Average Error: 35.3 → 0.9
Time: 3.2s
Precision: binary64
\[[x, y, z] = \mathsf{sort}([x, y, z]) \\]
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}} \]
↓
\[\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(x, z\right) \]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(x, z\right)

(FPCore (x y z)
 :precision binary64
 (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))
↓
(FPCore (x y z) :precision binary64 (* (sqrt 0.3333333333333333) (hypot x z)))
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 sqrt(0.3333333333333333) * hypot(x, z);
}

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

Original35.3
Target19.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot z\\ \end{array} \]

Derivation

  1. Initial program 35.3

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

  2. Simplified35.3

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

  3. Taylor expanded in y around 0 35.7

    \[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}} \cdot \sqrt{0.3333333333333333}} \]

  4. Simplified0.9

    \[\leadsto \color{blue}{\sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(x, z\right)} \]

  5. Final simplification0.9

    \[\leadsto \sqrt{0.3333333333333333} \cdot \mathsf{hypot}\left(x, z\right) \]

Reproduce

herbie shell --seed 2022088 
(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)))