Average Error: 38.3 → 26.4
Time: 3.2s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.65392376411012471 \cdot 10^{96}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 1.96522650657321481 \cdot 10^{59}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.65392376411012471 \cdot 10^{96}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 1.96522650657321481 \cdot 10^{59}:\\
\;\;\;\;\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 <= -2.6539237641101247e+96)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= 1.9652265065732148e+59)) {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
		} else {
			VAR_1 = x;
		}
		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

Original38.3
Target25.4
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.65392376411012471e96

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 18.4

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

    if -2.65392376411012471e96 < x < 1.96522650657321481e59

    1. Initial program 30.1

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

    if 1.96522650657321481e59 < x

    1. Initial program 50.5

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

    2. Taylor expanded around inf 21.6

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.65392376411012471 \cdot 10^{96}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 1.96522650657321481 \cdot 10^{59}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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))))