Average Error: 38.2 → 26.0
Time: 2.2s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.5222599376717584 \cdot 10^{+75}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3.35155786581593 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.5222599376717584 \cdot 10^{+75}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 3.35155786581593 \cdot 10^{+89}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.5222599376717584e+75)
   (- x)
   (if (<= x 3.35155786581593e+89) (sqrt (+ (+ (* x x) (* y y)) (* z z))) x)))
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 tmp;
	if ((x <= -3.5222599376717584e+75)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= 3.35155786581593e+89)) {
			tmp_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
		} else {
			tmp_1 = x;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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

Original38.2
Target25.4
Herbie26.0
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.5222599376717584e75

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 19.9

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

    3. Simplified19.9

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

    if -3.5222599376717584e75 < x < 3.35155786581592986e89

    1. Initial program 29.8

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

    if 3.35155786581592986e89 < x

    1. Initial program 53.1

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

    2. Taylor expanded around inf 18.9

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

  4. Final simplification26.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5222599376717584 \cdot 10^{+75}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3.35155786581593 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))