Average Error: 25.0 → 8.8
Time: 6.7s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.4280022663401162 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{z}{\frac{t}{\sqrt[3]{y}}}\right) + \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.4280022663401162 \cdot 10^{-49}:\\
\;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))) / t))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -4.428002266340116e-49)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (((double) cbrt(y)) * ((double) exp(z)))))))))) / t))));
	} else {
		VAR = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (z / ((double) (t / ((double) cbrt(y)))))))))) + ((double) (((double) log(1.0)) / t))))));
	}
	return VAR;
}

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
Target16.4
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.428002266340116e-49

    1. Initial program 13.6

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

    3. Applied add-cube-cbrt13.4

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

    4. Applied associate-*l*13.4

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

    if -4.428002266340116e-49 < z

    1. Initial program 30.9

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

    2. Taylor expanded around 0 6.8

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

    3. Simplified6.8

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

    4. Taylor expanded around 0 6.9

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

    6. Applied associate-/l*8.3

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

    8. Applied add-cube-cbrt8.4

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

    9. Applied *-un-lft-identity8.4

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

    10. Applied times-frac8.4

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

    11. Applied *-un-lft-identity8.4

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

    12. Applied times-frac6.4

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

    13. Simplified6.4

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.4280022663401162 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{\log \left(\left(1 - y\right) + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot e^{z}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{z}{\frac{t}{\sqrt[3]{y}}}\right) + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+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)))