Average Error: 11.6 → 6.8
Time: 4.1s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \left(y \cdot 2\right) \cdot \frac{z}{2 \cdot {z}^{2} - t \cdot y}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \left(y \cdot 2\right) \cdot \frac{z}{2 \cdot {z}^{2} - t \cdot y}
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 - ((y * 2.0) * (z / ((2.0 * pow(z, 2.0)) - (t * y)))));
}

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
Herbie6.8
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

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 *-un-lft-identity11.6

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

  4. Applied times-frac6.8

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

  5. Simplified6.8

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

  6. Simplified6.8

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

  7. Final simplification6.8

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

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(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)))))