Average Error: 29.6 → 1.3
Time: 27.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9.944691288244345289145222424598280497321 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot \left({\left(\sqrt[3]{x}\right)}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \end{array}\]
\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 9.944691288244345289145222424598280497321 \cdot 10^{-7}:\\
\;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot \left({\left(\sqrt[3]{x}\right)}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right) + 2}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r37786 = 1.0;
        double r37787 = eps;
        double r37788 = r37786 / r37787;
        double r37789 = r37786 + r37788;
        double r37790 = r37786 - r37787;
        double r37791 = x;
        double r37792 = r37790 * r37791;
        double r37793 = -r37792;
        double r37794 = exp(r37793);
        double r37795 = r37789 * r37794;
        double r37796 = r37788 - r37786;
        double r37797 = r37786 + r37787;
        double r37798 = r37797 * r37791;
        double r37799 = -r37798;
        double r37800 = exp(r37799);
        double r37801 = r37796 * r37800;
        double r37802 = r37795 - r37801;
        double r37803 = 2.0;
        double r37804 = r37802 / r37803;
        return r37804;
}
↓
double f(double x, double eps) {
        double r37805 = x;
        double r37806 = 9.944691288244345e-07;
        bool r37807 = r37805 <= r37806;
        double r37808 = cbrt(r37805);
        double r37809 = r37808 * r37808;
        double r37810 = 2.0;
        double r37811 = pow(r37809, r37810);
        double r37812 = pow(r37808, r37810);
        double r37813 = 0.6666666666666667;
        double r37814 = r37813 * r37805;
        double r37815 = 1.0;
        double r37816 = r37814 - r37815;
        double r37817 = r37812 * r37816;
        double r37818 = r37811 * r37817;
        double r37819 = 2.0;
        double r37820 = r37818 + r37819;
        double r37821 = r37820 / r37819;
        double r37822 = eps;
        double r37823 = r37815 / r37822;
        double r37824 = r37823 + r37815;
        double r37825 = r37815 - r37822;
        double r37826 = r37825 * r37805;
        double r37827 = exp(r37826);
        double r37828 = r37824 / r37827;
        double r37829 = r37823 - r37815;
        double r37830 = cbrt(r37829);
        double r37831 = r37830 * r37830;
        double r37832 = r37815 + r37822;
        double r37833 = r37832 * r37805;
        double r37834 = exp(r37833);
        double r37835 = cbrt(r37834);
        double r37836 = r37835 * r37835;
        double r37837 = r37831 / r37836;
        double r37838 = r37830 / r37835;
        double r37839 = r37837 * r37838;
        double r37840 = r37828 - r37839;
        double r37841 = r37840 / r37819;
        double r37842 = r37807 ? r37821 : r37841;
        return r37842;
}

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 < 9.944691288244345e-07

    1. Initial program 38.9

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

    2. Simplified38.9

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right) + 2}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt1.1

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

    7. Applied unpow-prod-down1.1

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

    8. Applied associate-*l*1.1

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

    if 9.944691288244345e-07 < x

    1. Initial program 1.8

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

    2. Simplified1.8

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

    4. Applied add-cube-cbrt1.8

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

    5. Applied add-cube-cbrt1.8

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

    6. Applied times-frac1.8

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 9.944691288244345289145222424598280497321 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot \left({\left(\sqrt[3]{x}\right)}^{2} \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \frac{\sqrt[3]{\frac{1}{\varepsilon} - 1}}{\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))