Average Error: 37.8 → 25.8
Time: 3.2s
Precision: binary64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.5532188233214463 \cdot 10^{+105}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.6874774532763033 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.5532188233214463 \cdot 10^{+105}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq 2.6874774532763033 \cdot 10^{+99}:\\
\;\;\;\;\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 -2.5532188233214463e+105)
   (- x)
   (if (<= x 2.6874774532763033e+99)
     (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 <= -2.5532188233214463e+105)) {
		tmp = ((double) -(x));
	} else {
		double tmp_1;
		if ((x <= 2.6874774532763033e+99)) {
			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

Original37.8
Target25.1
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 3 regimes

  2. if x < -2.55321882332144627e105

    1. Initial program Error: 54.0 bits

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

    2. Taylor expanded around -inf Error: 18.6 bits

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

    3. SimplifiedError: 18.6 bits

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

    if -2.55321882332144627e105 < x < 2.6874774532763033e99

    1. Initial program Error: 29.3 bits

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

    if 2.6874774532763033e99 < x

    1. Initial program Error: 54.1 bits

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

    2. Taylor expanded around inf Error: 19.2 bits

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

  4. Final simplificationError: 25.8 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5532188233214463 \cdot 10^{+105}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.6874774532763033 \cdot 10^{+99}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))