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

Average Error: 12.0 → 6.5

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.57127147346290313 \cdot 10^{197}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \end{array}\]

C

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) {
	double temp;
	if ((z <= -2.571271473462903e+197)) {
		temp = (x - ((y * 2.0) * 0.0));
	} else {
		temp = (x - (((y * 2.0) * ((cbrt(z) * cbrt(z)) / (cbrt(((2.0 * pow(z, 2.0)) - (t * y))) * cbrt(((2.0 * pow(z, 2.0)) - (t * y)))))) * (cbrt(z) / cbrt(((2.0 * pow(z, 2.0)) - (t * y))))));
	}
	return temp;
}

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	6.5

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

Derivation

1. Split input into 2 regimes

2. if z < -2.571271473462903e+197

   1. Initial program 27.9

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

   3. Applied *-un-lft-identity 27.9

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

   4. Applied times-frac 14.4

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

   5. Simplified 14.4

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

   6. Simplified 14.4

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

   7. Taylor expanded around 0 12.0

      \[\leadsto x - \left(y \cdot 2\right) \cdot \color{blue}{0}\]

   if -2.571271473462903e+197 < z

   1. Initial program 10.5

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

   3. Applied *-un-lft-identity 10.5

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

   4. Applied times-frac 6.2

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

   5. Simplified 6.2

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

   6. Simplified 6.2

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

   8. Applied add-cube-cbrt 6.4

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

   9. Applied add-cube-cbrt 6.5

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

   10. Applied times-frac 6.5

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

   11. Applied associate-*r* 6.0

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.57127147346290313 \cdot 10^{197}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))))