Average Error: 24.8 → 8.4
Time: 1.7m
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1479152.58950692159123718738555908203125:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(y \cdot e^{z} + \left(1 - y\right)\right)}}\\ \mathbf{elif}\;z \le -4.564121524611550729112059443357839172796 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{\log \left(\left(z + \frac{1}{2} \cdot \left(z \cdot z\right)\right) \cdot y + 1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(0.5 \cdot \frac{\left(z \cdot z\right) \cdot y}{t} + \left(\frac{\log 1}{t} + \left(\sqrt[3]{\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right)} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right) \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot y\right)\right)\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1479152.58950692159123718738555908203125:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(y \cdot e^{z} + \left(1 - y\right)\right)}}\\

\mathbf{elif}\;z \le -4.564121524611550729112059443357839172796 \cdot 10^{-64}:\\
\;\;\;\;x - \frac{\log \left(\left(z + \frac{1}{2} \cdot \left(z \cdot z\right)\right) \cdot y + 1\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \left(0.5 \cdot \frac{\left(z \cdot z\right) \cdot y}{t} + \left(\frac{\log 1}{t} + \left(\sqrt[3]{\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right)} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right) \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot y\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r14824907 = x;
        double r14824908 = 1.0;
        double r14824909 = y;
        double r14824910 = r14824908 - r14824909;
        double r14824911 = z;
        double r14824912 = exp(r14824911);
        double r14824913 = r14824909 * r14824912;
        double r14824914 = r14824910 + r14824913;
        double r14824915 = log(r14824914);
        double r14824916 = t;
        double r14824917 = r14824915 / r14824916;
        double r14824918 = r14824907 - r14824917;
        return r14824918;
}
↓
double f(double x, double y, double z, double t) {
        double r14824919 = z;
        double r14824920 = -1479152.5895069216;
        bool r14824921 = r14824919 <= r14824920;
        double r14824922 = x;
        double r14824923 = 1.0;
        double r14824924 = t;
        double r14824925 = y;
        double r14824926 = exp(r14824919);
        double r14824927 = r14824925 * r14824926;
        double r14824928 = 1.0;
        double r14824929 = r14824928 - r14824925;
        double r14824930 = r14824927 + r14824929;
        double r14824931 = log(r14824930);
        double r14824932 = r14824924 / r14824931;
        double r14824933 = r14824923 / r14824932;
        double r14824934 = r14824922 - r14824933;
        double r14824935 = -4.564121524611551e-64;
        bool r14824936 = r14824919 <= r14824935;
        double r14824937 = 0.5;
        double r14824938 = r14824919 * r14824919;
        double r14824939 = r14824937 * r14824938;
        double r14824940 = r14824919 + r14824939;
        double r14824941 = r14824940 * r14824925;
        double r14824942 = r14824941 + r14824928;
        double r14824943 = log(r14824942);
        double r14824944 = r14824943 / r14824924;
        double r14824945 = r14824922 - r14824944;
        double r14824946 = 0.5;
        double r14824947 = r14824938 * r14824925;
        double r14824948 = r14824947 / r14824924;
        double r14824949 = r14824946 * r14824948;
        double r14824950 = log(r14824928);
        double r14824951 = r14824950 / r14824924;
        double r14824952 = r14824928 * r14824919;
        double r14824953 = r14824952 / r14824924;
        double r14824954 = cbrt(r14824953);
        double r14824955 = r14824954 * r14824954;
        double r14824956 = r14824954 * r14824955;
        double r14824957 = cbrt(r14824956);
        double r14824958 = r14824957 * r14824954;
        double r14824959 = r14824954 * r14824925;
        double r14824960 = r14824958 * r14824959;
        double r14824961 = r14824951 + r14824960;
        double r14824962 = r14824949 + r14824961;
        double r14824963 = r14824922 - r14824962;
        double r14824964 = r14824936 ? r14824945 : r14824963;
        double r14824965 = r14824921 ? r14824934 : r14824964;
        return r14824965;
}

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

Original24.8
Target16.2
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 < -1479152.5895069216

    1. Initial program 11.4

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

    3. Applied clear-num11.4

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

    if -1479152.5895069216 < z < -4.564121524611551e-64

    1. Initial program 25.4

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

    2. Taylor expanded around 0 14.9

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

    3. Simplified14.9

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

    if -4.564121524611551e-64 < z

    1. Initial program 30.9

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified6.2

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

    5. Applied add-cube-cbrt6.4

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

    6. Applied associate-*l*6.4

      \[\leadsto x - \left(0.5 \cdot \frac{\left(z \cdot z\right) \cdot y}{t} + \left(\color{blue}{\left(\sqrt[3]{\frac{z \cdot 1}{t}} \cdot \sqrt[3]{\frac{z \cdot 1}{t}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot 1}{t}} \cdot y\right)} + \frac{\log 1}{t}\right)\right)\]
    7. Using strategy rm

    8. Applied add-cbrt-cube6.4

      \[\leadsto x - \left(0.5 \cdot \frac{\left(z \cdot z\right) \cdot y}{t} + \left(\left(\sqrt[3]{\frac{z \cdot 1}{t}} \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\frac{z \cdot 1}{t}} \cdot \sqrt[3]{\frac{z \cdot 1}{t}}\right) \cdot \sqrt[3]{\frac{z \cdot 1}{t}}}}\right) \cdot \left(\sqrt[3]{\frac{z \cdot 1}{t}} \cdot y\right) + \frac{\log 1}{t}\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1479152.58950692159123718738555908203125:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(y \cdot e^{z} + \left(1 - y\right)\right)}}\\ \mathbf{elif}\;z \le -4.564121524611550729112059443357839172796 \cdot 10^{-64}:\\ \;\;\;\;x - \frac{\log \left(\left(z + \frac{1}{2} \cdot \left(z \cdot z\right)\right) \cdot y + 1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(0.5 \cdot \frac{\left(z \cdot z\right) \cdot y}{t} + \left(\frac{\log 1}{t} + \left(\sqrt[3]{\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right)} \cdot \sqrt[3]{\frac{1 \cdot z}{t}}\right) \cdot \left(\sqrt[3]{\frac{1 \cdot z}{t}} \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))