Average Error: 29.6 → 1.1
Time: 8.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.19467781707589359996291022980585694313:\\ \;\;\;\;\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 2.19467781707589359996291022980585694313:\\
\;\;\;\;\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\

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

\end{array}
double f(double x, double eps) {
        double r43707 = 1.0;
        double r43708 = eps;
        double r43709 = r43707 / r43708;
        double r43710 = r43707 + r43709;
        double r43711 = r43707 - r43708;
        double r43712 = x;
        double r43713 = r43711 * r43712;
        double r43714 = -r43713;
        double r43715 = exp(r43714);
        double r43716 = r43710 * r43715;
        double r43717 = r43709 - r43707;
        double r43718 = r43707 + r43708;
        double r43719 = r43718 * r43712;
        double r43720 = -r43719;
        double r43721 = exp(r43720);
        double r43722 = r43717 * r43721;
        double r43723 = r43716 - r43722;
        double r43724 = 2.0;
        double r43725 = r43723 / r43724;
        return r43725;
}
↓
double f(double x, double eps) {
        double r43726 = x;
        double r43727 = 2.1946778170758936;
        bool r43728 = r43726 <= r43727;
        double r43729 = 0.33333333333333337;
        double r43730 = 3.0;
        double r43731 = pow(r43726, r43730);
        double r43732 = r43729 * r43731;
        double r43733 = 1.0;
        double r43734 = r43732 + r43733;
        double r43735 = 0.5;
        double r43736 = 2.0;
        double r43737 = pow(r43726, r43736);
        double r43738 = r43735 * r43737;
        double r43739 = r43734 - r43738;
        double r43740 = eps;
        double r43741 = r43733 / r43740;
        double r43742 = r43733 + r43741;
        double r43743 = r43733 - r43740;
        double r43744 = r43743 * r43726;
        double r43745 = exp(r43744);
        double r43746 = r43742 / r43745;
        double r43747 = 2.0;
        double r43748 = r43746 / r43747;
        double r43749 = r43733 + r43740;
        double r43750 = r43749 * r43726;
        double r43751 = exp(r43750);
        double r43752 = r43741 / r43751;
        double r43753 = r43752 / r43747;
        double r43754 = r43748 - r43753;
        double r43755 = r43733 / r43751;
        double r43756 = r43755 / r43747;
        double r43757 = r43754 + r43756;
        double r43758 = r43728 ? r43739 : r43757;
        return r43758;
}

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 < 2.1946778170758936

    1. Initial program 38.8

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

    2. Simplified38.8

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

    3. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]

    if 2.1946778170758936 < x

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied div-sub0.5

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

    5. Applied div-sub0.5

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

    6. Applied associate--r-0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.19467781707589359996291022980585694313:\\ \;\;\;\;\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))