Average Error: 11.3 → 0.1
Time: 4.8s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x + \frac{1}{\frac{0.5}{\frac{z}{t}} - \frac{z}{y}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x + \frac{1}{\frac{0.5}{\frac{z}{t}} - \frac{z}{y}}
(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))
↓
(FPCore (x y z t)
 :precision binary64
 (+ x (/ 1.0 (- (/ 0.5 (/ z t)) (/ z 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 + (1.0 / ((0.5 / (z / t)) - (z / 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

Original11.3
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified2.7

    \[\leadsto \color{blue}{x + \frac{y}{\frac{y \cdot t}{2 \cdot z} - z}}\]
  3. Using strategy rm

  4. Applied associate-/l*_binary64_135040.9

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

  5. Simplified1.0

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

  7. Applied div-inv_binary64_135541.0

    \[\leadsto x + \frac{y}{\frac{y}{\color{blue}{2 \cdot \frac{1}{\frac{t}{z}}}} - z}\]

  8. Simplified0.9

    \[\leadsto x + \frac{y}{\frac{y}{2 \cdot \color{blue}{\frac{z}{t}}} - z}\]
  9. Using strategy rm

  10. Applied clear-num_binary64_135561.0

    \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{y}{2 \cdot \frac{z}{t}} - z}{y}}}\]

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020280 
(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)))))