Average Error: 37.5 → 25.1
Time: 3.3s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6174687293654252 \cdot 10^{118}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.3566371815103393 \cdot 10^{105}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6174687293654252 \cdot 10^{118}:\\
\;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\

\mathbf{elif}\;x \le 1.3566371815103393 \cdot 10^{105}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)} \cdot \sqrt{\frac{1}{3}}\\

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

\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 <= -1.6174687293654252e+118)) {
		VAR = ((-1.0 * x) / sqrt(3.0));
	} else {
		double VAR_1;
		if ((x <= 1.3566371815103393e+105)) {
			VAR_1 = (sqrt(fma(x, x, fma(y, y, (z * z)))) * sqrt((1.0 / 3.0)));
		} else {
			VAR_1 = (x * 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

Original37.5
Target25.2
Herbie25.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.6174687293654252e+118

    1. Initial program 57.3

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

    3. Applied *-un-lft-identity57.3

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

    4. Applied associate-/r*57.3

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

    5. Simplified57.3

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)}}{3}}\]
    6. Using strategy rm

    7. Applied sqrt-div57.3

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

    8. Taylor expanded around -inf 16.8

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

    if -1.6174687293654252e+118 < x < 1.3566371815103393e+105

    1. Initial program 28.6

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

    3. Applied *-un-lft-identity28.6

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

    4. Applied associate-/r*28.6

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

    5. Simplified28.6

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)}}{3}}\]
    6. Using strategy rm

    7. Applied div-inv28.6

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

    8. Applied sqrt-prod28.7

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

    if 1.3566371815103393e+105 < x

    1. Initial program 55.5

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

    2. Taylor expanded around inf 18.4

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

  4. Final simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6174687293654252 \cdot 10^{118}:\\ \;\;\;\;\frac{-1 \cdot x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 1.3566371815103393 \cdot 10^{105}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, z \cdot z\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))