Average Error: 11.6 → 2.8
Time: 3.5s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - y \cdot \frac{2}{2 \cdot z - \frac{t \cdot y}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - y \cdot \frac{2}{2 \cdot z - \frac{t \cdot y}{z}}
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) (2.0 / ((double) (((double) (2.0 * z)) - ((double) (((double) (t * y)) / 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

Original11.6
Target0.1
Herbie2.8
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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*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 *-un-lft-identity6.7

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

  6. Applied *-un-lft-identity6.7

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

  7. Applied times-frac6.7

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

  8. Applied times-frac6.7

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

  9. Simplified6.7

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

  10. Simplified2.8

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

  11. Final simplification2.8

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

Reproduce

herbie shell --seed 2020128 
(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)))))