Average Error: 37.8 → 0.4
Time: 4.8s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\frac{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)}{\left|\sqrt[3]{3}\right|}}{\sqrt{\sqrt[3]{3}}}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\frac{\frac{\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)}{\left|\sqrt[3]{3}\right|}}{\sqrt{\sqrt[3]{3}}}
double f(double x, double y, double z) {
        double r973262 = x;
        double r973263 = r973262 * r973262;
        double r973264 = y;
        double r973265 = r973264 * r973264;
        double r973266 = r973263 + r973265;
        double r973267 = z;
        double r973268 = r973267 * r973267;
        double r973269 = r973266 + r973268;
        double r973270 = 3.0;
        double r973271 = r973269 / r973270;
        double r973272 = sqrt(r973271);
        return r973272;
}
↓
double f(double x, double y, double z) {
        double r973273 = x;
        double r973274 = y;
        double r973275 = hypot(r973273, r973274);
        double r973276 = z;
        double r973277 = hypot(r973275, r973276);
        double r973278 = 3.0;
        double r973279 = cbrt(r973278);
        double r973280 = fabs(r973279);
        double r973281 = r973277 / r973280;
        double r973282 = sqrt(r973279);
        double r973283 = r973281 / r973282;
        return r973283;
}

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

Original37.8
Target25.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 37.8

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

  3. Applied sqrt-div37.9

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

  5. Applied add-sqr-sqrt37.9

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

  6. Applied hypot-def28.8

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

  8. Applied hypot-def0.4

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

  10. Applied add-cube-cbrt0.4

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

  11. Applied sqrt-prod0.4

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

  12. Applied associate-/r*0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 +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)))