Average Error: 29.5 → 1.0
Time: 6.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 417.18899044172827:\\ \;\;\;\;\frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - \left(12 \cdot \left(x \cdot \sqrt[3]{x}\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]
\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 417.18899044172827:\\
\;\;\;\;\frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - \left(12 \cdot \left(x \cdot \sqrt[3]{x}\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r34798 = 1.0;
        double r34799 = eps;
        double r34800 = r34798 / r34799;
        double r34801 = r34798 + r34800;
        double r34802 = r34798 - r34799;
        double r34803 = x;
        double r34804 = r34802 * r34803;
        double r34805 = -r34804;
        double r34806 = exp(r34805);
        double r34807 = r34801 * r34806;
        double r34808 = r34800 - r34798;
        double r34809 = r34798 + r34799;
        double r34810 = r34809 * r34803;
        double r34811 = -r34810;
        double r34812 = exp(r34811);
        double r34813 = r34808 * r34812;
        double r34814 = r34807 - r34813;
        double r34815 = 2.0;
        double r34816 = r34814 / r34815;
        return r34816;
}
↓
double f(double x, double eps) {
        double r34817 = x;
        double r34818 = 417.18899044172827;
        bool r34819 = r34817 <= r34818;
        double r34820 = 8.0;
        double r34821 = 3.0;
        double r34822 = pow(r34817, r34821);
        double r34823 = r34820 * r34822;
        double r34824 = r34823 + r34820;
        double r34825 = 12.0;
        double r34826 = cbrt(r34817);
        double r34827 = r34817 * r34826;
        double r34828 = r34825 * r34827;
        double r34829 = 2.0;
        double r34830 = pow(r34826, r34829);
        double r34831 = r34828 * r34830;
        double r34832 = r34824 - r34831;
        double r34833 = cbrt(r34832);
        double r34834 = 2.0;
        double r34835 = r34833 / r34834;
        double r34836 = 1.0;
        double r34837 = eps;
        double r34838 = r34836 / r34837;
        double r34839 = r34836 + r34838;
        double r34840 = r34836 - r34837;
        double r34841 = r34840 * r34817;
        double r34842 = -r34841;
        double r34843 = exp(r34842);
        double r34844 = r34839 * r34843;
        double r34845 = r34817 * r34837;
        double r34846 = r34836 * r34817;
        double r34847 = r34845 + r34846;
        double r34848 = -r34847;
        double r34849 = exp(r34848);
        double r34850 = r34849 / r34837;
        double r34851 = r34850 - r34849;
        double r34852 = r34836 * r34851;
        double r34853 = r34844 - r34852;
        double r34854 = r34853 / r34834;
        double r34855 = r34819 ? r34835 : r34854;
        return r34855;
}

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 < 417.18899044172827

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cbrt-cube1.3

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

    5. Simplified1.3

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}}}{2}\]

    9. Applied unpow-prod-down1.3

      \[\leadsto \frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot \color{blue}{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2} \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)}}}{2}\]

    10. Applied associate-*r*1.3

      \[\leadsto \frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - \color{blue}{\left(12 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{2}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}}}}{2}\]

    11. Simplified1.3

      \[\leadsto \frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - \color{blue}{\left(12 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)} \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{2}\]

    if 417.18899044172827 < 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. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 417.18899044172827:\\ \;\;\;\;\frac{\sqrt[3]{\left(8 \cdot {x}^{3} + 8\right) - \left(12 \cdot \left(x \cdot \sqrt[3]{x}\right)\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))