Average Error: 37.5 → 25.2
Time: 4.2s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4.387595707132531 \cdot 10^{+117}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 5.427644600415484 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4.387595707132531 \cdot 10^{+117}:\\
\;\;\;\;\frac{-x}{\sqrt{3}}\\

\mathbf{elif}\;x \leq 5.427644600415484 \cdot 10^{+80}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

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

\end{array}
(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -4.387595707132531e+117)
   (/ (- x) (sqrt 3.0))
   (if (<= x 5.427644600415484e+80)
     (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0))
     (/ x (sqrt 3.0)))))
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 tmp;
	if (x <= -4.387595707132531e+117) {
		tmp = -x / sqrt(3.0);
	} else if (x <= 5.427644600415484e+80) {
		tmp = sqrt((((x * x) + (y * y)) + (z * z)) / 3.0);
	} else {
		tmp = x / sqrt(3.0);
	}
	return tmp;
}

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
Target25.4
Herbie25.2
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.38759570713253083e117

    1. Initial program 56.9

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

    3. Applied sqrt-div_binary64_2260156.9

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

    4. Taylor expanded around -inf 16.7

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

    5. Simplified16.7

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

    if -4.38759570713253083e117 < x < 5.4276446004154839e80

    1. Initial program 28.8

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

    if 5.4276446004154839e80 < x

    1. Initial program 52.2

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

    3. Applied sqrt-div_binary64_2260152.3

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

    4. Taylor expanded around inf 19.2

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

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.387595707132531 \cdot 10^{+117}:\\ \;\;\;\;\frac{-x}{\sqrt{3}}\\ \mathbf{elif}\;x \leq 5.427644600415484 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(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 3.0)) (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)))