Average Error: 11.6 → 1.4
Time: 3.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}
double f(double x, double y, double z, double t) {
        double r509426 = x;
        double r509427 = y;
        double r509428 = 2.0;
        double r509429 = r509427 * r509428;
        double r509430 = z;
        double r509431 = r509429 * r509430;
        double r509432 = r509430 * r509428;
        double r509433 = r509432 * r509430;
        double r509434 = t;
        double r509435 = r509427 * r509434;
        double r509436 = r509433 - r509435;
        double r509437 = r509431 / r509436;
        double r509438 = r509426 - r509437;
        return r509438;
}
↓
double f(double x, double y, double z, double t) {
        double r509439 = x;
        double r509440 = y;
        double r509441 = 2.0;
        double r509442 = r509440 * r509441;
        double r509443 = z;
        double r509444 = r509443 * r509441;
        double r509445 = t;
        double r509446 = cbrt(r509443);
        double r509447 = r509446 * r509446;
        double r509448 = r509445 / r509447;
        double r509449 = r509440 / r509446;
        double r509450 = r509448 * r509449;
        double r509451 = r509444 - r509450;
        double r509452 = r509442 / r509451;
        double r509453 = r509439 - r509452;
        return r509453;
}

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

Original11.6
Target0.1
Herbie1.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  3. Applied associate-/l*6.6

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

  5. Applied div-sub6.6

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

  6. Simplified2.8

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

  7. Simplified2.8

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

  9. Applied add-cube-cbrt2.9

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

  10. Applied times-frac1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2020064 
(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)))))