Average Error: 11.9 → 1.0
Time: 4.9s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[x + \frac{y}{\mathsf{fma}\left(-1, z, \frac{y}{2 \cdot \frac{z}{t}}\right)} \]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x + \frac{y}{\mathsf{fma}\left(-1, z, \frac{y}{2 \cdot \frac{z}{t}}\right)}

(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 (fma -1.0 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 / fma(-1.0, z, (y / (2.0 * (z / t)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.9
Target0.1
Herbie1.0
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

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. Simplified2.7

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

  3. Applied frac-2neg_binary642.7

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

  4. Simplified1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2021329 
(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)))))