Average Error: 38.2 → 0.4
Time: 4.5s
Precision: binary64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
↓
\[\frac{1}{\frac{\sqrt{3}}{\mathsf{hypot}\left(z, \mathsf{hypot}\left(x, y\right)\right)}} \]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\frac{1}{\frac{\sqrt{3}}{\mathsf{hypot}\left(z, \mathsf{hypot}\left(x, y\right)\right)}}

(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
↓
(FPCore (x y z)
 :precision binary64
 (/ 1.0 (/ (sqrt 3.0) (hypot z (hypot x y)))))
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) {
	return 1.0 / (sqrt(3.0) / hypot(z, hypot(x, y)));
}

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
Target26.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \]

Derivation

  1. Initial program 38.2

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

  2. Simplified38.2

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

  3. Applied egg-rr0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{\mathsf{hypot}\left(z, \mathsf{hypot}\left(x, y\right)\right)}}} \]

  4. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\sqrt{3}}{\mathsf{hypot}\left(z, \mathsf{hypot}\left(x, y\right)\right)}} \]

Reproduce

herbie shell --seed 2022130 
(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)))