Average Error: 25.2 → 8.7
Time: 6.9s
Precision: binary64
\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.73879058066647328 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{elif}\;z \le 5.4143196385504612 \cdot 10^{-116}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{\log \left(e^{{z}^{2} \cdot y}\right)}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \end{array}\]
x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.73879058066647328 \cdot 10^{-14}:\\
\;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\

\mathbf{elif}\;z \le 5.4143196385504612 \cdot 10^{-116}:\\
\;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{\log \left(e^{{z}^{2} \cdot y}\right)}{t}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{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 ((z <= -9.738790580666473e-14)) {
		VAR = ((double) (x - ((double) (1.0 / ((double) (t / ((double) log(((double) (((double) (1.0 - y)) + ((double) (y * ((double) exp(z))))))))))))));
	} else {
		double VAR_1;
		if ((z <= 5.414319638550461e-116)) {
			VAR_1 = ((double) (x - ((double) (((double) (1.0 * ((double) (((double) (z * y)) / t)))) + ((double) (((double) (((double) log(1.0)) / t)) + ((double) (0.5 * ((double) (((double) log(((double) exp(((double) (((double) pow(z, 2.0)) * y)))))) / t))))))))));
		} else {
			VAR_1 = ((double) (x - ((double) (((double) log(((double) (1.0 + ((double) (((double) (((double) (((double) pow(z, 2.0)) * ((double) (0.5 + ((double) (z * 0.16666666666666666)))))) + z)) * y)))))) / t))));
		}
		VAR = VAR_1;
	}
	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

Original25.2
Target16.2
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.88746230882079466 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -9.73879058066647328e-14

    1. Initial program 11.6

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

    3. Applied clear-num11.6

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

    if -9.73879058066647328e-14 < z < 5.4143196385504612e-116

    1. Initial program 31.5

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

    2. Taylor expanded around 0 5.8

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

    4. Applied add-log-exp6.7

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

    if 5.4143196385504612e-116 < z

    1. Initial program 30.6

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

    3. Applied sub-neg30.6

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

    4. Applied associate-+l+20.9

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

    5. Simplified20.9

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

    6. Taylor expanded around 0 11.0

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)} \cdot y\right)}{t}\]

    7. Simplified11.0

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right)} \cdot y\right)}{t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.73879058066647328 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}}\\ \mathbf{elif}\;z \le 5.4143196385504612 \cdot 10^{-116}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \left(\frac{\log 1}{t} + 0.5 \cdot \frac{\log \left(e^{{z}^{2} \cdot y}\right)}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(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)))