Average Error: 37.7 → 25.9
Time: 1.4s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.5178428645091481 \cdot 10^{115}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -9.87565315450708303 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le -4.3526816421108561 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 5.18720922429771752 \cdot 10^{129}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.5178428645091481 \cdot 10^{115}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le -9.87565315450708303 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{elif}\;x \le -4.3526816421108561 \cdot 10^{-53}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \le 5.18720922429771752 \cdot 10^{129}:\\
\;\;\;\;\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 <= -1.517842864509148e+115)) {
		VAR = (-1.0 * x);
	} else {
		double VAR_1;
		if ((x <= -9.875653154507083e-11)) {
			VAR_1 = sqrt((((x * x) + (y * y)) + (z * z)));
		} else {
			double VAR_2;
			if ((x <= -4.352681642110856e-53)) {
				VAR_2 = z;
			} else {
				double VAR_3;
				if ((x <= 5.1872092242977175e+129)) {
					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

Original37.7
Target25.1
Herbie25.9
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.517842864509148e+115

    1. Initial program 56.6

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

    2. Taylor expanded around -inf 16.5

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

    if -1.517842864509148e+115 < x < -9.875653154507083e-11 or -4.352681642110856e-53 < x < 5.1872092242977175e+129

    1. Initial program 29.1

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

    if -9.875653154507083e-11 < x < -4.352681642110856e-53

    1. Initial program 30.5

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

    2. Taylor expanded around 0 52.0

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

    if 5.1872092242977175e+129 < x

    1. Initial program 58.9

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

    2. Taylor expanded around inf 15.4

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

  4. Final simplification25.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5178428645091481 \cdot 10^{115}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -9.87565315450708303 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \le -4.3526816421108561 \cdot 10^{-53}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \le 5.18720922429771752 \cdot 10^{129}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))