Average Error: 11.6 → 0.9
Time: 7.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{y}{1 \cdot z - \frac{y \cdot 0.5}{\frac{z}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{1 \cdot z - \frac{y \cdot 0.5}{\frac{z}{t}}}
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) (y / ((double) (((double) (1.0 * z)) - ((double) (((double) (y * 0.5)) / ((double) (z / t))))))))));
}

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
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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*11.6

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

  4. Applied associate-/l*6.5

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

  5. Taylor expanded around 0 2.5

    \[\leadsto x - \frac{y}{\color{blue}{1 \cdot z - 0.5 \cdot \frac{t \cdot y}{z}}}\]
  6. Using strategy rm

  7. Applied *-commutative2.5

    \[\leadsto x - \frac{y}{1 \cdot z - 0.5 \cdot \frac{\color{blue}{y \cdot t}}{z}}\]

  8. Applied associate-/l*0.9

    \[\leadsto x - \frac{y}{1 \cdot z - 0.5 \cdot \color{blue}{\frac{y}{\frac{z}{t}}}}\]

  9. Applied associate-*r/0.9

    \[\leadsto x - \frac{y}{1 \cdot z - \color{blue}{\frac{0.5 \cdot y}{\frac{z}{t}}}}\]

  10. Simplified0.9

    \[\leadsto x - \frac{y}{1 \cdot z - \frac{\color{blue}{y \cdot 0.5}}{\frac{z}{t}}}\]

  11. Final simplification0.9

    \[\leadsto x - \frac{y}{1 \cdot z - \frac{y \cdot 0.5}{\frac{z}{t}}}\]

Reproduce

herbie shell --seed 2020113 
(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)))))