Average Error: 38.8 → 11.5
Time: 6.5s
Precision: binary64
\[[x, y, z]=\mathsf{sort}([x, y, z])\]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 1.0147606538952986 \cdot 10^{-102}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.8070760612209178 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(y \cdot \frac{y}{z}\right)\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq 1.0147606538952986 \cdot 10^{-102}:\\
\;\;\;\;-x\\

\mathbf{elif}\;z \leq 1.8070760612209178 \cdot 10^{+108}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;z + 0.5 \cdot \left(y \cdot \frac{y}{z}\right)\\

\end{array}
(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0147606538952986e-102)
   (- x)
   (if (<= z 1.8070760612209178e+108)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     (+ z (* 0.5 (* y (/ y 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 <= 1.0147606538952986e-102) {
		tmp = -x;
	} else if (z <= 1.8070760612209178e+108) {
		tmp = sqrt(((x * x) + (y * y)) + (z * z));
	} else {
		tmp = z + (0.5 * (y * (y / 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

Original38.8
Target20.1
Herbie11.5
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 1.01476065389529856e-102

    1. Initial program 34.1

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

    2. Taylor expanded around -inf 4.5

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

    3. Simplified4.5

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

    if 1.01476065389529856e-102 < z < 1.80707606122091784e108

    1. Initial program 19.9

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

    if 1.80707606122091784e108 < z

    1. Initial program 55.4

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

    2. Taylor expanded around inf 23.0

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

    3. Simplified23.0

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

    4. Taylor expanded around 0 16.8

      \[\leadsto z + 0.5 \cdot \color{blue}{\frac{{y}^{2}}{z}}\]

    5. Simplified16.8

      \[\leadsto z + 0.5 \cdot \color{blue}{\frac{y \cdot y}{z}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity_binary64_1542316.8

      \[\leadsto z + 0.5 \cdot \frac{y \cdot y}{\color{blue}{1 \cdot z}}\]

    8. Applied times-frac_binary64_1542910.4

      \[\leadsto z + 0.5 \cdot \color{blue}{\left(\frac{y}{1} \cdot \frac{y}{z}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.0147606538952986 \cdot 10^{-102}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1.8070760612209178 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(y \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021097 
(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))))