Average Error: 11.8 → 0.1
Time: 5.6s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{2 \cdot \frac{z}{y} - \frac{t}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{2 \cdot \frac{z}{y} - \frac{t}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r508905 = x;
        double r508906 = y;
        double r508907 = 2.0;
        double r508908 = r508906 * r508907;
        double r508909 = z;
        double r508910 = r508908 * r508909;
        double r508911 = r508909 * r508907;
        double r508912 = r508911 * r508909;
        double r508913 = t;
        double r508914 = r508906 * r508913;
        double r508915 = r508912 - r508914;
        double r508916 = r508910 / r508915;
        double r508917 = r508905 - r508916;
        return r508917;
}
↓
double f(double x, double y, double z, double t) {
        double r508918 = x;
        double r508919 = 1.0;
        double r508920 = 2.0;
        double r508921 = z;
        double r508922 = y;
        double r508923 = r508921 / r508922;
        double r508924 = r508920 * r508923;
        double r508925 = t;
        double r508926 = r508925 / r508921;
        double r508927 = r508924 - r508926;
        double r508928 = r508927 / r508920;
        double r508929 = r508919 / r508928;
        double r508930 = r508918 - r508929;
        return r508930;
}

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

Original11.8
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.8

    \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
  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 *-un-lft-identity6.8

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

  6. Applied *-un-lft-identity6.8

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

  7. Applied times-frac6.8

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

  8. Simplified6.8

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

  9. Simplified2.9

    \[\leadsto x - \frac{y \cdot 2}{1 \cdot \color{blue}{\left(2 \cdot z - \frac{t \cdot y}{z}\right)}}\]
  10. Using strategy rm

  11. Applied clear-num3.0

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

  12. Simplified2.9

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

  13. Taylor expanded around 0 0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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)))))