Average Error: 11.9 → 2.8
Time: 4.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t \cdot y}{z}}{2}}\]
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 r427849 = x;
        double r427850 = y;
        double r427851 = 2.0;
        double r427852 = r427850 * r427851;
        double r427853 = z;
        double r427854 = r427852 * r427853;
        double r427855 = r427853 * r427851;
        double r427856 = r427855 * r427853;
        double r427857 = t;
        double r427858 = r427850 * r427857;
        double r427859 = r427856 - r427858;
        double r427860 = r427854 / r427859;
        double r427861 = r427849 - r427860;
        return r427861;
}
↓
double f(double x, double y, double z, double t) {
        double r427862 = x;
        double r427863 = y;
        double r427864 = 2.0;
        double r427865 = z;
        double r427866 = r427864 * r427865;
        double r427867 = t;
        double r427868 = r427867 * r427863;
        double r427869 = r427868 / r427865;
        double r427870 = r427866 - r427869;
        double r427871 = r427870 / r427864;
        double r427872 = r427863 / r427871;
        double r427873 = r427862 - r427872;
        return r427873;
}

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 +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)))))