Average Error: 25.1 → 8.4
Time: 31.9s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.175834169922838207843406872122183162579 \cdot 10^{-23}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\ \mathbf{elif}\;z \le 3.613492309110387389235464214276966761737 \cdot 10^{-87}:\\ \;\;\;\;x - \left(\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) \cdot 1 + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\left(z \cdot z\right) \cdot \frac{1}{2} + z\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.175834169922838207843406872122183162579 \cdot 10^{-23}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\

\mathbf{elif}\;z \le 3.613492309110387389235464214276966761737 \cdot 10^{-87}:\\
\;\;\;\;x - \left(\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) \cdot 1 + \frac{\log 1}{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r13399827 = x;
        double r13399828 = 1.0;
        double r13399829 = y;
        double r13399830 = r13399828 - r13399829;
        double r13399831 = z;
        double r13399832 = exp(r13399831);
        double r13399833 = r13399829 * r13399832;
        double r13399834 = r13399830 + r13399833;
        double r13399835 = log(r13399834);
        double r13399836 = t;
        double r13399837 = r13399835 / r13399836;
        double r13399838 = r13399827 - r13399837;
        return r13399838;
}
↓
double f(double x, double y, double z, double t) {
        double r13399839 = z;
        double r13399840 = -2.1758341699228382e-23;
        bool r13399841 = r13399839 <= r13399840;
        double r13399842 = x;
        double r13399843 = 1.0;
        double r13399844 = y;
        double r13399845 = r13399843 - r13399844;
        double r13399846 = cbrt(r13399844);
        double r13399847 = r13399846 * r13399846;
        double r13399848 = exp(r13399839);
        double r13399849 = r13399848 * r13399846;
        double r13399850 = r13399847 * r13399849;
        double r13399851 = r13399845 + r13399850;
        double r13399852 = log(r13399851);
        double r13399853 = t;
        double r13399854 = r13399852 / r13399853;
        double r13399855 = r13399842 - r13399854;
        double r13399856 = 3.6134923091103874e-87;
        bool r13399857 = r13399839 <= r13399856;
        double r13399858 = cbrt(r13399853);
        double r13399859 = r13399858 * r13399858;
        double r13399860 = r13399839 / r13399859;
        double r13399861 = r13399844 / r13399858;
        double r13399862 = r13399860 * r13399861;
        double r13399863 = r13399862 * r13399843;
        double r13399864 = log(r13399843);
        double r13399865 = r13399864 / r13399853;
        double r13399866 = r13399863 + r13399865;
        double r13399867 = r13399842 - r13399866;
        double r13399868 = r13399839 * r13399839;
        double r13399869 = 0.5;
        double r13399870 = r13399868 * r13399869;
        double r13399871 = r13399870 + r13399839;
        double r13399872 = r13399844 * r13399871;
        double r13399873 = r13399843 + r13399872;
        double r13399874 = log(r13399873);
        double r13399875 = r13399874 / r13399853;
        double r13399876 = r13399842 - r13399875;
        double r13399877 = r13399857 ? r13399867 : r13399876;
        double r13399878 = r13399841 ? r13399855 : r13399877;
        return r13399878;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.1
Target16.4
Herbie8.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.1758341699228382e-23

    1. Initial program 12.5

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

    3. Applied add-cube-cbrt12.5

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

    4. Applied associate-*l*12.5

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

    if -2.1758341699228382e-23 < z < 3.6134923091103874e-87

    1. Initial program 31.6

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

    2. Taylor expanded around 0 5.8

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

    3. Simplified5.8

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

    4. Taylor expanded around 0 5.8

      \[\leadsto \color{blue}{x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt6.0

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

    7. Applied times-frac5.5

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

    if 3.6134923091103874e-87 < z

    1. Initial program 28.7

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

    2. Taylor expanded around 0 11.8

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

    3. Simplified11.8

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.175834169922838207843406872122183162579 \cdot 10^{-23}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(e^{z} \cdot \sqrt[3]{y}\right)\right)}{t}\\ \mathbf{elif}\;z \le 3.613492309110387389235464214276966761737 \cdot 10^{-87}:\\ \;\;\;\;x - \left(\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) \cdot 1 + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\left(z \cdot z\right) \cdot \frac{1}{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))