Average Error: 11.6 → 2.7
Time: 9.3s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.3609181127969462 \cdot 10^{-177} \lor \neg \left(z \le 2.2241990321314231 \cdot 10^{-191}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \left(z \cdot 2\right)\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.3609181127969462 \cdot 10^{-177} \lor \neg \left(z \le 2.2241990321314231 \cdot 10^{-191}\right):\\
\;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\

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

\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 (((z <= -3.360918112796946e-177) || !(z <= 2.224199032131423e-191))) {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t / ((double) (z / y))))))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (y / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) * ((double) (z * 2.0))))));
	}
	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

Original11.6
Target0.1
Herbie2.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.3609181127969462e-177 or 2.2241990321314231e-191 < z

    1. Initial program 12.6

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

    3. Applied associate-/l*6.1

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

    5. Applied div-sub6.1

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

    6. Simplified2.3

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

    7. Simplified2.3

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

    9. Applied associate-/l*1.7

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

    if -3.3609181127969462e-177 < z < 2.2241990321314231e-191

    1. Initial program 7.9

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

    3. Applied associate-/l*7.9

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

    5. Applied div-inv7.9

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

    6. Applied times-frac6.4

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

    7. Simplified6.4

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.3609181127969462 \cdot 10^{-177} \lor \neg \left(z \le 2.2241990321314231 \cdot 10^{-191}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \left(z \cdot 2\right)\\ \end{array}\]

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)))))