Average Error: 37.6 → 25.9
Time: 6.6s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.225117955444305 \cdot 10^{79}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 3.509581298916955 \cdot 10^{90}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.225117955444305 \cdot 10^{79}:\\
\;\;\;\;\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 3.509581298916955 \cdot 10^{90}:\\
\;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r856163 = x;
        double r856164 = r856163 * r856163;
        double r856165 = y;
        double r856166 = r856165 * r856165;
        double r856167 = r856164 + r856166;
        double r856168 = z;
        double r856169 = r856168 * r856168;
        double r856170 = r856167 + r856169;
        double r856171 = 3.0;
        double r856172 = r856170 / r856171;
        double r856173 = sqrt(r856172);
        return r856173;
}
↓
double f(double x, double y, double z) {
        double r856174 = x;
        double r856175 = -2.225117955444305e+79;
        bool r856176 = r856174 <= r856175;
        double r856177 = 1.0;
        double r856178 = 3.0;
        double r856179 = cbrt(r856178);
        double r856180 = r856179 * r856179;
        double r856181 = r856177 / r856180;
        double r856182 = sqrt(r856181);
        double r856183 = -1.0;
        double r856184 = r856177 / r856179;
        double r856185 = sqrt(r856184);
        double r856186 = r856185 * r856174;
        double r856187 = r856183 * r856186;
        double r856188 = r856182 * r856187;
        double r856189 = 3.509581298916955e+90;
        bool r856190 = r856174 <= r856189;
        double r856191 = r856174 * r856174;
        double r856192 = y;
        double r856193 = r856192 * r856192;
        double r856194 = r856191 + r856193;
        double r856195 = z;
        double r856196 = r856195 * r856195;
        double r856197 = r856194 + r856196;
        double r856198 = r856197 / r856178;
        double r856199 = sqrt(r856198);
        double r856200 = r856174 * r856185;
        double r856201 = r856182 * r856200;
        double r856202 = r856190 ? r856199 : r856201;
        double r856203 = r856176 ? r856188 : r856202;
        return r856203;
}

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

Original37.6
Target25.4
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;\frac{-z}{\sqrt{3}}\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\frac{\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}}{\sqrt{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.333333333333333315} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.225117955444305e+79

    1. Initial program 50.8

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

    3. Applied add-cube-cbrt50.8

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

    4. Applied *-un-lft-identity50.8

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

    5. Applied times-frac50.8

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

    6. Applied sqrt-prod50.8

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

    7. Taylor expanded around -inf 19.5

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

    if -2.225117955444305e+79 < x < 3.509581298916955e+90

    1. Initial program 29.7

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

    if 3.509581298916955e+90 < x

    1. Initial program 53.6

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

    3. Applied add-cube-cbrt53.6

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

    4. Applied *-un-lft-identity53.6

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

    5. Applied times-frac53.6

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

    6. Applied sqrt-prod53.6

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

    8. Applied div-inv53.6

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

    9. Applied sqrt-prod53.6

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

    10. Taylor expanded around inf 18.7

      \[\leadsto \sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(\color{blue}{x} \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.225117955444305 \cdot 10^{79}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{3}}} \cdot x\right)\right)\\ \mathbf{elif}\;x \le 3.509581298916955 \cdot 10^{90}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))