Average Error: 11.9 → 1.1
Time: 44.9s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}
double f(double x, double y, double z, double t) {
        double r18660831 = x;
        double r18660832 = y;
        double r18660833 = 2.0;
        double r18660834 = r18660832 * r18660833;
        double r18660835 = z;
        double r18660836 = r18660834 * r18660835;
        double r18660837 = r18660835 * r18660833;
        double r18660838 = r18660837 * r18660835;
        double r18660839 = t;
        double r18660840 = r18660832 * r18660839;
        double r18660841 = r18660838 - r18660840;
        double r18660842 = r18660836 / r18660841;
        double r18660843 = r18660831 - r18660842;
        return r18660843;
}
↓
double f(double x, double y, double z, double t) {
        double r18660844 = x;
        double r18660845 = y;
        double r18660846 = z;
        double r18660847 = t;
        double r18660848 = r18660847 / r18660846;
        double r18660849 = 2.0;
        double r18660850 = r18660845 / r18660849;
        double r18660851 = r18660848 * r18660850;
        double r18660852 = r18660846 - r18660851;
        double r18660853 = r18660845 / r18660852;
        double r18660854 = r18660844 - r18660853;
        return r18660854;
}

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
Herbie1.1
\[x - \frac{1.0}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.9

    \[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]

  2. Simplified1.1

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{2.0} \cdot \frac{t}{z}}}\]

  3. Final simplification1.1

    \[\leadsto x - \frac{y}{z - \frac{t}{z} \cdot \frac{y}{2.0}}\]

Reproduce

herbie shell --seed 2019165 +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)))))