Average Error: 37.7 → 25.4
Time: 3.2s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.97378979562191825 \cdot 10^{148}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 7.0809856866698759 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.97378979562191825 \cdot 10^{148}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 7.0809856866698759 \cdot 10^{84}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r814770 = x;
        double r814771 = r814770 * r814770;
        double r814772 = y;
        double r814773 = r814772 * r814772;
        double r814774 = r814771 + r814773;
        double r814775 = z;
        double r814776 = r814775 * r814775;
        double r814777 = r814774 + r814776;
        double r814778 = sqrt(r814777);
        return r814778;
}
↓
double f(double x, double y, double z) {
        double r814779 = x;
        double r814780 = -7.973789795621918e+148;
        bool r814781 = r814779 <= r814780;
        double r814782 = -1.0;
        double r814783 = r814782 * r814779;
        double r814784 = 7.080985686669876e+84;
        bool r814785 = r814779 <= r814784;
        double r814786 = r814779 * r814779;
        double r814787 = y;
        double r814788 = r814787 * r814787;
        double r814789 = r814786 + r814788;
        double r814790 = z;
        double r814791 = r814790 * r814790;
        double r814792 = r814789 + r814791;
        double r814793 = sqrt(r814792);
        double r814794 = r814785 ? r814793 : r814779;
        double r814795 = r814781 ? r814783 : r814794;
        return r814795;
}

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

Original37.7
Target25.1
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -7.973789795621918e+148

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 14.1

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

    if -7.973789795621918e+148 < x < 7.080985686669876e+84

    1. Initial program 29.2

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

    if 7.080985686669876e+84 < x

    1. Initial program 52.9

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

    2. Taylor expanded around inf 18.8

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

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.97378979562191825 \cdot 10^{148}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 7.0809856866698759 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :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))))