Average Error: 11.7 → 2.4
Time: 3.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}
double f(double x, double y, double z, double t) {
        double r498318 = x;
        double r498319 = y;
        double r498320 = 2.0;
        double r498321 = r498319 * r498320;
        double r498322 = z;
        double r498323 = r498321 * r498322;
        double r498324 = r498322 * r498320;
        double r498325 = r498324 * r498322;
        double r498326 = t;
        double r498327 = r498319 * r498326;
        double r498328 = r498325 - r498327;
        double r498329 = r498323 / r498328;
        double r498330 = r498318 - r498329;
        return r498330;
}
↓
double f(double x, double y, double z, double t) {
        double r498331 = x;
        double r498332 = y;
        double r498333 = 2.0;
        double r498334 = r498332 * r498333;
        double r498335 = z;
        double r498336 = r498335 * r498333;
        double r498337 = t;
        double r498338 = r498335 / r498332;
        double r498339 = r498337 / r498338;
        double r498340 = r498336 - r498339;
        double r498341 = r498334 / r498340;
        double r498342 = r498331 - r498341;
        return r498342;
}

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

Original11.7
Target0.1
Herbie2.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.7

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

  3. Applied associate-/l*6.9

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

  5. Applied div-sub6.9

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

  6. Simplified3.1

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

  7. Simplified3.1

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

  9. Applied associate-/l*2.4

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

  10. Final simplification2.4

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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)))))