Average Error: 38.3 → 26.0
Time: 4.4s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.725859237515929 \cdot 10^{+142}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 5.22004744817066 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.725859237515929 \cdot 10^{+142}:\\
\;\;\;\;\frac{-x}{\sqrt{3}}\\

\mathbf{elif}\;x \le 5.22004744817066 \cdot 10^{+101}:\\
\;\;\;\;\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.3333333333333333}\\

\end{array}
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) {
	double VAR;
	if ((x <= -9.725859237515929e+142)) {
		VAR = ((double) (((double) -(x)) / ((double) sqrt(3.0))));
	} else {
		double VAR_1;
		if ((x <= 5.22004744817066e+101)) {
			VAR_1 = ((double) (((double) sqrt(((double) (((double) (x * x)) + ((double) (((double) (y * y)) + ((double) (z * z)))))))) / ((double) sqrt(3.0))));
		} else {
			VAR_1 = ((double) (x * ((double) sqrt(0.3333333333333333))));
		}
		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

Original38.3
Target25.6
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -9.72585923751592876e142

    1. Initial program 61.0

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

    3. Applied sqrt-div61.0

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

    4. Simplified61.0

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

    5. Taylor expanded around -inf 15.4

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

    6. Simplified15.4

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

    if -9.72585923751592876e142 < x < 5.2200474481706602e101

    1. Initial program 29.8

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

    3. Applied sqrt-div29.9

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

    4. Simplified29.9

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

    if 5.2200474481706602e101 < x

    1. Initial program 55.2

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

    2. Taylor expanded around inf 18.1

      \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.725859237515929 \cdot 10^{+142}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 5.22004744817066 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))