Average Error: 38.5 → 0.4
Time: 4.3s
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 r745654 = x;
        double r745655 = r745654 * r745654;
        double r745656 = y;
        double r745657 = r745656 * r745656;
        double r745658 = r745655 + r745657;
        double r745659 = z;
        double r745660 = r745659 * r745659;
        double r745661 = r745658 + r745660;
        double r745662 = 3.0;
        double r745663 = r745661 / r745662;
        double r745664 = sqrt(r745663);
        return r745664;
}
↓
double f(double x, double y, double z) {
        double r745665 = x;
        double r745666 = y;
        double r745667 = hypot(r745665, r745666);
        double r745668 = z;
        double r745669 = hypot(r745667, r745668);
        double r745670 = 1.0;
        double r745671 = 3.0;
        double r745672 = r745670 / r745671;
        double r745673 = sqrt(r745672);
        double r745674 = r745669 * r745673;
        return r745674;
}

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

Original38.5
Target26.3
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 38.5

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

  3. Applied div-inv38.5

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

  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-def29.5

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)} \cdot \sqrt{\frac{1}{3}}\]
  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 2020034 +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)))