Average Error: 30.0 → 1.5
Time: 26.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 - \left(1 \cdot x\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)\right)\right)}{2}\\ \end{array}\]
\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 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\
\;\;\;\;\frac{2 - \left(1 \cdot x\right) \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)\right)\right)}{2}\\

\end{array}
double f(double x, double eps) {
        double r1559844 = 1.0;
        double r1559845 = eps;
        double r1559846 = r1559844 / r1559845;
        double r1559847 = r1559844 + r1559846;
        double r1559848 = r1559844 - r1559845;
        double r1559849 = x;
        double r1559850 = r1559848 * r1559849;
        double r1559851 = -r1559850;
        double r1559852 = exp(r1559851);
        double r1559853 = r1559847 * r1559852;
        double r1559854 = r1559846 - r1559844;
        double r1559855 = r1559844 + r1559845;
        double r1559856 = r1559855 * r1559849;
        double r1559857 = -r1559856;
        double r1559858 = exp(r1559857);
        double r1559859 = r1559854 * r1559858;
        double r1559860 = r1559853 - r1559859;
        double r1559861 = 2.0;
        double r1559862 = r1559860 / r1559861;
        return r1559862;
}
↓
double f(double x, double eps) {
        double r1559863 = x;
        double r1559864 = 1.1486676066584133e-09;
        bool r1559865 = r1559863 <= r1559864;
        double r1559866 = 2.0;
        double r1559867 = 1.0;
        double r1559868 = r1559867 * r1559863;
        double r1559869 = r1559868 * r1559863;
        double r1559870 = r1559866 - r1559869;
        double r1559871 = r1559870 / r1559866;
        double r1559872 = eps;
        double r1559873 = r1559872 - r1559867;
        double r1559874 = r1559863 * r1559873;
        double r1559875 = exp(r1559874);
        double r1559876 = r1559867 / r1559872;
        double r1559877 = r1559867 + r1559876;
        double r1559878 = r1559867 - r1559876;
        double r1559879 = r1559867 + r1559872;
        double r1559880 = r1559863 * r1559879;
        double r1559881 = exp(r1559880);
        double r1559882 = r1559878 / r1559881;
        double r1559883 = fma(r1559875, r1559877, r1559882);
        double r1559884 = log1p(r1559883);
        double r1559885 = expm1(r1559884);
        double r1559886 = r1559885 / r1559866;
        double r1559887 = r1559865 ? r1559871 : r1559886;
        return r1559887;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.1486676066584133e-09

    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. Simplified39.6

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

    3. Taylor expanded around 0 7.2

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77555756156289135105907917022705078125 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.2

      \[\leadsto \frac{\color{blue}{2 - \mathsf{fma}\left(1, x \cdot x, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\left(x \cdot x\right) \cdot x}}\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-log-exp1.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\color{blue}{\log \left(e^{\left(x \cdot x\right) \cdot x}\right)}}}\right)}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\log \color{blue}{\left(\left(\sqrt[3]{e^{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{e^{\left(x \cdot x\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(x \cdot x\right) \cdot x}}\right)}}}\right)}{2}\]

    9. Applied log-prod1.5

      \[\leadsto \frac{2 - \mathsf{fma}\left(1, x \cdot x, \frac{2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\frac{\varepsilon}{\color{blue}{\log \left(\sqrt[3]{e^{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{e^{\left(x \cdot x\right) \cdot x}}\right) + \log \left(\sqrt[3]{e^{\left(x \cdot x\right) \cdot x}}\right)}}}\right)}{2}\]

    10. Taylor expanded around inf 1.2

      \[\leadsto \frac{2 - \color{blue}{1 \cdot {x}^{2}}}{2}\]

    11. Simplified1.2

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

    if 1.1486676066584133e-09 < x

    1. Initial program 2.2

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

    2. Simplified2.2

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

    4. Applied expm1-log1p-u2.2

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, \frac{1}{\varepsilon} + 1, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 - \left(1 \cdot x\right) \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(e^{x \cdot \left(\varepsilon - 1\right)}, 1 + \frac{1}{\varepsilon}, \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))