Average Error: 24.7 → 8.5
Time: 12.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le \frac{8963761139109503}{9007199254740992}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log 1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + 1 \cdot z\right)}}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le \frac{8963761139109503}{9007199254740992}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r186681 = x;
        double r186682 = 1.0;
        double r186683 = y;
        double r186684 = r186682 - r186683;
        double r186685 = z;
        double r186686 = exp(r186685);
        double r186687 = r186683 * r186686;
        double r186688 = r186684 + r186687;
        double r186689 = log(r186688);
        double r186690 = t;
        double r186691 = r186689 / r186690;
        double r186692 = r186681 - r186691;
        return r186692;
}
↓
double f(double x, double y, double z, double t) {
        double r186693 = z;
        double r186694 = exp(r186693);
        double r186695 = 8963761139109503.0;
        double r186696 = 9007199254740992.0;
        double r186697 = r186695 / r186696;
        bool r186698 = r186694 <= r186697;
        double r186699 = x;
        double r186700 = 1.0;
        double r186701 = t;
        double r186702 = 1.0;
        double r186703 = y;
        double r186704 = r186702 - r186703;
        double r186705 = r186703 * r186694;
        double r186706 = r186704 + r186705;
        double r186707 = log(r186706);
        double r186708 = r186701 / r186707;
        double r186709 = r186700 / r186708;
        double r186710 = r186699 - r186709;
        double r186711 = log(r186702);
        double r186712 = 2.0;
        double r186713 = r186702 / r186712;
        double r186714 = 2.0;
        double r186715 = pow(r186693, r186714);
        double r186716 = r186713 * r186715;
        double r186717 = r186702 * r186693;
        double r186718 = r186716 + r186717;
        double r186719 = r186703 * r186718;
        double r186720 = r186711 + r186719;
        double r186721 = r186701 / r186720;
        double r186722 = r186700 / r186721;
        double r186723 = r186699 - r186722;
        double r186724 = r186698 ? r186710 : r186723;
        return r186724;
}

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

Original24.7
Target15.9
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.9951774003879591

    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 clear-num11.2

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

    if 0.9951774003879591 < (exp z)

    1. Initial program 30.6

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified7.3

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

    5. Applied clear-num7.3

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le \frac{8963761139109503}{9007199254740992}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log 1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + 1 \cdot z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.88746230882079466e119) (- (- 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)))