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

Average Error: 11.1 → 4.7

Time: 5.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le +inf.0:\\ \;\;\;\;x - \frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

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 ((((double) (x - ((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))))) <= +inf.0)) {
		VAR = ((double) (x - ((double) (((double) (((double) (y * 2.0)) / ((double) (((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) * ((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))))))) * ((double) (z / ((double) cbrt(((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))))))));
	} else {
		VAR = x;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.1
Target	0.1
Herbie	4.7

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

Derivation

1. Split input into 2 regimes

2. if (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))) < +inf.0

   1. Initial program 3.4

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

   3. Applied add-cube-cbrt 3.6

      \[\leadsto x - \frac{\left(y \cdot 2\right) \cdot z}{\color{blue}{\left(\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}\right) \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}}\]

   4. Applied times-frac 1.7

      \[\leadsto x - \color{blue}{\frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}}\]

   if +inf.0 < (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))))

   1. Initial program 64.0

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

   2. Taylor expanded around 0 25.9

      \[\leadsto \color{blue}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le +inf.0:\\ \;\;\;\;x - \frac{y \cdot 2}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t} \cdot \sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}} \cdot \frac{z}{\sqrt[3]{\left(z \cdot 2\right) \cdot z - y \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(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)))))