Average Error: 29.5 → 1.0
Time: 34.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 171.1161917239734:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}}{2}\\ \end{array}\]
\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 171.1161917239734:\\
\;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2 - x \cdot x\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}}{2}\\

\end{array}
double f(double x, double eps) {
        double r1725908 = 1.0;
        double r1725909 = eps;
        double r1725910 = r1725908 / r1725909;
        double r1725911 = r1725908 + r1725910;
        double r1725912 = r1725908 - r1725909;
        double r1725913 = x;
        double r1725914 = r1725912 * r1725913;
        double r1725915 = -r1725914;
        double r1725916 = exp(r1725915);
        double r1725917 = r1725911 * r1725916;
        double r1725918 = r1725910 - r1725908;
        double r1725919 = r1725908 + r1725909;
        double r1725920 = r1725919 * r1725913;
        double r1725921 = -r1725920;
        double r1725922 = exp(r1725921);
        double r1725923 = r1725918 * r1725922;
        double r1725924 = r1725917 - r1725923;
        double r1725925 = 2.0;
        double r1725926 = r1725924 / r1725925;
        return r1725926;
}
↓
double f(double x, double eps) {
        double r1725927 = x;
        double r1725928 = 171.1161917239734;
        bool r1725929 = r1725927 <= r1725928;
        double r1725930 = r1725927 * r1725927;
        double r1725931 = 0.6666666666666666;
        double r1725932 = r1725931 * r1725927;
        double r1725933 = 2.0;
        double r1725934 = r1725933 - r1725930;
        double r1725935 = fma(r1725930, r1725932, r1725934);
        double r1725936 = r1725935 / r1725933;
        double r1725937 = 1.0;
        double r1725938 = eps;
        double r1725939 = r1725937 - r1725938;
        double r1725940 = -r1725927;
        double r1725941 = r1725939 * r1725940;
        double r1725942 = exp(r1725941);
        double r1725943 = r1725937 / r1725938;
        double r1725944 = r1725943 + r1725937;
        double r1725945 = r1725942 * r1725944;
        double r1725946 = r1725943 - r1725937;
        double r1725947 = r1725938 + r1725937;
        double r1725948 = r1725940 * r1725947;
        double r1725949 = exp(r1725948);
        double r1725950 = r1725946 * r1725949;
        double r1725951 = r1725945 - r1725950;
        double r1725952 = cbrt(r1725951);
        double r1725953 = r1725952 * r1725952;
        double r1725954 = r1725953 * r1725952;
        double r1725955 = r1725954 / r1725933;
        double r1725956 = r1725929 ? r1725936 : r1725955;
        return r1725956;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 171.1161917239734

    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. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

    4. Taylor expanded around 0 1.2

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

    5. Simplified1.2

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

    if 171.1161917239734 < x

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 171.1161917239734:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))