Average Error: 29.6 → 1.1
Time: 7.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.19467781707589359996291022980585694313:\\ \;\;\;\;\left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 r43724 = 1.0;
        double r43725 = eps;
        double r43726 = r43724 / r43725;
        double r43727 = r43724 + r43726;
        double r43728 = r43724 - r43725;
        double r43729 = x;
        double r43730 = r43728 * r43729;
        double r43731 = -r43730;
        double r43732 = exp(r43731);
        double r43733 = r43727 * r43732;
        double r43734 = r43726 - r43724;
        double r43735 = r43724 + r43725;
        double r43736 = r43735 * r43729;
        double r43737 = -r43736;
        double r43738 = exp(r43737);
        double r43739 = r43734 * r43738;
        double r43740 = r43733 - r43739;
        double r43741 = 2.0;
        double r43742 = r43740 / r43741;
        return r43742;
}
↓
double f(double x, double eps) {
        double r43743 = x;
        double r43744 = 2.1946778170758936;
        bool r43745 = r43743 <= r43744;
        double r43746 = 0.33333333333333337;
        double r43747 = 3.0;
        double r43748 = pow(r43743, r43747);
        double r43749 = r43746 * r43748;
        double r43750 = 1.0;
        double r43751 = r43749 + r43750;
        double r43752 = 0.5;
        double r43753 = 2.0;
        double r43754 = pow(r43743, r43753);
        double r43755 = r43752 * r43754;
        double r43756 = r43751 - r43755;
        double r43757 = eps;
        double r43758 = r43750 / r43757;
        double r43759 = r43750 + r43758;
        double r43760 = r43750 - r43757;
        double r43761 = r43760 * r43743;
        double r43762 = exp(r43761);
        double r43763 = r43759 / r43762;
        double r43764 = 2.0;
        double r43765 = r43763 / r43764;
        double r43766 = r43750 + r43757;
        double r43767 = r43766 * r43743;
        double r43768 = exp(r43767);
        double r43769 = r43758 / r43768;
        double r43770 = r43769 / r43764;
        double r43771 = r43765 - r43770;
        double r43772 = r43750 / r43768;
        double r43773 = r43772 / r43764;
        double r43774 = r43771 + r43773;
        double r43775 = r43745 ? r43756 : r43774;
        return r43775;
}

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))