Average Error: 25.0 → 8.1
Time: 6.6s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.95399322317807933:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} \cdot y - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(1 \cdot \frac{z}{t} + \left(z \cdot \frac{z}{t}\right) \cdot 0.5\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 0.95399322317807933:\\
\;\;\;\;x - \frac{\log \left(1 + \left(e^{z} \cdot y - y\right)\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;x - y \cdot \left(1 \cdot \frac{z}{t} + \left(z \cdot \frac{z}{t}\right) \cdot 0.5\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)) <= 0.9539932231780793)) {
		VAR = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (((double) (((double) exp(z)) * y)) - y)))))) / t))));
	} else {
		VAR = ((double) (x - ((double) (y * ((double) (((double) (1.0 * ((double) (z / t)))) + ((double) (((double) (z * ((double) (z / t)))) * 0.5))))))));
	}
	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
Target16.5
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.95399322317807933

    1. Initial program 11.4

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

    2. Simplified11.4

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

    if 0.95399322317807933 < (exp z)

    1. Initial program 30.9

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

    2. Simplified16.3

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

    3. Taylor expanded around 0 7.5

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

    4. Simplified6.7

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

    5. Taylor expanded around inf 7.5

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

    6. Simplified6.7

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.95399322317807933:\\ \;\;\;\;x - \frac{\log \left(1 + \left(e^{z} \cdot y - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(1 \cdot \frac{z}{t} + \left(z \cdot \frac{z}{t}\right) \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))