Average Error: 38.2 → 26.4
Time: 3.1s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.123597247650645 \cdot 10^{+88}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333}\\ \mathbf{elif}\;x \leq 2.608580135401658 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.123597247650645 \cdot 10^{+88}:\\
\;\;\;\;-x \cdot \sqrt{0.3333333333333333}\\

\mathbf{elif}\;x \leq 2.608580135401658 \cdot 10^{+75}:\\
\;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\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 -1.123597247650645e+88)
   (- (* x (sqrt 0.3333333333333333)))
   (if (<= x 2.608580135401658e+75)
     (* (sqrt 0.3333333333333333) (sqrt (+ (+ (* x x) (* y y)) (* z z))))
     (/ 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 <= -1.123597247650645e+88) {
		tmp = -(x * sqrt(0.3333333333333333));
	} else if (x <= 2.608580135401658e+75) {
		tmp = sqrt(0.3333333333333333) * sqrt(((x * x) + (y * y)) + (z * z));
	} 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

Original38.2
Target25.7
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.123597247650645e88

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 19.2

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

    3. Simplified19.2

      \[\leadsto \color{blue}{-x \cdot \sqrt{0.3333333333333333}}\]

    if -1.123597247650645e88 < x < 2.60858013540165785e75

    1. Initial program 30.2

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

    3. Applied div-inv_binary6430.2

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

    4. Applied sqrt-prod_binary6430.3

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

    5. Simplified30.3

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

    if 2.60858013540165785e75 < x

    1. Initial program 51.3

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

    3. Applied sqrt-div_binary6451.3

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

    4. Taylor expanded around inf 20.4

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.123597247650645 \cdot 10^{+88}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333}\\ \mathbf{elif}\;x \leq 2.608580135401658 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))