Average Error: 38.2 → 26.4
Time: 3.9s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.36083027610352862 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 6.32055311910495607 \cdot 10^{79}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.36083027610352862 \cdot 10^{110}:\\
\;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\

\mathbf{elif}\;x \le 6.32055311910495607 \cdot 10^{79}:\\
\;\;\;\;\sqrt{0.333333333333333315 \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r770816 = x;
        double r770817 = r770816 * r770816;
        double r770818 = y;
        double r770819 = r770818 * r770818;
        double r770820 = r770817 + r770819;
        double r770821 = z;
        double r770822 = r770821 * r770821;
        double r770823 = r770820 + r770822;
        double r770824 = 3.0;
        double r770825 = r770823 / r770824;
        double r770826 = sqrt(r770825);
        return r770826;
}
↓
double f(double x, double y, double z) {
        double r770827 = x;
        double r770828 = -4.3608302761035286e+110;
        bool r770829 = r770827 <= r770828;
        double r770830 = -1.0;
        double r770831 = 0.3333333333333333;
        double r770832 = sqrt(r770831);
        double r770833 = r770827 * r770832;
        double r770834 = r770830 * r770833;
        double r770835 = 6.320553119104956e+79;
        bool r770836 = r770827 <= r770835;
        double r770837 = 2.0;
        double r770838 = pow(r770827, r770837);
        double r770839 = y;
        double r770840 = pow(r770839, r770837);
        double r770841 = z;
        double r770842 = pow(r770841, r770837);
        double r770843 = r770840 + r770842;
        double r770844 = r770838 + r770843;
        double r770845 = r770831 * r770844;
        double r770846 = sqrt(r770845);
        double r770847 = r770836 ? r770846 : r770833;
        double r770848 = r770829 ? r770834 : r770847;
        return r770848;
}

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

Original38.2
Target26.0
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 < -4.3608302761035286e+110

    1. Initial program 56.1

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

    2. Taylor expanded around -inf 18.2

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

    if -4.3608302761035286e+110 < x < 6.320553119104956e+79

    1. Initial program 29.9

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

    2. Taylor expanded around 0 29.9

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot {x}^{2} + \left(0.333333333333333315 \cdot {y}^{2} + 0.333333333333333315 \cdot {z}^{2}\right)}}\]

    3. Simplified29.9

      \[\leadsto \sqrt{\color{blue}{0.333333333333333315 \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}}\]

    if 6.320553119104956e+79 < x

    1. Initial program 52.1

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

    2. Taylor expanded around inf 21.0

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.36083027610352862 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \left(x \cdot \sqrt{0.333333333333333315}\right)\\ \mathbf{elif}\;x \le 6.32055311910495607 \cdot 10^{79}:\\ \;\;\;\;\sqrt{0.333333333333333315 \cdot \left({x}^{2} + \left({y}^{2} + {z}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{0.333333333333333315}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))