Average Error: 38.3 → 25.1
Time: 1.4s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.5755889766509834 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 8.13942122541697528 \cdot 10^{121}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.5755889766509834 \cdot 10^{141}:\\
\;\;\;\;-1 \cdot x\\

\mathbf{elif}\;x \le 8.13942122541697528 \cdot 10^{121}:\\
\;\;\;\;\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 <= -9.575588976650983e+141)) {
		VAR = (-1.0 * x);
	} else {
		double VAR_1;
		if ((x <= 8.139421225416975e+121)) {
			VAR_1 = sqrt((((x * x) + (y * y)) + (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
Target26.0
Herbie25.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -9.575588976650983e+141

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 14.7

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

    if -9.575588976650983e+141 < x < 8.139421225416975e+121

    1. Initial program 29.4

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

    if 8.139421225416975e+121 < x

    1. Initial program 57.9

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

    2. Taylor expanded around inf 15.0

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

  4. Final simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.5755889766509834 \cdot 10^{141}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le 8.13942122541697528 \cdot 10^{121}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))