Average Error: 30.1 → 1.1
Time: 30.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 381.9625649006933940654562320560216903687:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} + \frac{1 - \frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}}}{2}\\ \end{array}\]
\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 381.9625649006933940654562320560216903687:\\
\;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\

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

\end{array}
double f(double x, double eps) {
        double r2733809 = 1.0;
        double r2733810 = eps;
        double r2733811 = r2733809 / r2733810;
        double r2733812 = r2733809 + r2733811;
        double r2733813 = r2733809 - r2733810;
        double r2733814 = x;
        double r2733815 = r2733813 * r2733814;
        double r2733816 = -r2733815;
        double r2733817 = exp(r2733816);
        double r2733818 = r2733812 * r2733817;
        double r2733819 = r2733811 - r2733809;
        double r2733820 = r2733809 + r2733810;
        double r2733821 = r2733820 * r2733814;
        double r2733822 = -r2733821;
        double r2733823 = exp(r2733822);
        double r2733824 = r2733819 * r2733823;
        double r2733825 = r2733818 - r2733824;
        double r2733826 = 2.0;
        double r2733827 = r2733825 / r2733826;
        return r2733827;
}
↓
double f(double x, double eps) {
        double r2733828 = x;
        double r2733829 = 381.9625649006934;
        bool r2733830 = r2733828 <= r2733829;
        double r2733831 = 2.0;
        double r2733832 = r2733828 * r2733828;
        double r2733833 = 1.0;
        double r2733834 = r2733832 * r2733833;
        double r2733835 = r2733831 - r2733834;
        double r2733836 = r2733832 * r2733828;
        double r2733837 = 0.6666666666666667;
        double r2733838 = r2733836 * r2733837;
        double r2733839 = r2733835 + r2733838;
        double r2733840 = r2733839 / r2733831;
        double r2733841 = cbrt(r2733840);
        double r2733842 = r2733841 * r2733841;
        double r2733843 = cbrt(r2733842);
        double r2733844 = cbrt(r2733841);
        double r2733845 = r2733843 * r2733844;
        double r2733846 = r2733845 * r2733841;
        double r2733847 = r2733846 * r2733841;
        double r2733848 = eps;
        double r2733849 = r2733833 / r2733848;
        double r2733850 = r2733833 + r2733849;
        double r2733851 = r2733848 - r2733833;
        double r2733852 = r2733851 * r2733828;
        double r2733853 = exp(r2733852);
        double r2733854 = r2733850 * r2733853;
        double r2733855 = r2733833 - r2733849;
        double r2733856 = r2733848 + r2733833;
        double r2733857 = r2733856 * r2733828;
        double r2733858 = exp(r2733857);
        double r2733859 = r2733855 / r2733858;
        double r2733860 = r2733854 + r2733859;
        double r2733861 = r2733860 / r2733831;
        double r2733862 = r2733830 ? r2733847 : r2733861;
        return r2733862;
}

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 < 381.9625649006934

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    5. Applied add-cube-cbrt1.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + 0.6666666666666667406815349750104360282421 \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt1.4

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

    8. Applied cbrt-prod1.4

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

    if 381.9625649006934 < 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. Simplified0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 381.9625649006933940654562320560216903687:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}} \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 - \left(x \cdot x\right) \cdot 1\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666667406815349750104360282421}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} + \frac{1 - \frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))