Average Error: 37.7 → 25.1
Time: 17.0s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.65689218952316206016746897628479098224 \cdot 10^{102}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 4.471707363468376084214192176846928336007 \cdot 10^{118}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(y \cdot y + x \cdot x\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.65689218952316206016746897628479098224 \cdot 10^{102}:\\
\;\;\;\;\left(-x\right) \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\mathbf{elif}\;x \le 4.471707363468376084214192176846928336007 \cdot 10^{118}:\\
\;\;\;\;\sqrt{\frac{z \cdot z + \left(y \cdot y + x \cdot x\right)}{3}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r53068158 = x;
        double r53068159 = r53068158 * r53068158;
        double r53068160 = y;
        double r53068161 = r53068160 * r53068160;
        double r53068162 = r53068159 + r53068161;
        double r53068163 = z;
        double r53068164 = r53068163 * r53068163;
        double r53068165 = r53068162 + r53068164;
        double r53068166 = 3.0;
        double r53068167 = r53068165 / r53068166;
        double r53068168 = sqrt(r53068167);
        return r53068168;
}
↓
double f(double x, double y, double z) {
        double r53068169 = x;
        double r53068170 = -8.656892189523162e+102;
        bool r53068171 = r53068169 <= r53068170;
        double r53068172 = -r53068169;
        double r53068173 = 0.3333333333333333;
        double r53068174 = sqrt(r53068173);
        double r53068175 = r53068172 * r53068174;
        double r53068176 = 4.471707363468376e+118;
        bool r53068177 = r53068169 <= r53068176;
        double r53068178 = z;
        double r53068179 = r53068178 * r53068178;
        double r53068180 = y;
        double r53068181 = r53068180 * r53068180;
        double r53068182 = r53068169 * r53068169;
        double r53068183 = r53068181 + r53068182;
        double r53068184 = r53068179 + r53068183;
        double r53068185 = 3.0;
        double r53068186 = r53068184 / r53068185;
        double r53068187 = sqrt(r53068186);
        double r53068188 = sqrt(r53068185);
        double r53068189 = r53068169 / r53068188;
        double r53068190 = r53068177 ? r53068187 : r53068189;
        double r53068191 = r53068171 ? r53068175 : r53068190;
        return r53068191;
}

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

Original37.7
Target25.5
Herbie25.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -8.656892189523162e+102

    1. Initial program 55.3

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

    2. Taylor expanded around -inf 17.9

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{0.3333333333333333148296162562473909929395}\right)}\]

    3. Simplified17.9

      \[\leadsto \color{blue}{\left(-x\right) \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]

    if -8.656892189523162e+102 < x < 4.471707363468376e+118

    1. Initial program 28.6

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

    if 4.471707363468376e+118 < x

    1. Initial program 57.1

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

    3. Applied sqrt-div57.1

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

    4. Taylor expanded around inf 18.6

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

  4. Final simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.65689218952316206016746897628479098224 \cdot 10^{102}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 4.471707363468376084214192176846928336007 \cdot 10^{118}:\\ \;\;\;\;\sqrt{\frac{z \cdot z + \left(y \cdot y + x \cdot x\right)}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.Pixel:doubleRmsOfRGB8 from repa-algorithms-3.4.0.1"

  :herbie-target
  (if (< z -6.396479394109776e+136) (/ (- z) (sqrt 3.0)) (if (< z 7.320293694404182e+117) (/ (sqrt (+ (+ (* z z) (* x x)) (* y y))) (sqrt 3.0)) (* (sqrt 0.3333333333333333) z)))

  (sqrt (/ (+ (+ (* x x) (* y y)) (* z z)) 3.0)))