Average Error: 38.0 → 25.8
Time: 3.2s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0029854725212742 \cdot 10^{+114}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1.4463278090717525 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \leq -9.142573482041666 \cdot 10^{-106}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 8.528180313170455 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0029854725212742 \cdot 10^{+114}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -1.4463278090717525 \cdot 10^{-86}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{elif}\;x \leq -9.142573482041666 \cdot 10^{-106}:\\
\;\;\;\;z\\

\mathbf{elif}\;x \leq 8.528180313170455 \cdot 10^{+134}:\\
\;\;\;\;\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 -1.0029854725212742e+114)
   (- x)
   (if (<= x -1.4463278090717525e-86)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     (if (<= x -9.142573482041666e-106)
       z
       (if (<= x 8.528180313170455e+134)
         (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 <= -1.0029854725212742e+114)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= -1.4463278090717525e-86)) {
			tmp_1 = ((double) sqrt(((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (z * z))))));
		} else {
			double tmp_2;
			if ((x <= -9.142573482041666e-106)) {
				tmp_2 = z;
			} else {
				double tmp_3;
				if ((x <= 8.528180313170455e+134)) {
					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

Original38.0
Target25.7
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.0029854725212742e114

    1. Initial program Error: 55.5 bits

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

    2. Taylor expanded around -inf Error: 16.9 bits

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

    3. SimplifiedError: 16.9 bits

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

    if -1.0029854725212742e114 < x < -1.4463278090717525e-86 or -9.14257348204166585e-106 < x < 8.52818031317045504e134

    1. Initial program Error: 29.6 bits

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

    if -1.4463278090717525e-86 < x < -9.14257348204166585e-106

    1. Initial program Error: 31.5 bits

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

    2. Taylor expanded around 0 Error: 44.7 bits

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

    if 8.52818031317045504e134 < x

    1. Initial program Error: 60.4 bits

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

    2. Taylor expanded around inf Error: 15.4 bits

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

  4. Final simplificationError: 25.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0029854725212742 \cdot 10^{+114}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -1.4463278090717525 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{elif}\;x \leq -9.142573482041666 \cdot 10^{-106}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 8.528180313170455 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))