Average Error: 11.7 → 2.9
Time: 3.8s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.6604623612953187 \cdot 10^{-286} \lor \neg \left(x \le 3.5940792936678314 \cdot 10^{-167}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{\left(y \cdot 2\right) \cdot z}}\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.6604623612953187 \cdot 10^{-286} \lor \neg \left(x \le 3.5940792936678314 \cdot 10^{-167}\right):\\
\;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if (((x <= 1.6604623612953187e-286) || !(x <= 3.5940792936678314e-167))) {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t / ((double) (z / y))))))))));
	} else {
		VAR = ((double) (x - ((double) (1.0 / ((double) (((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))) / ((double) (((double) (y * 2.0)) * z))))))));
	}
	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

Original11.7
Target0.1
Herbie2.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.6604623612953187e-286 or 3.5940792936678314e-167 < x

    1. Initial program 11.5

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

    3. Applied associate-/l*6.3

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\]
    4. Using strategy rm

    5. Applied div-sub6.3

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

    6. Simplified2.5

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2} - \frac{y \cdot t}{z}}\]

    7. Simplified2.5

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t \cdot y}{z}}}\]
    8. Using strategy rm

    9. Applied associate-/l*1.8

      \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{t}{\frac{z}{y}}}}\]

    if 1.6604623612953187e-286 < x < 3.5940792936678314e-167

    1. Initial program 13.6

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

    3. Applied clear-num13.6

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.6604623612953187 \cdot 10^{-286} \lor \neg \left(x \le 3.5940792936678314 \cdot 10^{-167}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{\left(y \cdot 2\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))