Average Error: 38.1 → 25.6
Time: 20.3s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.74693859576126102544728952505913367202 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{-z}{\left|\sqrt[3]{3}\right|}}{\sqrt{\sqrt[3]{3}}}\\ \mathbf{elif}\;z \le 4.17182871699966700452276153071931598076 \cdot 10^{130}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.74693859576126102544728952505913367202 \cdot 10^{111}:\\
\;\;\;\;\frac{\frac{-z}{\left|\sqrt[3]{3}\right|}}{\sqrt{\sqrt[3]{3}}}\\

\mathbf{elif}\;z \le 4.17182871699966700452276153071931598076 \cdot 10^{130}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{z}{\sqrt{3}}\\

\end{array}
double f(double x, double y, double z) {
        double r564279 = x;
        double r564280 = r564279 * r564279;
        double r564281 = y;
        double r564282 = r564281 * r564281;
        double r564283 = r564280 + r564282;
        double r564284 = z;
        double r564285 = r564284 * r564284;
        double r564286 = r564283 + r564285;
        double r564287 = 3.0;
        double r564288 = r564286 / r564287;
        double r564289 = sqrt(r564288);
        return r564289;
}
↓
double f(double x, double y, double z) {
        double r564290 = z;
        double r564291 = -3.746938595761261e+111;
        bool r564292 = r564290 <= r564291;
        double r564293 = -r564290;
        double r564294 = 3.0;
        double r564295 = cbrt(r564294);
        double r564296 = fabs(r564295);
        double r564297 = r564293 / r564296;
        double r564298 = sqrt(r564295);
        double r564299 = r564297 / r564298;
        double r564300 = 4.171828716999667e+130;
        bool r564301 = r564290 <= r564300;
        double r564302 = x;
        double r564303 = y;
        double r564304 = r564303 * r564303;
        double r564305 = fma(r564302, r564302, r564304);
        double r564306 = fma(r564290, r564290, r564305);
        double r564307 = sqrt(r564306);
        double r564308 = 1.0;
        double r564309 = r564308 / r564294;
        double r564310 = sqrt(r564309);
        double r564311 = r564307 * r564310;
        double r564312 = sqrt(r564294);
        double r564313 = r564290 / r564312;
        double r564314 = r564301 ? r564311 : r564313;
        double r564315 = r564292 ? r564299 : r564314;
        return r564315;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original38.1
Target25.6
Herbie25.6
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.746938595761261e+111

    1. Initial program 56.5

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

    2. Simplified56.5

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

    4. Applied sqrt-div56.5

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

    5. Taylor expanded around -inf 18.0

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

    6. Simplified18.0

      \[\leadsto \frac{\color{blue}{-z}}{\sqrt{3}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt18.0

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

    9. Applied sqrt-prod18.0

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

    10. Applied associate-/r*18.0

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

    11. Simplified18.0

      \[\leadsto \frac{\color{blue}{\frac{-z}{\left|\sqrt[3]{3}\right|}}}{\sqrt{\sqrt[3]{3}}}\]

    if -3.746938595761261e+111 < z < 4.171828716999667e+130

    1. Initial program 29.4

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

    2. Simplified29.4

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

    4. Applied div-inv29.4

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

    5. Applied sqrt-prod29.5

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

    if 4.171828716999667e+130 < z

    1. Initial program 59.0

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

    2. Simplified59.0

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

    4. Applied sqrt-div59.0

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

    5. Taylor expanded around inf 15.8

      \[\leadsto \frac{\color{blue}{z}}{\sqrt{3}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.74693859576126102544728952505913367202 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{-z}{\left|\sqrt[3]{3}\right|}}{\sqrt{\sqrt[3]{3}}}\\ \mathbf{elif}\;z \le 4.17182871699966700452276153071931598076 \cdot 10^{130}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)} \cdot \sqrt{\frac{1}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))