Average Error: 11.3 → 0.9
Time: 3.9s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[x - \frac{y}{z - \frac{\frac{y}{2}}{\frac{z}{t}}} \]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{y}{z - \frac{\frac{y}{2}}{\frac{z}{t}}}

(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 (- z (/ (/ y 2.0) (/ z 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 - (y / (z - ((y / 2.0) / (z / 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

Original11.3
Target0.1
Herbie0.9
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

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

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

  3. Applied egg-rr0.9

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

  4. Applied egg-rr0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2022130 
(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)))))