Average Error: 38.0 → 25.8
Time: 4.5s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.0987829915356215 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \mathbf{elif}\;x \leq 3.114260434348677 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.0987829915356215 \cdot 10^{+116}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\

\mathbf{elif}\;x \leq 3.114260434348677 \cdot 10^{+87}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\

\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 -3.0987829915356215e+116)
   (*
    (sqrt (/ 1.0 (* (cbrt 3.0) (cbrt 3.0))))
    (- (* x (sqrt (/ 1.0 (cbrt 3.0))))))
   (if (<= x 3.114260434348677e+87)
     (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0))
     (*
      (sqrt (/ 1.0 (* (cbrt 3.0) (cbrt 3.0))))
      (* x (sqrt (/ 1.0 (cbrt 3.0))))))))
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 <= -3.0987829915356215e+116)) {
		tmp = ((double) (((double) sqrt((1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0))))))) * ((double) -(((double) (x * ((double) sqrt((1.0 / ((double) cbrt(3.0)))))))))));
	} else {
		double tmp_1;
		if ((x <= 3.114260434348677e+87)) {
			tmp_1 = ((double) sqrt((((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z)))) / 3.0)));
		} else {
			tmp_1 = ((double) (((double) sqrt((1.0 / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0))))))) * ((double) (x * ((double) sqrt((1.0 / ((double) cbrt(3.0)))))))));
		}
		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

Original38.0
Target26.0
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 < -3.098782991535621e116

    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 add-cube-cbrt_binary6457.3

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

    4. Applied *-un-lft-identity_binary6457.3

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

    5. Applied times-frac_binary6457.3

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

    6. Applied sqrt-prod_binary6457.3

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

    7. Taylor expanded around -inf 17.1

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

    8. Simplified17.1

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

    if -3.098782991535621e116 < x < 3.1142604343486769e87

    1. Initial program 29.6

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

    if 3.1142604343486769e87 < x

    1. Initial program 53.0

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

    3. Applied add-cube-cbrt_binary6453.0

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

    4. Applied *-un-lft-identity_binary6453.0

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

    5. Applied times-frac_binary6453.0

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

    6. Applied sqrt-prod_binary6453.0

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

    7. Taylor expanded around inf 18.8

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

    8. Simplified18.8

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.0987829915356215 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \mathbf{elif}\;x \leq 3.114260434348677 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))