Average Error: 12.0 → 2.3
Time: 3.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{\frac{2 \cdot z - t \cdot \frac{y}{z}}{2}}
double f(double x, double y, double z, double t) {
        double r601120 = x;
        double r601121 = y;
        double r601122 = 2.0;
        double r601123 = r601121 * r601122;
        double r601124 = z;
        double r601125 = r601123 * r601124;
        double r601126 = r601124 * r601122;
        double r601127 = r601126 * r601124;
        double r601128 = t;
        double r601129 = r601121 * r601128;
        double r601130 = r601127 - r601129;
        double r601131 = r601125 / r601130;
        double r601132 = r601120 - r601131;
        return r601132;
}
↓
double f(double x, double y, double z, double t) {
        double r601133 = x;
        double r601134 = y;
        double r601135 = 2.0;
        double r601136 = z;
        double r601137 = r601135 * r601136;
        double r601138 = t;
        double r601139 = r601134 / r601136;
        double r601140 = r601138 * r601139;
        double r601141 = r601137 - r601140;
        double r601142 = r601141 / r601135;
        double r601143 = r601134 / r601142;
        double r601144 = r601133 - r601143;
        return r601144;
}

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

Original12.0
Target0.1
Herbie2.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.0

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

  3. Applied associate-/l*6.9

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

  5. Applied associate-/l*6.9

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

  6. Simplified2.9

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

  8. Applied *-un-lft-identity2.9

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

  9. Applied times-frac2.3

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

  10. Simplified2.3

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

  11. Final simplification2.3

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

Reproduce

herbie shell --seed 2020057 
(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)))))