Average Error: 38.4 → 0.0
Time: 1.7s
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 r1351291 = x;
        double r1351292 = r1351291 * r1351291;
        double r1351293 = y;
        double r1351294 = r1351293 * r1351293;
        double r1351295 = r1351292 + r1351294;
        double r1351296 = z;
        double r1351297 = r1351296 * r1351296;
        double r1351298 = r1351295 + r1351297;
        double r1351299 = sqrt(r1351298);
        return r1351299;
}
↓
double f(double x, double y, double z) {
        double r1351300 = 1.0;
        double r1351301 = x;
        double r1351302 = y;
        double r1351303 = hypot(r1351301, r1351302);
        double r1351304 = r1351300 * r1351303;
        double r1351305 = z;
        double r1351306 = hypot(r1351304, r1351305);
        return r1351306;
}

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

Original38.4
Target26.1
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 38.4

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

  3. Applied add-sqr-sqrt38.4

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

  4. Applied hypot-def29.3

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

  6. Applied *-un-lft-identity29.3

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

  7. Applied sqrt-prod29.3

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

  8. Simplified29.3

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

  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 2020034 +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))))