Average Error: 11.3 → 0.1
Time: 11.9s
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 r373685 = x;
        double r373686 = y;
        double r373687 = 2.0;
        double r373688 = r373686 * r373687;
        double r373689 = z;
        double r373690 = r373688 * r373689;
        double r373691 = r373689 * r373687;
        double r373692 = r373691 * r373689;
        double r373693 = t;
        double r373694 = r373686 * r373693;
        double r373695 = r373692 - r373694;
        double r373696 = r373690 / r373695;
        double r373697 = r373685 - r373696;
        return r373697;
}
↓
double f(double x, double y, double z, double t) {
        double r373698 = x;
        double r373699 = 1.0;
        double r373700 = z;
        double r373701 = y;
        double r373702 = r373700 / r373701;
        double r373703 = 0.5;
        double r373704 = t;
        double r373705 = r373704 / r373700;
        double r373706 = r373703 * r373705;
        double r373707 = r373702 - r373706;
        double r373708 = r373699 / r373707;
        double r373709 = r373698 - r373708;
        return r373709;
}

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

Original11.3
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified2.5

    \[\leadsto \color{blue}{x - \frac{y}{\frac{z}{1} - \frac{\frac{y \cdot t}{z}}{2}}}\]
  3. Using strategy rm

  4. Applied associate-/l*0.9

    \[\leadsto x - \frac{y}{\frac{z}{1} - \frac{\color{blue}{\frac{y}{\frac{z}{t}}}}{2}}\]
  5. Using strategy rm

  6. Applied clear-num0.9

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

  7. Simplified0.9

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

  8. Taylor expanded around 0 0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 
(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)))))