Average Error: 37.7 → 0.0
Time: 1.8s
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 r667168 = x;
        double r667169 = r667168 * r667168;
        double r667170 = y;
        double r667171 = r667170 * r667170;
        double r667172 = r667169 + r667171;
        double r667173 = z;
        double r667174 = r667173 * r667173;
        double r667175 = r667172 + r667174;
        double r667176 = sqrt(r667175);
        return r667176;
}
↓
double f(double x, double y, double z) {
        double r667177 = 1.0;
        double r667178 = sqrt(r667177);
        double r667179 = x;
        double r667180 = y;
        double r667181 = hypot(r667179, r667180);
        double r667182 = r667178 * r667181;
        double r667183 = z;
        double r667184 = hypot(r667182, r667183);
        return r667184;
}

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
Target25.5
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 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-def28.6

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\sqrt{x \cdot x + y \cdot y}, z\right)}\]
  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. 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 2019353 +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))))