Average Error: 34.9 → 25.4
Time: 24.3s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.208442960829875 \cdot 10^{+137}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -1.9111814054112648 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\ \mathbf{elif}\;x \le -8.598460374864875 \cdot 10^{-202}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 9.794009169674087 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.208442960829875 \cdot 10^{+137}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \le -1.9111814054112648 \cdot 10^{-98}:\\
\;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\

\mathbf{elif}\;x \le -8.598460374864875 \cdot 10^{-202}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \le 9.794009169674087 \cdot 10^{+109}:\\
\;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y, double z) {
        double r43275773 = x;
        double r43275774 = r43275773 * r43275773;
        double r43275775 = y;
        double r43275776 = r43275775 * r43275775;
        double r43275777 = r43275774 + r43275776;
        double r43275778 = z;
        double r43275779 = r43275778 * r43275778;
        double r43275780 = r43275777 + r43275779;
        double r43275781 = sqrt(r43275780);
        return r43275781;
}
↓
double f(double x, double y, double z) {
        double r43275782 = x;
        double r43275783 = -6.208442960829875e+137;
        bool r43275784 = r43275782 <= r43275783;
        double r43275785 = -r43275782;
        double r43275786 = -1.9111814054112648e-98;
        bool r43275787 = r43275782 <= r43275786;
        double r43275788 = z;
        double r43275789 = r43275788 * r43275788;
        double r43275790 = y;
        double r43275791 = r43275790 * r43275790;
        double r43275792 = r43275782 * r43275782;
        double r43275793 = r43275791 + r43275792;
        double r43275794 = r43275789 + r43275793;
        double r43275795 = sqrt(r43275794);
        double r43275796 = -8.598460374864875e-202;
        bool r43275797 = r43275782 <= r43275796;
        double r43275798 = 9.794009169674087e+109;
        bool r43275799 = r43275782 <= r43275798;
        double r43275800 = r43275799 ? r43275795 : r43275782;
        double r43275801 = r43275797 ? r43275788 : r43275800;
        double r43275802 = r43275787 ? r43275795 : r43275801;
        double r43275803 = r43275784 ? r43275785 : r43275802;
        return r43275803;
}

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

Original34.9
Target23.3
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -6.208442960829875e+137

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 15.0

      \[\leadsto \color{blue}{-1 \cdot x}\]

    3. Simplified15.0

      \[\leadsto \color{blue}{-x}\]

    if -6.208442960829875e+137 < x < -1.9111814054112648e-98 or -8.598460374864875e-202 < x < 9.794009169674087e+109

    1. Initial program 27.0

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

    if -1.9111814054112648e-98 < x < -8.598460374864875e-202

    1. Initial program 28.1

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

    2. Taylor expanded around 0 46.6

      \[\leadsto \color{blue}{z}\]

    if 9.794009169674087e+109 < x

    1. Initial program 52.0

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

    2. Taylor expanded around inf 16.5

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.208442960829875 \cdot 10^{+137}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \le -1.9111814054112648 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\ \mathbf{elif}\;x \le -8.598460374864875 \cdot 10^{-202}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 9.794009169674087 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{z \cdot z + \left(y \cdot y + x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))