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

Average Error: 37.6 → 0.4

Time: 11.5s

Precision: 64

Math

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

↓

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

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) {
	return ((double) (((double) hypot(((double) hypot(y, x)), z)) / ((double) sqrt(((double) (((double) -(((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))) * ((double) -(((double) cbrt(3.0))))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	37.6
Target	25.4
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 37.6

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

3. Applied add-cube-cbrt 37.6

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

4. Applied add-sqr-sqrt 37.6

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

5. Applied times-frac 37.6

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

7. Applied frac-2neg 37.6

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

8. Applied frac-2neg 37.6

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

9. Applied frac-times 37.6

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

10. Applied sqrt-div 37.7

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

11. Simplified 37.7

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

13. Applied add-sqr-sqrt 37.7

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

14. Applied hypot-def 28.2

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

16. Applied +-commutative 28.2

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

17. Applied hypot-def 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020113 +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)))