Average Error: 11.1 → 0.1
Time: 18.9s
Precision: 64
\[x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}\]
x - \frac{\left(y \cdot 2.0\right) \cdot z}{\left(z \cdot 2.0\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}
double f(double x, double y, double z, double t) {
        double r18406769 = x;
        double r18406770 = y;
        double r18406771 = 2.0;
        double r18406772 = r18406770 * r18406771;
        double r18406773 = z;
        double r18406774 = r18406772 * r18406773;
        double r18406775 = r18406773 * r18406771;
        double r18406776 = r18406775 * r18406773;
        double r18406777 = t;
        double r18406778 = r18406770 * r18406777;
        double r18406779 = r18406776 - r18406778;
        double r18406780 = r18406774 / r18406779;
        double r18406781 = r18406769 - r18406780;
        return r18406781;
}
↓
double f(double x, double y, double z, double t) {
        double r18406782 = x;
        double r18406783 = 1.0;
        double r18406784 = z;
        double r18406785 = y;
        double r18406786 = r18406784 / r18406785;
        double r18406787 = t;
        double r18406788 = r18406787 / r18406784;
        double r18406789 = 0.5;
        double r18406790 = r18406788 * r18406789;
        double r18406791 = r18406786 - r18406790;
        double r18406792 = r18406783 / r18406791;
        double r18406793 = r18406782 - r18406792;
        return r18406793;
}

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

Original11.1
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2.0}}{z}}\]

Derivation

  1. Initial program 11.1

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

  2. Simplified1.2

    \[\leadsto \color{blue}{x - \frac{y}{z - \frac{y}{2.0} \cdot \frac{t}{z}}}\]
  3. Using strategy rm

  4. Applied clear-num1.2

    \[\leadsto x - \color{blue}{\frac{1}{\frac{z - \frac{y}{2.0} \cdot \frac{t}{z}}{y}}}\]

  5. Taylor expanded around 0 0.1

    \[\leadsto x - \frac{1}{\color{blue}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}}\]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{z}{y} - \frac{t}{z} \cdot 0.5}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))