Average Error: 11.8 → 0.1
Time: 14.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{2}{\frac{2 \cdot y}{z}} - \frac{t}{z} \cdot \frac{y}{2 \cdot y}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{2}{\frac{2 \cdot y}{z}} - \frac{t}{z} \cdot \frac{y}{2 \cdot y}}
double f(double x, double y, double z, double t) {
        double r374743 = x;
        double r374744 = y;
        double r374745 = 2.0;
        double r374746 = r374744 * r374745;
        double r374747 = z;
        double r374748 = r374746 * r374747;
        double r374749 = r374747 * r374745;
        double r374750 = r374749 * r374747;
        double r374751 = t;
        double r374752 = r374744 * r374751;
        double r374753 = r374750 - r374752;
        double r374754 = r374748 / r374753;
        double r374755 = r374743 - r374754;
        return r374755;
}
↓
double f(double x, double y, double z, double t) {
        double r374756 = x;
        double r374757 = 1.0;
        double r374758 = 2.0;
        double r374759 = y;
        double r374760 = r374758 * r374759;
        double r374761 = z;
        double r374762 = r374760 / r374761;
        double r374763 = r374758 / r374762;
        double r374764 = t;
        double r374765 = r374764 / r374761;
        double r374766 = r374759 / r374760;
        double r374767 = r374765 * r374766;
        double r374768 = r374763 - r374767;
        double r374769 = r374757 / r374768;
        double r374770 = r374756 - r374769;
        return r374770;
}

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. Simplified11.8

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

  6. Applied div-sub16.4

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

  7. Simplified6.4

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 
(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)))))