Average Error: 25.5 → 9.0
Time: 8.9s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.999999999999998779:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t}}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.999999999999998779:\\
\;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t}}\\

\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 ((((double) exp(z)) <= 0.9999999999999988)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))) * ((double) (1.0 / t))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (((double) (((double) cbrt(((double) (((double) log(1.0)) + ((double) (y * ((double) (((double) (0.5 * ((double) pow(z, 2.0)))) + ((double) (1.0 * z)))))))))) * ((double) cbrt(((double) (((double) log(1.0)) + ((double) (y * ((double) (((double) (0.5 * ((double) pow(z, 2.0)))) + ((double) (1.0 * z)))))))))))) / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))) * ((double) (((double) cbrt(((double) (((double) log(1.0)) + ((double) (y * ((double) (((double) (0.5 * ((double) pow(z, 2.0)))) + ((double) (1.0 * z)))))))))) / ((double) cbrt(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

Original25.5
Target16.4
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.999999999999998779

    1. Initial program 12.7

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

    3. Applied div-inv12.7

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

    if 0.999999999999998779 < (exp z)

    1. Initial program 31.2

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

    2. Taylor expanded around 0 7.1

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

    3. Simplified7.1

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

    5. Applied add-cube-cbrt7.3

      \[\leadsto x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

    6. Applied add-cube-cbrt7.3

      \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}\right) \cdot \sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

    7. Applied times-frac7.3

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.999999999999998779:\\ \;\;\;\;x - \log \left(\left(1 - y\right) + y \cdot e^{z}\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)} \cdot \sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}{\sqrt[3]{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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)))