Average Error: 11.6 → 0.1
Time: 1.1m
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}
double f(double x, double y, double z, double t) {
        double r15622855 = x;
        double r15622856 = y;
        double r15622857 = 2.0;
        double r15622858 = r15622856 * r15622857;
        double r15622859 = z;
        double r15622860 = r15622858 * r15622859;
        double r15622861 = r15622859 * r15622857;
        double r15622862 = r15622861 * r15622859;
        double r15622863 = t;
        double r15622864 = r15622856 * r15622863;
        double r15622865 = r15622862 - r15622864;
        double r15622866 = r15622860 / r15622865;
        double r15622867 = r15622855 - r15622866;
        return r15622867;
}
↓
double f(double x, double y, double z, double t) {
        double r15622868 = x;
        double r15622869 = 1.0;
        double r15622870 = z;
        double r15622871 = y;
        double r15622872 = r15622870 / r15622871;
        double r15622873 = 0.5;
        double r15622874 = t;
        double r15622875 = r15622870 / r15622874;
        double r15622876 = r15622873 / r15622875;
        double r15622877 = r15622872 - r15622876;
        double r15622878 = r15622869 / r15622877;
        double r15622879 = r15622868 - r15622878;
        return r15622879;
}

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
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Simplified3.5

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

  3. Taylor expanded around 0 3.5

    \[\leadsto x - \frac{z}{\color{blue}{\frac{{z}^{2}}{y} - 0.5 \cdot t}}\]

  4. Simplified1.4

    \[\leadsto x - \frac{z}{\color{blue}{\frac{z}{\frac{y}{z}} - t \cdot 0.5}}\]
  5. Using strategy rm

  6. Applied clear-num1.4

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{z}{\frac{y}{z}} - t \cdot 0.5}{z}}}\]

  7. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}}\]

  8. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \frac{0.5}{\frac{z}{t}}}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))