Average Error: 37.5 → 11.2
Time: 4.7s
Precision: binary64
\[[x, y, z]=\mathsf{sort}([x, y, z])\]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0642758757813286 \cdot 10^{+133}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -4.966071065003465 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(\frac{x \cdot x}{z} + y \cdot \frac{y}{z}\right)\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.0642758757813286 \cdot 10^{+133}:\\
\;\;\;\;-x\\

\mathbf{elif}\;x \leq -4.966071065003465 \cdot 10^{-133}:\\
\;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;z + 0.5 \cdot \left(\frac{x \cdot x}{z} + 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 (<= x -1.0642758757813286e+133)
   (- x)
   (if (<= x -4.966071065003465e-133)
     (sqrt (+ (+ (* x x) (* y y)) (* z z)))
     (+ z (* 0.5 (+ (/ (* x x) z) (* 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 (x <= -1.0642758757813286e+133) {
		tmp = -x;
	} else if (x <= -4.966071065003465e-133) {
		tmp = sqrt(((x * x) + (y * y)) + (z * z));
	} else {
		tmp = z + (0.5 * (((x * x) / z) + (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

Original37.5
Target19.6
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.06427587578132864e133

    1. Initial program 59.3

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

    2. Taylor expanded around -inf 9.5

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

    3. Simplified9.5

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

    if -1.06427587578132864e133 < x < -4.9660710650034648e-133

    1. Initial program 18.3

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

    if -4.9660710650034648e-133 < x

    1. Initial program 32.8

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

    2. Taylor expanded around inf 8.8

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

    3. Simplified8.8

      \[\leadsto \color{blue}{z + 0.5 \cdot \left(\frac{x \cdot x}{z} + \frac{y \cdot y}{z}\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity_binary648.8

      \[\leadsto z + 0.5 \cdot \left(\frac{x \cdot x}{z} + \frac{y \cdot y}{\color{blue}{1 \cdot z}}\right)\]

    6. Applied times-frac_binary644.0

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0642758757813286 \cdot 10^{+133}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -4.966071065003465 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(\frac{x \cdot x}{z} + y \cdot \frac{y}{z}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))