Average Error: 29.1 → 1.0
Time: 8.9m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.5819201758168828:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]
\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 1.5819201758168828:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r8028883 = 1.0;
        double r8028884 = eps;
        double r8028885 = r8028883 / r8028884;
        double r8028886 = r8028883 + r8028885;
        double r8028887 = r8028883 - r8028884;
        double r8028888 = x;
        double r8028889 = r8028887 * r8028888;
        double r8028890 = -r8028889;
        double r8028891 = exp(r8028890);
        double r8028892 = r8028886 * r8028891;
        double r8028893 = r8028885 - r8028883;
        double r8028894 = r8028883 + r8028884;
        double r8028895 = r8028894 * r8028888;
        double r8028896 = -r8028895;
        double r8028897 = exp(r8028896);
        double r8028898 = r8028893 * r8028897;
        double r8028899 = r8028892 - r8028898;
        double r8028900 = 2.0;
        double r8028901 = r8028899 / r8028900;
        return r8028901;
}
↓
double f(double x, double eps) {
        double r8028902 = x;
        double r8028903 = 1.5819201758168828;
        bool r8028904 = r8028902 <= r8028903;
        double r8028905 = r8028902 * r8028902;
        double r8028906 = r8028905 * r8028902;
        double r8028907 = 0.6666666666666666;
        double r8028908 = 2.0;
        double r8028909 = r8028908 - r8028905;
        double r8028910 = fma(r8028906, r8028907, r8028909);
        double r8028911 = r8028910 / r8028908;
        double r8028912 = eps;
        double r8028913 = -1.0;
        double r8028914 = r8028912 + r8028913;
        double r8028915 = r8028914 * r8028902;
        double r8028916 = exp(r8028915);
        double r8028917 = r8028916 / r8028912;
        double r8028918 = r8028916 + r8028917;
        double r8028919 = r8028913 - r8028912;
        double r8028920 = r8028902 * r8028919;
        double r8028921 = exp(r8028920);
        double r8028922 = r8028921 / r8028912;
        double r8028923 = r8028918 - r8028922;
        double r8028924 = r8028923 + r8028921;
        double r8028925 = r8028924 / r8028908;
        double r8028926 = r8028904 ? r8028911 : r8028925;
        return r8028926;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.5819201758168828

    1. Initial program 38.8

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

    2. Simplified38.8

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    6. Applied add-cbrt-cube1.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right) \cdot \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)\right) \cdot \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}}}{2}\]

    7. Taylor expanded around 0 1.2

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

    8. Simplified1.2

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

    if 1.5819201758168828 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied associate--r-0.5

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.5819201758168828:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot x\right), \frac{2}{3}, \left(2 - x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(\varepsilon + -1\right) \cdot x} + \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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))