Average Error: 38.1 → 26.9
Time: 4.7s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.81464981654202998 \cdot 10^{95}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -2.12884303097519742 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le 1.0143191331303366 \cdot 10^{-281}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 1.28679613135987034 \cdot 10^{63}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.81464981654202998 \cdot 10^{95}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le -2.12884303097519742 \cdot 10^{-249}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{elif}\;x \le 1.0143191331303366 \cdot 10^{-281}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \le 1.28679613135987034 \cdot 10^{63}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

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

\end{array}
double code(double x, double y, double z) {
	return ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((x <= -3.81464981654203e+95)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= -2.1288430309751974e-249)) {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
		} else {
			double VAR_2;
			if ((x <= 1.0143191331303366e-281)) {
				VAR_2 = z;
			} else {
				double VAR_3;
				if ((x <= 1.2867961313598703e+63)) {
					VAR_3 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
				} else {
					VAR_3 = x;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Target25.4
Herbie26.9
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -3.81464981654202998e95

    1. Initial program 53.7

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

    2. Taylor expanded around -inf 18.9

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

    if -3.81464981654202998e95 < x < -2.12884303097519742e-249 or 1.0143191331303366e-281 < x < 1.28679613135987034e63

    1. Initial program 29.0

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

    if -2.12884303097519742e-249 < x < 1.0143191331303366e-281

    1. Initial program 32.7

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

    2. Taylor expanded around 0 48.2

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

    if 1.28679613135987034e63 < x

    1. Initial program 50.7

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

    2. Taylor expanded around inf 20.9

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

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.81464981654202998 \cdot 10^{95}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -2.12884303097519742 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le 1.0143191331303366 \cdot 10^{-281}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 1.28679613135987034 \cdot 10^{63}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020175 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

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

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