Average Error: 25.1 → 8.6
Time: 51.7s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.556401213883932276978059049285856740204 \cdot 10^{-32}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) + \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.556401213883932276978059049285856740204 \cdot 10^{-32}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) + \frac{\log 1}{t}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r67969534 = x;
        double r67969535 = 1.0;
        double r67969536 = y;
        double r67969537 = r67969535 - r67969536;
        double r67969538 = z;
        double r67969539 = exp(r67969538);
        double r67969540 = r67969536 * r67969539;
        double r67969541 = r67969537 + r67969540;
        double r67969542 = log(r67969541);
        double r67969543 = t;
        double r67969544 = r67969542 / r67969543;
        double r67969545 = r67969534 - r67969544;
        return r67969545;
}
↓
double f(double x, double y, double z, double t) {
        double r67969546 = z;
        double r67969547 = -9.556401213883932e-32;
        bool r67969548 = r67969546 <= r67969547;
        double r67969549 = x;
        double r67969550 = 1.0;
        double r67969551 = y;
        double r67969552 = r67969550 - r67969551;
        double r67969553 = exp(r67969546);
        double r67969554 = r67969551 * r67969553;
        double r67969555 = cbrt(r67969554);
        double r67969556 = r67969555 * r67969555;
        double r67969557 = r67969556 * r67969555;
        double r67969558 = r67969552 + r67969557;
        double r67969559 = log(r67969558);
        double r67969560 = t;
        double r67969561 = r67969559 / r67969560;
        double r67969562 = r67969549 - r67969561;
        double r67969563 = cbrt(r67969560);
        double r67969564 = r67969563 * r67969563;
        double r67969565 = r67969546 / r67969564;
        double r67969566 = r67969551 / r67969563;
        double r67969567 = r67969565 * r67969566;
        double r67969568 = r67969550 * r67969567;
        double r67969569 = log(r67969550);
        double r67969570 = r67969569 / r67969560;
        double r67969571 = r67969568 + r67969570;
        double r67969572 = r67969549 - r67969571;
        double r67969573 = r67969548 ? r67969562 : r67969572;
        return r67969573;
}

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

Original25.1
Target16.3
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -9.556401213883932e-32

    1. Initial program 13.2

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

    3. Applied add-cube-cbrt13.1

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

    if -9.556401213883932e-32 < z

    1. Initial program 31.0

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

    2. Taylor expanded around 0 6.7

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

    3. Simplified6.7

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

    4. Taylor expanded around 0 6.7

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

    6. Applied add-cube-cbrt6.9

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

    7. Applied times-frac6.4

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.556401213883932276978059049285856740204 \cdot 10^{-32}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y \cdot e^{z}} \cdot \sqrt[3]{y \cdot e^{z}}\right) \cdot \sqrt[3]{y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right) + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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)))