Average Error: 37.6 → 10.9
Time: 8.0s
Precision: binary64
\[[x, y, z]=\mathsf{sort}([x, y, z])\]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 9.069523159749288 \cdot 10^{-44}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.3237415513114882 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq 9.069523159749288 \cdot 10^{-44}:\\
\;\;\;\;-x\\

\mathbf{elif}\;z \leq 1.3237415513114882 \cdot 10^{+120}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;z\\

\end{array}
(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z 9.069523159749288e-44)
   (- x)
   (if (<= z 1.3237415513114882e+120)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     z)))
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 tmp;
	if (z <= 9.069523159749288e-44) {
		tmp = -x;
	} else if (z <= 1.3237415513114882e+120) {
		tmp = sqrt(((x * x) + (y * y)) + (z * z));
	} else {
		tmp = z;
	}
	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

Original37.6
Target18.8
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 9.0695231597492883e-44

    1. Initial program 29.4

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

    2. Taylor expanded around -inf 7.3

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

    if 9.0695231597492883e-44 < z < 1.3237415513114882e120

    1. Initial program 18.7

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

    if 1.3237415513114882e120 < z

    1. Initial program 56.5

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

    2. Taylor expanded around inf 22.2

      \[\leadsto \color{blue}{z + \left(0.5 \cdot \frac{{x}^{2}}{z} + 0.5 \cdot \frac{{y}^{2}}{z}\right)}\]

    3. Simplified22.2

      \[\leadsto \color{blue}{z + 0.5 \cdot \left(\frac{x \cdot x}{z} + \frac{y \cdot y}{z}\right)}\]

    4. Taylor expanded around inf 9.4

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.069523159749288 \cdot 10^{-44}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.3237415513114882 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))))