Average Error: 25.2 → 9.0
Time: 23.7s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.9999993813411593990281289734411984682083:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)\right) \cdot \frac{1}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.9999993813411593990281289734411984682083:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \left(\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)\right) \cdot \frac{1}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r219508 = x;
        double r219509 = 1.0;
        double r219510 = y;
        double r219511 = r219509 - r219510;
        double r219512 = z;
        double r219513 = exp(r219512);
        double r219514 = r219510 * r219513;
        double r219515 = r219511 + r219514;
        double r219516 = log(r219515);
        double r219517 = t;
        double r219518 = r219516 / r219517;
        double r219519 = r219508 - r219518;
        return r219519;
}
↓
double f(double x, double y, double z, double t) {
        double r219520 = z;
        double r219521 = exp(r219520);
        double r219522 = 0.9999993813411594;
        bool r219523 = r219521 <= r219522;
        double r219524 = x;
        double r219525 = 1.0;
        double r219526 = y;
        double r219527 = r219525 - r219526;
        double r219528 = cbrt(r219526);
        double r219529 = r219528 * r219528;
        double r219530 = r219528 * r219521;
        double r219531 = r219529 * r219530;
        double r219532 = r219527 + r219531;
        double r219533 = log(r219532);
        double r219534 = t;
        double r219535 = r219533 / r219534;
        double r219536 = r219524 - r219535;
        double r219537 = log(r219525);
        double r219538 = 0.5;
        double r219539 = 2.0;
        double r219540 = pow(r219520, r219539);
        double r219541 = r219538 * r219540;
        double r219542 = r219525 * r219520;
        double r219543 = r219541 + r219542;
        double r219544 = r219526 * r219543;
        double r219545 = r219537 + r219544;
        double r219546 = 1.0;
        double r219547 = r219546 / r219534;
        double r219548 = r219545 * r219547;
        double r219549 = r219524 - r219548;
        double r219550 = r219523 ? r219536 : r219549;
        return r219550;
}

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

Original25.2
Target16.4
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.9999993813411594

    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 0.9999993813411594 < (exp 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 7.4

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

    3. Simplified7.4

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

    5. Applied div-inv7.5

      \[\leadsto x - \color{blue}{\left(\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)\right) \cdot \frac{1}{t}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.9999993813411593990281289734411984682083:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)\right) \cdot \frac{1}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))