Average Error: 25.0 → 8.8
Time: 8.8s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{t}}\right)\right) + \frac{\log 1}{t}\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.0:\\
\;\;\;\;x - \frac{\log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{t}}\right)\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 ((((double) exp(z)) <= 0.0)) {
		VAR = ((double) (x - ((double) (((double) (((double) log(((double) sqrt(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))))) + ((double) log(((double) sqrt(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z)))))))))))) / t))));
	} else {
		VAR = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (((double) cbrt(((double) (z / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))))) * ((double) cbrt(((double) (z / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))))))) * ((double) (((double) cbrt(((double) (z / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))))) * ((double) (y / ((double) cbrt(t)))))))))) + ((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
Target16.0
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 (exp z) < 0.0

    1. Initial program 11.4

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

    3. Applied add-sqr-sqrt11.4

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

    4. Applied log-prod11.4

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

    if 0.0 < (exp z)

    1. Initial program 30.7

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

    2. Taylor expanded around 0 8.0

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

    4. Applied add-cube-cbrt8.1

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

    5. Applied times-frac7.6

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

    7. Applied add-cube-cbrt7.6

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

    8. Applied associate-*l*7.6

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.0:\\ \;\;\;\;x - \frac{\log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right) + \log \left(\sqrt{\left(1 - y\right) + y \cdot e^{z}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(\left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\sqrt[3]{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{y}{\sqrt[3]{t}}\right)\right) + \frac{\log 1}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(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 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t)))

  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))