Average Error: 11.9 → 6.6
Time: 5.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \left(\frac{y}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\right) \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \left(\frac{y}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\right) \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}
double f(double x, double y, double z, double t) {
        double r496433 = x;
        double r496434 = y;
        double r496435 = 2.0;
        double r496436 = r496434 * r496435;
        double r496437 = z;
        double r496438 = r496436 * r496437;
        double r496439 = r496437 * r496435;
        double r496440 = r496439 * r496437;
        double r496441 = t;
        double r496442 = r496434 * r496441;
        double r496443 = r496440 - r496442;
        double r496444 = r496438 / r496443;
        double r496445 = r496433 - r496444;
        return r496445;
}
↓
double f(double x, double y, double z, double t) {
        double r496446 = x;
        double r496447 = y;
        double r496448 = z;
        double r496449 = 2.0;
        double r496450 = r496448 * r496449;
        double r496451 = r496450 * r496448;
        double r496452 = t;
        double r496453 = r496447 * r496452;
        double r496454 = r496451 - r496453;
        double r496455 = cbrt(r496454);
        double r496456 = r496447 / r496455;
        double r496457 = r496449 / r496455;
        double r496458 = r496456 * r496457;
        double r496459 = r496448 / r496455;
        double r496460 = r496458 * r496459;
        double r496461 = r496446 - r496460;
        return r496461;
}

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

Original11.9
Target0.1
Herbie6.6
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  3. Applied add-cube-cbrt12.1

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

  4. Applied times-frac6.6

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

  6. Applied times-frac6.6

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

  7. Final simplification6.6

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

Reproduce

herbie shell --seed 2020034 
(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)))))