Average Error: 11.6 → 1.4
Time: 3.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{\frac{2 \cdot z - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}{2}}\]
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 r440696 = x;
        double r440697 = y;
        double r440698 = 2.0;
        double r440699 = r440697 * r440698;
        double r440700 = z;
        double r440701 = r440699 * r440700;
        double r440702 = r440700 * r440698;
        double r440703 = r440702 * r440700;
        double r440704 = t;
        double r440705 = r440697 * r440704;
        double r440706 = r440703 - r440705;
        double r440707 = r440701 / r440706;
        double r440708 = r440696 - r440707;
        return r440708;
}
↓
double f(double x, double y, double z, double t) {
        double r440709 = x;
        double r440710 = y;
        double r440711 = 2.0;
        double r440712 = z;
        double r440713 = r440711 * r440712;
        double r440714 = t;
        double r440715 = cbrt(r440712);
        double r440716 = r440715 * r440715;
        double r440717 = r440714 / r440716;
        double r440718 = r440710 / r440715;
        double r440719 = r440717 * r440718;
        double r440720 = r440713 - r440719;
        double r440721 = r440720 / r440711;
        double r440722 = r440710 / r440721;
        double r440723 = r440709 - r440722;
        return r440723;
}

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
Herbie1.4
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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. Using strategy rm

  5. Applied associate-/l*6.6

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

  6. Simplified2.7

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

  8. Applied add-cube-cbrt2.8

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

  9. Applied times-frac1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2020064 +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)))))