Average Error: 11.4 → 1.0
Time: 17.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \frac{\frac{y}{\frac{z}{t}}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \frac{\frac{y}{\frac{z}{t}}}{2}}
double f(double x, double y, double z, double t) {
        double r340565 = x;
        double r340566 = y;
        double r340567 = 2.0;
        double r340568 = r340566 * r340567;
        double r340569 = z;
        double r340570 = r340568 * r340569;
        double r340571 = r340569 * r340567;
        double r340572 = r340571 * r340569;
        double r340573 = t;
        double r340574 = r340566 * r340573;
        double r340575 = r340572 - r340574;
        double r340576 = r340570 / r340575;
        double r340577 = r340565 - r340576;
        return r340577;
}
↓
double f(double x, double y, double z, double t) {
        double r340578 = x;
        double r340579 = y;
        double r340580 = z;
        double r340581 = t;
        double r340582 = r340580 / r340581;
        double r340583 = r340579 / r340582;
        double r340584 = 2.0;
        double r340585 = r340583 / r340584;
        double r340586 = r340580 - r340585;
        double r340587 = r340579 / r340586;
        double r340588 = r340578 - r340587;
        return r340588;
}

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

Original11.4
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.4

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

  2. Simplified2.7

    \[\leadsto \color{blue}{x - \frac{y}{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}}\]
  3. Using strategy rm

  4. Applied associate-/l*1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2019305 
(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)))))