Average Error: 25.3 → 7.8
Time: 8.9s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(e^{z} \cdot y + \left(1 - y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \leq 0:\\
\;\;\;\;x + \frac{-1}{\frac{t}{\log \left(e^{z} \cdot y + \left(1 - y\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;x - \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)\\

\end{array}
(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 (/ t (log (+ (* (exp z) y) (- 1.0 y))))))
   (-
    x
    (+
     (* 1.0 (* y (/ z t)))
     (+ (/ (log 1.0) t) (* 0.5 (* y (/ z (/ t z)))))))))
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 tmp;
	if ((((double) exp(z)) <= 0.0)) {
		tmp = ((double) (x + (-1.0 / (t / ((double) log(((double) (((double) (((double) exp(z)) * y)) + ((double) (1.0 - y))))))))));
	} else {
		tmp = ((double) (x - ((double) (((double) (1.0 * ((double) (y * (z / t))))) + ((double) ((((double) log(1.0)) / t) + ((double) (0.5 * ((double) (y * (z / (t / z))))))))))));
	}
	return tmp;
}

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

Original25.3
Target16.4
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 0.0

    1. Initial program Error: 11.5 bits

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

    2. SimplifiedError: 11.5 bits

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

    4. Applied clear-numError: 11.6 bits

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

    5. SimplifiedError: 11.6 bits

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

    if 0.0 < (exp z)

    1. Initial program Error: 30.8 bits

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

    2. SimplifiedError: 16.1 bits

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

    3. Taylor expanded around 0 Error: 7.2 bits

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

    4. SimplifiedError: 6.3 bits

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

  4. Final simplificationError: 7.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(e^{z} \cdot y + \left(1 - y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(1 \cdot \left(y \cdot \frac{z}{t}\right) + \left(\frac{\log 1}{t} + 0.5 \cdot \left(y \cdot \frac{z}{\frac{t}{z}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))