Average Error: 29.7 → 1.1
Time: 35.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 129.10958091941424:\\ \;\;\;\;\frac{2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{{e}^{\left(\varepsilon \cdot x + \left(-x\right)\right)}}{\varepsilon} + {e}^{\left(\varepsilon \cdot x + \left(-x\right)\right)}\right) - \left(\frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon} - e^{\left(-x\right) - \varepsilon \cdot x}\right)}{2}\\ \end{array}\]
\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 129.10958091941424:\\
\;\;\;\;\frac{2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2253827 = 1.0;
        double r2253828 = eps;
        double r2253829 = r2253827 / r2253828;
        double r2253830 = r2253827 + r2253829;
        double r2253831 = r2253827 - r2253828;
        double r2253832 = x;
        double r2253833 = r2253831 * r2253832;
        double r2253834 = -r2253833;
        double r2253835 = exp(r2253834);
        double r2253836 = r2253830 * r2253835;
        double r2253837 = r2253829 - r2253827;
        double r2253838 = r2253827 + r2253828;
        double r2253839 = r2253838 * r2253832;
        double r2253840 = -r2253839;
        double r2253841 = exp(r2253840);
        double r2253842 = r2253837 * r2253841;
        double r2253843 = r2253836 - r2253842;
        double r2253844 = 2.0;
        double r2253845 = r2253843 / r2253844;
        return r2253845;
}
↓
double f(double x, double eps) {
        double r2253846 = x;
        double r2253847 = 129.10958091941424;
        bool r2253848 = r2253846 <= r2253847;
        double r2253849 = 2.0;
        double r2253850 = 0.6666666666666666;
        double r2253851 = r2253850 * r2253846;
        double r2253852 = r2253846 * r2253846;
        double r2253853 = r2253851 * r2253852;
        double r2253854 = r2253853 - r2253852;
        double r2253855 = r2253849 + r2253854;
        double r2253856 = r2253855 / r2253849;
        double r2253857 = exp(1.0);
        double r2253858 = eps;
        double r2253859 = r2253858 * r2253846;
        double r2253860 = -r2253846;
        double r2253861 = r2253859 + r2253860;
        double r2253862 = pow(r2253857, r2253861);
        double r2253863 = r2253862 / r2253858;
        double r2253864 = r2253863 + r2253862;
        double r2253865 = r2253860 - r2253859;
        double r2253866 = exp(r2253865);
        double r2253867 = r2253866 / r2253858;
        double r2253868 = r2253867 - r2253866;
        double r2253869 = r2253864 - r2253868;
        double r2253870 = r2253869 / r2253849;
        double r2253871 = r2253848 ? r2253856 : r2253870;
        return r2253871;
}

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 < 129.10958091941424

    1. Initial program 39.4

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

    2. Simplified39.4

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

    6. Applied sub-neg1.3

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

    7. Applied associate-+l+1.3

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

    8. Simplified1.3

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

    if 129.10958091941424 < x

    1. Initial program 0.2

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

    2. Simplified0.2

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

    4. Applied *-un-lft-identity0.2

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

    5. Applied exp-prod0.2

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

    6. Simplified0.2

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

    8. Applied *-un-lft-identity0.2

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

    9. Applied exp-prod0.2

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

    10. Simplified0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 129.10958091941424:\\ \;\;\;\;\frac{2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{{e}^{\left(\varepsilon \cdot x + \left(-x\right)\right)}}{\varepsilon} + {e}^{\left(\varepsilon \cdot x + \left(-x\right)\right)}\right) - \left(\frac{e^{\left(-x\right) - \varepsilon \cdot x}}{\varepsilon} - e^{\left(-x\right) - \varepsilon \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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))