Average Error: 11.2 → 0.1
Time: 14.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z \cdot 2}{y} - \frac{t}{z}} \cdot 2\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z \cdot 2}{y} - \frac{t}{z}} \cdot 2
double f(double x, double y, double z, double t) {
        double r391912 = x;
        double r391913 = y;
        double r391914 = 2.0;
        double r391915 = r391913 * r391914;
        double r391916 = z;
        double r391917 = r391915 * r391916;
        double r391918 = r391916 * r391914;
        double r391919 = r391918 * r391916;
        double r391920 = t;
        double r391921 = r391913 * r391920;
        double r391922 = r391919 - r391921;
        double r391923 = r391917 / r391922;
        double r391924 = r391912 - r391923;
        return r391924;
}
↓
double f(double x, double y, double z, double t) {
        double r391925 = x;
        double r391926 = 1.0;
        double r391927 = z;
        double r391928 = 2.0;
        double r391929 = r391927 * r391928;
        double r391930 = y;
        double r391931 = r391929 / r391930;
        double r391932 = t;
        double r391933 = r391932 / r391927;
        double r391934 = r391931 - r391933;
        double r391935 = r391926 / r391934;
        double r391936 = r391935 * r391928;
        double r391937 = r391925 - r391936;
        return r391937;
}

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

Original11.2
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.2

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

  2. Simplified2.5

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

  4. Applied clear-num2.5

    \[\leadsto x - \color{blue}{\frac{1}{\frac{2 \cdot z - \frac{y \cdot t}{z}}{y}}} \cdot 2\]

  5. Simplified2.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{2 \cdot z - \frac{t}{\frac{z}{y}}}{y}}} \cdot 2\]

  6. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{2 \cdot \frac{z}{y} - \frac{t}{z}}} \cdot 2\]

  7. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z \cdot 2}{y} - \frac{t}{z}}} \cdot 2\]

  8. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z \cdot 2}{y} - \frac{t}{z}} \cdot 2\]

Reproduce

herbie shell --seed 2019194 
(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)))))