Average Error: 25.2 → 9.1
Time: 8.1s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.49844384939791808:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 9.2741206798617679 \cdot 10^{-221}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, \left({z}^{2} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.49844384939791808:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\

\mathbf{elif}\;z \le 9.2741206798617679 \cdot 10^{-221}:\\
\;\;\;\;x - \frac{\mathsf{fma}\left(0.5, \left({z}^{2} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r271873 = x;
        double r271874 = 1.0;
        double r271875 = y;
        double r271876 = r271874 - r271875;
        double r271877 = z;
        double r271878 = exp(r271877);
        double r271879 = r271875 * r271878;
        double r271880 = r271876 + r271879;
        double r271881 = log(r271880);
        double r271882 = t;
        double r271883 = r271881 / r271882;
        double r271884 = r271873 - r271883;
        return r271884;
}
↓
double f(double x, double y, double z, double t) {
        double r271885 = z;
        double r271886 = -0.4984438493979181;
        bool r271887 = r271885 <= r271886;
        double r271888 = x;
        double r271889 = 1.0;
        double r271890 = y;
        double r271891 = r271889 - r271890;
        double r271892 = cbrt(r271890);
        double r271893 = r271892 * r271892;
        double r271894 = exp(r271885);
        double r271895 = r271892 * r271894;
        double r271896 = r271893 * r271895;
        double r271897 = r271891 + r271896;
        double r271898 = log(r271897);
        double r271899 = t;
        double r271900 = r271898 / r271899;
        double r271901 = r271888 - r271900;
        double r271902 = 9.274120679861768e-221;
        bool r271903 = r271885 <= r271902;
        double r271904 = 0.5;
        double r271905 = 2.0;
        double r271906 = pow(r271885, r271905);
        double r271907 = r271906 * r271893;
        double r271908 = r271907 * r271892;
        double r271909 = r271885 * r271890;
        double r271910 = log(r271889);
        double r271911 = fma(r271889, r271909, r271910);
        double r271912 = fma(r271904, r271908, r271911);
        double r271913 = r271912 / r271899;
        double r271914 = r271888 - r271913;
        double r271915 = 0.5;
        double r271916 = r271906 * r271890;
        double r271917 = fma(r271885, r271890, r271889);
        double r271918 = fma(r271915, r271916, r271917);
        double r271919 = log(r271918);
        double r271920 = r271919 / r271899;
        double r271921 = r271888 - r271920;
        double r271922 = r271903 ? r271914 : r271921;
        double r271923 = r271887 ? r271901 : r271922;
        return r271923;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original25.2
Target16.4
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -0.4984438493979181

    1. Initial program 11.2

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

    3. Applied add-cube-cbrt11.2

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot e^{z}\right)}{t}\]

    4. Applied associate-*l*11.2

      \[\leadsto x - \frac{\log \left(\left(1 - y\right) + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)}\right)}{t}\]

    if -0.4984438493979181 < z < 9.274120679861768e-221

    1. Initial program 31.5

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 6.2

      \[\leadsto x - \frac{\color{blue}{0.5 \cdot \left({z}^{2} \cdot y\right) + \left(1 \cdot \left(z \cdot y\right) + \log 1\right)}}{t}\]

    3. Simplified6.2

      \[\leadsto x - \frac{\color{blue}{\mathsf{fma}\left(0.5, {z}^{2} \cdot y, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}}{t}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt6.2

      \[\leadsto x - \frac{\mathsf{fma}\left(0.5, {z}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\]

    6. Applied associate-*r*6.2

      \[\leadsto x - \frac{\mathsf{fma}\left(0.5, \color{blue}{\left({z}^{2} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}}, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\]

    if 9.274120679861768e-221 < z

    1. Initial program 31.5

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 11.9

      \[\leadsto x - \frac{\log \color{blue}{\left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + \left(z \cdot y + 1\right)\right)}}{t}\]

    3. Simplified11.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.49844384939791808:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 9.2741206798617679 \cdot 10^{-221}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(0.5, \left({z}^{2} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}, \mathsf{fma}\left(1, z \cdot y, \log 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\frac{1}{2}, {z}^{2} \cdot y, \mathsf{fma}\left(z, y, 1\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t)))

  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))