Average Error: 11.5 → 1.2
Time: 7.4s
Precision: 64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - \frac{z}{1 \cdot \frac{z}{\frac{y}{z}} - 0.5 \cdot t}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{z}{1 \cdot \frac{z}{\frac{y}{z}} - 0.5 \cdot t}
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 - (z / ((1.0 * (z / (y / z))) - (0.5 * 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

Original11.5
Target0.1
Herbie1.2
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.5

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

  3. Applied *-commutative11.5

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

  4. Applied associate-/l*6.2

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

  5. Taylor expanded around 0 3.6

    \[\leadsto x - \frac{z}{\color{blue}{1 \cdot \frac{{z}^{2}}{y} - 0.5 \cdot t}}\]
  6. Using strategy rm

  7. Applied unpow23.6

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

  8. Applied associate-/l*1.2

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

  9. Final simplification1.2

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

Reproduce

herbie shell --seed 2020066 +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)))))