Average Error: 37.8 → 25.5
Time: 1.3s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\
\;\;\;\;\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 r771002 = x;
        double r771003 = r771002 * r771002;
        double r771004 = y;
        double r771005 = r771004 * r771004;
        double r771006 = r771003 + r771005;
        double r771007 = z;
        double r771008 = r771007 * r771007;
        double r771009 = r771006 + r771008;
        double r771010 = sqrt(r771009);
        return r771010;
}
↓
double f(double x, double y, double z) {
        double r771011 = x;
        double r771012 = -4.241172532402712e+119;
        bool r771013 = r771011 <= r771012;
        double r771014 = -1.0;
        double r771015 = r771014 * r771011;
        double r771016 = 6.601942283666501e+109;
        bool r771017 = r771011 <= r771016;
        double r771018 = r771011 * r771011;
        double r771019 = y;
        double r771020 = r771019 * r771019;
        double r771021 = r771018 + r771020;
        double r771022 = z;
        double r771023 = r771022 * r771022;
        double r771024 = r771021 + r771023;
        double r771025 = sqrt(r771024);
        double r771026 = r771017 ? r771025 : r771011;
        double r771027 = r771013 ? r771015 : r771026;
        return r771027;
}

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

Original37.8
Target25.3
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -4.241172532402712e+119

    1. Initial program 56.7

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

    2. Taylor expanded around -inf 17.8

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

    if -4.241172532402712e+119 < x < 6.601942283666501e+109

    1. Initial program 29.2

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

    if 6.601942283666501e+109 < x

    1. Initial program 55.8

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

    2. Taylor expanded around inf 17.5

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.24117253240271209 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 6.6019422836665007 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))