Average Error: 11.6 → 1.8
Time: 4.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{2 \cdot z - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{2 \cdot z - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}
double f(double x, double y, double z, double t) {
        double r495129 = x;
        double r495130 = y;
        double r495131 = 2.0;
        double r495132 = r495130 * r495131;
        double r495133 = z;
        double r495134 = r495132 * r495133;
        double r495135 = r495133 * r495131;
        double r495136 = r495135 * r495133;
        double r495137 = t;
        double r495138 = r495130 * r495137;
        double r495139 = r495136 - r495138;
        double r495140 = r495134 / r495139;
        double r495141 = r495129 - r495140;
        return r495141;
}
↓
double f(double x, double y, double z, double t) {
        double r495142 = x;
        double r495143 = y;
        double r495144 = 2.0;
        double r495145 = r495143 * r495144;
        double r495146 = z;
        double r495147 = r495144 * r495146;
        double r495148 = t;
        double r495149 = cbrt(r495146);
        double r495150 = r495149 * r495149;
        double r495151 = r495148 / r495150;
        double r495152 = r495143 / r495149;
        double r495153 = r495151 * r495152;
        double r495154 = r495147 - r495153;
        double r495155 = r495145 / r495154;
        double r495156 = r495142 - r495155;
        return r495156;
}

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

Original11.6
Target0.1
Herbie1.8
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.6

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

  3. Applied associate-/l*6.6

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

  4. Taylor expanded around 0 3.1

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

  6. Applied add-cube-cbrt3.2

    \[\leadsto x - \frac{y \cdot 2}{2 \cdot z - \frac{t \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}}\]

  7. Applied times-frac1.8

    \[\leadsto x - \frac{y \cdot 2}{2 \cdot z - \color{blue}{\frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}}\]

  8. Final simplification1.8

    \[\leadsto x - \frac{y \cdot 2}{2 \cdot z - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]

Reproduce

herbie shell --seed 2020065 +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)))))