Average Error: 29.6 → 1.0
Time: 36.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 7.315544741537903:\\ \;\;\;\;\frac{\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}\right) - x \cdot x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{\sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)}}\right)}^{\left(\sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]
\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 7.315544741537903:\\
\;\;\;\;\frac{\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}\right) - x \cdot x\right) + 2}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1136686 = 1.0;
        double r1136687 = eps;
        double r1136688 = r1136686 / r1136687;
        double r1136689 = r1136686 + r1136688;
        double r1136690 = r1136686 - r1136687;
        double r1136691 = x;
        double r1136692 = r1136690 * r1136691;
        double r1136693 = -r1136692;
        double r1136694 = exp(r1136693);
        double r1136695 = r1136689 * r1136694;
        double r1136696 = r1136688 - r1136686;
        double r1136697 = r1136686 + r1136687;
        double r1136698 = r1136697 * r1136691;
        double r1136699 = -r1136698;
        double r1136700 = exp(r1136699);
        double r1136701 = r1136696 * r1136700;
        double r1136702 = r1136695 - r1136701;
        double r1136703 = 2.0;
        double r1136704 = r1136702 / r1136703;
        return r1136704;
}
↓
double f(double x, double eps) {
        double r1136705 = x;
        double r1136706 = 7.315544741537903;
        bool r1136707 = r1136705 <= r1136706;
        double r1136708 = r1136705 * r1136705;
        double r1136709 = r1136708 * r1136705;
        double r1136710 = 0.6666666666666666;
        double r1136711 = r1136709 * r1136710;
        double r1136712 = exp(r1136711);
        double r1136713 = log(r1136712);
        double r1136714 = r1136713 - r1136708;
        double r1136715 = 2.0;
        double r1136716 = r1136714 + r1136715;
        double r1136717 = r1136716 / r1136715;
        double r1136718 = 1.0;
        double r1136719 = eps;
        double r1136720 = r1136718 / r1136719;
        double r1136721 = r1136720 + r1136718;
        double r1136722 = r1136718 - r1136719;
        double r1136723 = -r1136705;
        double r1136724 = r1136722 * r1136723;
        double r1136725 = cbrt(r1136724);
        double r1136726 = r1136725 * r1136725;
        double r1136727 = exp(r1136726);
        double r1136728 = pow(r1136727, r1136725);
        double r1136729 = r1136721 * r1136728;
        double r1136730 = r1136720 - r1136718;
        double r1136731 = r1136718 + r1136719;
        double r1136732 = r1136723 * r1136731;
        double r1136733 = exp(r1136732);
        double r1136734 = r1136730 * r1136733;
        double r1136735 = r1136729 - r1136734;
        double r1136736 = r1136735 / r1136715;
        double r1136737 = r1136707 ? r1136717 : r1136736;
        return r1136737;
}

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 < 7.315544741537903

    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 1.2

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

    3. Simplified1.2

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

    5. Applied add-log-exp1.2

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

    if 7.315544741537903 < 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. Using strategy rm

    3. Applied add-cube-cbrt0.4

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

    4. Applied exp-prod0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 7.315544741537903:\\ \;\;\;\;\frac{\left(\log \left(e^{\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}\right) - x \cdot x\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot {\left(e^{\sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)}}\right)}^{\left(\sqrt[3]{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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))