Average Error: 29.3 → 1.0
Time: 6.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 241.297036783070638:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 241.297036783070638:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r43722 = 1.0;
        double r43723 = eps;
        double r43724 = r43722 / r43723;
        double r43725 = r43722 + r43724;
        double r43726 = r43722 - r43723;
        double r43727 = x;
        double r43728 = r43726 * r43727;
        double r43729 = -r43728;
        double r43730 = exp(r43729);
        double r43731 = r43725 * r43730;
        double r43732 = r43724 - r43722;
        double r43733 = r43722 + r43723;
        double r43734 = r43733 * r43727;
        double r43735 = -r43734;
        double r43736 = exp(r43735);
        double r43737 = r43732 * r43736;
        double r43738 = r43731 - r43737;
        double r43739 = 2.0;
        double r43740 = r43738 / r43739;
        return r43740;
}
↓
double f(double x, double eps) {
        double r43741 = x;
        double r43742 = 241.29703678307064;
        bool r43743 = r43741 <= r43742;
        double r43744 = 0.6666666666666667;
        double r43745 = 3.0;
        double r43746 = pow(r43741, r43745);
        double r43747 = r43744 * r43746;
        double r43748 = 2.0;
        double r43749 = r43747 + r43748;
        double r43750 = 1.0;
        double r43751 = 2.0;
        double r43752 = pow(r43741, r43751);
        double r43753 = r43750 * r43752;
        double r43754 = r43749 - r43753;
        double r43755 = pow(r43754, r43745);
        double r43756 = cbrt(r43755);
        double r43757 = r43756 / r43748;
        double r43758 = eps;
        double r43759 = r43750 / r43758;
        double r43760 = r43750 + r43759;
        double r43761 = r43750 - r43758;
        double r43762 = r43761 * r43741;
        double r43763 = -r43762;
        double r43764 = exp(r43763);
        double r43765 = r43760 * r43764;
        double r43766 = r43759 - r43750;
        double r43767 = r43750 + r43758;
        double r43768 = r43767 * r43741;
        double r43769 = -r43768;
        double r43770 = exp(r43769);
        double r43771 = sqrt(r43770);
        double r43772 = r43766 * r43771;
        double r43773 = r43772 * r43771;
        double r43774 = r43765 - r43773;
        double r43775 = r43774 / r43748;
        double r43776 = r43743 ? r43757 : r43775;
        return r43776;
}

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 < 241.29703678307064

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.2

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

    4. Applied add-cbrt-cube1.2

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

    5. Simplified1.2

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

    if 241.29703678307064 < 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-sqr-sqrt0.1

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

    4. Applied associate-*r*0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 241.297036783070638:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))