Average Error: 11.9 → 0.1
Time: 12.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{z}{y} - 0.5 \cdot \frac{t}{z}}\]
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} - 0.5 \cdot \frac{t}{z}}
double f(double x, double y, double z, double t) {
        double r730668 = x;
        double r730669 = y;
        double r730670 = 2.0;
        double r730671 = r730669 * r730670;
        double r730672 = z;
        double r730673 = r730671 * r730672;
        double r730674 = r730672 * r730670;
        double r730675 = r730674 * r730672;
        double r730676 = t;
        double r730677 = r730669 * r730676;
        double r730678 = r730675 - r730677;
        double r730679 = r730673 / r730678;
        double r730680 = r730668 - r730679;
        return r730680;
}
↓
double f(double x, double y, double z, double t) {
        double r730681 = x;
        double r730682 = 1.0;
        double r730683 = z;
        double r730684 = y;
        double r730685 = r730683 / r730684;
        double r730686 = 0.5;
        double r730687 = t;
        double r730688 = r730687 / r730683;
        double r730689 = r730686 * r730688;
        double r730690 = r730685 - r730689;
        double r730691 = r730682 / r730690;
        double r730692 = r730681 - r730691;
        return r730692;
}

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
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  2. Simplified1.0

    \[\leadsto \color{blue}{x - \frac{y}{\mathsf{fma}\left(\frac{t}{z}, -\frac{y}{2}, z\right)}}\]
  3. Using strategy rm

  4. Applied clear-num1.1

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

  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

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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)))))