Average Error: 37.8 → 25.4
Time: 4.0s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.66577969947544324 \cdot 10^{120}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.3162058554559041 \cdot 10^{110}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.66577969947544324 \cdot 10^{120}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\

\mathbf{elif}\;x \le 1.3162058554559041 \cdot 10^{110}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

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

\end{array}
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 VAR;
	if ((x <= -4.665779699475443e+120)) {
		VAR = (-1.0 * (x * sqrt(0.3333333333333333)));
	} else {
		double VAR_1;
		if ((x <= 1.316205855455904e+110)) {
			VAR_1 = sqrt(((((x * x) + (y * y)) + (z * z)) / 3.0));
		} else {
			VAR_1 = (x / 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

Original37.8
Target25.6
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.665779699475443e+120

    1. Initial program 58.3

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

    2. Taylor expanded around -inf 17.5

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)}\]

    if -4.665779699475443e+120 < x < 1.316205855455904e+110

    1. Initial program 28.9

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

    if 1.316205855455904e+110 < x

    1. Initial program 55.9

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

    3. Applied sqrt-div55.9

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

    4. Taylor expanded around inf 17.5

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

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.66577969947544324 \cdot 10^{120}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.3162058554559041 \cdot 10^{110}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(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)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3)))