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

Average Error: 11.5 → 2.1

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 1853.961211351907:\\ \;\;\;\;x - y \cdot \frac{2}{2 \cdot z - \frac{t \cdot y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \end{array}\]

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 <= 1853.961211351907)) {
		VAR = ((double) (x - ((double) (y * ((double) (2.0 / ((double) (((double) (2.0 * z)) - ((double) (((double) (t * y)) / z))))))))));
	} else {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t / ((double) (z / y))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.5
Target	0.1
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if z < 1853.961211351907

   1. Initial program 9.4

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

   3. Applied associate-/l* 5.8

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

   5. Applied *-un-lft-identity 5.8

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

   6. Applied *-un-lft-identity 5.8

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

   7. Applied times-frac 5.8

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

   8. Applied times-frac 5.7

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

   9. Simplified 5.7

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

   10. Simplified 2.6

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

   if 1853.961211351907 < z

   1. Initial program 17.9

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

   3. Applied associate-/l* 8.2

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

   5. Applied div-sub 8.2

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

   6. Simplified 2.6

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

   7. Simplified 2.6

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

   9. Applied associate-/l* 0.5

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1853.961211351907:\\ \;\;\;\;x - y \cdot \frac{2}{2 \cdot z - \frac{t \cdot y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - \frac{t}{\frac{z}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(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)))))