Average Error: 11.8 → 0.1
Time: 40.8s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{t}{2 \cdot z}}\]
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{t}{2 \cdot z}}
double f(double x, double y, double z, double t) {
        double r21465823 = x;
        double r21465824 = y;
        double r21465825 = 2.0;
        double r21465826 = r21465824 * r21465825;
        double r21465827 = z;
        double r21465828 = r21465826 * r21465827;
        double r21465829 = r21465827 * r21465825;
        double r21465830 = r21465829 * r21465827;
        double r21465831 = t;
        double r21465832 = r21465824 * r21465831;
        double r21465833 = r21465830 - r21465832;
        double r21465834 = r21465828 / r21465833;
        double r21465835 = r21465823 - r21465834;
        return r21465835;
}
↓
double f(double x, double y, double z, double t) {
        double r21465836 = x;
        double r21465837 = 1.0;
        double r21465838 = z;
        double r21465839 = y;
        double r21465840 = r21465838 / r21465839;
        double r21465841 = t;
        double r21465842 = 2.0;
        double r21465843 = r21465842 * r21465838;
        double r21465844 = r21465841 / r21465843;
        double r21465845 = r21465840 - r21465844;
        double r21465846 = r21465837 / r21465845;
        double r21465847 = r21465836 - r21465846;
        return r21465847;
}

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 clear-num11.8

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +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)))))