Average Error: 38.1 → 27.1
Time: 3.2s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9.804340525800049 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \mathbf{elif}\;x \leq 8.163804980003783 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;x \leq 2.514104751174992 \cdot 10^{-146}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 2.0671969795246543 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9.804340525800049 \cdot 10^{+73}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\

\mathbf{elif}\;x \leq 8.163804980003783 \cdot 10^{-264}:\\
\;\;\;\;\sqrt{\left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right) \cdot 0.3333333333333333}\\

\mathbf{elif}\;x \leq 2.514104751174992 \cdot 10^{-146}:\\
\;\;\;\;\frac{z}{\sqrt{3}}\\

\mathbf{elif}\;x \leq 2.0671969795246543 \cdot 10^{+127}:\\
\;\;\;\;\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\sqrt{3}}\\

\end{array}
(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -9.804340525800049e+73)
   (*
    (sqrt (/ 1.0 (* (cbrt 3.0) (cbrt 3.0))))
    (- (* x (sqrt (/ 1.0 (cbrt 3.0))))))
   (if (<= x 8.163804980003783e-264)
     (sqrt (* (+ (+ (* x x) (* y y)) (* z z)) 0.3333333333333333))
     (if (<= x 2.514104751174992e-146)
       (/ z (sqrt 3.0))
       (if (<= x 2.0671969795246543e+127)
         (/ (sqrt (+ (+ (* x x) (* y y)) (* z z))) (sqrt 3.0))
         (/ x (sqrt 3.0)))))))
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 tmp;
	if (x <= -9.804340525800049e+73) {
		tmp = sqrt(1.0 / (cbrt(3.0) * cbrt(3.0))) * -(x * sqrt(1.0 / cbrt(3.0)));
	} else if (x <= 8.163804980003783e-264) {
		tmp = sqrt((((x * x) + (y * y)) + (z * z)) * 0.3333333333333333);
	} else if (x <= 2.514104751174992e-146) {
		tmp = z / sqrt(3.0);
	} else if (x <= 2.0671969795246543e+127) {
		tmp = sqrt(((x * x) + (y * y)) + (z * z)) / sqrt(3.0);
	} else {
		tmp = x / sqrt(3.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target25.7
Herbie27.1
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if x < -9.8043405258000489e73

    1. Initial program 51.5

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

    3. Applied add-cube-cbrt_binary6451.5

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

    4. Applied *-un-lft-identity_binary6451.5

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

    5. Applied times-frac_binary6451.5

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

    6. Applied sqrt-prod_binary6451.5

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

    7. Taylor expanded around -inf 20.0

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

    8. Simplified20.0

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

    if -9.8043405258000489e73 < x < 8.16380498000378273e-264

    1. Initial program 29.5

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

    3. Applied div-inv_binary6429.5

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

    4. Simplified29.5

      \[\leadsto \sqrt{\left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right) \cdot \color{blue}{0.3333333333333333}}\]

    if 8.16380498000378273e-264 < x < 2.51410475117499204e-146

    1. Initial program 32.5

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

    3. Applied sqrt-div_binary6432.6

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

    4. Taylor expanded around 0 47.1

      \[\leadsto \frac{\color{blue}{z}}{\sqrt{3}}\]

    if 2.51410475117499204e-146 < x < 2.0671969795246543e127

    1. Initial program 28.5

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

    3. Applied sqrt-div_binary6428.6

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

    if 2.0671969795246543e127 < x

    1. Initial program 59.4

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

    3. Applied sqrt-div_binary6459.4

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

    4. Taylor expanded around inf 16.0

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{3}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.804340525800049 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \mathbf{elif}\;x \leq 8.163804980003783 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{\left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right) \cdot 0.3333333333333333}\\ \mathbf{elif}\;x \leq 2.514104751174992 \cdot 10^{-146}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 2.0671969795246543 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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)))