Average Error: 29.7 → 1.0
Time: 27.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{2 + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3} - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \left(\sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}}\right) - \left(\frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon} - e^{\left(-1 - \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]
\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 100.79533204800275:\\
\;\;\;\;\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]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \left(\sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}}\right) - \left(\frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon} - e^{\left(-1 - \varepsilon\right) \cdot x}\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r773903 = 1.0;
        double r773904 = eps;
        double r773905 = r773903 / r773904;
        double r773906 = r773903 + r773905;
        double r773907 = r773903 - r773904;
        double r773908 = x;
        double r773909 = r773907 * r773908;
        double r773910 = -r773909;
        double r773911 = exp(r773910);
        double r773912 = r773906 * r773911;
        double r773913 = r773905 - r773903;
        double r773914 = r773903 + r773904;
        double r773915 = r773914 * r773908;
        double r773916 = -r773915;
        double r773917 = exp(r773916);
        double r773918 = r773913 * r773917;
        double r773919 = r773912 - r773918;
        double r773920 = 2.0;
        double r773921 = r773919 / r773920;
        return r773921;
}
↓
double f(double x, double eps) {
        double r773922 = x;
        double r773923 = 100.79533204800275;
        bool r773924 = r773922 <= r773923;
        double r773925 = 2.0;
        double r773926 = r773922 * r773922;
        double r773927 = r773926 * r773922;
        double r773928 = 0.6666666666666666;
        double r773929 = r773927 * r773928;
        double r773930 = r773929 - r773926;
        double r773931 = r773925 + r773930;
        double r773932 = r773931 / r773925;
        double r773933 = eps;
        double r773934 = -1.0;
        double r773935 = r773933 + r773934;
        double r773936 = r773935 * r773922;
        double r773937 = exp(r773936);
        double r773938 = r773937 / r773933;
        double r773939 = r773937 + r773938;
        double r773940 = cbrt(r773939);
        double r773941 = r773940 * r773940;
        double r773942 = r773940 * r773941;
        double r773943 = r773934 - r773933;
        double r773944 = r773943 * r773922;
        double r773945 = exp(r773944);
        double r773946 = r773945 / r773933;
        double r773947 = r773946 - r773945;
        double r773948 = r773942 - r773947;
        double r773949 = r773948 / r773925;
        double r773950 = r773924 ? r773932 : r773949;
        return r773950;
}

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 < 100.79533204800275

    1. Initial program 39.6

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

    2. Simplified39.6

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

    5. Taylor expanded around inf 1.3

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

    6. Simplified1.3

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

    if 100.79533204800275 < 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. Simplified0.1

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

    4. Applied add-cube-cbrt0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{2 + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3} - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \left(\sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}} \cdot \sqrt[3]{e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}}\right) - \left(\frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon} - e^{\left(-1 - \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))