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

Average Error: 38.2 → 25.6

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.90401259865907417 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 7.4023060657315993 \cdot 10^{90}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot \frac{\sqrt{1}}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double temp;
	if ((x <= -4.904012598659074e+153)) {
		temp = (-1.0 * (x * sqrt(0.3333333333333333)));
	} else {
		double temp_1;
		if ((x <= 7.402306065731599e+90)) {
			temp_1 = sqrt((fma(x, x, fma(y, y, (z * z))) * (sqrt(1.0) / 3.0)));
		} else {
			temp_1 = (x / sqrt(3.0));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.2
Target	26.3
Herbie	25.6

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

Derivation

1. Split input into 3 regimes

2. if x < -4.904012598659074e+153

   1. Initial program 63.8

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

   2. Taylor expanded around -inf 14.2

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

   if -4.904012598659074e+153 < x < 7.402306065731599e+90

   1. Initial program 29.3

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

   3. Applied div-inv 29.4

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

   5. Applied *-un-lft-identity 29.4

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

   6. Applied add-sqr-sqrt 29.4

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

   7. Applied times-frac 29.4

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

   8. Applied associate-*r* 29.4

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

   9. Simplified 29.4

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

   if 7.402306065731599e+90 < x

   1. Initial program 54.6

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

   3. Applied div-inv 54.6

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

   5. Applied *-un-lft-identity 54.6

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

   6. Applied add-sqr-sqrt 54.6

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

   7. Applied times-frac 54.6

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

   8. Applied associate-*r* 54.6

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

   9. Simplified 54.6

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)} \cdot \frac{\sqrt{1}}{3}}\]
   10. Using strategy rm

   11. Applied associate-*r/ 54.6

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

   12. Applied sqrt-div 54.6

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

   13. Simplified 54.6

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

   14. Taylor expanded around inf 19.1

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

4. Final simplification 25.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.90401259865907417 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 7.4023060657315993 \cdot 10^{90}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right) \cdot \frac{\sqrt{1}}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3)))