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

Average Error: 38.2 → 25.5

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.729724887074998 \cdot 10^{+100}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(x \cdot \left(-{\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\right)\\ \mathbf{elif}\;x \leq 2.142628107794844 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(x \cdot {\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) sqrt((((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 <= -3.729724887074998e+100)) {
		VAR = ((double) (((double) (((double) sqrt((1.0 / ((double) cbrt(((double) sqrt(3.0))))))) * ((double) sqrt((1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0))))))))) * ((double) (x * ((double) -(((double) pow((1.0 / ((double) sqrt(3.0))), 0.16666666666666666))))))));
	} else {
		double VAR_1;
		if ((x <= 2.142628107794844e+138)) {
			VAR_1 = ((double) sqrt((((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0)));
		} else {
			VAR_1 = ((double) (((double) (((double) sqrt((1.0 / ((double) cbrt(((double) sqrt(3.0))))))) * ((double) sqrt((1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0))))))))) * ((double) (x * ((double) pow((1.0 / ((double) sqrt(3.0))), 0.16666666666666666))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	38.2
Target	26.0
Herbie	25.5

\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -3.72972488707499779e100

   1. Initial program 54.8

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

   3. Applied add-cube-cbrt 54.8

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

   4. Applied *-un-lft-identity 54.8

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

   5. Applied times-frac 54.8

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

   6. Applied sqrt-prod 54.8

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

   7. Simplified 54.8

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

   9. Applied add-sqr-sqrt 54.9

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

   10. Applied cbrt-prod 54.9

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

   11. Applied *-un-lft-identity 54.9

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

   12. Applied times-frac 54.8

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

   13. Applied sqrt-prod 54.8

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

   14. Applied associate-*r* 54.8

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

   15. Simplified 54.8

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

   16. Taylor expanded around -inf 18.3

       \[\leadsto \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \color{blue}{\left(-1 \cdot \left(x \cdot {\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\right)}\]

   17. Simplified 18.3

       \[\leadsto \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \color{blue}{\left(x \cdot \left(-{\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\right)}\]

   if -3.72972488707499779e100 < x < 2.142628107794844e138

   1. Initial program 29.4

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

   if 2.142628107794844e138 < x

   1. Initial program 60.9

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

   3. Applied add-cube-cbrt 60.9

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

   4. Applied *-un-lft-identity 60.9

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

   5. Applied times-frac 60.9

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

   6. Applied sqrt-prod 60.9

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

   7. Simplified 60.9

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

   9. Applied add-sqr-sqrt 60.9

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

   10. Applied cbrt-prod 60.9

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

   11. Applied *-un-lft-identity 60.9

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

   12. Applied times-frac 60.9

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

   13. Applied sqrt-prod 60.9

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

   14. Applied associate-*r* 60.9

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

   15. Simplified 60.9

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

   16. Taylor expanded around inf 15.3

       \[\leadsto \left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \color{blue}{\left(x \cdot {\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.729724887074998 \cdot 10^{+100}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(x \cdot \left(-{\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\right)\\ \mathbf{elif}\;x \leq 2.142628107794844 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{3}}}} \cdot \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}\right) \cdot \left(x \cdot {\left(\frac{1}{\sqrt{3}}\right)}^{0.16666666666666666}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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)))