Average Error: 11.2 → 0.1
Time: 4.4s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[x + \frac{-1}{-0.5 \cdot \frac{t}{z} + \frac{z}{y}} \]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x + \frac{-1}{-0.5 \cdot \frac{t}{z} + \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 (/ t z)) (/ 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 * (t / z)) + (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

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

Derivation

  1. Initial program 11.2

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

  2. Simplified2.7

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

  4. Applied clear-num_binary642.8

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

  5. Simplified1.0

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

  7. Applied sub-neg_binary641.0

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2021196 
(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)))))