Average Error: 29.0 → 1.0
Time: 7.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 80.092290002178601:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 80.092290002178601:\\
\;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r42907 = 1.0;
        double r42908 = eps;
        double r42909 = r42907 / r42908;
        double r42910 = r42907 + r42909;
        double r42911 = r42907 - r42908;
        double r42912 = x;
        double r42913 = r42911 * r42912;
        double r42914 = -r42913;
        double r42915 = exp(r42914);
        double r42916 = r42910 * r42915;
        double r42917 = r42909 - r42907;
        double r42918 = r42907 + r42908;
        double r42919 = r42918 * r42912;
        double r42920 = -r42919;
        double r42921 = exp(r42920);
        double r42922 = r42917 * r42921;
        double r42923 = r42916 - r42922;
        double r42924 = 2.0;
        double r42925 = r42923 / r42924;
        return r42925;
}
↓
double f(double x, double eps) {
        double r42926 = x;
        double r42927 = 80.0922900021786;
        bool r42928 = r42926 <= r42927;
        double r42929 = 0.6666666666666667;
        double r42930 = 3.0;
        double r42931 = pow(r42926, r42930);
        double r42932 = r42929 * r42931;
        double r42933 = 2.0;
        double r42934 = r42932 + r42933;
        double r42935 = 1.0;
        double r42936 = 2.0;
        double r42937 = pow(r42926, r42936);
        double r42938 = r42935 * r42937;
        double r42939 = r42934 - r42938;
        double r42940 = r42939 / r42933;
        double r42941 = eps;
        double r42942 = r42926 * r42941;
        double r42943 = r42935 * r42926;
        double r42944 = r42942 - r42943;
        double r42945 = exp(r42944);
        double r42946 = r42945 / r42941;
        double r42947 = r42946 + r42945;
        double r42948 = r42935 * r42947;
        double r42949 = r42935 / r42941;
        double r42950 = r42949 - r42935;
        double r42951 = r42935 + r42941;
        double r42952 = r42951 * r42926;
        double r42953 = -r42952;
        double r42954 = exp(r42953);
        double r42955 = r42950 * r42954;
        double r42956 = r42948 - r42955;
        double r42957 = r42956 / r42933;
        double r42958 = r42928 ? r42940 : r42957;
        return r42958;
}

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 < 80.0922900021786

    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

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

    if 80.0922900021786 < x

    1. Initial program 0.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 80.092290002178601:\\ \;\;\;\;\frac{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))