Average Error: 11.6 → 0.1
Time: 3.2s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[x - \frac{1}{\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)} \]
x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - \frac{1}{\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\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 (/ 1.0 (fma -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 / fma(-0.5, (t / z), (z / y)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.6
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

  1. Initial program 11.6

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

  2. Simplified3.0

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

  3. Applied clear-num_binary643.0

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

  4. Taylor expanded in z around 0 0.1

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

  5. Simplified0.1

    \[\leadsto x - \frac{1}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)}} \]

  6. Final simplification0.1

    \[\leadsto x - \frac{1}{\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)} \]

Reproduce

herbie shell --seed 2022019 
(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)))))