Average Error: 29.9 → 1.1
Time: 32.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.224816812712469:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)}\right) \cdot \log \left(e^{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)}}\right)}{2}\\ \end{array}\]
\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 2.224816812712469:\\
\;\;\;\;\frac{2 + \left(x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2380567 = 1.0;
        double r2380568 = eps;
        double r2380569 = r2380567 / r2380568;
        double r2380570 = r2380567 + r2380569;
        double r2380571 = r2380567 - r2380568;
        double r2380572 = x;
        double r2380573 = r2380571 * r2380572;
        double r2380574 = -r2380573;
        double r2380575 = exp(r2380574);
        double r2380576 = r2380570 * r2380575;
        double r2380577 = r2380569 - r2380567;
        double r2380578 = r2380567 + r2380568;
        double r2380579 = r2380578 * r2380572;
        double r2380580 = -r2380579;
        double r2380581 = exp(r2380580);
        double r2380582 = r2380577 * r2380581;
        double r2380583 = r2380576 - r2380582;
        double r2380584 = 2.0;
        double r2380585 = r2380583 / r2380584;
        return r2380585;
}
↓
double f(double x, double eps) {
        double r2380586 = x;
        double r2380587 = 2.224816812712469;
        bool r2380588 = r2380586 <= r2380587;
        double r2380589 = 2.0;
        double r2380590 = 0.6666666666666666;
        double r2380591 = r2380586 * r2380586;
        double r2380592 = r2380590 * r2380591;
        double r2380593 = r2380586 * r2380592;
        double r2380594 = r2380593 - r2380591;
        double r2380595 = r2380589 + r2380594;
        double r2380596 = r2380595 / r2380589;
        double r2380597 = 1.0;
        double r2380598 = eps;
        double r2380599 = r2380597 / r2380598;
        double r2380600 = r2380599 + r2380597;
        double r2380601 = r2380597 - r2380598;
        double r2380602 = -r2380586;
        double r2380603 = r2380601 * r2380602;
        double r2380604 = exp(r2380603);
        double r2380605 = r2380600 * r2380604;
        double r2380606 = r2380598 * r2380602;
        double r2380607 = r2380606 + r2380602;
        double r2380608 = exp(r2380607);
        double r2380609 = r2380608 / r2380598;
        double r2380610 = r2380609 - r2380608;
        double r2380611 = r2380605 - r2380610;
        double r2380612 = cbrt(r2380611);
        double r2380613 = r2380612 * r2380612;
        double r2380614 = exp(r2380612);
        double r2380615 = log(r2380614);
        double r2380616 = r2380613 * r2380615;
        double r2380617 = r2380616 / r2380589;
        double r2380618 = r2380588 ? r2380596 : r2380617;
        return r2380618;
}

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 < 2.224816812712469

    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. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    if 2.224816812712469 < x

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.4

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

    4. Applied add-cube-cbrt0.4

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

    6. Applied add-log-exp0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.224816812712469:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)}\right) \cdot \log \left(e^{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}}{\varepsilon} - e^{\varepsilon \cdot \left(-x\right) + \left(-x\right)}\right)}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))