Average Error: 28.9 → 1.0
Time: 17.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 249.63027630533077:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(x \cdot 0.66666666666666674 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\\ \end{array}\]
\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 249.63027630533077:\\
\;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(x \cdot 0.66666666666666674 - 1\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r39724 = 1.0;
        double r39725 = eps;
        double r39726 = r39724 / r39725;
        double r39727 = r39724 + r39726;
        double r39728 = r39724 - r39725;
        double r39729 = x;
        double r39730 = r39728 * r39729;
        double r39731 = -r39730;
        double r39732 = exp(r39731);
        double r39733 = r39727 * r39732;
        double r39734 = r39726 - r39724;
        double r39735 = r39724 + r39725;
        double r39736 = r39735 * r39729;
        double r39737 = -r39736;
        double r39738 = exp(r39737);
        double r39739 = r39734 * r39738;
        double r39740 = r39733 - r39739;
        double r39741 = 2.0;
        double r39742 = r39740 / r39741;
        return r39742;
}
↓
double f(double x, double eps) {
        double r39743 = x;
        double r39744 = 249.63027630533077;
        bool r39745 = r39743 <= r39744;
        double r39746 = 2.0;
        double r39747 = r39743 * r39743;
        double r39748 = 0.6666666666666667;
        double r39749 = r39743 * r39748;
        double r39750 = 1.0;
        double r39751 = r39749 - r39750;
        double r39752 = r39747 * r39751;
        double r39753 = r39746 + r39752;
        double r39754 = r39753 / r39746;
        double r39755 = eps;
        double r39756 = r39750 / r39755;
        double r39757 = r39756 + r39750;
        double r39758 = r39750 - r39755;
        double r39759 = r39758 * r39743;
        double r39760 = exp(r39759);
        double r39761 = r39757 / r39760;
        double r39762 = r39756 - r39750;
        double r39763 = r39750 + r39755;
        double r39764 = r39763 * r39743;
        double r39765 = exp(r39764);
        double r39766 = r39762 / r39765;
        double r39767 = r39761 - r39766;
        double r39768 = log(r39767);
        double r39769 = exp(r39768);
        double r39770 = r39769 / r39746;
        double r39771 = r39745 ? r39754 : r39770;
        return r39771;
}

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 < 249.63027630533077

    1. Initial program 38.7

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

    2. Simplified38.7

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

    3. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    4. Simplified1.3

      \[\leadsto \frac{\color{blue}{2 + \left(x \cdot x\right) \cdot \left(x \cdot 0.66666666666666674 - 1\right)}}{2}\]

    if 249.63027630533077 < 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. Simplified0.1

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

    4. Applied add-exp-log0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 249.63027630533077:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(x \cdot 0.66666666666666674 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))