Average Error: 38.4 → 0.4
Time: 3.5s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right) \cdot \sqrt{\frac{1}{3}}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right) \cdot \sqrt{\frac{1}{3}}
double f(double x, double y, double z) {
        double r987716 = x;
        double r987717 = r987716 * r987716;
        double r987718 = y;
        double r987719 = r987718 * r987718;
        double r987720 = r987717 + r987719;
        double r987721 = z;
        double r987722 = r987721 * r987721;
        double r987723 = r987720 + r987722;
        double r987724 = 3.0;
        double r987725 = r987723 / r987724;
        double r987726 = sqrt(r987725);
        return r987726;
}
↓
double f(double x, double y, double z) {
        double r987727 = x;
        double r987728 = y;
        double r987729 = hypot(r987727, r987728);
        double r987730 = z;
        double r987731 = hypot(r987729, r987730);
        double r987732 = 1.0;
        double r987733 = 3.0;
        double r987734 = r987732 / r987733;
        double r987735 = sqrt(r987734);
        double r987736 = r987731 * r987735;
        return r987736;
}

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

Original38.4
Target26.0
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 38.4

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

  3. Applied div-inv38.4

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

  4. Applied sqrt-prod38.5

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

  6. Applied add-sqr-sqrt38.5

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

  7. Applied hypot-def29.2

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)} \cdot \sqrt{\frac{1}{3}}\]
  8. Using strategy rm

  9. Applied hypot-def0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3)))