Average Error: 38.1 → 25.8
Time: 3.8s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.355766780126997 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;x \leq 1.322520265236586 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.355766780126997 \cdot 10^{+128}:\\
\;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\

\mathbf{elif}\;x \leq 1.322520265236586 \cdot 10^{+124}:\\
\;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.3333333333333333}\\

\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 -1.355766780126997e+128)
   (* x (- (sqrt 0.3333333333333333)))
   (if (<= x 1.322520265236586e+124)
     (* (sqrt (+ (* x x) (+ (* y y) (* z z)))) (sqrt (/ 1.0 3.0)))
     (* x (sqrt 0.3333333333333333)))))
double code(double x, double y, double z) {
	return ((double) sqrt((((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.355766780126997e+128)) {
		tmp = ((double) (x * ((double) -(((double) sqrt(0.3333333333333333))))));
	} else {
		double tmp_1;
		if ((x <= 1.322520265236586e+124)) {
			tmp_1 = ((double) (((double) sqrt(((double) (((double) (x * x)) + ((double) (((double) (y * y)) + ((double) (z * z)))))))) * ((double) sqrt((1.0 / 3.0)))));
		} else {
			tmp_1 = ((double) (x * ((double) sqrt(0.3333333333333333))));
		}
		tmp = tmp_1;
	}
	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

Original38.1
Target25.7
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.35576678012699705e128

    1. Initial program Error: 58.4 bits

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

    2. Taylor expanded around -inf Error: 15.2 bits

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.3333333333333333}\right)}\]

    3. SimplifiedError: 15.2 bits

      \[\leadsto \color{blue}{x \cdot \left(-\sqrt{0.3333333333333333}\right)}\]

    if -1.35576678012699705e128 < x < 1.32252026523658605e124

    1. Initial program Error: 29.7 bits

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

    3. Applied div-invError: 29.7 bits

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

    4. Applied sqrt-prodError: 29.8 bits

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

    5. SimplifiedError: 29.8 bits

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

    if 1.32252026523658605e124 < x

    1. Initial program Error: 58.3 bits

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

    2. Taylor expanded around inf Error: 17.2 bits

      \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 25.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.355766780126997 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(-\sqrt{0.3333333333333333}\right)\\ \mathbf{elif}\;x \leq 1.322520265236586 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))