Average Error: 38.6 → 0.0
Time: 3.4s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\mathsf{hypot}\left(\sqrt{1} \cdot \mathsf{hypot}\left(x, y\right), z\right)\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\mathsf{hypot}\left(\sqrt{1} \cdot \mathsf{hypot}\left(x, y\right), z\right)
double f(double x, double y, double z) {
        double r626266 = x;
        double r626267 = r626266 * r626266;
        double r626268 = y;
        double r626269 = r626268 * r626268;
        double r626270 = r626267 + r626269;
        double r626271 = z;
        double r626272 = r626271 * r626271;
        double r626273 = r626270 + r626272;
        double r626274 = sqrt(r626273);
        return r626274;
}
↓
double f(double x, double y, double z) {
        double r626275 = 1.0;
        double r626276 = sqrt(r626275);
        double r626277 = x;
        double r626278 = y;
        double r626279 = hypot(r626277, r626278);
        double r626280 = r626276 * r626279;
        double r626281 = z;
        double r626282 = hypot(r626280, r626281);
        return r626282;
}

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

Original38.6
Target26.1
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.3964793941097758 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.3202936944041821 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Initial program 38.6

    \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt38.6

    \[\leadsto \sqrt{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}} + z \cdot z}\]

  4. Applied hypot-def29.3

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity29.3

    \[\leadsto \mathsf{hypot}\left(\sqrt{\color{blue}{1 \cdot \left(x \cdot x + y \cdot y\right)}}, z\right)\]

  7. Applied sqrt-prod29.3

    \[\leadsto \mathsf{hypot}\left(\color{blue}{\sqrt{1} \cdot \sqrt{x \cdot x + y \cdot y}}, z\right)\]

  8. Simplified0.0

    \[\leadsto \mathsf{hypot}\left(\sqrt{1} \cdot \color{blue}{\mathsf{hypot}\left(x, y\right)}, z\right)\]

  9. Final simplification0.0

    \[\leadsto \mathsf{hypot}\left(\sqrt{1} \cdot \mathsf{hypot}\left(x, y\right), z\right)\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))