Average Error: 38.0 → 0.0
Time: 3.2s
Precision: 64
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}\]
↓
\[\mathsf{hypot}\left(1 \cdot \mathsf{hypot}\left(x, y\right), z\right)\]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\mathsf{hypot}\left(1 \cdot \mathsf{hypot}\left(x, y\right), z\right)
double f(double x, double y, double z) {
        double r656349 = x;
        double r656350 = r656349 * r656349;
        double r656351 = y;
        double r656352 = r656351 * r656351;
        double r656353 = r656350 + r656352;
        double r656354 = z;
        double r656355 = r656354 * r656354;
        double r656356 = r656353 + r656355;
        double r656357 = sqrt(r656356);
        return r656357;
}
↓
double f(double x, double y, double z) {
        double r656358 = 1.0;
        double r656359 = x;
        double r656360 = y;
        double r656361 = hypot(r656359, r656360);
        double r656362 = r656358 * r656361;
        double r656363 = z;
        double r656364 = hypot(r656362, r656363);
        return r656364;
}

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

Original38.0
Target25.8
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 38.0

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

  3. Applied add-sqr-sqrt38.0

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

  4. Applied hypot-def28.8

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

  6. Applied *-un-lft-identity28.8

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

  7. Applied sqrt-prod28.8

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

  8. Simplified28.8

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))))