Average Error: 37.1 → 16.2
Time: 5.4s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.1682637110415909 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 3.2268796450835897 \cdot 10^{137}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.1682637110415909 \cdot 10^{141}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\

\mathbf{elif}\;x \le 3.2268796450835897 \cdot 10^{137}:\\
\;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{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 temp;
	if ((x <= -3.168263711041591e+141)) {
		temp = (-1.0 * (x * sqrt(0.3333333333333333)));
	} else {
		double temp_1;
		if ((x <= 3.2268796450835897e+137)) {
			temp_1 = (hypot(sqrt(((x * x) + (y * y))), z) / sqrt(3.0));
		} else {
			temp_1 = (x / sqrt(3.0));
		}
		temp = temp_1;
	}
	return temp;
}

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

Original37.1
Target25.4
Herbie16.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.168263711041591e+141

    1. Initial program 61.0

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

    2. Taylor expanded around -inf 14.1

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

    if -3.168263711041591e+141 < x < 3.2268796450835897e+137

    1. Initial program 28.2

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

    3. Applied div-inv28.2

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

    5. Applied un-div-inv28.2

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

    6. Applied sqrt-div28.3

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

    8. Applied add-sqr-sqrt28.3

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

    9. Applied hypot-def16.6

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

    if 3.2268796450835897e+137 < x

    1. Initial program 60.3

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

    3. Applied div-inv60.3

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

    5. Applied un-div-inv60.3

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

    6. Applied sqrt-div60.3

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

    7. Taylor expanded around inf 16.4

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.1682637110415909 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 3.2268796450835897 \cdot 10^{137}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))