Average Error: 29.4 → 0.9
Time: 30.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 54.31805746173307:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)\right) \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]
\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 54.31805746173307:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)\right) \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r988826 = 1.0;
        double r988827 = eps;
        double r988828 = r988826 / r988827;
        double r988829 = r988826 + r988828;
        double r988830 = r988826 - r988827;
        double r988831 = x;
        double r988832 = r988830 * r988831;
        double r988833 = -r988832;
        double r988834 = exp(r988833);
        double r988835 = r988829 * r988834;
        double r988836 = r988828 - r988826;
        double r988837 = r988826 + r988827;
        double r988838 = r988837 * r988831;
        double r988839 = -r988838;
        double r988840 = exp(r988839);
        double r988841 = r988836 * r988840;
        double r988842 = r988835 - r988841;
        double r988843 = 2.0;
        double r988844 = r988842 / r988843;
        return r988844;
}
↓
double f(double x, double eps) {
        double r988845 = x;
        double r988846 = 54.31805746173307;
        bool r988847 = r988845 <= r988846;
        double r988848 = 0.6666666666666666;
        double r988849 = r988845 * r988845;
        double r988850 = r988849 * r988845;
        double r988851 = 2.0;
        double r988852 = r988851 - r988849;
        double r988853 = fma(r988848, r988850, r988852);
        double r988854 = r988853 / r988851;
        double r988855 = -r988845;
        double r988856 = 1.0;
        double r988857 = eps;
        double r988858 = r988856 - r988857;
        double r988859 = r988855 * r988858;
        double r988860 = exp(r988859);
        double r988861 = r988856 / r988857;
        double r988862 = r988861 + r988856;
        double r988863 = r988860 * r988862;
        double r988864 = r988861 - r988856;
        double r988865 = r988857 + r988856;
        double r988866 = r988855 * r988865;
        double r988867 = exp(r988866);
        double r988868 = r988864 * r988867;
        double r988869 = r988863 - r988868;
        double r988870 = cbrt(r988869);
        double r988871 = log1p(r988870);
        double r988872 = expm1(r988871);
        double r988873 = r988870 * r988870;
        double r988874 = r988872 * r988873;
        double r988875 = r988874 / r988851;
        double r988876 = r988847 ? r988854 : r988875;
        return r988876;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 54.31805746173307

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}}{2}\]

    if 54.31805746173307 < 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

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

    5. Applied expm1-log1p-u0.3

      \[\leadsto \frac{\left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 54.31805746173307:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)\right) \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(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))