Average Error: 25.2 → 9.1
Time: 7.8s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.49844384939791808:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 9.2741206798617679 \cdot 10^{-221}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.49844384939791808:\\
\;\;\;\;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 9.2741206798617679 \cdot 10^{-221}:\\
\;\;\;\;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(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r335450 = x;
        double r335451 = 1.0;
        double r335452 = y;
        double r335453 = r335451 - r335452;
        double r335454 = z;
        double r335455 = exp(r335454);
        double r335456 = r335452 * r335455;
        double r335457 = r335453 + r335456;
        double r335458 = log(r335457);
        double r335459 = t;
        double r335460 = r335458 / r335459;
        double r335461 = r335450 - r335460;
        return r335461;
}
↓
double f(double x, double y, double z, double t) {
        double r335462 = z;
        double r335463 = -0.4984438493979181;
        bool r335464 = r335462 <= r335463;
        double r335465 = x;
        double r335466 = 1.0;
        double r335467 = y;
        double r335468 = r335466 - r335467;
        double r335469 = cbrt(r335467);
        double r335470 = r335469 * r335469;
        double r335471 = exp(r335462);
        double r335472 = r335469 * r335471;
        double r335473 = r335470 * r335472;
        double r335474 = r335468 + r335473;
        double r335475 = log(r335474);
        double r335476 = t;
        double r335477 = r335475 / r335476;
        double r335478 = r335465 - r335477;
        double r335479 = 9.274120679861768e-221;
        bool r335480 = r335462 <= r335479;
        double r335481 = log(r335466);
        double r335482 = 0.5;
        double r335483 = 2.0;
        double r335484 = pow(r335462, r335483);
        double r335485 = r335482 * r335484;
        double r335486 = r335466 * r335462;
        double r335487 = r335485 + r335486;
        double r335488 = r335467 * r335487;
        double r335489 = r335481 + r335488;
        double r335490 = r335489 / r335476;
        double r335491 = r335465 - r335490;
        double r335492 = 0.5;
        double r335493 = r335492 * r335484;
        double r335494 = r335493 + r335462;
        double r335495 = r335467 * r335494;
        double r335496 = r335466 + r335495;
        double r335497 = log(r335496);
        double r335498 = r335497 / r335476;
        double r335499 = r335465 - r335498;
        double r335500 = r335480 ? r335491 : r335499;
        double r335501 = r335464 ? r335478 : r335500;
        return r335501;
}

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

Derivation

  1. Split input into 3 regimes

  2. if z < -0.4984438493979181

    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 add-cube-cbrt11.2

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

    4. Applied associate-*l*11.2

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

    if -0.4984438493979181 < z < 9.274120679861768e-221

    1. Initial program 31.5

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

    3. Applied add-cube-cbrt26.0

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

    4. Applied associate-*l*26.0

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

    5. Taylor expanded around 0 6.2

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

    6. Simplified6.2

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

    if 9.274120679861768e-221 < z

    1. Initial program 31.5

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

    2. Taylor expanded around 0 11.9

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

    3. Simplified11.9

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.49844384939791808:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le 9.2741206798617679 \cdot 10^{-221}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + y \cdot \left(\frac{1}{2} \cdot {z}^{2} + z\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))