Average Error: 38.1 → 0.0
Time: 6.1s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\mathsf{hypot}\left(\mathsf{hypot}\left(y, x\right), z\right)\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\mathsf{hypot}\left(\mathsf{hypot}\left(y, x\right), z\right)
double f(double x, double y, double z) {
        double r606777 = x;
        double r606778 = r606777 * r606777;
        double r606779 = y;
        double r606780 = r606779 * r606779;
        double r606781 = r606778 + r606780;
        double r606782 = z;
        double r606783 = r606782 * r606782;
        double r606784 = r606781 + r606783;
        double r606785 = sqrt(r606784);
        return r606785;
}
↓
double f(double x, double y, double z) {
        double r606786 = y;
        double r606787 = x;
        double r606788 = hypot(r606786, r606787);
        double r606789 = z;
        double r606790 = hypot(r606788, r606789);
        return r606790;
}

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

Original38.1
Target25.2
Herbie0.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.396479394109775845820908799933348003545 \cdot 10^{136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \lt 7.320293694404182125923160810847974073098 \cdot 10^{117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array}\]

Derivation

  1. Initial program 38.1

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

  3. Applied add-sqr-sqrt38.1

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

  4. Applied hypot-def28.6

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

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

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

  7. Applied sqrt-prod28.6

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

  8. Simplified28.6

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"

  :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))))