Average Error: 29.3 → 1.1
Time: 35.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.9639489834403893731007428868906572461128:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, \left(x \cdot x\right) \cdot x, 2 - x \cdot \left(1 \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{-\log \left(e^{x \cdot \left(\varepsilon + 1\right)}\right)}, \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \frac{e^{-x \cdot \left(\varepsilon + 1\right)}}{\varepsilon}\right) \cdot 1\right)}{2}\\ \end{array}\]
\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 0.9639489834403893731007428868906572461128:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, \left(x \cdot x\right) \cdot x, 2 - x \cdot \left(1 \cdot x\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2317837 = 1.0;
        double r2317838 = eps;
        double r2317839 = r2317837 / r2317838;
        double r2317840 = r2317837 + r2317839;
        double r2317841 = r2317837 - r2317838;
        double r2317842 = x;
        double r2317843 = r2317841 * r2317842;
        double r2317844 = -r2317843;
        double r2317845 = exp(r2317844);
        double r2317846 = r2317840 * r2317845;
        double r2317847 = r2317839 - r2317837;
        double r2317848 = r2317837 + r2317838;
        double r2317849 = r2317848 * r2317842;
        double r2317850 = -r2317849;
        double r2317851 = exp(r2317850);
        double r2317852 = r2317847 * r2317851;
        double r2317853 = r2317846 - r2317852;
        double r2317854 = 2.0;
        double r2317855 = r2317853 / r2317854;
        return r2317855;
}
↓
double f(double x, double eps) {
        double r2317856 = x;
        double r2317857 = 0.9639489834403894;
        bool r2317858 = r2317856 <= r2317857;
        double r2317859 = 0.6666666666666667;
        double r2317860 = r2317856 * r2317856;
        double r2317861 = r2317860 * r2317856;
        double r2317862 = 2.0;
        double r2317863 = 1.0;
        double r2317864 = r2317863 * r2317856;
        double r2317865 = r2317856 * r2317864;
        double r2317866 = r2317862 - r2317865;
        double r2317867 = fma(r2317859, r2317861, r2317866);
        double r2317868 = r2317867 / r2317862;
        double r2317869 = eps;
        double r2317870 = r2317869 + r2317863;
        double r2317871 = r2317856 * r2317870;
        double r2317872 = exp(r2317871);
        double r2317873 = log(r2317872);
        double r2317874 = -r2317873;
        double r2317875 = exp(r2317874);
        double r2317876 = r2317869 - r2317863;
        double r2317877 = r2317856 * r2317876;
        double r2317878 = exp(r2317877);
        double r2317879 = r2317878 / r2317869;
        double r2317880 = r2317879 + r2317878;
        double r2317881 = -r2317871;
        double r2317882 = exp(r2317881);
        double r2317883 = r2317882 / r2317869;
        double r2317884 = r2317880 - r2317883;
        double r2317885 = r2317884 * r2317863;
        double r2317886 = fma(r2317863, r2317875, r2317885);
        double r2317887 = r2317886 / r2317862;
        double r2317888 = r2317858 ? r2317868 : r2317887;
        return r2317888;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 0.9639489834403894

    1. Initial program 38.8

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

    2. 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

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, x \cdot \left(x \cdot x\right), 2 - x \cdot \left(x \cdot 1\right)\right)}}{2}\]

    if 0.9639489834403894 < x

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    5. Applied add-log-exp0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.9639489834403893731007428868906572461128:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, \left(x \cdot x\right) \cdot x, 2 - x \cdot \left(1 \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{-\log \left(e^{x \cdot \left(\varepsilon + 1\right)}\right)}, \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \frac{e^{-x \cdot \left(\varepsilon + 1\right)}}{\varepsilon}\right) \cdot 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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))