Average Error: 11.9 → 1.1
Time: 3.6s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]
↓
\[x - y \cdot \frac{1}{z - \frac{y}{2} \cdot \frac{t}{z}}\]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - y \cdot \frac{1}{z - \frac{y}{2} \cdot \frac{t}{z}}
(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 (* y (/ 1.0 (- z (* (/ y 2.0) (/ t z)))))))
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 * (1.0 / (z - ((y / 2.0) * (t / 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

Original11.9
Target0.1
Herbie1.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

  1. Initial program 11.9

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

  2. Simplified2.9

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

  4. Applied times-frac_binary641.0

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

  6. Applied div-inv_binary641.1

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

  7. Final simplification1.1

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

Reproduce

herbie shell --seed 2020231 
(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)))))