Average Error: 24.2 → 8.5
Time: 34.4s
Precision: 64
\[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -804280505547.8507:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -804280505547.8507:\\
\;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\

\mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\
\;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r18511559 = x;
        double r18511560 = 1.0;
        double r18511561 = y;
        double r18511562 = r18511560 - r18511561;
        double r18511563 = z;
        double r18511564 = exp(r18511563);
        double r18511565 = r18511561 * r18511564;
        double r18511566 = r18511562 + r18511565;
        double r18511567 = log(r18511566);
        double r18511568 = t;
        double r18511569 = r18511567 / r18511568;
        double r18511570 = r18511559 - r18511569;
        return r18511570;
}
↓
double f(double x, double y, double z, double t) {
        double r18511571 = z;
        double r18511572 = -804280505547.8507;
        bool r18511573 = r18511571 <= r18511572;
        double r18511574 = x;
        double r18511575 = exp(r18511571);
        double r18511576 = y;
        double r18511577 = r18511575 * r18511576;
        double r18511578 = 1.0;
        double r18511579 = r18511578 - r18511576;
        double r18511580 = r18511577 + r18511579;
        double r18511581 = log(r18511580);
        double r18511582 = 1.0;
        double r18511583 = t;
        double r18511584 = r18511582 / r18511583;
        double r18511585 = r18511581 * r18511584;
        double r18511586 = r18511574 - r18511585;
        double r18511587 = 2.5098281878527637e-138;
        bool r18511588 = r18511571 <= r18511587;
        double r18511589 = 0.5;
        double r18511590 = r18511571 * r18511589;
        double r18511591 = r18511590 + r18511578;
        double r18511592 = r18511591 * r18511571;
        double r18511593 = r18511592 * r18511576;
        double r18511594 = log(r18511578);
        double r18511595 = r18511593 + r18511594;
        double r18511596 = r18511595 / r18511583;
        double r18511597 = r18511574 - r18511596;
        double r18511598 = 0.5;
        double r18511599 = r18511598 * r18511571;
        double r18511600 = r18511571 * r18511599;
        double r18511601 = r18511600 + r18511571;
        double r18511602 = r18511576 * r18511601;
        double r18511603 = r18511602 + r18511578;
        double r18511604 = log(r18511603);
        double r18511605 = r18511604 / r18511583;
        double r18511606 = r18511574 - r18511605;
        double r18511607 = r18511588 ? r18511597 : r18511606;
        double r18511608 = r18511573 ? r18511586 : r18511607;
        return r18511608;
}

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

Original24.2
Target15.9
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2.0}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1.0 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -804280505547.8507

    1. Initial program 10.8

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]
    2. Using strategy rm

    3. Applied div-inv10.8

      \[\leadsto x - \color{blue}{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}}\]

    if -804280505547.8507 < z < 2.5098281878527637e-138

    1. Initial program 29.2

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 6.5

      \[\leadsto x - \frac{\color{blue}{\log 1.0 + \left(1.0 \cdot \left(z \cdot y\right) + 0.5 \cdot \left({z}^{2} \cdot y\right)\right)}}{t}\]

    3. Simplified6.5

      \[\leadsto x - \frac{\color{blue}{\log 1.0 + \left(z \cdot \left(1.0 + z \cdot 0.5\right)\right) \cdot y}}{t}\]

    if 2.5098281878527637e-138 < z

    1. Initial program 29.4

      \[x - \frac{\log \left(\left(1.0 - y\right) + y \cdot e^{z}\right)}{t}\]

    2. Taylor expanded around 0 12.1

      \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot y + \left(\frac{1}{2} \cdot \left({z}^{2} \cdot y\right) + 1.0\right)\right)}}{t}\]

    3. Simplified12.1

      \[\leadsto x - \frac{\log \color{blue}{\left(y \cdot \left(\left(z \cdot \frac{1}{2}\right) \cdot z + z\right) + 1.0\right)}}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -804280505547.8507:\\ \;\;\;\;x - \log \left(e^{z} \cdot y + \left(1.0 - y\right)\right) \cdot \frac{1}{t}\\ \mathbf{elif}\;z \le 2.5098281878527637 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{\left(\left(z \cdot 0.5 + 1.0\right) \cdot z\right) \cdot y + \log 1.0}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(y \cdot \left(z \cdot \left(\frac{1}{2} \cdot z\right) + z\right) + 1.0\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))