Average Error: 11.3 → 1.2
Time: 19.0s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \left(\frac{t}{z} \cdot \sqrt[3]{\frac{y}{2.0}}\right) \cdot \left(\sqrt[3]{\frac{y}{2.0}} \cdot \sqrt[3]{\frac{y}{2.0}}\right)}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \left(\frac{t}{z} \cdot \sqrt[3]{\frac{y}{2.0}}\right) \cdot \left(\sqrt[3]{\frac{y}{2.0}} \cdot \sqrt[3]{\frac{y}{2.0}}\right)}
double f(double x, double y, double z, double t) {
        double r20530343 = x;
        double r20530344 = y;
        double r20530345 = 2.0;
        double r20530346 = r20530344 * r20530345;
        double r20530347 = z;
        double r20530348 = r20530346 * r20530347;
        double r20530349 = r20530347 * r20530345;
        double r20530350 = r20530349 * r20530347;
        double r20530351 = t;
        double r20530352 = r20530344 * r20530351;
        double r20530353 = r20530350 - r20530352;
        double r20530354 = r20530348 / r20530353;
        double r20530355 = r20530343 - r20530354;
        return r20530355;
}
↓
double f(double x, double y, double z, double t) {
        double r20530356 = x;
        double r20530357 = y;
        double r20530358 = z;
        double r20530359 = t;
        double r20530360 = r20530359 / r20530358;
        double r20530361 = 2.0;
        double r20530362 = r20530357 / r20530361;
        double r20530363 = cbrt(r20530362);
        double r20530364 = r20530360 * r20530363;
        double r20530365 = r20530363 * r20530363;
        double r20530366 = r20530364 * r20530365;
        double r20530367 = r20530358 - r20530366;
        double r20530368 = r20530357 / r20530367;
        double r20530369 = r20530356 - r20530368;
        return r20530369;
}

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

Original11.3
Target0.1
Herbie1.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.3

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.1

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{2.0} \cdot \frac{t}{z}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt1.2

    \[\leadsto x - \frac{y}{z - \color{blue}{\left(\left(\sqrt[3]{\frac{y}{2.0}} \cdot \sqrt[3]{\frac{y}{2.0}}\right) \cdot \sqrt[3]{\frac{y}{2.0}}\right)} \cdot \frac{t}{z}}\]

  5. Applied associate-*l*1.2

    \[\leadsto x - \frac{y}{z - \color{blue}{\left(\sqrt[3]{\frac{y}{2.0}} \cdot \sqrt[3]{\frac{y}{2.0}}\right) \cdot \left(\sqrt[3]{\frac{y}{2.0}} \cdot \frac{t}{z}\right)}}\]

  6. Final simplification1.2

    \[\leadsto x - \frac{y}{z - \left(\frac{t}{z} \cdot \sqrt[3]{\frac{y}{2.0}}\right) \cdot \left(\sqrt[3]{\frac{y}{2.0}} \cdot \sqrt[3]{\frac{y}{2.0}}\right)}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))