Average Error: 11.1 → 0.1
Time: 14.1s
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 r21778784 = x;
        double r21778785 = y;
        double r21778786 = 2.0;
        double r21778787 = r21778785 * r21778786;
        double r21778788 = z;
        double r21778789 = r21778787 * r21778788;
        double r21778790 = r21778788 * r21778786;
        double r21778791 = r21778790 * r21778788;
        double r21778792 = t;
        double r21778793 = r21778785 * r21778792;
        double r21778794 = r21778791 - r21778793;
        double r21778795 = r21778789 / r21778794;
        double r21778796 = r21778784 - r21778795;
        return r21778796;
}
↓
double f(double x, double y, double z, double t) {
        double r21778797 = x;
        double r21778798 = 1.0;
        double r21778799 = z;
        double r21778800 = y;
        double r21778801 = r21778799 / r21778800;
        double r21778802 = t;
        double r21778803 = r21778802 / r21778799;
        double r21778804 = 0.5;
        double r21778805 = r21778803 * r21778804;
        double r21778806 = r21778801 - r21778805;
        double r21778807 = r21778798 / r21778806;
        double r21778808 = r21778797 - r21778807;
        return r21778808;
}

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

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

  4. Applied clear-num1.2

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

  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 
(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)))))