Average Error: 11.7 → 2.4
Time: 4.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{t}{\frac{z}{y}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - \frac{t}{\frac{z}{y}}}
double f(double x, double y, double z, double t) {
        double r385880 = x;
        double r385881 = y;
        double r385882 = 2.0;
        double r385883 = r385881 * r385882;
        double r385884 = z;
        double r385885 = r385883 * r385884;
        double r385886 = r385884 * r385882;
        double r385887 = r385886 * r385884;
        double r385888 = t;
        double r385889 = r385881 * r385888;
        double r385890 = r385887 - r385889;
        double r385891 = r385885 / r385890;
        double r385892 = r385880 - r385891;
        return r385892;
}
↓
double f(double x, double y, double z, double t) {
        double r385893 = x;
        double r385894 = y;
        double r385895 = 2.0;
        double r385896 = r385894 * r385895;
        double r385897 = z;
        double r385898 = r385895 * r385897;
        double r385899 = t;
        double r385900 = r385897 / r385894;
        double r385901 = r385899 / r385900;
        double r385902 = r385898 - r385901;
        double r385903 = r385896 / r385902;
        double r385904 = r385893 - r385903;
        return r385904;
}

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. Taylor expanded around 0 3.1

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

  6. Applied associate-/l*2.4

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

  7. Final simplification2.4

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

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)))))