Average Error: 37.3 → 11.1
Time: 3.6s
Precision: binary64
\[[x, y, z]=\mathsf{sort}([x, y, z])\]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 1.0148214255014566 \cdot 10^{-99}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.165928133677629 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{{z}^{2} + {x}^{2}}\\ \mathbf{elif}\;z \leq 2.0391049608848342 \cdot 10^{-27}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7.85671152855867 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(y \cdot \frac{y}{z}\right)\\ \end{array}\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq 1.0148214255014566 \cdot 10^{-99}:\\
\;\;\;\;-x\\

\mathbf{elif}\;z \leq 3.165928133677629 \cdot 10^{-60}:\\
\;\;\;\;\sqrt{{z}^{2} + {x}^{2}}\\

\mathbf{elif}\;z \leq 2.0391049608848342 \cdot 10^{-27}:\\
\;\;\;\;-x\\

\mathbf{elif}\;z \leq 7.85671152855867 \cdot 10^{+125}:\\
\;\;\;\;\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 1.0148214255014566e-99)
   (- x)
   (if (<= z 3.165928133677629e-60)
     (sqrt (+ (pow z 2.0) (pow x 2.0)))
     (if (<= z 2.0391049608848342e-27)
       (- x)
       (if (<= z 7.85671152855867e+125)
         (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 <= 1.0148214255014566e-99) {
		tmp = -x;
	} else if (z <= 3.165928133677629e-60) {
		tmp = sqrt(pow(z, 2.0) + pow(x, 2.0));
	} else if (z <= 2.0391049608848342e-27) {
		tmp = -x;
	} else if (z <= 7.85671152855867e+125) {
		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

Original37.3
Target19.2
Herbie11.1
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < 1.0148214255014566e-99 or 3.1659281336776289e-60 < z < 2.03910496088483418e-27

    1. Initial program 30.3

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

    2. Taylor expanded around -inf 7.0

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

    if 1.0148214255014566e-99 < z < 3.1659281336776289e-60

    1. Initial program 22.1

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

    2. Taylor expanded around 0 22.2

      \[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}}\]

    if 2.03910496088483418e-27 < z < 7.8567115285586695e125

    1. Initial program 18.4

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

    if 7.8567115285586695e125 < z

    1. Initial program 56.8

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

    2. Taylor expanded around inf 22.5

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

    3. Simplified22.5

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

    4. Taylor expanded around inf 16.0

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

    5. Simplified16.0

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

    7. Applied *-un-lft-identity_binary6416.0

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

    8. Applied times-frac_binary649.0

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

    9. Simplified9.0

      \[\leadsto z + 0.5 \cdot \left(\color{blue}{y} \cdot \frac{y}{z}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.0148214255014566 \cdot 10^{-99}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 3.165928133677629 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{{z}^{2} + {x}^{2}}\\ \mathbf{elif}\;z \leq 2.0391049608848342 \cdot 10^{-27}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7.85671152855867 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;z + 0.5 \cdot \left(y \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021139 
(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))))