Average Error: 24.9 → 9.1
Time: 22.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -55420076014679.5:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -55420076014679.5:\\
\;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r234696 = x;
        double r234697 = 1.0;
        double r234698 = y;
        double r234699 = r234697 - r234698;
        double r234700 = z;
        double r234701 = exp(r234700);
        double r234702 = r234698 * r234701;
        double r234703 = r234699 + r234702;
        double r234704 = log(r234703);
        double r234705 = t;
        double r234706 = r234704 / r234705;
        double r234707 = r234696 - r234706;
        return r234707;
}
↓
double f(double x, double y, double z, double t) {
        double r234708 = z;
        double r234709 = -55420076014679.5;
        bool r234710 = r234708 <= r234709;
        double r234711 = x;
        double r234712 = 1.0;
        double r234713 = y;
        double r234714 = r234712 - r234713;
        double r234715 = exp(r234708);
        double r234716 = r234713 * r234715;
        double r234717 = r234714 + r234716;
        double r234718 = log(r234717);
        double r234719 = 1.0;
        double r234720 = t;
        double r234721 = r234719 / r234720;
        double r234722 = r234718 * r234721;
        double r234723 = r234711 - r234722;
        double r234724 = r234708 * r234713;
        double r234725 = r234724 / r234720;
        double r234726 = r234712 * r234725;
        double r234727 = log(r234712);
        double r234728 = r234727 / r234720;
        double r234729 = 0.5;
        double r234730 = 2.0;
        double r234731 = pow(r234708, r234730);
        double r234732 = r234731 * r234713;
        double r234733 = r234732 / r234720;
        double r234734 = r234729 * r234733;
        double r234735 = r234728 + r234734;
        double r234736 = r234726 + r234735;
        double r234737 = r234711 - r234736;
        double r234738 = r234710 ? r234723 : r234737;
        return r234738;
}

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

Original24.9
Target16.2
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -55420076014679.5

    1. Initial program 11.3

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

    3. Applied div-inv11.3

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

    if -55420076014679.5 < z

    1. Initial program 30.4

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

    2. Taylor expanded around 0 8.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -55420076014679.5:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{{z}^{2} \cdot y}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.88746230882079466e119) (- (- 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)))