Average Error: 37.6 → 25.7
Time: 4.3s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.0968830859253667 \cdot 10^{+151}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 3.7492675251181064 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt{3}}} \cdot \frac{x}{\sqrt{\sqrt{3}}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.0968830859253667 \cdot 10^{+151}:\\
\;\;\;\;\frac{-x}{\sqrt{3}}\\

\mathbf{elif}\;x \leq 3.7492675251181064 \cdot 10^{+108}:\\
\;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\sqrt{3}}} \cdot \frac{x}{\sqrt{\sqrt{3}}}\\

\end{array}
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 <= -3.0968830859253667e+151)) {
		VAR = (((double) -(x)) / ((double) sqrt(3.0)));
	} else {
		double VAR_1;
		if ((x <= 3.7492675251181064e+108)) {
			VAR_1 = ((double) (((double) sqrt(((double) (((double) (x * x)) + ((double) (((double) (y * y)) + ((double) (z * z)))))))) * ((double) sqrt((1.0 / 3.0)))));
		} else {
			VAR_1 = ((double) ((1.0 / ((double) sqrt(((double) sqrt(3.0))))) * (x / ((double) sqrt(((double) sqrt(3.0)))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Original37.6
Target25.3
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.0968830859253667e151

    1. Initial program 63.4

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

    3. Applied sqrt-div63.4

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

    4. Simplified63.4

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

    5. Taylor expanded around -inf 15.5

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\sqrt{3}}\]

    6. Simplified15.5

      \[\leadsto \frac{\color{blue}{-x}}{\sqrt{3}}\]

    if -3.0968830859253667e151 < x < 3.7492675251181064e108

    1. Initial program 29.1

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

    3. Applied div-inv29.1

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

    4. Applied sqrt-prod29.2

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

    5. Simplified29.2

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

    if 3.7492675251181064e108 < x

    1. Initial program 55.4

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

    3. Applied sqrt-div55.4

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

    4. Simplified55.4

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

    5. Taylor expanded around inf 18.2

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{3}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt18.2

      \[\leadsto \frac{x}{\sqrt{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}}\]

    8. Applied sqrt-prod18.7

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{3}} \cdot \sqrt{\sqrt{3}}}}\]

    9. Applied *-un-lft-identity18.7

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\sqrt{3}} \cdot \sqrt{\sqrt{3}}}\]

    10. Applied times-frac18.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\sqrt{3}}} \cdot \frac{x}{\sqrt{\sqrt{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.0968830859253667 \cdot 10^{+151}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 3.7492675251181064 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt{3}}} \cdot \frac{x}{\sqrt{\sqrt{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))