Average Error: 37.4 → 0.0
Time: 3.5s
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 r652007 = x;
        double r652008 = r652007 * r652007;
        double r652009 = y;
        double r652010 = r652009 * r652009;
        double r652011 = r652008 + r652010;
        double r652012 = z;
        double r652013 = r652012 * r652012;
        double r652014 = r652011 + r652013;
        double r652015 = sqrt(r652014);
        return r652015;
}
↓
double f(double x, double y, double z) {
        double r652016 = 1.0;
        double r652017 = sqrt(r652016);
        double r652018 = x;
        double r652019 = y;
        double r652020 = hypot(r652018, r652019);
        double r652021 = r652017 * r652020;
        double r652022 = z;
        double r652023 = hypot(r652021, r652022);
        return r652023;
}

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

Original37.4
Target25.4
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 37.4

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

  3. Applied add-sqr-sqrt37.4

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

  4. Applied hypot-def28.4

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

  6. Applied *-un-lft-identity28.4

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

  7. Applied sqrt-prod28.4

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

  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 2020056 +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))))