Average Error: 11.4 → 1.5
Time: 4.3s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}
double code(double x, double y, double z, double t) {
	return (x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t))));
}
↓
double code(double x, double y, double z, double t) {
	return (x - ((y * 2.0) / ((z * 2.0) - ((t / (cbrt(z) * cbrt(z))) * (y / cbrt(z))))));
}

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

Original11.4
Target0.1
Herbie1.5
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.4

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

  3. Applied associate-/l*6.7

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

  5. Applied div-sub6.7

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

  6. Simplified2.8

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

  7. Simplified2.8

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

  9. Applied add-cube-cbrt2.9

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

  10. Applied times-frac1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2020092 +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)))))