Average Error: 29.3 → 1.0
Time: 25.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(e^{-x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-\left(\varepsilon + 1\right)\right) \cdot x}\right)}}{2}\\ \end{array}\]
\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 183.1502607165994:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r619831 = 1.0;
        double r619832 = eps;
        double r619833 = r619831 / r619832;
        double r619834 = r619831 + r619833;
        double r619835 = r619831 - r619832;
        double r619836 = x;
        double r619837 = r619835 * r619836;
        double r619838 = -r619837;
        double r619839 = exp(r619838);
        double r619840 = r619834 * r619839;
        double r619841 = r619833 - r619831;
        double r619842 = r619831 + r619832;
        double r619843 = r619842 * r619836;
        double r619844 = -r619843;
        double r619845 = exp(r619844);
        double r619846 = r619841 * r619845;
        double r619847 = r619840 - r619846;
        double r619848 = 2.0;
        double r619849 = r619847 / r619848;
        return r619849;
}
↓
double f(double x, double eps) {
        double r619850 = x;
        double r619851 = 183.1502607165994;
        bool r619852 = r619850 <= r619851;
        double r619853 = 0.6666666666666666;
        double r619854 = r619850 * r619850;
        double r619855 = r619854 * r619850;
        double r619856 = 2.0;
        double r619857 = r619856 - r619854;
        double r619858 = fma(r619853, r619855, r619857);
        double r619859 = r619858 / r619856;
        double r619860 = 1.0;
        double r619861 = eps;
        double r619862 = r619860 - r619861;
        double r619863 = r619850 * r619862;
        double r619864 = -r619863;
        double r619865 = exp(r619864);
        double r619866 = r619860 / r619861;
        double r619867 = r619866 + r619860;
        double r619868 = r619865 * r619867;
        double r619869 = r619866 - r619860;
        double r619870 = r619861 + r619860;
        double r619871 = -r619870;
        double r619872 = r619871 * r619850;
        double r619873 = exp(r619872);
        double r619874 = r619869 * r619873;
        double r619875 = r619868 - r619874;
        double r619876 = log(r619875);
        double r619877 = exp(r619876);
        double r619878 = r619877 / r619856;
        double r619879 = r619852 ? r619859 : r619878;
        return r619879;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 183.1502607165994

    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. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    5. Applied *-un-lft-identity1.3

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot \color{blue}{\left(1 \cdot x\right)}, 2 - x \cdot x\right)}{2}\]

    6. Applied associate-*r*1.3

      \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{3}, \color{blue}{\left(\left(x \cdot x\right) \cdot 1\right) \cdot x}, 2 - x \cdot x\right)}{2}\]

    7. Simplified1.3

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

    if 183.1502607165994 < 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-exp-log0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(e^{-x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-\left(\varepsilon + 1\right)\right) \cdot x}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))