Average Error: 24.8 → 8.3
Time: 9.0s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\
\;\;\;\;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{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\
\;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r339415 = x;
        double r339416 = 1.0;
        double r339417 = y;
        double r339418 = r339416 - r339417;
        double r339419 = z;
        double r339420 = exp(r339419);
        double r339421 = r339417 * r339420;
        double r339422 = r339418 + r339421;
        double r339423 = log(r339422);
        double r339424 = t;
        double r339425 = r339423 / r339424;
        double r339426 = r339415 - r339425;
        return r339426;
}
↓
double f(double x, double y, double z, double t) {
        double r339427 = z;
        double r339428 = -5.050049854843211e-06;
        bool r339429 = r339427 <= r339428;
        double r339430 = x;
        double r339431 = 1.0;
        double r339432 = y;
        double r339433 = r339431 - r339432;
        double r339434 = cbrt(r339432);
        double r339435 = r339434 * r339434;
        double r339436 = exp(r339427);
        double r339437 = r339434 * r339436;
        double r339438 = r339435 * r339437;
        double r339439 = r339433 + r339438;
        double r339440 = log(r339439);
        double r339441 = t;
        double r339442 = r339440 / r339441;
        double r339443 = r339430 - r339442;
        double r339444 = 1.8264381599139905e-113;
        bool r339445 = r339427 <= r339444;
        double r339446 = r339427 * r339432;
        double r339447 = r339446 / r339441;
        double r339448 = r339431 * r339447;
        double r339449 = log(r339431);
        double r339450 = r339449 / r339441;
        double r339451 = 0.5;
        double r339452 = 2.0;
        double r339453 = pow(r339427, r339452);
        double r339454 = r339453 * r339432;
        double r339455 = r339454 / r339441;
        double r339456 = r339451 * r339455;
        double r339457 = r339450 + r339456;
        double r339458 = r339448 + r339457;
        double r339459 = r339430 - r339458;
        double r339460 = 0.5;
        double r339461 = r339460 * r339453;
        double r339462 = r339461 + r339427;
        double r339463 = r339432 * r339462;
        double r339464 = r339431 + r339463;
        double r339465 = log(r339464);
        double r339466 = r339465 / r339441;
        double r339467 = r339430 - r339466;
        double r339468 = r339445 ? r339459 : r339467;
        double r339469 = r339429 ? r339443 : r339468;
        return r339469;
}

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
Target15.8
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -5.050049854843211e-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 add-cube-cbrt11.9

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

    4. Applied associate-*l*11.9

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

    if -5.050049854843211e-06 < z < 1.8264381599139905e-113

    1. Initial program 30.6

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

    2. Taylor expanded around 0 5.8

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

    3. Simplified5.8

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

    5. Applied add-cube-cbrt5.9

      \[\leadsto x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

    6. Applied *-un-lft-identity5.9

      \[\leadsto x - \frac{\color{blue}{1 \cdot \left(\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)\right)}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

    7. Applied times-frac5.9

      \[\leadsto x - \color{blue}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{\sqrt[3]{t}}}\]

    8. Taylor expanded around inf 5.8

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

    if 1.8264381599139905e-113 < z

    1. Initial program 28.7

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

    2. Taylor expanded around 0 11.8

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

    3. Simplified11.8

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.050049854843210856672648667586855708578 \cdot 10^{-6}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 1.826438159913990508737942027076897606014 \cdot 10^{-113}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))