(r441862);
        double r441864 = r441863 * r441863;
        double r441865 = r441864 * r441863;
        double r441866 = r441857 - r441865;
        double r441867 = cbrt(r441866);
        double r441868 = r441867 * r441867;
        double r441869 = r441867 * r441868;
        double r441870 = r441869 / r441843;
        double r441871 = r441838 ? r441849 : r441870;
        return r441871;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 5.301278737300282

    1. Initial program 39.2

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{{x}^{3} \cdot 0.6666666666666667406815349750104360282421 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}}{2}\]

    if 5.301278737300282 < x

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt0.5

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

    4. Simplified0.5

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

    5. Simplified0.5

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

    7. Applied add-cube-cbrt0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.301278737300282095645798108307644724846:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.6666666666666667406815349750104360282421 + \left(2 - \left(x \cdot x\right) \cdot 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))