Average Error: 25.1 → 8.9
Time: 8.2s
Precision: 64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.0074356716971061 \cdot 10^{21}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.0074356716971061 \cdot 10^{21}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return (x - (log(((1.0 - y) + (y * exp(z)))) / t));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -3.007435671697106e+21)) {
		VAR = (x - (1.0 / (t / log(((1.0 - y) + (y * exp(z)))))));
	} else {
		VAR = (x - ((log(1.0) + (y * ((0.5 * pow(z, 2.0)) + (1.0 * z)))) / 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
Target16.4
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.007435671697106e+21

    1. Initial program 10.7

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

    3. Applied clear-num10.8

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

    if -3.007435671697106e+21 < z

    1. Initial program 30.6

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

    2. Taylor expanded around 0 8.2

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

    3. Simplified8.2

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.0074356716971061 \cdot 10^{21}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(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)))