Average Error: 38.6 → 16.4
Time: 5.1s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.3742010521062343 \cdot 10^{153}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.3321670395294099 \cdot 10^{154}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.3742010521062343 \cdot 10^{153}:\\
\;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\

\mathbf{elif}\;x \le 1.3321670395294099 \cdot 10^{154}:\\
\;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\

\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 <= -7.374201052106234e+153)) {
		VAR = ((double) (((double) (-1.0 * x)) / ((double) sqrt(3.0))));
	} else {
		double VAR_1;
		if ((x <= 1.3321670395294099e+154)) {
			VAR_1 = ((double) (((double) hypot(((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y)))))), 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

Original38.6
Target26.0
Herbie16.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -7.374201052106234e+153

    1. Initial program 64.0

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

    3. Applied sqrt-div64.0

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

    4. Taylor expanded around -inf 14.3

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

    if -7.374201052106234e+153 < x < 1.3321670395294099e+154

    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. Using strategy rm

    5. Applied add-sqr-sqrt29.9

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

    6. Applied hypot-def17.1

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

    if 1.3321670395294099e+154 < x

    1. Initial program 64.0

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

    2. Taylor expanded around inf 14.7

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.3742010521062343 \cdot 10^{153}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.3321670395294099 \cdot 10^{154}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(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)))