Average Error: 11.5 → 1.7
Time: 3.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -5.4087609003385163 \cdot 10^{-201}:\\ \;\;\;\;x - \frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -5.4087609003385163 \cdot 10^{-201}:\\
\;\;\;\;x - \frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{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 ((((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) <= -5.408760900338516e-201)) {
		VAR = ((double) (x - ((double) (((double) (((double) (y * 2.0)) / ((double) (((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) * ((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))))))) * ((double) (z / ((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t * ((double) (y / 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

Original11.5
Target0.1
Herbie1.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < -5.408760900338516e-201

    1. Initial program 3.7

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

    3. Applied add-cube-cbrt4.1

      \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}\right) \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}}\]

    4. Applied times-frac1.0

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}}\]

    if -5.408760900338516e-201 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))

    1. Initial program 13.3

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
    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-sub7.9

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

    6. Simplified3.2

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

    7. Simplified3.2

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

    9. Applied *-un-lft-identity3.2

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

    10. Applied times-frac1.8

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

    11. Simplified1.8

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -5.4087609003385163 \cdot 10^{-201}:\\ \;\;\;\;x - \frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))