Average Error: 29.6 → 0.8
Time: 47.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 357.09970468422745:\\ \;\;\;\;\frac{2 - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-2}{3} + x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2}\\ \end{array}\]
\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 357.09970468422745:\\
\;\;\;\;\frac{2 - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-2}{3} + x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2}\\

\end{array}
double f(double x, double eps) {
        double r3359894 = 1.0;
        double r3359895 = eps;
        double r3359896 = r3359894 / r3359895;
        double r3359897 = r3359894 + r3359896;
        double r3359898 = r3359894 - r3359895;
        double r3359899 = x;
        double r3359900 = r3359898 * r3359899;
        double r3359901 = -r3359900;
        double r3359902 = exp(r3359901);
        double r3359903 = r3359897 * r3359902;
        double r3359904 = r3359896 - r3359894;
        double r3359905 = r3359894 + r3359895;
        double r3359906 = r3359905 * r3359899;
        double r3359907 = -r3359906;
        double r3359908 = exp(r3359907);
        double r3359909 = r3359904 * r3359908;
        double r3359910 = r3359903 - r3359909;
        double r3359911 = 2.0;
        double r3359912 = r3359910 / r3359911;
        return r3359912;
}
↓
double f(double x, double eps) {
        double r3359913 = x;
        double r3359914 = 357.09970468422745;
        bool r3359915 = r3359913 <= r3359914;
        double r3359916 = 2.0;
        double r3359917 = r3359913 * r3359913;
        double r3359918 = r3359917 * r3359913;
        double r3359919 = -0.6666666666666666;
        double r3359920 = r3359918 * r3359919;
        double r3359921 = r3359920 + r3359917;
        double r3359922 = r3359916 - r3359921;
        double r3359923 = r3359922 / r3359916;
        double r3359924 = -1.0;
        double r3359925 = eps;
        double r3359926 = r3359924 + r3359925;
        double r3359927 = r3359926 * r3359913;
        double r3359928 = exp(r3359927);
        double r3359929 = r3359928 / r3359925;
        double r3359930 = r3359928 + r3359929;
        double r3359931 = r3359924 - r3359925;
        double r3359932 = r3359913 * r3359931;
        double r3359933 = exp(r3359932);
        double r3359934 = r3359933 / r3359925;
        double r3359935 = r3359934 - r3359933;
        double r3359936 = r3359930 - r3359935;
        double r3359937 = r3359936 * r3359936;
        double r3359938 = r3359937 * r3359936;
        double r3359939 = cbrt(r3359938);
        double r3359940 = r3359939 / r3359916;
        double r3359941 = r3359915 ? r3359923 : r3359940;
        return r3359941;
}

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 < 357.09970468422745

    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. Simplified39.3

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    5. Taylor expanded around 0 1.1

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

    6. Simplified1.1

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

    8. Applied associate--l-1.1

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

    if 357.09970468422745 < 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. Simplified0.0

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

    4. Applied add-cbrt-cube0.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 357.09970468422745:\\ \;\;\;\;\frac{2 - \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-2}{3} + x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \cdot \left(\left(e^{\left(-1 + \varepsilon\right) \cdot x} + \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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))