Average Error: 11.2 → 1.0
Time: 4.0s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{\frac{t}{z}}{\frac{1}{y}}}\]
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{\frac{t}{z}}{\frac{1}{y}}}
double f(double x, double y, double z, double t) {
        double r522901 = x;
        double r522902 = y;
        double r522903 = 2.0;
        double r522904 = r522902 * r522903;
        double r522905 = z;
        double r522906 = r522904 * r522905;
        double r522907 = r522905 * r522903;
        double r522908 = r522907 * r522905;
        double r522909 = t;
        double r522910 = r522902 * r522909;
        double r522911 = r522908 - r522910;
        double r522912 = r522906 / r522911;
        double r522913 = r522901 - r522912;
        return r522913;
}
↓
double f(double x, double y, double z, double t) {
        double r522914 = x;
        double r522915 = y;
        double r522916 = 2.0;
        double r522917 = r522915 * r522916;
        double r522918 = z;
        double r522919 = r522916 * r522918;
        double r522920 = t;
        double r522921 = r522920 / r522918;
        double r522922 = 1.0;
        double r522923 = r522922 / r522915;
        double r522924 = r522921 / r522923;
        double r522925 = r522919 - r522924;
        double r522926 = r522917 / r522925;
        double r522927 = r522914 - r522926;
        return r522927;
}

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
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Using strategy rm

  3. Applied associate-/l*6.3

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

  4. Taylor expanded around 0 2.3

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

  6. Applied associate-/l*2.2

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

  8. Applied div-inv2.3

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

  9. Applied associate-/r*1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 
(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)))))