Average Error: 11.8 → 1.1
Time: 6.9s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{\left(y \cdot 2\right) \cdot 1}{2 \cdot z - \frac{y}{\frac{z}{t}}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{\left(y \cdot 2\right) \cdot 1}{2 \cdot z - \frac{y}{\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) (((double) (((double) (y * 2.0)) * 1.0)) / ((double) (((double) (2.0 * z)) - ((double) (y / ((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

Original11.8
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.8

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

  3. Applied *-un-lft-identity11.8

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

  4. Applied associate-*r*11.8

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

  5. Applied associate-/l*6.6

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

  6. Taylor expanded around 0 3.0

    \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot 1}{\color{blue}{2 \cdot z - \frac{t \cdot y}{z}}}\]
  7. Using strategy rm

  8. Applied *-commutative3.0

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

  9. Applied associate-/l*1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2020114 
(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)))))