Average Error: 29.3 → 1.0
Time: 25.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 183.1502607165994:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{e^{\left(-x\right) + \varepsilon \cdot \left(-x\right)}}{\varepsilon} - e^{\left(-x\right) + \varepsilon \cdot \left(-x\right)}\right)}{2}\\ \end{array}\]
\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) + \frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r967639 = 1.0;
        double r967640 = eps;
        double r967641 = r967639 / r967640;
        double r967642 = r967639 + r967641;
        double r967643 = r967639 - r967640;
        double r967644 = x;
        double r967645 = r967643 * r967644;
        double r967646 = -r967645;
        double r967647 = exp(r967646);
        double r967648 = r967642 * r967647;
        double r967649 = r967641 - r967639;
        double r967650 = r967639 + r967640;
        double r967651 = r967650 * r967644;
        double r967652 = -r967651;
        double r967653 = exp(r967652);
        double r967654 = r967649 * r967653;
        double r967655 = r967648 - r967654;
        double r967656 = 2.0;
        double r967657 = r967655 / r967656;
        return r967657;
}
↓
double f(double x, double eps) {
        double r967658 = x;
        double r967659 = 183.1502607165994;
        bool r967660 = r967658 <= r967659;
        double r967661 = 2.0;
        double r967662 = r967658 * r967658;
        double r967663 = r967661 - r967662;
        double r967664 = 0.6666666666666666;
        double r967665 = r967662 * r967658;
        double r967666 = r967664 * r967665;
        double r967667 = r967663 + r967666;
        double r967668 = r967667 / r967661;
        double r967669 = 1.0;
        double r967670 = eps;
        double r967671 = r967669 / r967670;
        double r967672 = r967671 + r967669;
        double r967673 = r967669 - r967670;
        double r967674 = -r967658;
        double r967675 = r967673 * r967674;
        double r967676 = exp(r967675);
        double r967677 = r967672 * r967676;
        double r967678 = r967670 * r967674;
        double r967679 = r967674 + r967678;
        double r967680 = exp(r967679);
        double r967681 = r967680 / r967670;
        double r967682 = r967681 - r967680;
        double r967683 = r967677 - r967682;
        double r967684 = r967683 / r967661;
        double r967685 = r967660 ? r967668 : r967684;
        return r967685;
}

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

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

    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. Taylor expanded around inf 0.1

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

  4. Final simplification1.0

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

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))