Average Error: 25.0 → 8.5
Time: 24.3s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.910423669767732067232418602387156170153 \cdot 10^{-4}:\\ \;\;\;\;x - \frac{\log \left(\left(y \cdot e^{z} - y\right) + 1\right)}{t}\\ \mathbf{elif}\;z \le -6.320708783077287419620562923534616061602 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{\log \left(\left(\left(z + \frac{1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(z \cdot z\right) \cdot \frac{1}{2}\right) \cdot y + 1\right)}{t}\\ \mathbf{elif}\;z \le 1.650990275674465671078628579622610486763 \cdot 10^{-159}:\\ \;\;\;\;x - \left(\frac{\log 1}{t} + \frac{y \cdot z}{t} \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\left(\left(z + \frac{1}{6} \cdot \left(z \cdot \left(z \cdot z\right)\right)\right) + \left(z \cdot z\right) \cdot \frac{1}{2}\right) \cdot y + 1\right)}{t}\\ \end{array}\]
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(\left(y \cdot e^{z} - y\right) + 1\right)}{t}\\

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

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

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

\end{array}
double f(double x, double y, double z, double t) {
        double r17436776 = x;
        double r17436777 = 1.0;
        double r17436778 = y;
        double r17436779 = r17436777 - r17436778;
        double r17436780 = z;
        double r17436781 = exp(r17436780);
        double r17436782 = r17436778 * r17436781;
        double r17436783 = r17436779 + r17436782;
        double r17436784 = log(r17436783);
        double r17436785 = t;
        double r17436786 = r17436784 / r17436785;
        double r17436787 = r17436776 - r17436786;
        return r17436787;
}
↓
double f(double x, double y, double z, double t) {
        double r17436788 = z;
        double r17436789 = -0.0001910423669767732;
        bool r17436790 = r17436788 <= r17436789;
        double r17436791 = x;
        double r17436792 = y;
        double r17436793 = exp(r17436788);
        double r17436794 = r17436792 * r17436793;
        double r17436795 = r17436794 - r17436792;
        double r17436796 = 1.0;
        double r17436797 = r17436795 + r17436796;
        double r17436798 = log(r17436797);
        double r17436799 = t;
        double r17436800 = r17436798 / r17436799;
        double r17436801 = r17436791 - r17436800;
        double r17436802 = -6.320708783077287e-143;
        bool r17436803 = r17436788 <= r17436802;
        double r17436804 = 0.16666666666666666;
        double r17436805 = r17436788 * r17436788;
        double r17436806 = r17436788 * r17436805;
        double r17436807 = r17436804 * r17436806;
        double r17436808 = r17436788 + r17436807;
        double r17436809 = 0.5;
        double r17436810 = r17436805 * r17436809;
        double r17436811 = r17436808 + r17436810;
        double r17436812 = r17436811 * r17436792;
        double r17436813 = r17436812 + r17436796;
        double r17436814 = log(r17436813);
        double r17436815 = r17436814 / r17436799;
        double r17436816 = r17436791 - r17436815;
        double r17436817 = 1.6509902756744657e-159;
        bool r17436818 = r17436788 <= r17436817;
        double r17436819 = log(r17436796);
        double r17436820 = r17436819 / r17436799;
        double r17436821 = r17436792 * r17436788;
        double r17436822 = r17436821 / r17436799;
        double r17436823 = r17436822 * r17436796;
        double r17436824 = r17436820 + r17436823;
        double r17436825 = r17436791 - r17436824;
        double r17436826 = r17436818 ? r17436825 : r17436816;
        double r17436827 = r17436803 ? r17436816 : r17436826;
        double r17436828 = r17436790 ? r17436801 : r17436827;
        return r17436828;
}

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 sub-neg11.5

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

    4. Applied associate-+l+11.5

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

    5. Simplified11.5

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

    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 sub-neg29.5

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

    4. Applied associate-+l+18.5

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

    5. Simplified18.5

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

    6. Taylor expanded around 0 11.5

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

    7. Simplified11.5

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

    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

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

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)))