Average Error: 12.1 → 1.3
Time: 3.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y \cdot 2}{z \cdot 2 - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{t}{\frac{z}{\sqrt[3]{y}}}}\]
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 - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \frac{t}{\frac{z}{\sqrt[3]{y}}}}
double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (t / ((double) (z / ((double) cbrt(y))))))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original12.1
Target0.1
Herbie1.3
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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*6.8

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

  5. Applied div-sub6.8

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

  6. Simplified3.0

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

  7. Simplified3.0

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

  9. Applied associate-/l*2.2

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

  11. Applied add-cube-cbrt2.3

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

  12. Applied *-un-lft-identity2.3

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

  13. Applied times-frac2.3

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\color{blue}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}}}}\]

  14. Applied *-un-lft-identity2.3

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \frac{\color{blue}{1 \cdot t}}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}}}\]

  15. Applied times-frac1.3

    \[\leadsto x - \frac{y \cdot 2}{z \cdot 2 - \color{blue}{\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{t}{\frac{z}{\sqrt[3]{y}}}}}\]

  16. Simplified1.3

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

  17. Final simplification1.3

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

Reproduce

herbie shell --seed 2020126 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))