Average Error: 38.2 → 26.4
Time: 17.5s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -18637040731758232623605069933264714596350:\\ \;\;\;\;-1 \cdot \frac{x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 4.376373199179285827713056036389642308458 \cdot 10^{91}:\\ \;\;\;\;\sqrt{\frac{6004799503160661}{18014398509481984} \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{6004799503160661}{18014398509481984}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -18637040731758232623605069933264714596350:\\
\;\;\;\;-1 \cdot \frac{x}{\sqrt{3}}\\

\mathbf{elif}\;x \le 4.376373199179285827713056036389642308458 \cdot 10^{91}:\\
\;\;\;\;\sqrt{\frac{6004799503160661}{18014398509481984} \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{\frac{6004799503160661}{18014398509481984}}\\

\end{array}
double f(double x, double y, double z) {
        double r544291 = x;
        double r544292 = r544291 * r544291;
        double r544293 = y;
        double r544294 = r544293 * r544293;
        double r544295 = r544292 + r544294;
        double r544296 = z;
        double r544297 = r544296 * r544296;
        double r544298 = r544295 + r544297;
        double r544299 = 3.0;
        double r544300 = r544298 / r544299;
        double r544301 = sqrt(r544300);
        return r544301;
}
↓
double f(double x, double y, double z) {
        double r544302 = x;
        double r544303 = -1.8637040731758233e+40;
        bool r544304 = r544302 <= r544303;
        double r544305 = -1.0;
        double r544306 = 3.0;
        double r544307 = sqrt(r544306);
        double r544308 = r544302 / r544307;
        double r544309 = r544305 * r544308;
        double r544310 = 4.376373199179286e+91;
        bool r544311 = r544302 <= r544310;
        double r544312 = 6004799503160661.0;
        double r544313 = 18014398509481984.0;
        double r544314 = r544312 / r544313;
        double r544315 = 2.0;
        double r544316 = pow(r544302, r544315);
        double r544317 = y;
        double r544318 = pow(r544317, r544315);
        double r544319 = z;
        double r544320 = pow(r544319, r544315);
        double r544321 = r544318 + r544320;
        double r544322 = r544316 + r544321;
        double r544323 = r544314 * r544322;
        double r544324 = sqrt(r544323);
        double r544325 = sqrt(r544314);
        double r544326 = r544302 * r544325;
        double r544327 = r544311 ? r544324 : r544326;
        double r544328 = r544304 ? r544309 : r544327;
        return r544328;
}

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

Original38.2
Target25.7
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.8637040731758233e+40

    1. Initial program 49.0

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

    3. Applied add-sqr-sqrt49.0

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

    4. Applied associate-/r*49.0

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

    5. Taylor expanded around -inf 23.5

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{\sqrt{3}}}\]

    if -1.8637040731758233e+40 < x < 4.376373199179286e+91

    1. Initial program 29.7

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

    2. Taylor expanded around 0 29.7

      \[\leadsto \sqrt{\color{blue}{0.3333333333333333148296162562473909929395 \cdot {x}^{2} + \left(0.3333333333333333148296162562473909929395 \cdot {y}^{2} + 0.3333333333333333148296162562473909929395 \cdot {z}^{2}\right)}}\]

    3. Simplified29.7

      \[\leadsto \sqrt{\color{blue}{\frac{6004799503160661}{18014398509481984} \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}}\]

    if 4.376373199179286e+91 < x

    1. Initial program 53.4

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

    2. Taylor expanded around inf 18.9

      \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]

    3. Simplified18.9

      \[\leadsto \color{blue}{x \cdot \sqrt{\frac{6004799503160661}{18014398509481984}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -18637040731758232623605069933264714596350:\\ \;\;\;\;-1 \cdot \frac{x}{\sqrt{3}}\\ \mathbf{elif}\;x \le 4.376373199179285827713056036389642308458 \cdot 10^{91}:\\ \;\;\;\;\sqrt{\frac{6004799503160661}{18014398509481984} \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{6004799503160661}{18014398509481984}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"
  :precision binary64

  :herbie-target
  (if (< z -6.3964793941097758e136) (/ (- z) (sqrt 3)) (if (< z 7.3202936944041821e117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3)) (* (sqrt 0.333333333333333315) z)))

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