Average Error: 11.9 → 2.8
Time: 3.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r495339 = x;
        double r495340 = y;
        double r495341 = 2.0;
        double r495342 = r495340 * r495341;
        double r495343 = z;
        double r495344 = r495342 * r495343;
        double r495345 = r495343 * r495341;
        double r495346 = r495345 * r495343;
        double r495347 = t;
        double r495348 = r495340 * r495347;
        double r495349 = r495346 - r495348;
        double r495350 = r495344 / r495349;
        double r495351 = r495339 - r495350;
        return r495351;
}
↓
double f(double x, double y, double z, double t) {
        double r495352 = x;
        double r495353 = y;
        double r495354 = 2.0;
        double r495355 = z;
        double r495356 = r495354 * r495355;
        double r495357 = t;
        double r495358 = r495357 * r495353;
        double r495359 = r495358 / r495355;
        double r495360 = r495356 - r495359;
        double r495361 = r495360 / r495354;
        double r495362 = r495353 / r495361;
        double r495363 = r495352 - r495362;
        return r495363;
}

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
Herbie2.8
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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 associate-/l*6.8

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

  5. Applied associate-/l*6.8

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

  6. Simplified2.8

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

  7. Final simplification2.8

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

Reproduce

herbie shell --seed 2020024 
(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)))))