Average Error: 38.2 → 26.5
Time: 11.6s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.51459440820440755 \cdot 10^{142}:\\ \;\;\;\;-\left(x \cdot \sqrt{\sqrt{0.333333333333333315}}\right) \cdot \sqrt{\sqrt{0.333333333333333315}}\\ \mathbf{elif}\;x \le -9.04578056600420164 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.7225630620991519 \cdot 10^{-270}:\\ \;\;\;\;\frac{y}{\sqrt{3}}\\ \mathbf{elif}\;x \le 4.24489976765657814 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.51459440820440755 \cdot 10^{142}:\\
\;\;\;\;-\left(x \cdot \sqrt{\sqrt{0.333333333333333315}}\right) \cdot \sqrt{\sqrt{0.333333333333333315}}\\

\mathbf{elif}\;x \le -9.04578056600420164 \cdot 10^{-238}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\

\mathbf{elif}\;x \le 1.7225630620991519 \cdot 10^{-270}:\\
\;\;\;\;\frac{y}{\sqrt{3}}\\

\mathbf{elif}\;x \le 4.24489976765657814 \cdot 10^{97}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r830475 = x;
        double r830476 = r830475 * r830475;
        double r830477 = y;
        double r830478 = r830477 * r830477;
        double r830479 = r830476 + r830478;
        double r830480 = z;
        double r830481 = r830480 * r830480;
        double r830482 = r830479 + r830481;
        double r830483 = 3.0;
        double r830484 = r830482 / r830483;
        double r830485 = sqrt(r830484);
        return r830485;
}
↓
double f(double x, double y, double z) {
        double r830486 = x;
        double r830487 = -7.514594408204408e+142;
        bool r830488 = r830486 <= r830487;
        double r830489 = 0.3333333333333333;
        double r830490 = sqrt(r830489);
        double r830491 = sqrt(r830490);
        double r830492 = r830486 * r830491;
        double r830493 = r830492 * r830491;
        double r830494 = -r830493;
        double r830495 = -9.045780566004202e-238;
        bool r830496 = r830486 <= r830495;
        double r830497 = z;
        double r830498 = y;
        double r830499 = r830498 * r830498;
        double r830500 = fma(r830486, r830486, r830499);
        double r830501 = fma(r830497, r830497, r830500);
        double r830502 = sqrt(r830501);
        double r830503 = 1.0;
        double r830504 = 3.0;
        double r830505 = r830503 / r830504;
        double r830506 = sqrt(r830505);
        double r830507 = r830502 * r830506;
        double r830508 = 1.722563062099152e-270;
        bool r830509 = r830486 <= r830508;
        double r830510 = sqrt(r830504);
        double r830511 = r830498 / r830510;
        double r830512 = 4.244899767656578e+97;
        bool r830513 = r830486 <= r830512;
        double r830514 = r830486 * r830490;
        double r830515 = r830513 ? r830507 : r830514;
        double r830516 = r830509 ? r830511 : r830515;
        double r830517 = r830496 ? r830507 : r830516;
        double r830518 = r830488 ? r830494 : r830517;
        return r830518;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original38.2
Target26.1
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -7.514594408204408e+142

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 14.1

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

    3. Simplified14.1

      \[\leadsto \color{blue}{-x \cdot \sqrt{0.333333333333333315}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt14.1

      \[\leadsto -x \cdot \sqrt{\color{blue}{\sqrt{0.333333333333333315} \cdot \sqrt{0.333333333333333315}}}\]

    6. Applied sqrt-prod14.1

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

    7. Applied associate-*r*14.1

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

    if -7.514594408204408e+142 < x < -9.045780566004202e-238 or 1.722563062099152e-270 < x < 4.244899767656578e+97

    1. Initial program 28.9

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

    3. Applied div-inv28.9

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

    4. Applied sqrt-prod29.0

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

    5. Simplified29.0

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

    if -9.045780566004202e-238 < x < 1.722563062099152e-270

    1. Initial program 31.0

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

    3. Applied sqrt-div31.1

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

    4. Simplified31.1

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

    5. Taylor expanded around 0 44.8

      \[\leadsto \frac{\color{blue}{y}}{\sqrt{3}}\]

    if 4.244899767656578e+97 < x

    1. Initial program 54.9

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

    2. Taylor expanded around inf 19.5

      \[\leadsto \color{blue}{x \cdot \sqrt{0.333333333333333315}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.51459440820440755 \cdot 10^{142}:\\ \;\;\;\;-\left(x \cdot \sqrt{\sqrt{0.333333333333333315}}\right) \cdot \sqrt{\sqrt{0.333333333333333315}}\\ \mathbf{elif}\;x \le -9.04578056600420164 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.7225630620991519 \cdot 10^{-270}:\\ \;\;\;\;\frac{y}{\sqrt{3}}\\ \mathbf{elif}\;x \le 4.24489976765657814 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))