Average Error: 29.4 → 0.9
Time: 26.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 54.31805746173307:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \left(\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)}\right)}{2}\\ \end{array}\]
\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 54.31805746173307:\\
\;\;\;\;\frac{\left(2 + x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) - x \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \left(\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\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 r1124988 = 1.0;
        double r1124989 = eps;
        double r1124990 = r1124988 / r1124989;
        double r1124991 = r1124988 + r1124990;
        double r1124992 = r1124988 - r1124989;
        double r1124993 = x;
        double r1124994 = r1124992 * r1124993;
        double r1124995 = -r1124994;
        double r1124996 = exp(r1124995);
        double r1124997 = r1124991 * r1124996;
        double r1124998 = r1124990 - r1124988;
        double r1124999 = r1124988 + r1124989;
        double r1125000 = r1124999 * r1124993;
        double r1125001 = -r1125000;
        double r1125002 = exp(r1125001);
        double r1125003 = r1124998 * r1125002;
        double r1125004 = r1124997 - r1125003;
        double r1125005 = 2.0;
        double r1125006 = r1125004 / r1125005;
        return r1125006;
}
↓
double f(double x, double eps) {
        double r1125007 = x;
        double r1125008 = 54.31805746173307;
        bool r1125009 = r1125007 <= r1125008;
        double r1125010 = 2.0;
        double r1125011 = 0.6666666666666666;
        double r1125012 = r1125007 * r1125007;
        double r1125013 = r1125011 * r1125012;
        double r1125014 = r1125007 * r1125013;
        double r1125015 = r1125010 + r1125014;
        double r1125016 = r1125015 - r1125012;
        double r1125017 = r1125016 / r1125010;
        double r1125018 = eps;
        double r1125019 = -1.0;
        double r1125020 = r1125018 + r1125019;
        double r1125021 = r1125020 * r1125007;
        double r1125022 = exp(r1125021);
        double r1125023 = r1125022 / r1125018;
        double r1125024 = r1125022 + r1125023;
        double r1125025 = r1125019 - r1125018;
        double r1125026 = r1125007 * r1125025;
        double r1125027 = exp(r1125026);
        double r1125028 = r1125027 / r1125018;
        double r1125029 = r1125028 - r1125027;
        double r1125030 = r1125024 - r1125029;
        double r1125031 = cbrt(r1125030);
        double r1125032 = r1125031 * r1125031;
        double r1125033 = r1125031 * r1125032;
        double r1125034 = r1125033 / r1125010;
        double r1125035 = r1125009 ? r1125017 : r1125034;
        return r1125035;
}

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 < 54.31805746173307

    1. Initial program 39.1

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

    2. Simplified39.1

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

    3. 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

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

    if 54.31805746173307 < x

    1. Initial program 0.3

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

    2. Simplified0.3

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

    4. Applied add-cube-cbrt0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 54.31805746173307:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \left(\sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \cdot \sqrt[3]{\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))