Average Error: 37.8 → 25.7
Time: 4.6s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.01637277483957589 \cdot 10^{88}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 9.156732467259482 \cdot 10^{100}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.01637277483957589 \cdot 10^{88}:\\
\;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\

\mathbf{elif}\;x \le 9.156732467259482 \cdot 10^{100}:\\
\;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.333333333333333315}\\

\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 <= -5.016372774839576e+88)) {
		VAR = ((double) (((double) (-1.0 * x)) / ((double) sqrt(3.0))));
	} else {
		double VAR_1;
		if ((x <= 9.156732467259482e+100)) {
			VAR_1 = ((double) sqrt(((double) (0.3333333333333333 * ((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))))));
		} 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

Original37.8
Target25.5
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.01637277483957589e88

    1. Initial program 53.7

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

    3. Applied sqrt-div53.7

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

    4. Taylor expanded around -inf 20.0

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

    if -5.01637277483957589e88 < x < 9.156732467259482e100

    1. Initial program 29.2

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

    2. Taylor expanded around 0 29.2

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot {x}^{2} + \left(0.333333333333333315 \cdot {y}^{2} + 0.333333333333333315 \cdot {z}^{2}\right)}}\]

    3. Simplified29.2

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}}\]

    if 9.156732467259482e100 < x

    1. Initial program 55.0

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

    2. Taylor expanded around inf 17.9

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.01637277483957589 \cdot 10^{88}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 9.156732467259482 \cdot 10^{100}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left(\left(x \cdot x + y \cdot y\right) + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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) (/ (neg 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)))