Average Error: 25.5 → 9.2
Time: 6.8s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\
\;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\

\mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\
\;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r311885 = x;
        double r311886 = 1.0;
        double r311887 = y;
        double r311888 = r311886 - r311887;
        double r311889 = z;
        double r311890 = exp(r311889);
        double r311891 = r311887 * r311890;
        double r311892 = r311888 + r311891;
        double r311893 = log(r311892);
        double r311894 = t;
        double r311895 = r311893 / r311894;
        double r311896 = r311885 - r311895;
        return r311896;
}
↓
double f(double x, double y, double z, double t) {
        double r311897 = z;
        double r311898 = -4.2011862993899546e-06;
        bool r311899 = r311897 <= r311898;
        double r311900 = x;
        double r311901 = 1.0;
        double r311902 = exp(r311897);
        double r311903 = 1.0;
        double r311904 = r311902 - r311903;
        double r311905 = y;
        double r311906 = r311904 * r311905;
        double r311907 = r311901 + r311906;
        double r311908 = log(r311907);
        double r311909 = t;
        double r311910 = r311908 / r311909;
        double r311911 = r311900 - r311910;
        double r311912 = 3.4446276387824773e-225;
        bool r311913 = r311897 <= r311912;
        double r311914 = log(r311901);
        double r311915 = 0.5;
        double r311916 = 2.0;
        double r311917 = pow(r311897, r311916);
        double r311918 = r311915 * r311917;
        double r311919 = r311901 * r311897;
        double r311920 = r311918 + r311919;
        double r311921 = r311905 * r311920;
        double r311922 = r311914 + r311921;
        double r311923 = r311922 / r311909;
        double r311924 = r311900 - r311923;
        double r311925 = 0.16666666666666666;
        double r311926 = 3.0;
        double r311927 = pow(r311897, r311926);
        double r311928 = r311925 * r311927;
        double r311929 = 0.5;
        double r311930 = r311929 * r311917;
        double r311931 = r311928 + r311930;
        double r311932 = r311897 + r311931;
        double r311933 = r311905 * r311932;
        double r311934 = r311901 + r311933;
        double r311935 = log(r311934);
        double r311936 = r311935 / r311909;
        double r311937 = r311900 - r311936;
        double r311938 = r311913 ? r311924 : r311937;
        double r311939 = r311899 ? r311911 : r311938;
        return r311939;
}

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

Original25.5
Target16.5
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -4.2011862993899546e-06

    1. Initial program 11.9

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

    3. Applied sub-neg11.9

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

    4. Applied associate-+l+11.9

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

    5. Simplified11.9

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

    if -4.2011862993899546e-06 < z < 3.4446276387824773e-225

    1. Initial program 31.7

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

    2. Taylor expanded around 0 6.1

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

    3. Simplified6.1

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

    if 3.4446276387824773e-225 < z

    1. Initial program 31.1

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

    3. Applied sub-neg31.1

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

    4. Applied associate-+l+18.1

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

    5. Simplified18.1

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

    6. Taylor expanded around 0 11.5

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left(\frac{1}{6} \cdot \left({z}^{3} \cdot y\right) + \left(z \cdot y + \frac{1}{2} \cdot \left({z}^{2} \cdot y\right)\right)\right)}\right)}{t}\]

    7. Simplified11.5

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)}\right)}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.2011862993899546 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} - 1\right) \cdot y\right)}{t}\\ \mathbf{elif}\;z \le 3.44462763878247732 \cdot 10^{-225}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))