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

Average Error: 37.5 → 0.4

Time: 3.8s

Precision: 64

Math

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

↓

\[\left|\frac{\mathsf{hypot}\left(1 \cdot \mathsf{hypot}\left(x, y\right), z\right)}{\sqrt{3}}\right|\]

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) {
	return fabs((hypot((1.0 * hypot(x, y)), z) / sqrt(3.0)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.5
Target	25.6
Herbie	0.4

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

Derivation

1. Initial program 37.5

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

3. Applied add-sqr-sqrt 37.6

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

4. Applied add-sqr-sqrt 37.6

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

5. Applied times-frac 37.6

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

6. Applied rem-sqrt-square 37.6

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

8. Applied add-sqr-sqrt 37.6

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

9. Applied hypot-def 28.6

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

11. Applied *-un-lft-identity 28.6

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

12. Applied sqrt-prod 28.6

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

13. Simplified 28.6

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

14. Simplified 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020056 +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)))