Average Error: 38.1 → 25.7
Time: 21.3s
Precision: 64
\[\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.562702489758357495152124455939761881698 \cdot 10^{103}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 2.33657074031230004125882966613984705911 \cdot 10^{138}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]
\sqrt{\frac{\left(x \cdot x + y \cdot y\right) + z \cdot z}{3}}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.562702489758357495152124455939761881698 \cdot 10^{103}:\\
\;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\

\mathbf{elif}\;x \le 2.33657074031230004125882966613984705911 \cdot 10^{138}:\\
\;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{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 r584566 = x;
        double r584567 = r584566 * r584566;
        double r584568 = y;
        double r584569 = r584568 * r584568;
        double r584570 = r584567 + r584569;
        double r584571 = z;
        double r584572 = r584571 * r584571;
        double r584573 = r584570 + r584572;
        double r584574 = 3.0;
        double r584575 = r584573 / r584574;
        double r584576 = sqrt(r584575);
        return r584576;
}
↓
double f(double x, double y, double z) {
        double r584577 = x;
        double r584578 = -1.5627024897583575e+103;
        bool r584579 = r584577 <= r584578;
        double r584580 = 0.3333333333333333;
        double r584581 = sqrt(r584580);
        double r584582 = r584577 * r584581;
        double r584583 = -r584582;
        double r584584 = 2.3365707403123e+138;
        bool r584585 = r584577 <= r584584;
        double r584586 = r584577 * r584577;
        double r584587 = y;
        double r584588 = r584587 * r584587;
        double r584589 = r584586 + r584588;
        double r584590 = z;
        double r584591 = r584590 * r584590;
        double r584592 = r584589 + r584591;
        double r584593 = sqrt(r584592);
        double r584594 = 3.0;
        double r584595 = cbrt(r584594);
        double r584596 = r584595 * r584595;
        double r584597 = r584593 / r584596;
        double r584598 = r584593 / r584595;
        double r584599 = r584597 * r584598;
        double r584600 = sqrt(r584599);
        double r584601 = 1.0;
        double r584602 = r584601 / r584596;
        double r584603 = sqrt(r584602);
        double r584604 = r584601 / r584595;
        double r584605 = sqrt(r584604);
        double r584606 = r584577 * r584605;
        double r584607 = r584603 * r584606;
        double r584608 = r584585 ? r584600 : r584607;
        double r584609 = r584579 ? r584583 : r584608;
        return r584609;
}

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
Target25.6
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 < -1.5627024897583575e+103

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 17.8

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

    3. Simplified17.8

      \[\leadsto \color{blue}{-x \cdot \sqrt{0.3333333333333333148296162562473909929395}}\]

    if -1.5627024897583575e+103 < x < 2.3365707403123e+138

    1. Initial program 29.7

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

    3. Applied add-cube-cbrt29.7

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

    4. Applied add-sqr-sqrt29.7

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

    5. Applied times-frac29.7

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

    if 2.3365707403123e+138 < x

    1. Initial program 60.4

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

    3. Applied add-cube-cbrt60.4

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

    4. Applied *-un-lft-identity60.4

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

    5. Applied times-frac60.4

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

    6. Applied sqrt-prod60.4

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

    8. Applied div-inv60.4

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

    9. Applied sqrt-prod60.4

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

    10. Taylor expanded around inf 15.3

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.562702489758357495152124455939761881698 \cdot 10^{103}:\\ \;\;\;\;-x \cdot \sqrt{0.3333333333333333148296162562473909929395}\\ \mathbf{elif}\;x \le 2.33657074031230004125882966613984705911 \cdot 10^{138}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}}{\sqrt[3]{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \left(x \cdot \sqrt{\frac{1}{\sqrt[3]{3}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))