Average Error: 11.5 → 6.4
Time: 4.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}} \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}} \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}
double f(double x, double y, double z, double t) {
        double r462258 = x;
        double r462259 = y;
        double r462260 = 2.0;
        double r462261 = r462259 * r462260;
        double r462262 = z;
        double r462263 = r462261 * r462262;
        double r462264 = r462262 * r462260;
        double r462265 = r462264 * r462262;
        double r462266 = t;
        double r462267 = r462259 * r462266;
        double r462268 = r462265 - r462267;
        double r462269 = r462263 / r462268;
        double r462270 = r462258 - r462269;
        return r462270;
}
↓
double f(double x, double y, double z, double t) {
        double r462271 = x;
        double r462272 = y;
        double r462273 = 2.0;
        double r462274 = r462272 * r462273;
        double r462275 = z;
        double r462276 = 2.0;
        double r462277 = pow(r462275, r462276);
        double r462278 = r462273 * r462277;
        double r462279 = t;
        double r462280 = r462279 * r462272;
        double r462281 = r462278 - r462280;
        double r462282 = cbrt(r462281);
        double r462283 = r462282 * r462282;
        double r462284 = r462274 / r462283;
        double r462285 = r462275 / r462282;
        double r462286 = r462284 * r462285;
        double r462287 = r462271 - r462286;
        return r462287;
}

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
Herbie6.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  3. Applied *-un-lft-identity11.5

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

  4. Applied times-frac6.7

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

  5. Simplified6.7

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

  6. Simplified6.7

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

  8. Applied add-cube-cbrt6.9

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

  9. Applied *-un-lft-identity6.9

    \[\leadsto x - \left(y \cdot 2\right) \cdot \frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}\right) \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\]

  10. Applied times-frac6.9

    \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}} \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right)}\]

  11. Applied associate-*r*6.4

    \[\leadsto x - \color{blue}{\left(\left(y \cdot 2\right) \cdot \frac{1}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}}\]

  12. Simplified6.4

    \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}} \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\]

  13. Final simplification6.4

    \[\leadsto x - \frac{y \cdot 2}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}} \cdot \frac{z}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\]

Reproduce

herbie shell --seed 2020002 
(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)))))