Average Error: 12.1 → 1.7
Time: 5.7s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}{2}}\]
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 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}{2}}
double f(double x, double y, double z, double t) {
        double r459060 = x;
        double r459061 = y;
        double r459062 = 2.0;
        double r459063 = r459061 * r459062;
        double r459064 = z;
        double r459065 = r459063 * r459064;
        double r459066 = r459064 * r459062;
        double r459067 = r459066 * r459064;
        double r459068 = t;
        double r459069 = r459061 * r459068;
        double r459070 = r459067 - r459069;
        double r459071 = r459065 / r459070;
        double r459072 = r459060 - r459071;
        return r459072;
}
↓
double f(double x, double y, double z, double t) {
        double r459073 = x;
        double r459074 = y;
        double r459075 = 2.0;
        double r459076 = z;
        double r459077 = r459075 * r459076;
        double r459078 = t;
        double r459079 = cbrt(r459076);
        double r459080 = r459079 * r459079;
        double r459081 = r459078 / r459080;
        double r459082 = r459074 / r459079;
        double r459083 = r459081 * r459082;
        double r459084 = r459077 - r459083;
        double r459085 = r459084 / r459075;
        double r459086 = r459074 / r459085;
        double r459087 = r459073 - r459086;
        return r459087;
}

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

Original12.1
Target0.1
Herbie1.7
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 12.1

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

  3. Applied associate-/l*7.0

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

  5. Applied associate-/l*7.0

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

  6. Simplified3.0

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

  8. Applied add-cube-cbrt3.1

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

  9. Applied times-frac1.7

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

  10. Final simplification1.7

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

Reproduce

herbie shell --seed 2019322 
(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)))))