Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Average Error: 12.0 → 2.6

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.93026976530572721 \cdot 10^{-54} \lor \neg \left(z \le 2.607693348704506 \cdot 10^{-81}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t}}{\frac{1}{z}}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (x - ((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((z <= -9.930269765305727e-54) || !(z <= 2.607693348704506e-81))) {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t / ((double) (z / y))))))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (((double) (y * 2.0)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) / ((double) (1.0 / z))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	12.0
Target	0.1
Herbie	2.6

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

Derivation

1. Split input into 2 regimes

2. if z < -9.930269765305727e-54 or 2.607693348704506e-81 < z

   1. Initial program 15.2

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

   3. Applied associate-/l* 7.0

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

   5. Applied div-sub 7.0

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

   6. Simplified 2.4

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

   7. Simplified 2.4

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

   9. Applied associate-/l* 1.0

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

   if -9.930269765305727e-54 < z < 2.607693348704506e-81

   1. Initial program 6.9

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

   3. Applied associate-/l* 6.6

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

   5. Applied div-inv 6.7

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

   6. Applied associate-/r* 5.2

      \[\leadsto x - \color{blue}{\frac{\frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t}}{\frac{1}{z}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.93026976530572721 \cdot 10^{-54} \lor \neg \left(z \le 2.607693348704506 \cdot 10^{-81}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y \cdot 2}{\left(z \cdot 2\right) \cdot z - y \cdot t}}{\frac{1}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1 (- (/ z y) (/ (/ t 2) z))))

  (- x (/ (* (* y 2) z) (- (* (* z 2) z) (* y t)))))