Average Error: 11.1 → 0.1
Time: 2.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{1}{\frac{\frac{z \cdot 2}{y}}{2} - \frac{\frac{t}{2}}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\frac{\frac{z \cdot 2}{y}}{2} - \frac{\frac{t}{2}}{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 - (1.0 / ((((z * 2.0) / y) / 2.0) - ((t / 2.0) / 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

Original11.1
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.1

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

  3. Applied *-commutative11.1

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

  4. Applied associate-/l*6.3

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

  6. Applied associate-/r*6.3

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

  7. Simplified1.4

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

  9. Applied clear-num1.5

    \[\leadsto x - \color{blue}{\frac{1}{\frac{\frac{\frac{z \cdot 2}{y} \cdot z - t}{2}}{z}}}\]
  10. Using strategy rm

  11. Applied div-sub1.5

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

  12. Applied div-sub1.5

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

  13. Simplified0.1

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

  14. Final simplification0.1

    \[\leadsto x - \frac{1}{\frac{\frac{z \cdot 2}{y}}{2} - \frac{\frac{t}{2}}{z}}\]

Reproduce

herbie shell --seed 2020078 
(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)))))