Average Error: 29.6 → 4.6
Time: 7.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.969928030300931088124740972008478830418 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot \log \left(e^{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}\right)\\ \end{array}\]
\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 2.969928030300931088124740972008478830418 \cdot 10^{-27}:\\
\;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r49829 = 1.0;
        double r49830 = eps;
        double r49831 = r49829 / r49830;
        double r49832 = r49829 + r49831;
        double r49833 = r49829 - r49830;
        double r49834 = x;
        double r49835 = r49833 * r49834;
        double r49836 = -r49835;
        double r49837 = exp(r49836);
        double r49838 = r49832 * r49837;
        double r49839 = r49831 - r49829;
        double r49840 = r49829 + r49830;
        double r49841 = r49840 * r49834;
        double r49842 = -r49841;
        double r49843 = exp(r49842);
        double r49844 = r49839 * r49843;
        double r49845 = r49838 - r49844;
        double r49846 = 2.0;
        double r49847 = r49845 / r49846;
        return r49847;
}
↓
double f(double x, double eps) {
        double r49848 = x;
        double r49849 = 2.969928030300931e-27;
        bool r49850 = r49848 <= r49849;
        double r49851 = 1.3877787807814457e-17;
        double r49852 = cbrt(r49848);
        double r49853 = exp(r49852);
        double r49854 = sqrt(r49853);
        double r49855 = log(r49854);
        double r49856 = r49855 + r49855;
        double r49857 = r49852 * r49856;
        double r49858 = 3.0;
        double r49859 = pow(r49857, r49858);
        double r49860 = eps;
        double r49861 = r49860 / r49848;
        double r49862 = r49859 / r49861;
        double r49863 = 1.0;
        double r49864 = 0.5;
        double r49865 = 2.0;
        double r49866 = pow(r49848, r49865);
        double r49867 = r49864 * r49866;
        double r49868 = r49863 - r49867;
        double r49869 = fma(r49851, r49862, r49868);
        double r49870 = r49863 + r49860;
        double r49871 = r49870 * r49848;
        double r49872 = -r49871;
        double r49873 = exp(r49872);
        double r49874 = 2.0;
        double r49875 = r49873 / r49874;
        double r49876 = r49863 / r49860;
        double r49877 = r49863 - r49876;
        double r49878 = r49863 + r49876;
        double r49879 = r49863 - r49860;
        double r49880 = r49879 * r49848;
        double r49881 = exp(r49880);
        double r49882 = exp(r49881);
        double r49883 = log(r49882);
        double r49884 = r49874 * r49883;
        double r49885 = r49878 / r49884;
        double r49886 = fma(r49875, r49877, r49885);
        double r49887 = r49850 ? r49869 : r49886;
        return r49887;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 2.969928030300931e-27

    1. Initial program 38.3

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

    2. Simplified38.3

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

    3. Taylor expanded around 0 6.1

      \[\leadsto \color{blue}{\left(1.387778780781445675529539585113525390625 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified6.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt6.1

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    7. Applied unpow-prod-down6.1

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    8. Applied associate-/l*6.1

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{{\left(\sqrt[3]{x}\right)}^{3}}}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    9. Simplified6.1

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    10. Using strategy rm

    11. Applied add-log-exp4.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt4.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \log \color{blue}{\left(\sqrt{e^{\sqrt[3]{x}}} \cdot \sqrt{e^{\sqrt[3]{x}}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    14. Applied log-prod4.3

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    if 2.969928030300931e-27 < x

    1. Initial program 5.5

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

    2. Simplified5.5

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

    4. Applied add-log-exp5.5

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.969928030300931088124740972008478830418 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot \log \left(e^{e^{\left(1 - \varepsilon\right) \cdot x}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))