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

Average Error: 12.1 → 5.6

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 7.3021372951627968 \cdot 10^{168}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

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) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) <= 7.302137295162797e+168)) {
		VAR = ((double) (x - ((double) (((double) (((double) (y * 2.0)) * ((double) (((double) (((double) cbrt(z)) * ((double) cbrt(z)))) / ((double) (((double) cbrt(((double) (((double) (2.0 * ((double) pow(z, 2.0)))) - ((double) (t * y)))))) * ((double) cbrt(((double) (((double) (2.0 * ((double) pow(z, 2.0)))) - ((double) (t * y)))))))))))) * ((double) (((double) cbrt(z)) / ((double) cbrt(((double) (((double) (2.0 * ((double) pow(z, 2.0)))) - ((double) (t * y))))))))))));
	} else {
		VAR = x;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	12.1
Target	0.1
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < 7.302137295162797e+168

   1. Initial program 3.2

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

   3. Applied *-un-lft-identity 3.2

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

   4. Applied times-frac 2.1

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

   5. Simplified 2.1

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

   6. Simplified 2.1

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

   8. Applied add-cube-cbrt 2.3

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

   9. Applied add-cube-cbrt 2.3

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

   10. Applied times-frac 2.3

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

   11. Applied associate-*r* 1.8

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

   if 7.302137295162797e+168 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))

   1. Initial program 61.6

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

   2. Taylor expanded around 0 26.7

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 7.3021372951627968 \cdot 10^{168}:\\ \;\;\;\;x - \left(\left(y \cdot 2\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y} \cdot \sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{2 \cdot {z}^{2} - t \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(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)))))