Average Error: 29.4 → 1.0
Time: 24.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 158.8018617543875:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)\right)}{2}\\ \end{array}\]
\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 158.8018617543875:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r792917 = 1.0;
        double r792918 = eps;
        double r792919 = r792917 / r792918;
        double r792920 = r792917 + r792919;
        double r792921 = r792917 - r792918;
        double r792922 = x;
        double r792923 = r792921 * r792922;
        double r792924 = -r792923;
        double r792925 = exp(r792924);
        double r792926 = r792920 * r792925;
        double r792927 = r792919 - r792917;
        double r792928 = r792917 + r792918;
        double r792929 = r792928 * r792922;
        double r792930 = -r792929;
        double r792931 = exp(r792930);
        double r792932 = r792927 * r792931;
        double r792933 = r792926 - r792932;
        double r792934 = 2.0;
        double r792935 = r792933 / r792934;
        return r792935;
}
↓
double f(double x, double eps) {
        double r792936 = x;
        double r792937 = 158.8018617543875;
        bool r792938 = r792936 <= r792937;
        double r792939 = r792936 * r792936;
        double r792940 = r792939 * r792936;
        double r792941 = 0.6666666666666666;
        double r792942 = 2.0;
        double r792943 = r792942 - r792939;
        double r792944 = fma(r792940, r792941, r792943);
        double r792945 = r792944 / r792942;
        double r792946 = 1.0;
        double r792947 = eps;
        double r792948 = r792946 - r792947;
        double r792949 = -r792936;
        double r792950 = r792948 * r792949;
        double r792951 = exp(r792950);
        double r792952 = r792946 / r792947;
        double r792953 = r792952 + r792946;
        double r792954 = r792951 * r792953;
        double r792955 = r792952 - r792946;
        double r792956 = r792947 + r792946;
        double r792957 = r792949 * r792956;
        double r792958 = exp(r792957);
        double r792959 = cbrt(r792958);
        double r792960 = r792959 * r792959;
        double r792961 = r792959 * r792960;
        double r792962 = r792955 * r792961;
        double r792963 = r792954 - r792962;
        double r792964 = r792963 / r792942;
        double r792965 = r792938 ? r792945 : r792964;
        return r792965;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 158.8018617543875

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    if 158.8018617543875 < 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. Using strategy rm

    3. Applied add-cube-cbrt0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 158.8018617543875:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \left(\sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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))