Average Error: 37.4 → 26.7
Time: 1.3s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.04694359430339678 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -5.1573652837714696 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le -4.6992675325234154 \cdot 10^{-229}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 4.9296672245653324 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.04694359430339678 \cdot 10^{129}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le -5.1573652837714696 \cdot 10^{-105}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{elif}\;x \le -4.6992675325234154 \cdot 10^{-229}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \le 4.9296672245653324 \cdot 10^{135}:\\
\;\;\;\;\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 sqrt((((x * x) + (y * y)) + (z * z)));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((x <= -2.0469435943033968e+129)) {
		VAR = (-1.0 * x);
	} else {
		double VAR_1;
		if ((x <= -5.1573652837714696e-105)) {
			VAR_1 = sqrt((((x * x) + (y * y)) + (z * z)));
		} else {
			double VAR_2;
			if ((x <= -4.6992675325234154e-229)) {
				VAR_2 = z;
			} else {
				double VAR_3;
				if ((x <= 4.9296672245653324e+135)) {
					VAR_3 = sqrt((((x * x) + (y * y)) + (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

Original37.4
Target25.0
Herbie26.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -2.0469435943033968e+129

    1. Initial program 58.8

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

    2. Taylor expanded around -inf 15.6

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

    if -2.0469435943033968e+129 < x < -5.1573652837714696e-105 or -4.6992675325234154e-229 < x < 4.9296672245653324e+135

    1. Initial program 28.4

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

    if -5.1573652837714696e-105 < x < -4.6992675325234154e-229

    1. Initial program 29.4

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

    2. Taylor expanded around 0 47.8

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

    if 4.9296672245653324e+135 < x

    1. Initial program 59.5

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

    2. Taylor expanded around inf 15.7

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.04694359430339678 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -5.1573652837714696 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le -4.6992675325234154 \cdot 10^{-229}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 4.9296672245653324 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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))))