Average Error: 38.1 → 25.6
Time: 17.1s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.247629272160778236302167026642172876392 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.3333333333333333148296162562473909929395}\right)\\ \mathbf{elif}\;x \le 1.879788545217995887059617832949467600343 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\frac{\left({x}^{2} + {y}^{2}\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.247629272160778236302167026642172876392 \cdot 10^{129}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.3333333333333333148296162562473909929395}\right)\\

\mathbf{elif}\;x \le 1.879788545217995887059617832949467600343 \cdot 10^{136}:\\
\;\;\;\;\sqrt{\frac{\left({x}^{2} + {y}^{2}\right) + z \cdot z}{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\end{array}
double f(double x, double y, double z) {
        double r682149 = x;
        double r682150 = r682149 * r682149;
        double r682151 = y;
        double r682152 = r682151 * r682151;
        double r682153 = r682150 + r682152;
        double r682154 = z;
        double r682155 = r682154 * r682154;
        double r682156 = r682153 + r682155;
        double r682157 = 3.0;
        double r682158 = r682156 / r682157;
        double r682159 = sqrt(r682158);
        return r682159;
}
↓
double f(double x, double y, double z) {
        double r682160 = x;
        double r682161 = -8.247629272160778e+129;
        bool r682162 = r682160 <= r682161;
        double r682163 = -1.0;
        double r682164 = 0.3333333333333333;
        double r682165 = sqrt(r682164);
        double r682166 = r682160 * r682165;
        double r682167 = r682163 * r682166;
        double r682168 = 1.879788545217996e+136;
        bool r682169 = r682160 <= r682168;
        double r682170 = 2.0;
        double r682171 = pow(r682160, r682170);
        double r682172 = y;
        double r682173 = pow(r682172, r682170);
        double r682174 = r682171 + r682173;
        double r682175 = z;
        double r682176 = r682175 * r682175;
        double r682177 = r682174 + r682176;
        double r682178 = 3.0;
        double r682179 = r682177 / r682178;
        double r682180 = sqrt(r682179);
        double r682181 = r682169 ? r682180 : r682166;
        double r682182 = r682162 ? r682167 : r682181;
        return r682182;
}

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

Original38.1
Target25.5
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 x < -8.247629272160778e+129

    1. Initial program 59.0

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

    2. Taylor expanded around -inf 16.7

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

    if -8.247629272160778e+129 < x < 1.879788545217996e+136

    1. Initial program 29.5

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

    2. Taylor expanded around 0 29.5

      \[\leadsto \sqrt{\frac{\color{blue}{\left({x}^{2} + {y}^{2}\right)} + z \cdot z}{3}}\]

    if 1.879788545217996e+136 < x

    1. Initial program 60.4

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

    2. Taylor expanded around inf 15.1

      \[\leadsto \color{blue}{x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.247629272160778236302167026642172876392 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.3333333333333333148296162562473909929395}\right)\\ \mathbf{elif}\;x \le 1.879788545217995887059617832949467600343 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\frac{\left({x}^{2} + {y}^{2}\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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)))