Average Error: 37.9 → 26.8
Time: 2.3s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.1944956675816954 \cdot 10^{+58}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.2027754573141866 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \leq 1.433782692190375 \cdot 10^{-265}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9385061330767246 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.1944956675816954 \cdot 10^{+58}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -3.2027754573141866 \cdot 10^{-302}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{elif}\;x \leq 1.433782692190375 \cdot 10^{-265}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \leq 1.9385061330767246 \cdot 10^{+66}:\\
\;\;\;\;\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 -1.1944956675816954e+58)
   (- x)
   (if (<= x -3.2027754573141866e-302)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     (if (<= x 1.433782692190375e-265)
       z
       (if (<= x 1.9385061330767246e+66)
         (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 <= -1.1944956675816954e+58)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= -3.2027754573141866e-302)) {
			tmp_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
		} else {
			double tmp_2;
			if ((x <= 1.433782692190375e-265)) {
				tmp_2 = z;
			} else {
				double tmp_3;
				if ((x <= 1.9385061330767246e+66)) {
					tmp_3 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
				} else {
					tmp_3 = x;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		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

Original37.9
Target25.2
Herbie26.8
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.1944956675816954e58

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 21.0

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

    3. Simplified21.0

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

    if -1.1944956675816954e58 < x < -3.2027754573141866e-302 or 1.4337826921903749e-265 < x < 1.93850613307672458e66

    1. Initial program 29.2

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

    if -3.2027754573141866e-302 < x < 1.4337826921903749e-265

    1. Initial program 34.0

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

    2. Taylor expanded around 0 47.3

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

    if 1.93850613307672458e66 < x

    1. Initial program 51.2

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

    2. Taylor expanded around inf 21.0

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

  4. Final simplification26.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1944956675816954 \cdot 10^{+58}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.2027754573141866 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \leq 1.433782692190375 \cdot 10^{-265}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 1.9385061330767246 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))