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

Average Error: 37.6 → 25.1

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.03150720873795742 \cdot 10^{131}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.25466368308091277 \cdot 10^{144}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

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 <= -3.0315072087379574e+131)) {
		VAR = ((double) (-1.0 * ((double) (x * ((double) sqrt(0.3333333333333333))))));
	} else {
		double VAR_1;
		if ((x <= 1.2546636830809128e+144)) {
			VAR_1 = ((double) sqrt(((double) (0.3333333333333333 * ((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))))));
		} else {
			VAR_1 = ((double) (x * ((double) sqrt(0.3333333333333333))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.6
Target	25.1
Herbie	25.1

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

Derivation

1. Split input into 3 regimes

2. if x < -3.0315072087379574e+131

   1. Initial program 59.5

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

   2. Taylor expanded around -inf 15.5

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)}\]

   if -3.0315072087379574e+131 < x < 1.2546636830809128e+144

   1. Initial program 28.9

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

   2. Taylor expanded around 0 28.9

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot {x}^{2} + \left(0.333333333333333315 \cdot {y}^{2} + 0.333333333333333315 \cdot {z}^{2}\right)}}\]

   3. Simplified 28.9

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}\]

   if 1.2546636830809128e+144 < x

   1. Initial program 61.9

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

   2. Taylor expanded around inf 14.7

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.03150720873795742 \cdot 10^{131}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.25466368308091277 \cdot 10^{144}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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)))