Average Error: 11.2 → 1.4
Time: 3.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 1.6508786123963074 \cdot 10^{108}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]
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 1.6508786123963074 \cdot 10^{108}:\\
\;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}\\

\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)))))) <= 1.6508786123963074e+108)) {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (((double) (t * y)) / z))))))));
	} 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

Original11.2
Target0.1
Herbie1.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < 1.6508786123963074e+108

    1. Initial program 2.8

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

    3. Applied associate-/l*2.1

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

    5. Applied div-sub2.1

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

    6. Simplified1.3

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

    7. Simplified1.3

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

    if 1.6508786123963074e+108 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))

    1. Initial program 56.0

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

    3. Applied associate-/l*30.9

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

    5. Applied div-sub30.9

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

    6. Simplified10.7

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

    7. Simplified10.7

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

    9. Applied *-un-lft-identity10.7

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

    10. Applied times-frac1.9

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

    11. Simplified1.9

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 1.6508786123963074 \cdot 10^{108}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t \cdot y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(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)))))