Average Error: 29.3 → 1.0
Time: 26.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]
\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 183.1502607165994:\\
\;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r693581 = 1.0;
        double r693582 = eps;
        double r693583 = r693581 / r693582;
        double r693584 = r693581 + r693583;
        double r693585 = r693581 - r693582;
        double r693586 = x;
        double r693587 = r693585 * r693586;
        double r693588 = -r693587;
        double r693589 = exp(r693588);
        double r693590 = r693584 * r693589;
        double r693591 = r693583 - r693581;
        double r693592 = r693581 + r693582;
        double r693593 = r693592 * r693586;
        double r693594 = -r693593;
        double r693595 = exp(r693594);
        double r693596 = r693591 * r693595;
        double r693597 = r693590 - r693596;
        double r693598 = 2.0;
        double r693599 = r693597 / r693598;
        return r693599;
}
↓
double f(double x, double eps) {
        double r693600 = x;
        double r693601 = 183.1502607165994;
        bool r693602 = r693600 <= r693601;
        double r693603 = 2.0;
        double r693604 = r693600 * r693600;
        double r693605 = r693603 - r693604;
        double r693606 = r693604 * r693600;
        double r693607 = 0.6666666666666666;
        double r693608 = r693606 * r693607;
        double r693609 = r693605 + r693608;
        double r693610 = r693609 / r693603;
        double r693611 = 1.0;
        double r693612 = eps;
        double r693613 = r693611 / r693612;
        double r693614 = r693613 + r693611;
        double r693615 = -r693600;
        double r693616 = r693611 - r693612;
        double r693617 = r693615 * r693616;
        double r693618 = exp(r693617);
        double r693619 = r693614 * r693618;
        double r693620 = r693612 + r693611;
        double r693621 = r693620 * r693615;
        double r693622 = exp(r693621);
        double r693623 = r693613 - r693611;
        double r693624 = r693622 * r693623;
        double r693625 = r693619 - r693624;
        double r693626 = log(r693625);
        double r693627 = exp(r693626);
        double r693628 = r693627 / r693603;
        double r693629 = r693602 ? r693610 : r693628;
        return r693629;
}

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 < 183.1502607165994

    1. Initial program 38.9

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

    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

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

    if 183.1502607165994 < 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. Using strategy rm

    3. Applied add-exp-log0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))