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

Average Error: 11.5 → 1.1

Time: 4.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -3.0499995990678665 \cdot 10^{-65}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot \frac{z}{2 \cdot {z}^{2} - t \cdot y}\\ \mathbf{elif}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -0.0:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \mathbf{elif}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 1.13501917837364353 \cdot 10^{123}:\\ \;\;\;\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]

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 ((((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) <= -3.0499995990678665e-65)) {
		VAR = ((double) (x - ((double) (((double) (y * 2.0)) * ((double) (z / ((double) (((double) (2.0 * ((double) pow(z, 2.0)))) - ((double) (t * y))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) <= -0.0)) {
			VAR_1 = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t * ((double) (y / z))))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t)))))) <= 1.1350191783736435e+123)) {
				VAR_2 = ((double) (x - ((double) (((double) (((double) (y * 2.0)) * z)) / ((double) (((double) (((double) (z * 2.0)) * z)) - ((double) (y * t))))))));
			} else {
				VAR_2 = ((double) (x - ((double) (((double) (y * 2.0)) / ((double) (((double) (z * 2.0)) - ((double) (t * ((double) (y / z))))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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	1.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < -3.0499995990678665e-65

   1. Initial program 5.8

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

   3. Applied *-un-lft-identity 5.8

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

   4. Applied times-frac 0.9

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

   5. Simplified 0.9

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

   6. Simplified 0.9

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

   if -3.0499995990678665e-65 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < -0.0 or 1.1350191783736435e+123 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))

   1. Initial program 15.9

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

   3. Applied associate-/l* 9.1

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

   5. Applied div-sub 9.1

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

   6. Simplified 3.7

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

   7. Simplified 3.7

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

   9. Applied *-un-lft-identity 3.7

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

   10. Applied times-frac 1.4

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

   11. Simplified 1.4

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

   if -0.0 < (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t))) < 1.1350191783736435e+123

   1. Initial program 0.2

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -3.0499995990678665 \cdot 10^{-65}:\\ \;\;\;\;x - \left(y \cdot 2\right) \cdot \frac{z}{2 \cdot {z}^{2} - t \cdot y}\\ \mathbf{elif}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le -0.0:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \mathbf{elif}\;\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \le 1.13501917837364353 \cdot 10^{123}:\\ \;\;\;\;x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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)))))