Average Error: 29.8 → 1.0
Time: 7.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 335.02122944782855:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]
\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 335.02122944782855:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r52834 = 1.0;
        double r52835 = eps;
        double r52836 = r52834 / r52835;
        double r52837 = r52834 + r52836;
        double r52838 = r52834 - r52835;
        double r52839 = x;
        double r52840 = r52838 * r52839;
        double r52841 = -r52840;
        double r52842 = exp(r52841);
        double r52843 = r52837 * r52842;
        double r52844 = r52836 - r52834;
        double r52845 = r52834 + r52835;
        double r52846 = r52845 * r52839;
        double r52847 = -r52846;
        double r52848 = exp(r52847);
        double r52849 = r52844 * r52848;
        double r52850 = r52843 - r52849;
        double r52851 = 2.0;
        double r52852 = r52850 / r52851;
        return r52852;
}
↓
double f(double x, double eps) {
        double r52853 = x;
        double r52854 = 335.02122944782855;
        bool r52855 = r52853 <= r52854;
        double r52856 = 3.0;
        double r52857 = pow(r52853, r52856);
        double r52858 = 0.6666666666666667;
        double r52859 = 2.0;
        double r52860 = 1.0;
        double r52861 = 2.0;
        double r52862 = pow(r52853, r52861);
        double r52863 = r52860 * r52862;
        double r52864 = r52859 - r52863;
        double r52865 = fma(r52857, r52858, r52864);
        double r52866 = r52865 / r52859;
        double r52867 = eps;
        double r52868 = r52860 / r52867;
        double r52869 = r52860 + r52868;
        double r52870 = r52860 - r52867;
        double r52871 = r52870 * r52853;
        double r52872 = -r52871;
        double r52873 = exp(r52872);
        double r52874 = r52869 * r52873;
        double r52875 = r52853 * r52867;
        double r52876 = r52860 * r52853;
        double r52877 = r52875 + r52876;
        double r52878 = -r52877;
        double r52879 = exp(r52878);
        double r52880 = r52879 / r52867;
        double r52881 = 1.0;
        double r52882 = fma(r52853, r52867, r52876);
        double r52883 = exp(r52882);
        double r52884 = r52881 / r52883;
        double r52885 = r52880 - r52884;
        double r52886 = r52860 * r52885;
        double r52887 = r52874 - r52886;
        double r52888 = r52887 / r52859;
        double r52889 = r52855 ? r52866 : r52888;
        return r52889;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 335.02122944782855

    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 1.4

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]

    if 335.02122944782855 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 335.02122944782855:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - \frac{1}{e^{\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(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))