Average Error: 24.2 → 8.5
Time: 32.8s
Precision: 64
\[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -804280505547.8507:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -804280505547.8507:\\
\;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\

\mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\
\;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r14825781 = x;
        double r14825782 = 1.0;
        double r14825783 = y;
        double r14825784 = r14825782 - r14825783;
        double r14825785 = z;
        double r14825786 = exp(r14825785);
        double r14825787 = r14825783 * r14825786;
        double r14825788 = r14825784 + r14825787;
        double r14825789 = log(r14825788);
        double r14825790 = t;
        double r14825791 = r14825789 / r14825790;
        double r14825792 = r14825781 - r14825791;
        return r14825792;
}
↓
double f(double x, double y, double z, double t) {
        double r14825793 = z;
        double r14825794 = -804280505547.8507;
        bool r14825795 = r14825793 <= r14825794;
        double r14825796 = x;
        double r14825797 = exp(r14825793);
        double r14825798 = y;
        double r14825799 = r14825797 * r14825798;
        double r14825800 = 1.0;
        double r14825801 = r14825800 - r14825798;
        double r14825802 = r14825799 + r14825801;
        double r14825803 = log(r14825802);
        double r14825804 = 1.0;
        double r14825805 = t;
        double r14825806 = r14825804 / r14825805;
        double r14825807 = r14825803 * r14825806;
        double r14825808 = r14825796 - r14825807;
        double r14825809 = 2.5098281878527637e-138;
        bool r14825810 = r14825793 <= r14825809;
        double r14825811 = 0.5;
        double r14825812 = r14825793 * r14825811;
        double r14825813 = r14825812 + r14825800;
        double r14825814 = r14825813 * r14825793;
        double r14825815 = r14825814 * r14825798;
        double r14825816 = log(r14825800);
        double r14825817 = r14825815 + r14825816;
        double r14825818 = r14825817 / r14825805;
        double r14825819 = r14825796 - r14825818;
        double r14825820 = 0.5;
        double r14825821 = r14825820 * r14825793;
        double r14825822 = r14825793 * r14825821;
        double r14825823 = r14825822 + r14825793;
        double r14825824 = r14825798 * r14825823;
        double r14825825 = r14825824 + r14825800;
        double r14825826 = log(r14825825);
        double r14825827 = r14825826 / r14825805;
        double r14825828 = r14825796 - r14825827;
        double r14825829 = r14825810 ? r14825819 : r14825828;
        double r14825830 = r14825795 ? r14825808 : r14825829;
        return r14825830;
}

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

Original24.2
Target15.9
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2.0}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1.0 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -804280505547.8507

    1. Initial program 10.8

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
    2. Using strategy rm

    3. Applied div-inv10.8

      \[\leadsto x - \color{blue}{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}}\]

    if -804280505547.8507 < z < 2.5098281878527637e-138

    1. Initial program 29.2

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 6.5

      \[\leadsto x - \frac{\color{blue}{\log 1.0 + \left(1.0 \cdot \left(z \cdot y\right) + 0.5 \cdot \left({z}^{2} \cdot y\right)\right)}}{t}\]

    3. Simplified6.5

      \[\leadsto x - \frac{\color{blue}{\log 1.0 + \left(z \cdot \left(1.0 + z \cdot 0.5\right)\right) \cdot y}}{t}\]

    if 2.5098281878527637e-138 < z

    1. Initial program 29.4

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 12.1

      \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot y + \left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + 1.0\right)\right)}}{t}\]

    3. Simplified12.1

      \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot \left(\left(z \cdot \frac{1}{2}\right) \cdot z + z\right) + 1.0\right)}}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -804280505547.8507:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))