Average Error: 37.7 → 0.0
Time: 3.1s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\mathsf{hypot}\left(\mathsf{hypot}\left(x, y\right), z\right)
double f(double x, double y, double z) {
        double r754923 = x;
        double r754924 = r754923 * r754923;
        double r754925 = y;
        double r754926 = r754925 * r754925;
        double r754927 = r754924 + r754926;
        double r754928 = z;
        double r754929 = r754928 * r754928;
        double r754930 = r754927 + r754929;
        double r754931 = sqrt(r754930);
        return r754931;
}
↓
double f(double x, double y, double z) {
        double r754932 = x;
        double r754933 = y;
        double r754934 = hypot(r754932, r754933);
        double r754935 = z;
        double r754936 = hypot(r754934, r754935);
        return r754936;
}

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

Original37.7
Target25.1
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 37.7

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

  3. Applied add-sqr-sqrt37.7

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

  4. Applied hypot-def28.3

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

  6. Applied hypot-def0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +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))))