Average Error: 25.1 → 8.7
Time: 13.6s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.998271172497884218:\\ \;\;\;\;x - \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)} \cdot \sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.998271172497884218:\\
\;\;\;\;x - \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)} \cdot \sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t}}\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (x - (((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 ((((double) exp(z)) <= 0.9982711724978842)) {
		VAR = ((double) (x - ((double) ((((double) (((double) cbrt(((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))))) * ((double) cbrt(((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))))))) / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * (((double) cbrt(((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))))) / ((double) cbrt(t)))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (1.0 * (((double) (z * y)) / t))) + (((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

Original25.1
Target16.0
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.998271172497884218

    1. Initial program 11.6

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

    3. Applied add-cube-cbrt11.7

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

    4. Applied add-cube-cbrt11.8

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

    5. Applied times-frac11.8

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

    if 0.998271172497884218 < (exp z)

    1. Initial program 31.1

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

    2. Taylor expanded around 0 7.3

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

    3. Simplified7.3

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

    4. Taylor expanded around 0 7.4

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.998271172497884218:\\ \;\;\;\;x - \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)} \cdot \sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}{\sqrt[3]{t}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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 (/ (/ (neg 0.5) (* y t)) (* z z))) (* (/ (neg 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)))