Average Error: 38.1 → 25.7
Time: 7.1s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.052922090158918936796058329983412299556 \cdot 10^{85}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 2.831723391366531742542950799194105898801 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\sqrt[3]{3}} \cdot \left|\sqrt[3]{3}\right|}\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.052922090158918936796058329983412299556 \cdot 10^{85}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\

\mathbf{elif}\;x \le 2.831723391366531742542950799194105898801 \cdot 10^{132}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\sqrt{\sqrt[3]{3}} \cdot \left|\sqrt[3]{3}\right|}\\

\end{array}
double f(double x, double y, double z) {
        double r934718 = x;
        double r934719 = r934718 * r934718;
        double r934720 = y;
        double r934721 = r934720 * r934720;
        double r934722 = r934719 + r934721;
        double r934723 = z;
        double r934724 = r934723 * r934723;
        double r934725 = r934722 + r934724;
        double r934726 = 3.0;
        double r934727 = r934725 / r934726;
        double r934728 = sqrt(r934727);
        return r934728;
}
↓
double f(double x, double y, double z) {
        double r934729 = x;
        double r934730 = -1.052922090158919e+85;
        bool r934731 = r934729 <= r934730;
        double r934732 = 1.0;
        double r934733 = 3.0;
        double r934734 = cbrt(r934733);
        double r934735 = r934734 * r934734;
        double r934736 = r934732 / r934735;
        double r934737 = sqrt(r934736);
        double r934738 = -1.0;
        double r934739 = r934732 / r934734;
        double r934740 = sqrt(r934739);
        double r934741 = r934740 * r934729;
        double r934742 = r934738 * r934741;
        double r934743 = r934737 * r934742;
        double r934744 = 2.8317233913665317e+132;
        bool r934745 = r934729 <= r934744;
        double r934746 = r934729 * r934729;
        double r934747 = y;
        double r934748 = r934747 * r934747;
        double r934749 = r934746 + r934748;
        double r934750 = z;
        double r934751 = r934750 * r934750;
        double r934752 = r934749 + r934751;
        double r934753 = r934752 / r934733;
        double r934754 = sqrt(r934753);
        double r934755 = sqrt(r934734);
        double r934756 = fabs(r934734);
        double r934757 = r934755 * r934756;
        double r934758 = r934729 / r934757;
        double r934759 = r934745 ? r934754 : r934758;
        double r934760 = r934731 ? r934743 : r934759;
        return r934760;
}

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
Target26.0
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333148296162562473909929395} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.052922090158919e+85

    1. Initial program 53.4

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

    3. Applied add-cube-cbrt53.4

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

    4. Applied *-un-lft-identity53.4

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

    5. Applied times-frac53.4

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

    6. Applied sqrt-prod53.4

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

    7. Taylor expanded around -inf 20.3

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

    if -1.052922090158919e+85 < x < 2.8317233913665317e+132

    1. Initial program 29.5

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

    if 2.8317233913665317e+132 < x

    1. Initial program 59.7

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

    3. Applied add-cube-cbrt59.7

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

    4. Applied *-un-lft-identity59.7

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

    5. Applied times-frac59.8

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

    6. Applied sqrt-prod59.8

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

    7. Taylor expanded around inf 14.1

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)}\]
    8. Using strategy rm

    9. Applied sqrt-div14.4

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

    10. Applied associate-*l/14.3

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

    11. Applied sqrt-div14.3

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

    12. Applied frac-times14.2

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

    13. Simplified14.2

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

    14. Simplified14.2

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.052922090158918936796058329983412299556 \cdot 10^{85}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 2.831723391366531742542950799194105898801 \cdot 10^{132}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\sqrt[3]{3}} \cdot \left|\sqrt[3]{3}\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))