Average Error: 29.4 → 5.9
Time: 25.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.434809865163426042522110932873457417314 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(2 - \frac{x \cdot x}{\frac{\varepsilon}{x}} \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}\right) - 1 \cdot \left(x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{\left(\varepsilon + 1\right) \cdot x}} - \left(\frac{\frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}} - \left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x}\right)}{2}\\ \end{array}\]
\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 3.434809865163426042522110932873457417314 \cdot 10^{-10}:\\
\;\;\;\;\frac{\left(2 - \frac{x \cdot x}{\frac{\varepsilon}{x}} \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}\right) - 1 \cdot \left(x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2445875 = 1.0;
        double r2445876 = eps;
        double r2445877 = r2445875 / r2445876;
        double r2445878 = r2445875 + r2445877;
        double r2445879 = r2445875 - r2445876;
        double r2445880 = x;
        double r2445881 = r2445879 * r2445880;
        double r2445882 = -r2445881;
        double r2445883 = exp(r2445882);
        double r2445884 = r2445878 * r2445883;
        double r2445885 = r2445877 - r2445875;
        double r2445886 = r2445875 + r2445876;
        double r2445887 = r2445886 * r2445880;
        double r2445888 = -r2445887;
        double r2445889 = exp(r2445888);
        double r2445890 = r2445885 * r2445889;
        double r2445891 = r2445884 - r2445890;
        double r2445892 = 2.0;
        double r2445893 = r2445891 / r2445892;
        return r2445893;
}
↓
double f(double x, double eps) {
        double r2445894 = x;
        double r2445895 = 3.434809865163426e-10;
        bool r2445896 = r2445894 <= r2445895;
        double r2445897 = 2.0;
        double r2445898 = r2445894 * r2445894;
        double r2445899 = eps;
        double r2445900 = r2445899 / r2445894;
        double r2445901 = r2445898 / r2445900;
        double r2445902 = 2.7755575615628914e-17;
        double r2445903 = r2445901 * r2445902;
        double r2445904 = r2445897 - r2445903;
        double r2445905 = 1.0;
        double r2445906 = r2445905 * r2445898;
        double r2445907 = r2445904 - r2445906;
        double r2445908 = r2445907 / r2445897;
        double r2445909 = r2445899 + r2445905;
        double r2445910 = r2445909 * r2445894;
        double r2445911 = exp(r2445910);
        double r2445912 = r2445905 / r2445911;
        double r2445913 = r2445905 / r2445899;
        double r2445914 = r2445913 / r2445911;
        double r2445915 = r2445913 + r2445905;
        double r2445916 = r2445899 - r2445905;
        double r2445917 = r2445916 * r2445894;
        double r2445918 = exp(r2445917);
        double r2445919 = r2445915 * r2445918;
        double r2445920 = r2445914 - r2445919;
        double r2445921 = r2445912 - r2445920;
        double r2445922 = r2445921 / r2445897;
        double r2445923 = r2445896 ? r2445908 : r2445922;
        return r2445923;
}

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 < 3.434809865163426e-10

    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. Simplified38.9

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

    3. Taylor expanded around 0 7.1

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77555756156289135105907917022705078125 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.1

      \[\leadsto \frac{\color{blue}{\left(2 - \frac{x \cdot x}{\frac{\varepsilon}{x}} \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}\right) - 1 \cdot \left(x \cdot x\right)}}{2}\]

    if 3.434809865163426e-10 < x

    1. Initial program 2.8

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

    2. Simplified2.8

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

    4. Applied div-sub2.8

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

    5. Applied associate-+l-2.6

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.434809865163426042522110932873457417314 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(2 - \frac{x \cdot x}{\frac{\varepsilon}{x}} \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}\right) - 1 \cdot \left(x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{\left(\varepsilon + 1\right) \cdot x}} - \left(\frac{\frac{1}{\varepsilon}}{e^{\left(\varepsilon + 1\right) \cdot x}} - \left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))