Average Error: 38.4 → 26.4
Time: 5.9s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.35612836383285588 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.04793530695557172 \cdot 10^{77}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.35612836383285588 \cdot 10^{108}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\

\mathbf{elif}\;x \le 1.04793530695557172 \cdot 10^{77}:\\
\;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3}}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.333333333333333315}\\

\end{array}
double f(double x, double y, double z) {
        double r853480 = x;
        double r853481 = r853480 * r853480;
        double r853482 = y;
        double r853483 = r853482 * r853482;
        double r853484 = r853481 + r853483;
        double r853485 = z;
        double r853486 = r853485 * r853485;
        double r853487 = r853484 + r853486;
        double r853488 = 3.0;
        double r853489 = r853487 / r853488;
        double r853490 = sqrt(r853489);
        return r853490;
}
↓
double f(double x, double y, double z) {
        double r853491 = x;
        double r853492 = -1.356128363832856e+108;
        bool r853493 = r853491 <= r853492;
        double r853494 = -1.0;
        double r853495 = 0.3333333333333333;
        double r853496 = sqrt(r853495);
        double r853497 = r853491 * r853496;
        double r853498 = r853494 * r853497;
        double r853499 = 1.0479353069555717e+77;
        bool r853500 = r853491 <= r853499;
        double r853501 = r853491 * r853491;
        double r853502 = y;
        double r853503 = r853502 * r853502;
        double r853504 = r853501 + r853503;
        double r853505 = z;
        double r853506 = r853505 * r853505;
        double r853507 = r853504 + r853506;
        double r853508 = sqrt(r853507);
        double r853509 = 3.0;
        double r853510 = cbrt(r853509);
        double r853511 = r853510 * r853510;
        double r853512 = r853508 / r853511;
        double r853513 = r853508 / r853510;
        double r853514 = r853512 * r853513;
        double r853515 = sqrt(r853514);
        double r853516 = r853500 ? r853515 : r853497;
        double r853517 = r853493 ? r853498 : r853516;
        return r853517;
}

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
Target25.9
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.356128363832856e+108

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 18.6

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)}\]

    if -1.356128363832856e+108 < x < 1.0479353069555717e+77

    1. Initial program 29.9

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

    3. Applied add-cube-cbrt29.9

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

    4. Applied add-sqr-sqrt29.9

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

    5. Applied times-frac30.0

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

    if 1.0479353069555717e+77 < x

    1. Initial program 53.1

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

    2. Taylor expanded around inf 20.7

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.35612836383285588 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 1.04793530695557172 \cdot 10^{77}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))