Average Error: 25.0 → 8.5
Time: 43.1s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.910423669767732067232418602387156170153 \cdot 10^{-4}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -6.320708783077287419620562923534616061602 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\ \mathbf{elif}\;z \le 1.650990275674465671078628579622610486763 \cdot 10^{-159}:\\ \;\;\;\;x - \left(\frac{\log 1}{t} + 1 \cdot \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.910423669767732067232418602387156170153 \cdot 10^{-4}:\\
\;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\

\mathbf{elif}\;z \le -6.320708783077287419620562923534616061602 \cdot 10^{-143}:\\
\;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\

\mathbf{elif}\;z \le 1.650990275674465671078628579622610486763 \cdot 10^{-159}:\\
\;\;\;\;x - \left(\frac{\log 1}{t} + 1 \cdot \frac{y \cdot z}{t}\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r15810741 = x;
        double r15810742 = 1.0;
        double r15810743 = y;
        double r15810744 = r15810742 - r15810743;
        double r15810745 = z;
        double r15810746 = exp(r15810745);
        double r15810747 = r15810743 * r15810746;
        double r15810748 = r15810744 + r15810747;
        double r15810749 = log(r15810748);
        double r15810750 = t;
        double r15810751 = r15810749 / r15810750;
        double r15810752 = r15810741 - r15810751;
        return r15810752;
}
↓
double f(double x, double y, double z, double t) {
        double r15810753 = z;
        double r15810754 = -0.0001910423669767732;
        bool r15810755 = r15810753 <= r15810754;
        double r15810756 = x;
        double r15810757 = 1.0;
        double r15810758 = y;
        double r15810759 = exp(r15810753);
        double r15810760 = r15810758 * r15810759;
        double r15810761 = r15810758 - r15810760;
        double r15810762 = r15810757 - r15810761;
        double r15810763 = log(r15810762);
        double r15810764 = t;
        double r15810765 = r15810763 / r15810764;
        double r15810766 = r15810756 - r15810765;
        double r15810767 = -6.320708783077287e-143;
        bool r15810768 = r15810753 <= r15810767;
        double r15810769 = r15810758 * r15810753;
        double r15810770 = -r15810769;
        double r15810771 = -0.5;
        double r15810772 = r15810753 * r15810769;
        double r15810773 = r15810771 * r15810772;
        double r15810774 = r15810753 * r15810753;
        double r15810775 = -0.16666666666666666;
        double r15810776 = r15810775 * r15810769;
        double r15810777 = r15810774 * r15810776;
        double r15810778 = r15810773 + r15810777;
        double r15810779 = r15810770 + r15810778;
        double r15810780 = r15810757 - r15810779;
        double r15810781 = log(r15810780);
        double r15810782 = r15810781 / r15810764;
        double r15810783 = r15810756 - r15810782;
        double r15810784 = 1.6509902756744657e-159;
        bool r15810785 = r15810753 <= r15810784;
        double r15810786 = log(r15810757);
        double r15810787 = r15810786 / r15810764;
        double r15810788 = r15810769 / r15810764;
        double r15810789 = r15810757 * r15810788;
        double r15810790 = r15810787 + r15810789;
        double r15810791 = r15810756 - r15810790;
        double r15810792 = r15810785 ? r15810791 : r15810783;
        double r15810793 = r15810768 ? r15810783 : r15810792;
        double r15810794 = r15810755 ? r15810766 : r15810793;
        return r15810794;
}

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

Original25.0
Target16.2
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -0.0001910423669767732

    1. Initial program 11.5

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

    3. Applied associate-+l-11.5

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

    if -0.0001910423669767732 < z < -6.320708783077287e-143 or 1.6509902756744657e-159 < z

    1. Initial program 29.5

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

    3. Applied associate-+l-18.5

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

    4. Taylor expanded around 0 11.5

      \[\leadsto x - \frac{\log \left(1 - \color{blue}{\left(-\left(\frac{1}{6} \cdot \left({z}^{3} \cdot y\right) + \left(z \cdot y + \frac{1}{2} \cdot \left({z}^{2} \cdot y\right)\right)\right)\right)}\right)}{t}\]

    5. Simplified11.5

      \[\leadsto x - \frac{\log \left(1 - \color{blue}{\left(-\left(y \cdot z + \left(\left(\left(y \cdot z\right) \cdot z\right) \cdot \frac{1}{2} + \left(\frac{1}{6} \cdot \left(y \cdot z\right)\right) \cdot \left(z \cdot z\right)\right)\right)\right)}\right)}{t}\]

    if -6.320708783077287e-143 < z < 1.6509902756744657e-159

    1. Initial program 31.8

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

    2. Taylor expanded around 0 3.8

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.910423669767732067232418602387156170153 \cdot 10^{-4}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(y - y \cdot e^{z}\right)\right)}{t}\\ \mathbf{elif}\;z \le -6.320708783077287419620562923534616061602 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\ \mathbf{elif}\;z \le 1.650990275674465671078628579622610486763 \cdot 10^{-159}:\\ \;\;\;\;x - \left(\frac{\log 1}{t} + 1 \cdot \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 - \left(\left(-y \cdot z\right) + \left(\frac{-1}{2} \cdot \left(z \cdot \left(y \cdot z\right)\right) + \left(z \cdot z\right) \cdot \left(\frac{-1}{6} \cdot \left(y \cdot z\right)\right)\right)\right)\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))