Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1

Average Error: 37.4 → 25.8

Time: 6.7s

Precision: 64

Math

\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.77107210764756 \cdot 10^{107}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le -1.8518021247462165 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{elif}\;x \le -4.6586170515265218 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot z\right)\\ \mathbf{elif}\;x \le 1.0270452633617047 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) sqrt(((double) (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((x <= -2.7710721076475597e+107)) {
		VAR = ((double) (((double) sqrt(((double) (1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))) * ((double) (-1.0 * ((double) (((double) sqrt(((double) (1.0 / ((double) cbrt(3.0)))))) * x))))));
	} else {
		double VAR_1;
		if ((x <= -1.8518021247462165e-219)) {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0))));
		} else {
			double VAR_2;
			if ((x <= -4.658617051526522e-281)) {
				VAR_2 = ((double) (((double) sqrt(((double) (1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))) * ((double) (((double) sqrt(((double) (1.0 / ((double) cbrt(3.0)))))) * z))));
			} else {
				double VAR_3;
				if ((x <= 1.0270452633617047e+135)) {
					VAR_3 = ((double) sqrt(((double) (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0))));
				} else {
					VAR_3 = ((double) (((double) sqrt(((double) (1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))) * ((double) (((double) sqrt(((double) (1.0 / ((double) cbrt(3.0)))))) * x))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.4
Target	25.7
Herbie	25.8

\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if x < -2.7710721076475597e+107

   1. Initial program 55.3

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 55.3

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

   4. Applied *-un-lft-identity 55.3

      \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\]

   5. Applied times-frac 55.3

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   6. Applied sqrt-prod 55.4

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   7. Taylor expanded around -inf 17.7

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)}\]

   if -2.7710721076475597e+107 < x < -1.8518021247462165e-219 or -4.658617051526522e-281 < x < 1.0270452633617047e+135

   1. Initial program 28.7

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]

   if -1.8518021247462165e-219 < x < -4.658617051526522e-281

   1. Initial program 30.5

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 30.5

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

   4. Applied *-un-lft-identity 30.5

      \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\]

   5. Applied times-frac 30.5

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   6. Applied sqrt-prod 30.5

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   7. Taylor expanded around 0 45.2

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot z\right)}\]

   if 1.0270452633617047e+135 < x

   1. Initial program 59.5

      \[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 59.5

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

   4. Applied *-un-lft-identity 59.5

      \[\leadsto \sqrt{\frac{\color{blue}{1 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\]

   5. Applied times-frac 59.5

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   6. Applied sqrt-prod 59.5

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{\sqrt[3]{3}}}}\]

   7. Taylor expanded around inf 15.3

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.77107210764756 \cdot 10^{107}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le -1.8518021247462165 \cdot 10^{-219}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{elif}\;x \le -4.6586170515265218 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot z\right)\\ \mathbf{elif}\;x \le 1.0270452633617047 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020140 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))