Average Error: 37.9 → 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 r882963 = x;
        double r882964 = r882963 * r882963;
        double r882965 = y;
        double r882966 = r882965 * r882965;
        double r882967 = r882964 + r882966;
        double r882968 = z;
        double r882969 = r882968 * r882968;
        double r882970 = r882967 + r882969;
        double r882971 = 3.0;
        double r882972 = r882970 / r882971;
        double r882973 = sqrt(r882972);
        return r882973;
}
↓
double f(double x, double y, double z) {
        double r882974 = x;
        double r882975 = y;
        double r882976 = hypot(r882974, r882975);
        double r882977 = z;
        double r882978 = hypot(r882976, r882977);
        double r882979 = 1.0;
        double r882980 = 3.0;
        double r882981 = r882979 / r882980;
        double r882982 = sqrt(r882981);
        double r882983 = r882978 * r882982;
        return r882983;
}

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

Original37.9
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 37.9

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

  3. Applied div-inv37.9

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

  4. Applied sqrt-prod38.0

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

  6. Applied add-sqr-sqrt38.0

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

  7. Applied hypot-def29.1

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)} \cdot \sqrt{\frac{1}{3}}\]
  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 2020039 +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)))